I find this ridiculous. Mr. Wildberger is confusing value with representation. Representing the base of the natural logarithm with the letter 'e' is no less valid than representing the sum of 1 and 2 with the symbol '3'. Both numbers exist and have definite values. Both can be represented in different ways. When we write '12.3' there's nothing canonical or necessary about that notation. In fact it's just shorthand for the longer expression "1x10^1+2x10^0+3x10^-1". The two expressions represent the same value, the same point on the number line. Neither is any more the "real" value than the other. There's no inherent contradiction in computing a derivative. There's no necessity to do an infinite set of calculations. Taking a limit isn't implying an infinite set of calculations. It's a way of saying something like "If we could do an infinite number of calculations, the resulting value would be..." To put it another way, we don't need to do an infinite number of calculations because we have another way to find the value. As a high school mathematics teacher I find this whole thing to be part of the distressing and dysfunctional belief that mathematics is a set of problems with correct answers. The standard curriculum starts out on the wrong foot by reducing the whole subject to a matter of calculating the value of an expression, and it never recovers. Normally you have to get past the first year or two of university mathematics to discover that that's not at all what mathematics is. What Mr. Wildberger is showing isn't that arithmetic is fake. He's showing that computers are bad at math.
@@keystoneperspectives OK, so I'm being a bit cute when I say computers are bad at math. I'm not sure what you mean by "the math". Do you mean computing the result of an arithmetic operation? Math is so much more than that, but even if we limit ourselves to arithmetic calculator, a computer can only represent an infinitesimally small proportion of all integers, and an even smaller proportion of real numbers. One way to show this is as follows: in a spreadsheet cell type "=(2^50+1)-(2^50)". That's obviously 1 but on my PC the result is 0. Given the set of all possible arithmetical computations, which can be shown by a simple diagonal construction to be uncountably infinite, a computer can only perform an infinitesimally small number of them.
Most of this is flat out wrong. For example: - Operations on the real numbers is well defined via Cauchy sequences. The proofs are not that hard. - Limits, integration and such don't require you to do infinite tasks. It's always just about being closer than a given epsilon in FINITE steps. Also, what's the obsession with writing out numbers in decimal about?
Consider the standard view that a derivative describes the tangent at a point. With the limit definition, every single case considered deals with the secant across some interval. Are we justified to say that the limit describes a point, even though we know we cannot bring that interval to zero?
@@keystoneperspectives That is exactly the problem we had to overcome while making the concept of limits. We can never bring it to 0, we can only bound it arbitrarily close to 0 in a neighbourhood close to it, which is the next best thing.
@@aviralsood8141 Are you sure the concept of limits did not create a new contradiction as fallacy/sophism? Derivatives can and should be described as varying coefficients of polynomial function without limits and any geometrical interpretations, pure algebraicaly. With limits all very not simple where in one expression we have three and more dimentional/variable, and sophism of limits becomes self-evident.
@@aviralsood8141 If we can never bring the denominator to zero, why do we think that the two points defining the secant can be brought together to produce the tangent?
19:25 You’re just flat out wrong. Given an algorithm for the nth digit of some number (rational or irrational) A and an algorithm for the nth digit of some number B- it’s actually quite trivial to create an algorithm for the nth digit of A+B. Let’s call the output of our algorithm for A given the input n A(n) and similarly for B. Then here is the algorithm S(n) for generating the nth digit of the sum of A and B. Take A(n) + B(n). If your result is greater than 9, subtract 10. Otherwise keep it. We’ll call this number s. Now take the sum of A(n+1) and B(n+1). This result will fall into three cases- Case 1- it’s greater than 9. Then add 1 to s or if s is 9, make s=0, and your done. You have the nth digit of A+B. Case #2 Your result is less than 9 and you’re done. s is the nth digit of A+B. Case #3 Your result is 9. So you move on to summing A(n+2) and B(n+2). This result will once again fall into three cases and you follow the same steps previously used only in the case of it being 9, you move on to the sum of A(n+3) and B(n+3). And so you continue this process until you reach a non-9 result, and you now have the nth digit of the sum of A and B. Now, I’ll acknowledge that it is possible this sum generates an infinite number of 9s at some point- so there may be some specific point where you’re summing algorithm won’t terminate. But if you’d argue that means you don’t truly have an algorithm for this- then I’d counter that the same reasoning can be applied to the decimal system in general when trying to calculate the sum of 1/3 and 2/3 (represented as decimals) and so you have to acknowledge that regular old decimal addition of the rationals ALSO is “fake” because there’s “no algorithm” for computing the sum of any two rational numbers.
Incidentally- this also means when you have the algorithms for the decimal expansion of any set of irrational numbers, you can generate an algorithm for their sum- which basically answers most of the “challenges” posed later in the video. For the multiplication and exponential problems- it’s still fairly trivial to develop general algorithms for those results although you have to go quite a bit further to the right in order to make sure your results aren’t being changed by carrying digits.
I didn't watch the whole video. I stopped at 20:11. Several things here: - Real numbers are usually defined together with the arithmetic operations on them. So arithmetic works in this case by defintion. - Your statement is basically "Because there is no algorithm which can give you any n-th digit of a sum of all two real numbers after finite time , arithmetics is ill defined". But it is quite a basic thing in math to show existence (that is logical consistency) of an object without actually providing an algorithm to construct it (which can be impossible) e.g. -- existence of a smallest sigma algebra -- existence of uncomputable numbers -- A functional analysis proof of the existence of continuous but nowhere differentiable functions. etc. But even if you'd demand an algorithm for the sake of practicability or whatever the following Algorithm (which stops after finite time) is completely sufficient: Input: a,b, e >0 Output: c s.t. |c-(a+b)|
"- Real numbers are usually defined together with the arithmetic operations on them. So arithmetic works in this case by defintion. " do you realize that this allows for anything? mouse + cheese = elephant by definition in my new mathematical system, how useful is it? "quite a basic thing in math to show existence (that is logical consistency) .." that is the heart of the topic, those are not the same thing. You can have logical consistency of whatever joke-based math. that in no way helps us model the real world. "I don't see why you'd need an algorithm which can give you the correct digit after finite time since this digits are just representations anyway." sorry, what? isn't it the purpose of math operations to obtains precise results? this thing ("real" numbers) are at best an useful engineering solution to math problems, it is not rigorous math.
@@jgmartinezmd6867 - Real numbers are useful as you know if you look at all kinds of sciences. My point is that arithmetics on real numbers are by definition well-defined. You cannot think of numbers without the arithmetic operations on them. Therefore it is a fruitless task to check if arithmetics works on real numbers.. it does. - As I understood he makes a mathematical point here.. I assumed he really thinks that arithmetics is ill defined (in a mathematical sense) on real numbers. This is not true. - It is not necessarily the point of math to provide an algorithm to calculate stuff. Often the task is too show properties of some objects which can be useful in real life applications. The arguments used in showing these properties can be quite abstract. Showing something by constructing it is one proof strategy but by far not the only one. But in this case you actually have an algorithn to provide arbitrary precise results. Note that getting arbitrary precise results is not equivalten to getting the n-th digit right. Consider the following toy example to get the point: a=1/3=0.333... b=2/3=0.666... obviously a+b=1.000... numerically you could e.g. take the truncated version of a und b up to the n-th digit and add them. that means: a_0= 0; a_1=0.3 ; a_2=0.33 ... a_n=0.333...333 (up to the n-th digit) a_0= 0; a_1=0.6 ; a_2=0.66 ... a_n=0.666...666(up to the n-th digit) therefore a_n+b_n = 0.999...999 (up to the n-th digit) As you see you never get any digits right but you get arbitrarly close to 1 Note that digits are just a representation of the number. It's not problematic that you are not getting a digit right as long as the results gets arbitrarly close
Hi Dr. Wildberger, This video really fascinated me, because I have been having similar thoughts without ever coming across other peoples views such as yours. I do not want to speak for you, but I believe in a fakeness that possibly goes way beyond what you are suggesting. I believe the whole real number line is completely misunderstood through a deep social conditioning of our understanding of reality. The 'order' of numbers on the real line is fake, it is an assumption of our reality that we believe in because we are deeply socially conditioned to do so. I actually believe that every number is infinitely dimensional (even natural numbers), and they live in a chaos space where there is no order or structure whatsoever. The apparent 'order' of the numbers on the real line is a contrivance of our reality that goes onto create our reality as we experience it. Put another way, the order we believe numbers sit in, create the type of reality we experience. If we believed that the numbers sit in a different order say, 1,2,3,4,6,5,7,8,9, this belief would create an entirely new reality for us all. I believe the fakeness goes way beyond this. For example the equals sign is the biggest fakeness of all, nothing is ever equal to something else that looks different, i.e. 1+2 is not the same as 3, they are fundamentally different things. I think we need to abandon 'equals', and replace it with 'transforms into'. I am also with you on connecting, sociology, psychology, and pure mathematics, again I take this to the extreme, because I believe that when we peel off all the layers of our ego, we will see all of truth, and this is not just a belief, as I have had mystical experiences where I have experienced exactly this. I would love to talk more with you, if this is of interest to you, I have also put up a few videos on these sort of ideas in my TH-cam Channel. Thank you so much for the wonderful videos you have published.
Not a mathematician here, merely an amateur, but I do have a thought I'd like to share. Aren't numbers merely labels we apply to abstract objects? Some of these objects are more agreable to our understanding and lend themselves to simpler, easier to manipulate labelling schemes, like the integers. From there, computation really amounts to finding the tidiest, shortest, cleanest label for an object. Computing 1/2 + 1/3 is not asking "What is this?' but rather "Which is a more agreable label for this object." '2' is a tidier label than '6 - 4', but both really represent the same thing. By "tidier" I really mean "more manageable for our human brains". Our most widely accepted/used labelling system for these awkward 'real numbers' does produce rather unwieldy labels for them but do all system not have the same kind of limitations? For instance, when it comes to large integers, we do rightly resort to 'computational' notation as a more practical label (ex: the googolplex), not to mention how complicated some basic comparisons like > become with these larger numbers. Again, we seem to have unwieldy labels for objects that are further removed from our human scales. In that sense that number systems are human inventions to make foreign objects manageable by our human brains, aren't we bound to hit a hard limit of complexity/unwieldyness whatever the system we come up with? As always your videos and the insights you kindly share with us are invaluable. Thank you so much for that!
The real numbers minus the rational numbers arguably don't even exist. Yes, including the irrational numbers, as a construct such as sqrt(2) is meaningless in nature. That they are impossible is easily proven: if there were an algorithm that can compute an infinite number, then either it must require an infinite amount of time, or an infinite amount of computing power to do so. Neither apply to anything that exists in nature., especially not physical objects with very much bounded dimensions. The very best you'd get is an approximation that terminates after n digits. If they don't actually exist in nature, they must only exist in the minds of people. Abstract objects don't exist. What is the locus of abstract objects? None, except the minds of people. Those numbers are not merely "more manageable" for our brains, they are a product of them wholesale. So what is the relevance of -- as you put it -- creating symbols to denote mathematical objects us humans have come up, except the study of said human constructs? You can have real, irrational, complex etc. numbers no problem. As long you accept they are fictional.
@@ekszentrik To be fair, Prof. Wildberger is enthusiastically willing to work with *algebraic* extensions to the rationals (or any other basic field), such as a symbolic sqrt(2), defined as a number with the property that when you square it, the result is 2. Similarly, he has no problem with the complex/imaginary numbers like i, or the i, j, and k of quaternions, etc. But the *underlying* field he'll use will be the rationals, rather than the 'reals'. And he won't mess around with trying to approximate the symbolic sqrt(2) unless/until it's needed for some practical approximate purpose at the end of whatever calculation it arises/remains in.
@@ekszentrik Interesting point, with which I'd be tempted to agree. But I fail to follow your meaning when you say that sqrt(2) is fictional. If you're correct wouldn't it follow that the right triangle with sides 1 is also fictional? If it is so for other reasons than simple physical imprecision in building one, doesn't it in turn render the number 1 fictional as well? Ad absurdum it feels like 'fictionality' as in having its locus in the human mind is a rather weak concept, if not entirely circular and devoid of any epistemological significance. What do you think?
@@ekszentrik Ok so your concept of existence is that something exists, if it is part of the "real" world right? How do you know that the real world even "exists"? What does it mean to be fictional and what does it mean to "calculate" something? Mathematicians have long pondered about these questions, and the best way to solve it is to not talk about "existence", but rather to formulate mathematics as a language (i.e. a set of rules how to but together signs on a peace of paper, where the grammatical rules how one is allowed to put these letters together are called the "axioms". Then a mathematical proof for a "Statement" (which is also, by definition, just a sequence of letters) is defined to be a Sequence of letters satisfying the grammatical rules such that the Statement is written at the end of this (finite) sequence. In this framework one can for example take the axiom System of ZFC (Zermelo Frenkel Choice), and then one can "talk about" mostly everything that is commonly done in pure mathematics and one can for example prove (in the sense I described above) that there exists something that we call the "real numbers" and it is unique up to unique isomorphism (when one makes this precise). In principle, for any mathematical proof (in the classical sense) to be "correct" (for example in ZFC) it should be possible to translate this proof in the language described above, so that when a computer parses this finite string, it satisfies all grammatical rules required in ZFC. Of course ZFC could have contradictions, maybe because it talks about infinitely big sets. However then the theory is not "proken" but rather any statement can be proven in the above sense and the problem is that the theory will just become uniteresting. I'm not an expert in logic, but if you want a reference for this look at "Combinatorial Set Theory" by "Lorenz Halbeisen"
@Jukke jukke Sorry I am not too adept at adding Cauchy sequences in their entirety. Perhaps you could illustrate the technique by adding the Cauchy sequences for pi and e together?
@@njwildberger Let a and b be two Cauchy sequences of rational numbers. Define a + b = { (i, s) | i in N and s = a(i) + b(i) }. This can be shown to be a Cauchy sequence of rational numbers. Then, pi + e = { x | (Ea)(Eb)( a in pi and b in e and x is an equivalent Cauchy sequence of rational numbers to a + b) }
@@billh17 Your definition is inconsistent. If a and b are Cauchy sequences given by some functions a(i) and b(i), why have you switched to set notation for their sum? You also concluded that "pi + e = { x | (Ea)(Eb)( a in pi and b in e and x is an equivalent Cauchy sequence of rational numbers to a + b) }". Hence I suppose you would suggest to a primary school student that "1/2 + 1/3 = { x | (Ea)(Eb)( a in 1/2 and b in /13 and x is an equivalent Cauchy sequence of rational numbers to a + b) }" in the "arithmetic of real numbers". Would such an answer get full marks? At what point are you going to admit that this is all just talking??
@@njwildberger functions are sets of ordered pairs. This is a completely consistent definition. Why would you bring up primary school? Should all of mathematics be simple enough to explain to an 8 year old? That's absurd
When I started in primary school, I used to ask questions about arithmetic , like how can 3x4 be the same as 4x3. My teacher had never of the commutative law. When I got to secondary school and began studying calculus , there wasn't time to ask any more questions, exams had to be got ready.
@Thomas Kember Nothing like a busy schedule to keep the unwanted questions at bay. I wonder how many analysis or calculus classes around the world are going to start on Monday morning with the prof saying: "There has been some interesting discussion coming from Wildberger's claim that our real number arithmetic is fake. Let's spend the hour actually investigating that claim, and see how it stands up." If this happens in your course, please let us know!
Wow, it looks like you were unlucky to get a bad teacher. The best and worst thing about teachers is that they form our attitude toward their subject. They make us experience it only from their point of view. If you don’t like a subject, your teacher failed to inspire you.
Hi Dr. Wildberger: I would offer one minor suggestion. Of the three academic disciplines of psychology, sociology, and anthropology the one, it seems to me, that you should be appealing to is anthropology not sociology. These three fields all ask similar questions, often, but they have very different methodologies. A psychologist researcher might try to conduct a controlled experiment to study your issue, a sociologist might try to go back through historical records and analyze data or maybe take a poll. An anthropologist will actually get in there and study mathematicians “in the field” and I suspect that is what you are after. However, such a study would be conducted without any judgement regarding the “correct” resolution of the issue. Their work may yield a lot of information about how and why the culture of mathematicians dismisses outlying ideas, or how TH-cam gives minority perspectives a stronger voice. Their work will not address “Why can’t’ most mathematicians see the obvious”. That is, anthropologists (or “sociologists” as you call for) will never assist you to shine light on this interesting issue. That burden will always reside with you and the merits/usefulness of your alternative ideas.
@Xyl... That is a good point --- I would love to include anthropologists and indeed psychologists in this investigation. But I am a bit more hopeful than you about getting reasonable social/cultural/psychological answers to the question "Why can’t’ most mathematicians see the obvious?”. And indeed there is another potentially interested cohort that I would like to get thinking about these issues --- the pure mathematicians themselves! I bet a lot of them would have some pretty interesting takes on all of this; once we get them over the hurdle of actually considering the possibility that they have been seriously duped, and are currently doing the same to the next generation.
@@njwildberger It's not that they don't *see* what you're saying-- they see it fine, they just *disagree* with you about it. As to _why_ they disagree, almost certainly because you haven't actually made a good argument. Your whole point seems to be that everything should be computable because they better corresponds to the real world, but pure mathematicians generally aren't concerned with modeling the real world anyway, so your point isn't really _relevant_ to their work. The thing that you don't seem to understand is that _all_ mathematical objects are purely abstract ideas. *None* of them "exist" in the usual sense of the word. That's as true for the rationals as it is for the reals. They "exist" in the same sense that magic "exists" in the world of _Harry Potter._ That is, they're just _ideas,_ and so, as long as they're logically consistent, they're all equally valid. You can say that the real numbers don't "exist" _in the constructable universe_ and that's fine, but it's the same kind of statement as, "There's no such thing as magic in the world of _Batman._" _True,_ but it does nothing to invalidate the statement, "In the world of _Harry Potter_ magic exists." It's all about what universe(s) of discourse we're working in. Just because _your_ preferred universe of discourse doesn't have real numbers doesn't justify claiming that there isn't another, equally valid, universe of discourse that does.
@@Lucky10279 it sounded like he argues that they aren't logically consistent. For example, the addition of some transcendentals can't be evaluated. Or, that they are treated in analysis like discreet numbers and ongoing processes interchangeably. In this 39 minute video he does not simply claim they don't exist- he gives several examples of their apparent logical incoherence. If you can clarify those examples we all might learn something.
@@Robinsonero Well I didn't watch the whole video, as I was already familiar with argument from an article of his. If you tell me a specific example I'd be happy to try to clarify it though.
I think the ironic thing about this lecture is that it presupposes that common arithmetic is “more real” simply because it’s the more common human experience. IMO, the universe doesn’t care enough to count things, implying that counting is a human abstraction itself.
19:45 : is the example correct? How can you tell these 2 numbers have infinite sequence of digit? Only by their actual algebraic definition - as the roots of an equation for instance. And it is starting from this definition, the only thing conveying all the information about them, that you can hope to create the right algorithm for calculating (any approximation of ) their sum. That there's no universal algorithm to calculate the sum of 2 real numbers doesn't invalidate the definition of the real numbers. It's a solution space like the complex numbers. You can deny this - but at the expense of staying stuck in trivia.
I have a doubt if the concept of infinity is non sensical then math should consist of a finite number of theorems , but according to godel incompleteness theorems math is incomplete, in other words no matter how many theorems we prove, there will be true statements which cannot be proved by our current system, and if math is complete and finite then it's inconsistent. So how to deal with this?
Hi Norman, Thanks for making a clear presentation outlining your view. I conceed that with your proposed questions, the best I could do probably arrive at re-expressing them as infinite sums etc. However, I think your notion of "definition" is much stronger than what most pure mathematians require a definition to be. Never in pure mathematics has it been the case that a definition must be some kind of algorithm from which we obtain explicit objects. Definitions need not provide explicit representations or "answers" or algorithms for allowing humans to digest them. The brilliance of the foundation of R uses definitions which don't give explicit representations of objects time and time again. You're absolutely right, barely any operations on real numbers give explicit answers but that doesn't mean they are not valid objects. Your qualm with infinite sums I think you already know is not quite right and highlights exactly the point I'm making here. It's defined as the limit of partial sums. Who said you have to evaluate explicitly all these partial sums? Or know explicitly what the limit is for that matter. Limits are defined also in a way which does not prescribe an algorithm or explicit represention of them but they are nonetheless well defined. Even if some limits will never be obtainable by humans in explicit form, they are valid objects nonetheless. I think what needs to be assessed here is the fact that explicit representations or "answers" of things is subjective. Here is a (degenerate) example of my point: 1+2=3 means: "the unique natural number 1+2 is equal to 3" As opposed to: "1+2 evaluates to 3" This latter interpretation makes no sense, it's like saying that 1+2 represents something that "is to be done". This is incorrect. The sum has already "happened" and it so happens to be equal to 3. 1+2 has an "explicit representation" or "answer" of 3. Apologies for rambling, I may edit my response to make the message clearer. Thanks, David
@ David Alexander Are you really confident that this is an approach to mathematics that is coherent, or that would be approved by the great mathematicians from days gone by, or that it actually is well-defined? Isn't there something solid for your Grade 3 student to learn that 6 + 7 = 13, and not some range of possible wafflings? Are you happy that if "real numbers" include natural numbers and integers, then over the real numbers all arithmetical questions posed by primary school teachers become trivial by your own standards? So for example 6 + 7 is equal to 6 + 7, because that is a "valid object". Full marks?
@@WildEggmathematicscourses it hardly seems like the approval of greats from days gone by makes for a good criterion of soundness. Pythagoreans were excellent geometricians but would have scoffed at the work of Galois for relying upon symbolic notation rather than construction, and yet his own work is powerful enough to demonstrate the limitations of their craft in a completely rigorous fashion. We can't blame them for it, but the greats of old are not so blessed as we with the benefit of hindsight
@@WildEggmathematicscourses you don't need the approval of past greats, we have living great experts today that are perfectly capable to evaluate any approach. Mathematics done by Gauss, Newton etc is not the only valid mathematics. This sort of reasoning is appeal to authority fallacy, it exposes a deficiency in your argument. Yes it is perfectly adequate to teach children that 6 + 7 is precisely, identically, unmistakably 6 + 7. However we also need to teach that 13 = 6 + 7. Although we only teach transitivity at university level, it is extremely practical to teach children the next step: 6 + 7 = 13. Don't forget school is not meant to teach only one side (theoretical or practical). You can finish a higher level mathematics course without doing a single computation (if you choose to not define demonstrations as computations), but you can't finish an applied mathematics course without doing computations, they are the whole point of these courses. So is the point of school, both teach you foundations and applications. 'So for example 6 + 7 is equal to 6 + 7, because that is a "valid object". ' - in an ideal world, children would leave primary school knowing this.
@@njwildberger You are entirely misreading that statement. That last statement was but a response to your last questions: "Are you happy that if 'real numbers' include natural numbers and integers, then over the real numbers all arithmetical questions posed by primary school teachers become trivial by your own standards? So for example 6 + 7 is equal to 6 + 7, because that is a 'valid object'. Full marks?". It was not meant in any way to justify real number arithmetic. I used to teach arithmetic to children (Kumon system) and one the main issues I've seen some children develop is failing to understand expressions as "surrogates" for numbers. This fault diminished their intuition. I fully expect the success case for primary mathematical education to include the notion that valid (closed) expressions and numbers are identical.
Speaking as a mathematician, it seems to me that while sociologists may be able to give an account of how mathematicians come to 'pretend that all is well', they can't actually fix the math foundational problem. Maybe it can't be, anyhow! - but otherwise, few sociologists are mathematicians by training or inclination. So it's up to the mathematical community to clean out its own stables - and Hercules the Sociologist, can take the week off! At a personal level, MJW, I'm glad to see you looking a whole lot better. Your apparent absence for a while had me worried! Whatever it is you're doing - keep right on doing it!
It’s funny how people have such affinity to what they’ve learned. Still, Norman is 100% correct from a pure mathematical sense. He has shown most trig and calc can be done in a better way, built on a stronger foundation. I believe the hope is that future minds can build on that. However, the current limitations must first be recognized before they can be overcome.
All he has shown is an alternative way which takes a few definitions and changes them to not rely so heavily on real numbers. I like those methods but still the fact remains that real numbers are part of a majority of mathematical topics and there has been no one who has come up with a 'better' way.
@@aviralsood8141 In this instance i would argue all he has done is incorrectly used the term arithmetic. rather than actually talk about binary operators on a field.
Dimensional Gauge Symmetry; my TOE explains the most with the least. It is God's combinatorial infinity. We know they are all stealing from me, and now God does too.
@@dsm5d723 so what does gauge symmetry have to with any of this? Interestingly, if one considers the reals as ill-defined, not a single Lie group survives, and not a single gauge theory survives.
Can we consider a real number X to be an algorithm that emits rational numbers that approximate X up to any finite level of accuracy that we input to the algo? Note this is different from asking for an algo that emits a certain number of correct decimal places, since it is immune to the waves-of-carries problem. It is straightforward to define such algorithms for pi and e, in fact they are quite compact. We can then easily generate an algorithm meeting the same requirement for pi + e. The algorithms are themselves finite artifacts, and we only ever ask for a finite (but arbitrary) level of precision.
I think there is another problem in the "program that returns i-th digit" formulation: with finite amount of memory and deterministic processor we will always end up with a sequence that starts to repeat itself starting from some position. Thus, we will be only able to get a rational numbers, which can be easily added.
@Edward Newell, Please have a very careful read of your first sentence. Can you see that it has an essentially self-referential nature? What if your algorithm seems to be outputing numbers that are close to the outputs of my algorithm. How do we conclude they are in fact the same? In any case trying to define "real numbers" via algorithms is tempting. But no-one has been able to make it work. Maybe for a more limited notion there is some hope. Part of the problem: you first have to wrestle with the general definition of "algorithm"including again the issue about when two algorithms are "the same". And defining the arithmetical operations is fraught with difficulty.
Imagine two different algorithms that both claim to produce pi+e. How would you be able to tell they are the same if all you can ever calculate is an approximation? Likewise for whether neither one, either one or both are correct?
The calculation of the sum of two numbers with infinite digits is defined although there is no algorithm to calculate all its digits. Mathematics is based on well-defined concepts rather than computable algorithms. It seems that your philosophical position is intuistism, which believes that mathematics must be intuitive and rejects all well-defined mathematical concepts when there is no algorithm to calculate the concepts. This position is too restrictive to do Maths.
Any time you introduce any kind of infinity into an expression, you've entered a whole different kind of math. Why should having an infinite number of non-repeating digits (even describing a finite number) be any less problematic than infinities of any other kind?
There are certain objects which are only represented infinitely in certain bases, like 1/3 in base ten. Also, 1/3 in decimal notation is a repeating decimal, so the problem of being cot computable doesn't exist. It's clear that 0.3333333.... plus 0.3333333... is 0.6666666. But transcendental numbers cannot be represented this way.
making it to the end of the video, I understand what the problem is: you're hung up on Zeno's paradox. Yes, it's that "trivial" of a problem. You see, in theoretical thinking we don't need time or space. Mathematics was never about the real world. In a theoretical world everything goes, at least everything the mind can conceive of. It only depends on your ability to create or choose axioms. We can readily have the result of an infinite number of operations. Admit to yourself that you CAN imagine it, and you'll find yourself to be able to. I claim to be able to solve the paradox. Proof? I didn't define my axioms first. Here they go (although this is all still very informal): 1. Peano arithmetic. 2. First-order logic (sorry, not trying to be aggressive, but is this "jargon" to you as you point out at the end? I mean, if you want to have an "adult" conversation about math you can't escape some rigor). [My personal axioms:] 3. (In our hypothetical reality) we can hold an infinite number of characters. 4. (In our hypothetical reality) we can perform an infinite number of tasks. [In other words, there is no time or space, they're unnecessary concepts] Proof: I can map Achilles' steps and the tortoise steps to infinite sequences of numbers. Using my axioms I can take the sums of these sequences, then I can verify they're identical. Now, this only solves the paradox in my make-believe theoretical world. Zeno's paradox obviously has applications to both theoretical worlds and the physical world. If you want to solve it physically you need to go beyond mathematics, into physics or philosophy or something. What I set out to do here is make an analogy between the paradox and the process of taking infinite sums, and show that it is easily doable if you have the diligence to build the appropriate foundations. As a token of good will, I will attempt to solve your computing challenges. The primary answer is this: your questions are ill-defined. If I just repeat the left-hand side expression you won't accept the answer, whereas the answer is valid and perfectly sound. You didn't specify what format you want your answer to be in. You only said in a hand-waving manner that all the obvious answers are unsatisfactory. So, in the absence of an explanation of what would be considered satisfactory, we can only make assumptions. I'll assume you don't want to see symbols that represent numbers "beyond" rational or infinities (sums). I'll further assume that if I can compute any digit, then I can claim I know the precise answer. Now go ahead and give a decimal place for any of these questions and I'll give you a digit. Mind the fact that, for your own convenience, farther digits will require a long time to compute. Even if this process extrapolates the age of the universe, it's in principle always doable and that's all that matters. But I'll grant you this. There are points of contention in mathematics. There are open questions. There are questionable axioms. Sets, which are to mathematics as the quarks are to Physics, are not defined in a way that makes everybody happy. But everything above these matters, everything that stems from these definitions is unquestionably sound, including real numbers.
Video Content 00:00 Introduction 02:36 Number systems 06:41 What system do scientists generally use? 10:09 What system do pure mathematicians use? 15:27 Arithmetic with 'real numbers' 24:43 Computing challenges for analysts
The key reason we programmers like "floating point" is because of time complexity. The floating point math, while inaccurate, will always run at the same speed, due to the truncation of lesser significant figures. This is rather important for certain applications that repeatedly do the same calculations (realtime systems), as you need the speed to be predicable. However, consider a rational number: as you add, sub, mul and div, the complexity of the output rapidly increases, i.e. both the numerator and denominator constantly get bigger. Eventually (albeit often quickly) this saturates your integer word size (usually 64 bit), meaning you have to use multiple words, which takes CPU time (you can only process 1 word at a time). This is highly problematic because you want calculations in realtime systems (such as video games, flight computers, etc) to run at a predictable speed, not slow down pseudo arbitrarily depending on input device data (well a lot of games do this anyway, so imagine how much worse it would be if the hardware calculations themselves could slow down in this way (well actually they can, due to CPU caching and branch prediction, but this isn't a problem with arithmetic)). Real arithmetic is fake because you don't actually do arithmetic? You just say what you did algebraically (if even that). Rather useless from a practical standpoint. Really just glorified philosophy. My problem with "real" number theory is it will get you stuck in "analysis paralysis" if you try to use it in any practical application. Things work much better when you just say "infinity is NaN (not a number)", and strangely enough, this doesn't seem to limit our capability in any meaningful way.
Is the square root of 2 a number? If so it is not a natural number, not an integer, and not a rational number. Or equivalently if you construct a square 1 unit of distance on each side and apply the Pythagorean Theorem then does the length of the diagonal exist or not or does the Pythagorean Theorem not work in that case - or any other case of a square for that matter. If you construct a rectangle with sides of length 3 and 4 then the diagonal is 5 and magically the Pythagorean Theorem works in cases like this.
Norman Wildberger is of the opinion that the metric concepts of length and angle are very problematic. In their place he uses the quasi metric concepts of quadrance and spread which are much more well behaved. Intuitively quadrance is the same as length squared and spread is the same as [sine angle] squared. Although it is better to think of length as square root of quadrance and of angle as arcus sine of square root of spread, which is generally ill defined, but works sometimes, inside the rational numbers. That said, as an *abstract symbolic* concept, there is a sense in which a square root of 2 exists (just like a square root of -1 exists), but this is a point on an axis dimensionally independent from the rational axis, it is not a metric quantity anymore than imaginary i is a metric quantity. There are algebraic extensions of rational numbers, of far higher dimensionality than the usual nonsense about "the complex numbers being an algebraically complete but metrically 2 dimensional extension of the real numbers". So forget about only using 2 dimensions for expressing field algebraic solutions to polynomial equations!
As a mathematician who now works with computer science, I understand your point. Basically, for you something should only be considered to exist if it is computable. And yes, in the computable universe, Real numbers do not exist. Heck, any infinity set does not exist. But the way you formulate your ideas - an attack to one of the pillars of mainstream mathematics - makes mathematicians on the internet complete disregard your video as a random internet madman, although it does make it more viral. Try to reformulate it as a movement for a new kind of mathematics that is closer to the computable world and the real world by avoiding infinity axioms and only dealing with objects which are computable or can be sufficiently approximated by computable things.
@GNavarro This is not an isolated video, although maybe you have just arrived here just now. I have more than 600 videos across a wide range of mathematics. The Math Foundations series presents a consistent critique of mainstream pure mathematics, but also lays out many new directions. The Algebraic Calculus One course is running now on openlearning, you can join for free, and start learning Calculus in the right way. Please have a look at the other playlists too. And mathematicians do not consider me a random internet madman, many know about what I am doing full well. They just don't like it, because it undermines the status quo upon which many people's life's work rests. But there is more at stake --- the education of a new generation of maths students!
@@njwildberger Norman, I love you because you never stop giving. You have added great value to my life. Thank you! I purchased your book, a fine product, excellent work my Brother.
Not all mathematicians ignore his work. He does an excellent job of explaining some very interesting ideas in an accessible way. Occasionally going into what a lot of us would disagree with. Still he offers a valuable service here that even a platonic can appreciate.
I live in leafy Surrey in England. It is autumn and many leaves have fallen. There are a finute number of leaves although nobody can tell me how many leaves have fallen at a particular instant. Or even if an odd or even number of leaves have fallen. This does not mean that the number of leaves does not exist. Similarly we cannot specify all of the digits of pi + e but I can give you any number of the digits that you want or better still, need. 2⅔ + 3⅗ is a perfectly good number. Just as 2+3 can be used in place of 5 in any computation.
@@njwildbergerThough we can treat two conjugate square roots of 2 in a very similar vein to the way we treat two conjugate square roots of -1. Purely algebraic field extensions (of rational numbers or prime fields) that is, polynumbers i think you call them. Adjoining such square roots of 2 (or of other positive numbers) gives us a kind of relativistic/hyperbolic geometry over the rationals, rather than the elliptic geometry we get by adjoining square roots of -1 or of -3 (or to a lesser extent of other negative numbers). This view of geometry as algebraic extensions of rational numbers or similar well defined base fields opens up the possibility of seriously studying 3 dimensional, 4 dimensional, 6 dimensional, 5 dimensional etc commutative field geometry (with conjugate roots in a sense determined by the Galois group of the defining polynomial of the Galois field extension) in a sense the possibility of which is generally masked by the illusion of complex numbers being a "complete" 2 dimensional field extension of the real numbers (both number systems being little more than illusions), and all higher dimensional algebraic geometries (over real numbers) demanding either zero divisors or non commutativity etc.
You seem to impose the condition that it all has to be algorithmic and computable. The current consensus in the (most of) mathematical community is the opposite. This is a difference of opinions, not facts. To say it differently - yes, the operations are not finitary or computable, but for the way pure mathematicians use them this is not a problem. I consider the use of the word 'fake' in this context to be misleading. Regarding your question about concrete values of something like e + pi. Before I comment on that, I offer a counter question - what is the circumference of a circle of radius one? Or it's area? Is it a number? If yes, tell me which, concretely, as a finitary, number-like expression. If it is not, please explain why it is unreasonable to have such a number. Or maybe it is the case that circles do not really exist, but only their rational approximations? I did not watch your videos on foundations of math, and I assume that somewhere there might be an answer, but I most likely won't go through those videos anytime soon. Now coming back to your question. There is no concrete object in the form of a 'palpable' rational-like finite expression and you know that. But what does it prove? It only proves the point that these things are not in general finite or computable - which is something that no one really disputes (but yes, the audience might find it strange and surprising). Say you have a motorbike, I have car and you send a bunch of sociologists to ask me how many wheels my vehicle has - of course the answer is 4, but that only proves that I have a car (and I never claimed otherwise), not that having a car is somehow wrong. The problem is that the way you set up the question makes it seem that any answer other than 2 is just bad. Yes, a friend of mine who thought I had a motorbike may be surprised by me having a car, but it does not make your point that having cars is bad stronger. The way mathematics is set up does not really need the objects it speaks of to be finitary or computable. Engineers need that, but not mathematicians. The problem you have with all this seems to be mainly philosophical - you just don't think that this is what mathematics should be, I guess because it is too distant from reality as we know it. I guess a lot of philosophical problems of this type come from the fact that mathematics actually does not work with _any_ objects - it is purely syntactical and the meaning/interpretation of its theorems and objects it speaks of is just in our heads. No actual construction of objects takes place, the only thing that happens is theorem proving (symbol manipulation). Sadly, this is something I have basically never seen explained properly, in a common sense fashion. I would really like to elaborate on this here, but this post would be just too long. I definitely agree with you that the foundations are usually not taught in enough detail. By the way, you say that rationals are ok. Fine, but what actually _is_ a rational number? If 2/3 and 4/6 are the same number, then neither of them actually _is_ the number, but they are both a representation of something. What actually _is_ this object they represent? Is it finite? Can you exhibit it fully? You can certainly write a computer program which tells you that two strings like 2/3 and 4/6 represent the same number, and you can also do arithmetic by a program. But do you actually want to postulate that only computable 'objects' are fine? Even then - what _are_ these objects anyway? Are they actually just the programs (finite strings of symbols) themselves or something like a collection of pairs of the form (input,output)? PS: I am actually a fan of your lectures, you have a very nice no-nonsense teaching style.
@Leonard Szekeres First of all you have a famous last name--- George Szekeres was a friend of mine at UNSW and one of the best mathematicians in Australia during his day. Are you perhaps related? Let me answer some of your questions: a circle of radius 1 in the usual affine plane does not have a well-defined circumference, nor a well-defined area. Archimedes knew this already more than 2000 years ago. What we can do is find approximate circumferences, and approximate areas, and that is exactly the kind of question Archimedes set out to answer. Modern pure mathematicians are uncomfortable following Archimedes --they want exact answers, even if they have to impose them artificially. As for pi+e+sqrt(2), I take it that your best answer is: there is no concrete form for it. Fine, thus we can all agree this is a fake arithmetic. Additional extensive discussion about what mathematics should or should not be is irrelevant. You can't add the basic numbers in your system, therefore it is a fake system. I know it is hard to swallow that you have been misled, perhaps for many years. Why not have a go at the MathFoundations series?
@@njwildberger No, I am fairly certain I am not related to him, even distantly. What I take issue with is your use of word 'fake'. One can also say the following "In your mathematics even the most basic geometrical objects like circles do not have some very basic attributes such as area or circumference. Fine, thus we can all agree this is a fake mathematics." Of course, you can object to the above and say "Well, if a circle of radius one has a circumference, then _what_ is it? Please give me a concrete answer". To which I can reply "Well if a circle of radius one exists, then _what_ is it? Concretely please". And you can tell me that it consists of all rational points with distance 1 from 0. But that's a description of the object, not the object itself. You can say that this description is at least computable, but there are many programs which describe it - so which one _is_ the circle? Or all of them? That's a pretty large object, and how is that exhibited concretely? "Additional extensive discussion about what mathematics should or should not be is irrelevant." - your entire argument is based on something that you think mathematics _should_ be (you say that arithmetical operations _should_ be computable) and at the same time you say that any discussion about what mathematics should or shouldn't be is irrelevant? Come on... " What we can do is find approximate circumferences, and approximate areas..." - but what is it that you are approximating? By using smaller and smaller angles and line elements you can get a number which is arbitrarily close to X, but what is X? If X does not exist then you cannot speak about approximating it. Of course, you can get by without having an actual object which you are approximating, but then you end up with a Cauchy sequence. And then it is convenient to at least pretend we have them and we can do operations with them, even if we do not believe they actually 'exist' in the real world (neither do circles with infinitely many points).
If I have a finite resolution to work in I can give you exact values for circumference and area, and at no point will I be invoking pi or any other irrational in my calculations.
"We're disturbing the priest, won't you please come to our feast Do we mind disturbing the priest, not at all, not at all, not in the least" -- Black Sabbath
For the final example shown between 24:42 and 31:43, each term in that symbolic statement and the sum can be expressed as finite numbers if you change the base accordingly. The least common multiple of the denominators 2, 3, and 5 is 30, so in base 30 the problem becomes 0.F + 0.A + 0.6 = 1.1. Left in base 10, the sum is simplified to 31/30 rather than leaving it as 1/2 + 1/3 + 1/5. Hypothetically, the other problems could be shown in finite form if expressed in base pi, base e, base sqrt(2) and so forth, but then that may create problems with expressing whole numbers from the decimal system in finite form, depending on the problem. Logarithms can be used to convert between bases as needed, though this would reintroduce those errors of finite calculations.
@Dorian, No-one has seriously suggested a number system with base pi or base e. With sqrt(2) we can't make such a sweeping statement, because algebraic approaches go someways in this direction. Once again the confusion arises with our terminology, of mixing up sqrt(2) as a magnitude somewhere between 1.4 and 1.5, and as a novel symbol involved in a quadratic extension field of the rationals. Please watch the Famous Math Problems 20, 21 and 22 videos (the latter is still on its way..)
30 years programming I have never used floating point outside of school. Counting numbers and addition work trillions upon trillions of times without error. Thus when you say this is not used, I am quite perplexed. I guess you have heard of floating point and somehow extrapolated that onto all of computer programming?
I agree. For geometry / statistics / simulation we often use float (maybe double), but for counting things we use int (maybe uint64 or occasionally even bigger) unless we are using a really crippling programming language that forces us to use float. He could have been more careful to draw this distinction.
@@WildEggmathematicscourses I have yet to imagine how this could be correct. It feels similar to arguing that everyone MUST wear life jackets 24/7 on the basis that some people might not know how to swim and then arguing that no one knows how to swim using the universal life jacket rule as evidence.
I still ask myself, how do we know there is foundation of mathematics? And if it is the foundation then how can we reach it with our modern Mathematics that is built on it? Maybe Mathematics is like Physics, maybe in this universe it exists in that way and knowing beyond the universe is impossible
To be able to calculate, or compute, the challenge expressions 1-4 at 24:44 in the video, we would have to define some precision first. The results, as reals, may be expressed as intervals of decimals, with a very high precision (to the 10th decimal digit), like this: 1) 7.2740880444 < π+e+√2 < 7.2740880445 2) 12.0770079567 < π×e×√2 < 12.0770079568 3) 1.6344452924 < π÷(e÷√2) < 1.6344452925 4) 110.8924304703 < π^(e^√2) < 110.8924304704 Perhaps we would like a perfect expression for each answer, but not by just repeating the expression in each challenge. Sometimes you are lucky, as in the expression 8 that results is the number 1, but most often you will probably end up with an irrational number and get, expressed as a decimal, a non-repeating string of digits. If we would like exact decimal expressions for irrational numbers like π, e and √2, we can't expect these to be expressed in their entirety, since by definition these expressions are infinite in size, but maybe the interval expressions are sufficient, since they can be extended to a precision of 100 digits, 1 000 digits, 1 000 000 digits or, in fact, any finite number you like (or has the time to compute). If we actually need exact expressions for irrational numbers, is this even possible in the real numbers? If we look at the definition of an irrational real number, we can see it is defined down to infinitesimal size, which means that if the constant has non-zero infinitesimal addition it can't be exactly matched by a real number to differentiate it from another constant that has no infinitesimal addition. This calls for another number system that is able to express infinitesimal additions, but this extension doesn't solve the whole problem, since a constant may differ with an infinitesimal to the infinitesimals in that system and so on. Surely, sometimes you are lucky enough to have an expression that entirely defines the irrational number, so maybe such an expression (an infinite series perhaps) can be used instead of a decimal interval if that is necessary in the context. If precision is really important, you could even work in an precision extended number system and then go back to reals (or rationals maybe) and in the process all non-zero infinitesimals vanishes. So, are we satisfied? It depends on what we really want, it also depends on how precise we want the constants to be defined and so on. There is simply no perfect number system that does all that, but we can at least pick one that is optimal for our needs, and when infinitesimal precision isn't necessary real number intervals will almost always do.
It’s better to think of running algorithms up to a certain point, and then terminating and having an approximate solution. That’s what applied mathematicians do. Pure mathematicians pretend that they can run algorithms to infinity to get exact solutions which they can’t write down. So they have an elaborate language of talking about all these objects which are not actually obtainable.
I think that how mathematics is framed has sociological (and economic) implications to how humans develop technologically. Therefore, questioning mathematical assumptions ( such as ability to perform infinite tasks) is sociologically relevant. I thank Prof. Wildberger for bringing this question into my attention. May his efforts continue to be a blessing to all of humanity like the mathematicians of the old.
I get your point. But the sociological relevance of questioning is not always positive. Take flat-earth, vaccination and global warming, for example. Every child and teenager has to question these concepts as they learn about them. But the questioning of these concepts openly in society is doing nothing but harm. The same way, this prof. does not have valid claims and for any reason (that I won't judge here) failed to check himself. All this is doing is confusing the public, essentially it's disinformation.
@@Kav107 No it is not. Mathematics describes real physical processes. How are the processes transformed into human knowledge? Through complex social processes. Ignoring the observer would be poor science.
@Ben Catechi mathematics doesn't describe -just- real physical processes. It also describes impossible physical processes and predicts physical processes in which are un-testable. In fact it's logic is self contained. Concepts of unity, zero and infinity remain constant at any point in the universe and bypasses society or viewer. The simple fact is, it doesn't matter what society you live in..here there or anywhere. Or if there was no society at all. The concept of 1+1=2 is just that.
@@Kav107 No. Those concepts refer to actual physical processes. The problem is you think that a physical process as newtonian mechanics, when really a physical process can be a lot more complicated than that, including human society itself. Ideas can be objectively correct, but the form those ideas take is highly subjective. I'm not arguing that there is no such thing as objective truth. I'm saying that your insistence that human mathematical and logical systems, capture that objective truth without being filtered through society only puts your knowledge farther away from actual objective truth.
Any engineer would ask but how do you explain that the values we calculate using these perhaps slightly flaky procedures yield accurate results. Its worse for the physicists who sum known divergent series that miraculously agree with measurements to many decimal places.
The answer is that you don't. Arithmetic with real numbers is literally impossible. What you do is convert numbers like pi and e into rational approximations at the 11th hour before doing the actual calculation.
Of course, a divergent series doesn't produce a specific value. What has been been done instead is to introduce extra operations, such as weights, that select only some terms within a filter, which amounts to admitting that outside that range the mathematical model is not a good fit. There are other similar "regularization" techniques, not all of them being logical. One misadventure comes from bosonic string theory, where one of Ramanjuan's sums has been used to replace the sum of all natural numbers by -1/12, and thus the spacetime manifold's dimension was "proved" to be D=26, whereas the replacement wasn't necessary in the first place since the unmodified model produced D=2 without applying any trickery. (The original hope was that the model ought to "prove" that D=4 and thereby validate the theory.)
In the replies to this comment, I will present an extensive breakdown of the video point by point and a conclusive analysis. Please bear with me since this is an extremely long breakdown.
0:32 - 0:37 Contention in the community? I have no idea of what you are talking about. Arithmetic is not at all controversial right now in the mathematical community. 0:46 - 0:48 I assume you will define what "fake arithmetic" is later, correct? 2:41 - 2:51 This is not really correct. To begin with, the phrase "number system" is completely meaningless and ambiguous as you have presented it. Also, arithmetic operators are defined axiomatically, thus it is inaccurate to present operators and their axioms separately. To further worsen your argument, in mathematics, we often perform arithmetic with objects that, generally, are not considered numbers, such as sets, vectors, matrices, higher-order arrays, and other types of formal abstract structures. Many of them have no absolutely no relationship to the real numbers. A more accurate presentation of the topic that is fit even for non-mathematicians, such as sociologists, for example, would be to say that an arithmetic underlies (1) a collection of objects (2) a collection of rules that prescribe how these objects behave. 3:37 - 3:52 No, this is completely incorrect. Firstly, there is no such a thing as "the set of decimal numbers," and pretending that such a set exists, when you are presenting this to an audience which you yourself acknowledged is not necessarily mathematically inclined, is very dishonest. All this does is create misconceptions and confusion for the sociologists and anger the mathematicians and logicians. The set should merely be presented as the set of real numbers, and then just provide some examples, such as the numbers π or the golden ratio φ, numbers a non-mathematician may be familiar with, and this will give them an idea of what you mean without being misleading. Secondly, those numbers are not contained in the set of rational numbers. To the contrary: the set of rational numbers is a subset of the set of real numbers. 5:26 - 5:34 Once again, this is misleading, because as I stated earlier, mathematical operations are defined based on certain properties, not the other way around. 6:20 - 6:41 You need to be careful with how you are wording what you say, because first you said these properties need to be proven, but then you stated that you need very precise definitions, not acknowledging that operations are themselves defined by properties. The problem is that you are starting this discussion without even having introduced the terminology properly to sociologists, and you have not explained to them what is the distinction between an axiom, a definition, and a theorem, and how those are all related. 9:01 - 9:23 I am not sure if this claim is supposed to be general about arithmetic, or this is only specific to floating-point systems and other similar things in applications. However, *just in case,* I need to clarify that this claim is only true precisely in an applied mathematics context. In general, though, this is not the case. Mathematicians generally work with exact, precise arithmetic, not approximations, unless they are studying approximation theory.
9:58 - 10:09 I am glad you acknowledge this, because a very popular misconception among non-mathematicians is that this is not the case. 10:45 - 10:54 Presenting this as the key point is somewhat misleading, though I understand what you were trying to get at. The set of real numbers are defined via limits, be it indirectly with Dedekind cuts, or directly as equivalence relations of Cauchy sequences. In either case, these numbers that are expressible as an infinite string in decimal digital representation are just the limit of a sequence, and it so happens that this limit has at least one decimal representation. 10:55 - 11:09 I seriously hope you are not going to present this video with the premise that infinity is a problematic concept. Such a premise is fundamentally flawed. 11:45 - 12:00 This is completely and beyond inaccurate. π & e as numbers are much older than what you are claiming them to be. The irrationality of numbers such as e & π was proven centuries ago, but in addition to that, you are neglecting to mention that *algebraic* irrational numbers, such as sqrt(2) and φ, were known to exist and to be rational by the Greek millennia ago. Constructible numbers in general include those irrational numbers, called quadratic irrational numbers, and they were of great importance to the Greek. I am no sociologist, but I do know also that in other ancient cultures, some of the metallic ratios were considered important as well for at least some applications in architecture and the visual arts. As for proving that their decimal digit representation required an infinite string, yes, this did occur later - though not that much later, but the existence itself of these numbers was known for a very long time. So saying that it was only now that we had to acknowledge that other types of numbers besides the irrational numbers exist is just false. 17:56 - 18:00 Challenging? Yes. Problematic? No. 18:05 - 18:14 This is false. Others in the comments section have already given examples of this, so I myself will not bother having to repeat what they said, but you should know you can also do a quick search on Google and see for yourself that such algorithms do exist, and applied mathematicians will tell you this. Many of these algorithms have already been implemented in computing, though to a limited degree, obviously. Stating there is no algorithm is inaccurate, and this is stemming from the fundamental misconception that there is only one valid sequence of steps to add any two numbers expressed in decimal notation with a finite string. Actually, you need not appeal to decimal strings to add real numbers at all. 19:19 - 19:24 Sure, for computable numbers, this is true. I fail to understand how this is problematic. At best, all this implies is that string representation is limited, and so are computers, which is a moot point, because computers are already limited anyway by virtue of physics. You can never accomplish with a computer all that a human can accomplish because that is just the nature of computers: they are different from humans. However, you need not know what the decimal string representation of a number is to be able to work with the number in the relevant context and understand it. Also, you already discussed earlier how, in applications, it is strictly necessary to have a flexibility for truncations and for approximations, so this is not a problem. These numbers are still definable, and from the definition alone, you can always make some amount of progress. Also, it should be noted that even the formal definitions of computable functions from any given computing model appeal to a concept of theoretical lack of limits about a given thing.
19:25 - 19:30 No, this is not true. There are plenty of real transcendental numbers which are computable numbers. For example, this is true of π and e. π + e is a computable number, as are π & e themselves. More impressively, πe & π/e are also computable numbers. Not only are they computable, they are efficiently computable, because there exist fast-converging sequences of partial sums of a function-generated sequence with which you can calculate these. Computing them is easier than proving their rationality or lack thereof. 19:31 - 19:39 Yes, you can. Your statement has been thoroughly debunked. The mathematical literature on the computational analysis and computer science, as well as the existence of irrational numbers that are also computable numbers is almost a century old and is quite enormous. I have nothing else to add here because, as I said, the literature is overwhelmingly large and comprehensive, and it does far more than just addressing the false claims you are presenting here. 19:44 - 19:51 What? Nonsense. Because uncomputable numbers exist and you fail to understand how they work, this means real number arithmetic is fake? That is just now how it works. An operation need not be defined algorithmically. In fact, they are not defined algorithmically. Some functions are computable, and some are not, but what makes a function satisfy the definition of a function has nothing to do with computability. All a funcion is, it is a subset of the Cartesian product of two sets, such that for every element of the first factor, there exists at most one element of the second factor which the former forms a pair that is an element of the subset. 19:53 - 19:56 They are properly defined. They are the supremum of a set of numbers and the infimum of its complement, they are the limit of a set of Cauchy sequences. The limit operator, the supremum operator, and infimum operator, as well as the sets they act on, are all properly defined. Cauchy sequences are well-defined. Therefore, the numbers are properly defined. There is nothing problematic here. I fail to understand what the problem is. Uncomputable numbers exist? What exactly is the problem with the existence of uncomputable numbers? Mathematics is a collection of formal theories, not a prescription or description of how the physical world operates. 19:57 - 20:03 Your presentation of real analysis as it is understood by mathematicians today is completely inaccurate, as is your presentation of the understanding that we have of set theory and group theory. I cannot tell if this is deliberate dishonesty, or deliberate ignorance and misunderstanding of the literature, but regardless, the fact that you are presenting these obviously incorrect claims to an audience you yourself acknowledged may not have the sufficient education to be able to discern as true or false for themselves in order for to present constructive criticism and logical refutations, is completely abhorrent. There is no denying that some of your other work is respectable, and being allowed to present skeptical viewpoints is necessary for the community to work, but when you actively try to mislead an unknowing audience that is seeking to be educated on a topic for the sake of getting a validation card, that is when the conversation stops being about skepticism and it starts being about your ethics. This is not a critical argument you are presenting about mathematics, you are merely trying to fill an agenda disguising it as a presentation on mathematics you happened to construct very poorly. 20:20 - 20:28 Yes. Sometimes, some things are too complicated to explain properly. You want to tell you how the Big Bang works? Well, I cannot explain that to you if you lack certain prerequisite knowledge to understand the explanation you are asking for. That is just how life works. You cannot run without first learning how to walk. This is supposed to be a flow of the educational system? The education system is completely broken, I agree with this, but not for this reason. This is also by no means a flaw with real analysis, this is ultimately just a mere inconvenience inherent in the way learning fundamentally works. I cannot teach you about computable numbers if you are still trying to learn how to add numbers, so of course I would have to tell you "that is for a more advanced course." 20:28 - 20:38 Such as?
20:38 - 20:41 You cannot seriously say that given the hundreds of thousands of number theoretic proofs that exist, not only on the theorems of arithmetic themselves, but even on the effective finite axiomatization of arithmetic or impossibility thereof. This is just more dishonesty. If you could at least bother to give examples, then that would somewhat help your case. 21:33 - 21:38 What does that actually mean? If by "evaluate," you mean "to compute the complete decimal digit string representation of the number," then sure, it is true that we cannot write such a number in a complete decimal digit string representation. However, this is an entirely arbitrary, unnecessarily restrictive, unhelpful, and misguided definition of the verb "to evaluate." Such a definition would also imply that numbers such as 10^(10^100) and TREE(3) are not actually numbers that exist, because it is impossible to write the complete decimal digit string representation on a paper. This is particularly true for uncomputably large numbers, such as the busy beaver numbers and the Rayo numbers. You may as well deny the existence of any natural number larger than 10^(10^80). Also, this claim seems to point to a fundamental conflation between the symbolic representation of a number and the number itself. The symbolic machinery you so despise is employed to represent *every* number, because numbers are fundamentally abstract conceptualizations not present in the physical world, so a physical actualization of the pure concept of the number is impossible. When I say every number, I really mean *every* number. The symbol "2" is just that: a symbol for the number. Other languages have other symbols for it as well. The symbol should not be confused for the number it represents. In this regard, if your logical deductions are consistent, then using the symbol "2" in Kindergarden arithmetic is no less problematic than using the symbol "log(3)" to represent the irrational numerical value it represents. In fact, why should you care about decimal digital string representations when computers operate in binary digital string representations? We should be representing the number as 10, not 2, at least if we want to maintain logical consistency with what your comments seem to imply. By any sane and reasonable definition of the verb "to evaluate," these symbolical representations are already trivially evaluated, because we already know what abstract object to identify the symbolic representation with. The equality relation, when written as part of a symbolically-written equation of two expressions, merely identifies a representation of a number with another representation. The number itself is already known from recognizing the value represented by either string of symbols, or from using definitions and axioms to simplify the expressions via identification with other expressions: other string-symbolic representations of the same number, or object, more generally, since this also applies for matrices and vectors, for examples. This process of identifying an expression written with a string of symbols representing an object with another expression whose corresponding number value it represents is already known and recognized, is what mathematicians call an evaluation. It has nothing to do with computing a string of digits in any partocular base, be it decimal or binary. Mathematicians do not think of digital strings as being numbers. These digital strings are merely how we choose to represent them because they are a very historically convenient way of actually keeping track of tallies and counts. You need to stop thinking as if numbers had to be fundamentally representable this way to be valid. Plenty of non-mathematicians, and especially younger students, think of the decimal representation as the number itself. So if you tell them 0.(9) = 1, they lose their mind, because they are incapable of conceptualizing the idea that a number can be represented by a decimal digital string in two different ways or more, because they think of each representation as a number, not as a representation. This would be akin to thinking that 2/4 and 2/1 are different rational numbers because the integers being divided are different. Strangely enough, they also may find themselves thinking that 1 and 1.0 are different numbers. This is a mistake that we need to address. 22:15 - 22:18 It is interesting that you mention γ in your list of examples, because it actually demonstrates why your previous claims are silly. Did you know that it is not known whether γ is rational or not? It has been proven that if γ were to be rational, then it would have an astronomically large denominator. Writing the number in quotient form would be completely unfeasible. It also would have a decimal digital string representation of infinite length. This would be an example of a rational number that, according to your argumentation, could not exist, it would be a "fake" number from a "fake" arithmetic. Also, independently of whether γ itself is rational or not, such rational numbers do exist, you even mentioned them earlier.
23:26 - 23:48 Wait, so you ARE arguing that these rational numbers are "fake" as you would call them because they cannot be written? This seems to contradict statements you made earlier in your presentation. Anyhow, I am glad you are at least now choosing to stay logically consistent, although you had to go quite late into the video to make the decision. 24:05 - 24:09 This infinite sum is an example of a computable real number, so it is inaccurate to say you cannot calculate it. Anyhow, yes, that symbol is the number's "name," whatever that means. Numbers do not have names. We assign them names because we humans need a way to refer to them, but the names depend on the language we choose to communicate with. As far as the conceptual abstract object that we call number itself is concerned, though, it has no name, and it does not care if it as a name or not. It just is what it is: an abstract object that has the value of a quantity. 24:39 - 24:42 To this moment, you have failed to successfully explain why it does not work. All you did was claim there is no algorithm to compute these numbers, which is false, and then proceeded to say that the entire system is ill-defined without any proof of this, and then began to proliferate arguments that are based on flawed premises and misunderstandings of concepts and of the current paradigm. A lot of finitist ideas are based on misconceptions and a lack of understanding of concepts as well, but here, as I stated earlier, I get the impression that you maybe do understand the concepts, and just pretend not to for the sake of an agenda. I still have not been given sufficient reason to forgive the amount of dishonesty in the video, so pardon me for continuing to repeatedly call you out on your intentions and your dishonest presentation of the concepts to an unknowing audience. Anyhow, this was all a rant just for to explain that, no, you have not proven the system does not work. All you did was present a few claims that can be easily shown to be false, and decided to move on from there. 25:14 - 25:21 So you are telling sociologists to prevent us from being dishonest in the exact same way to decided to be dishonest throughout most of the video by completely dodging the question and failing to present a proof of concept? Yes, I approve of that, although it makes you seem like a hypocrite. 25:35 - 25:39 Hearing you say this is quite hilarious, when in reality, the entire premise of finitism is reliant on philosophical obfuscation and not any sort of operational formal theory of logic. 26:03 - 26:09 π + e + sqrt(2) = 7.274088044421933522624619578841863460524088368452013469118591958 What? You wanted more digits? You never said how many digits you wanted the answer to have. This is what you get for asking a loaded dishonest question and not giving any room to explain why it is a silly question. You wanted an answer? I gave an answer, and you *have* to accept it, because you said you would not allow for any jargon or philosophical arguments about the validity of the question or the answers
If you adding/multiplying real numbers then you are an engineer. That also means you know how to calculate the margin of error in the bottom line so that the bridges would not fail due to rounding errors. If you are a pure mathematician you don’t really need to deal with such busy work.
I think strictly speaking engineers use rational approximations of reals, since reals can't be computed. Maybe some are computing pi exactly, but it might be a long while before those bridges get built.
@@Robinsonero Irrational numbers do not fit in finite number of bits but that is no reason for not building bridges, skyscrapers or spaceships. We can calculate the upper bounds in our calculations and we always use a safety factor appropriate for the application in engineering.
@@EnginAtik I agree, we can rely on rational numbers, including rational approximations of pi, phi or e whenever we need to compute or manipulate numbers (a process known as math). Of course the pure complete transcendental notion of these numbers can't ever be obtained by mathematical operations, and they can't be used mathematically either. We can philosophize about the nature of transcendental values but we must always retreat to the rational numbers whenever we want to do actual math.
You ended with the sociological question "how did we get here in the first place?" and I say we got here because the Industrial Revolution needed bodies and the sensibilities of the Industrial Revolution valued the Engineer above all else. Whether consciously devised or simply iteratively emperically driven, it was accepted that the gross human capacity for rote memorization vastly outweighed gross human capacity for abstract thought. In the absence of omniscience required to know which 6 year olds were the best "investments" of effort, mathematics were framed with infinitums and inherently unanswerable impossibilities to discourage people from asking "Why?" and encouraging them to just move onto their next task on the educational assembly line; if the output was good enough for government work, it was deemed sufficient and it never mattered to them that they were discouraging people of seeker deeper understanding by presenting them the illusion of results commensurate with a "Higher Power."
Prof. Wildberger: The problem is that you are removing the real numbers from the number line without considering what's left in their place. This leaves your math to be far less powerful than infinity-based math, which is why I suspect that it hasn't become mainstream. When you remove the real numbers from the number line, what you are left with are continua in their place. When you do not complete infinite summations, what you are left with is a potentially infinite process. Only by incorporating continua and potentially infinite processes into your philosophy can finite math compete with infinite math. If you want to see what I'm talking about, check out this video on a finite perspective on the dartboard paradox: th-cam.com/video/LQmwZZNUMNA/w-d-xo.html&t I'd love to discuss potentially-infinite math with you if you're interested.
@@ThePallidor By incorporating "potentially infinite process", I am not suggesting that we complete them. I'm saying that "potentially infinite processes" have value in themselves even if they cannot be completed. Pi and e may not be numbers, but they're still mathematical entities which can be finitely manipulated. Rejecting them leaves the resulting mathematics with a big void.
@Gennady Arshad Notowidigdo I view these 'potentially infinite processes' as computer programs which do not terminate. If we interrupt it, then we produce a rational number. This is what you are talking about in how it has value in applied math. I agree, but I am not talking about this. I am talking about studying the programs themselves and seeing how the programs relate. Although these programs cannot complete their output, they can often be written with finite lines of code and manipulated with a finite number of operations. I see no issue with rigour in this pursuit. The issue is only when we believe that these programs can be executed in completion. While I do believe that it is extremely hard to make the establishment change to an entirely finite mathematics (I'm fighting this fight too and I feel the pain - I've made many TH-cam videos in this area and I struggle to get views), I think Norman's primary problem is that his version of finite mathematics is lacking. Specifically, it's lacking because his mathematics is based on points (which are inherently linked with infinity) when it should be based on continua.
@@ThePallidor Yes pi and e are processes, let's call them algorithms. But who says an algorithm is not a valid object? Who says there is no value in seeing how these algorithms relate? The problem is not in working with the algorithms, the problem is in assuming that we can work with the output of the algorithms.
@@ThePallidor With a few lines of code, I can write a program to spit out the terms corresponding to a Cauchy sequence, one by one. The *execution of the program* is dynamic and can never be completed. It is meaningless to talk about the complete output of this program. But the program itself (the few lines of code) is static, finite, and a valid mathematical object. It's just not a number. Numbers are not the only valid mathematical objects. On points: Points are 0-dimensional. What's your point? On pure math: Math that is in principle beyond the realm of any possible computer is not math, it's fiction. But it is a stretch to say that math needs to be tied to some application. There are well defined finite numbers that have no connection to anything physical. Are these "nothing"?
@@ThePallidor "The problem comes when they are presented as numbers" Agreed. "Points are dots of definite but negligible width" Are you saying that a point has non-zero width?
I'm not sure I understand what the problem is--I watched the video and it seems like "lazy evaluation" is a solution; I've always thought of real numbers as "recipes" for cooking up decimals, rather than decimal objects in and of themselves. If you combine two different recipes using a procedure which results in a third recipe, then that seems like a satisfactory way to combine recipes. For instance, we'll take the case of pi + e. For simplicity, I'll leave out the sqrt(2), but it will be clear how to re-incorporate it once I have explained the procedure. "Pi" and "e" are recipes for cooking up decimals--the real numbers are the recipes, not the decimals. We start cooking: First, we add the 3 + 2 = 5. Then, we mix in the 0.1 + 0.7 = 0.8, sift out the 5 from 5.8, and set it aside: we are now confident that 5 is the ones digit in pi + e. We then carry on in this fashion to obtain more and more digits of the decimal number--that is, we have obtained a recipe for calculating the real number which is result of "+(e,pi) = e + pi" (which we should not expect to have a better symbolic identity, because lists of symbols in any alphabet are a countable set, and we know that the cardinality of the real numbers is uncountable). We took two recipes and combined them to cook up a third recipe--a perfectly reasonable addition of two real numbers. I think the (much) more interesting problem which isn't really talked about that much in this video is that determining if two real numbers are equal, in general, is an undecidable problem (which is why we need to provide proofs that two numbers are equal--because it is a small miracle when it happens!). If you have two recipes for decimals, then you can compare them digit by digit. If they ever differ, then you can say if one is different from the other (moreover, you can say which one is bigger). However, if they really are the same, then your procedure doesn't halt. That is, you can never purely computationally prove that two real numbers are equal in finite time. Instead, you need metacomputational methods, including proofs and abstract reasoning.
@Alexander Sanchez Trying to get a theory of "real numbers" using recipes/algorithms/computer programs is tempting and it is a reasonable thing to try to do. Sadly no-one has done it yet. It appears to be much easier to just suppose that such a thing could be done, and then move on to more "advanced and interesting topics". I will be touching on the computational aspects of a much more restricted and tractable problem --- how to do arithmetic with repeating decimals. Turns out this has not been worked out carefully either, as far as I know. It is absolutely remarkable how many important foundational issues have been abandoned to scrabble after the ever-diminishing returns of yet more sophisticated, narrow and incomprehensible investigations.
@@njwildberger I appreciate your thoughtful response, professor! I'll be interested to see what you have to present; it seems like one potential problem is the fact that you can get repeating 9's and never get anywhere. One way I know that actually fixes this issue is to use "bijective numeration" instead of the standard alphabet of symbols--that at least makes it so that you're always guaranteed to make some progress. However, there seems to be a difference between the question of the existence of an algorithm and the practicality of such an algorithm. I wonder: what are your goals for a satisfactory arithmetic with repeating decimals? Take care!
There’s no problem with algorithmic addition of Cauchy sequences of rationals to arbitrary specified precision, as long as we can query the Cauchy sequences to arbitrary precision (an epsilon/2 argument). This is unrelated to the ambiguity in the decimal representation of real numbers.
That would only be an approximate answer, not a true, exact answer, in the pure mathematics sense. 1/2 + 1/3 + 1/5 = 31/30. That's the complete, true, exact answer. It can be done for rational numbers. What's the complete, true, exact answer to π + e + √2, using whatever definition of real numbers you please? That's his point. You can't do the same thing with so-called 'real' numbers that you can do with rationals. Especially when you need to actually write down the answer, rather than just talk about the 'answer' as if it were already written down. For example, what is the explicit, written down "algorithmic addition of Cauchy sequences of rationals to arbitrary specified precision" answer to that calculation? Now suppose the 'arbitrary precision' we're specifying is 'exact precision', for usage in pure mathematics, rather than an approximation; can you actually exhibit such an explicit, written down answer?
@Rob Harwood "Especially when you need to actually write down the answer, rather than just talk about the 'answer' as if it were already written down." {Sorry for my English, I speak Spanish.} This comment strikes me... We are not carefully distinguishing the Experience from the "talk about the Experience". This is very common in our present way of relating to our observations, ideas, speech... We pretend, and many many times we don't know we are pretending. We confuse, take our beliefs for granted... instead of checking them. We finally must learn to listen, deep listening to do Math & Science. Otherwise we sell "pescado podrido" (rotten fish, idiom) to our students. Thanks for this video, and all the material. Thanks Prof. Wildberger. Diego Vallejo / Prof. Cálculo / Facultad Ingeniería UNLP/Arg
Could you explain why computational short cuts are to be expected? For example, in a free magma there are no non-trivial identities. Is it fruitful to assume that all expressions are equal to a "simpler" expression? It seems that mathematics is in the identities, not the actual computation. (btw I kind of agree with you - I'm just playing devil's advocate)
Here is how the defense might respond: That two unknowns should be equal does not determine what either of them is; it only determines that they are not diverse according to quantity.
The definition of a Dedekind cut seems to be circular since every number is defined but all the numbers it is not, put into two infinite sets of those numbers larger and smaller respectively. Is this a paradox or am I missing something?
It is a common technique/principle which is not bad in and of itself. Imagine you have two approaches C and D to a single ostensible phenomenon P. Here P is "real numbers" and C is some vague conception of them, which nonetheless is meaningful to you personally. D on the other hand is the axiomatic notion of a Dedekind cut, which does not reveal the purpose of its being, but which is hopefully more strict, less vague, more communicable. Now let's agree that sqrt(2), pi and e ought to be examples of the phenomenon P. You are assumed to have some understanding of this within C. What you can attempt to do is you characterize these examples by abstract statements involving C. Then you reinterpret these as statements involving D instead of C. If you are lucky, they are still true when reinterpreted in D, and if you are very lucky, they characterize certain things uniquely (which we should call sqrt(2), pi and e) even according to approach D. In this way of analogy, knowledge is transferred from C to D! (Furthermore, if a new approach E, still to P, is discovered later, you should reinterpret the abstract statements above as statements involving E instead, to test this approach and hopefully learn what sqrt(2), pi and e must mean according to E.) Not sure if that makes sense the way I have put it. I learned this in category theory, which offers more insights into abstraction than set theory, but which does not provide a good (powerful) foundation for real numbers either, as far as I can tell. A simple example of an "abstract statement" (as used above) in category theory is the "universal property of the product", which you can look up if you want. This "abstract statement" allows you to decide what the analogue of the cartesian product should be in various niches of mathematics where set theory is not the primary object of study.
It is a bit weird but if you plot y=x²-2 then if the x axis consists of only rational numbers then the graph does not cut the x axis for any value. i.e. the x axis has infinitely many gaps. The presentation seems to ignore these points.
@@joecotter6803Yes this is a necessary consequence of working over the rational numbers rather than over the (Archimedean) metric algebraic "real" numbers. This problem can be somewhat "solved" in a certain sense however, without admitting transcendental numbers of any kind, by just providing "real places" (or "complex places") for various algebraic numbers. If we still insist on working over just the rational numbers, then we get counter intuitive results about under which circumstances various conics intersect, or whether certain points or lines are inside or outside a geometric figure. Iirc i have mentioned this to prof. Wildberger in the yt comments of some of his (highly promising!) lectures about geometry, especially the universal hyperbolic geometry and rational trigonometry lectures.
So, professor Wildberger, you want to know what pi+e+sqrt(2) is? Well, pi=3.1416±0.0001; e=2.7183±0.0001; sqrt(2)=1.4142±0.0001, therefore pi+e+sqrt(2)=7.2741±0.0003. One may never represent an irrational number with a nice fraction, but we can get the error as small as we want, given time to compute. I have been teached about real analysis and IEEE-754 computer floating point arithmetic system, physics &c., in my time in university. It amuses me, that more than often, we have to break an integral formula to finite sums to give the task to computer i.e program it. But these integral formulas have been already achieved by (Riemann) summing infinitely. But even if it amuses me, I certainly wouldn't want the equations start with summation symbol, capital sigma. The curve of the integral sign, the dx,dV,dxdy, they talk to me. They pleasure my eye.
Hello, just to recap are you saying that addition for real numbers doesn't work because it will take infinite time to finish ? then what about computing natural numbers that takes a long time to compute ? Like Ackermann function which deals basically with natural numbers. For example it is estimated that A(4, 2) will take roughly 2^(65536) steps to compute. That's why I'm interested of your insight for those (natural) numbers
Doesn't one have the same problem with fractions? 6/3 evaluates to 2. But if you apply the division algorithm to 1/3 it never terminates. So we simply keep the symbol "1/3" and claim that's the real number. So if one can claim that 1 can be divided into 3 equal parts, why cannot be claimed that a unit circle has a circumference?
@otraguardia Our use of fractions, or rational numbers, to seemingly avoid division is interesting and warrants discussion. When we write 1/3 we are not so much dividing 1 by 3 as introducing an entirely new symbol that stands for that operation in our minds, but is logically separate from it. Is this yet another example of empty manipulation with symbols? No, because we define the arithmetic with fractions consistently in terms of the symbols we introduce. Thus for example we define a/b+c/d = (ad+bc) / bd . However one problem, or rather challenge, with this is to deal with the multiplicity of forms of a fraction ie 1/3 = 2/6 = (-5) / (-15) etc. More generally we declare a notion of equality a/b = c/d precisely when ad-bc=0. But that does mean that we have to check that our operations and statements are well-defined --- that is independent of the forms used in the definition. Doing this for the addition above is an important exercise that I bet 90% of math majors have never done. So yes, our arithmetic with fractions certainly warrants some criticism. But it is able to respond! Note how different this is from the "real number" symbolics. With the "real numbers" there is no set restriction to the number of new symbols invoked. And crucially there is no clearly defined test for equality. How do we tell whether my algorithm / app for an infinite decimal is the same as your algorithm /app? If we run both to a certain extent and discover some difference in outputs, then we can confidently declare the "real numbers" different. But how to tell if they are the same if they actually are? This turns out to be a theoretical impossibility in general! Yet another point where analysts instinctively know: the less said the better.
From watching most of his videos on this topic, I think it's safe to say that he considers all forms of the 'reals', i.e. all the forms that have been proposed so far, to be deficient in one way or another. Usually it boils down to the necessity to actually 'finish' an 'infinite' computation in one way or another. (Although that is not his only objection. Some definitions of 'reals' have other problems as well, for example, and in particular, anything involving the 'Axiom of Choice' or an equivalent.)
@@robharwood3538 said "Some definitions of 'reals' have other problems as well, for example, and in particular, anything involving the 'Axiom of Choice' or an equivalent.)" I don't know of any definition of the "reals" that use the axiom of choice. Do you have an example?
@@billh17 The ZFC axioms. I guess you're right in the sense that they don't *actually* 'define' what a set is, and so -- since reals are supposedly founded on set theory -- they don't actually define what a real number is, either. But that's just another problem regarding the fakeness of real number arithmetic then, right? 😉
@@robharwood3538 answered "The ZFC axioms." Ok, but I was assuming that we were in the framework of ZFC. The standard definitions of "reals" don't depend upon the axiom of choice. @@robharwood3538 said "they don't actually define what a real number is, either" But this video itself says "real" numbers are defined in ZFC by Dedekind cuts or by sets of equivalent Cauchy sequences.
@Account Name; Actually the p-adics are much more tractable than the "real numbers". Their arithmetic is much cleaner and actually more interesting. Have a look at th-cam.com/video/XXRwlo_MHnI/w-d-xo.html where I call them "reversimals" for reasons that are closely related to the superiority of their arithmetic. But you still have to be careful!!
Computers use floating point because real number arithmetic is impossible to actually implement. Yup. You can say that the approximations are good enough. But in some uses, if something is exactly 0.0, or exactly 1.0 it determines what TYPE it is (ie: of a multivector, you calculate the type: scalar, vector, bivector, trivector... based on what coefficients are 0.0). Two unit vectors are parallel if their dot product squared is 1. They are perpendicular if their cross product squared is 1.0. So, if you do a calculation and get 0.000000001 for a vector part, you can't decide whether it's supposed to be a vector. In floating point, you CANT use the "==" operator reliably. Of course, floating point is excellent for engineering approximations. But you cant use it for proofs, in general. But I would also say something that NJW might object to.... You can't really implement exact FRACTIONAL arithmetic either; though you can know when you DO have an exact answer - which will be the case in most practical situations. If have a sequence that multiplies a whole bunch of rational numbers; you reach a point where you may not have enough digits for the numerator and the denominator in practice. If you don't run out of SPACE, calculations get so slow that you run out of TIME. Anything that has O(2^n) running time is not calculable in practice when n actually gets large. An example with Fourier transforms: As long as the best known algorithm is O(n^2), you can't really do anything useful with it in practice. An O(n lg[n]) algorithm was discovered for it, so suddenly it's useful in practice. In fact, these RUNNING times, and RUNNING space are MORE relevant than numerical accuracy for doing engineering work. When we go back to symbols, you can almost think of "pi" as an irreduceable "unit"; as is "e", etc. ("i" is a unit, but it's an ambiguous unit, as it doesn't specify the plane or orientation that it's in; which may or may not matter, depending on what you are doing; it is a thing that squares to -1, without identifying which thing it is. Clifford Alegebra (ie: Geometric Alegebra) fixes this.) Other units are algebraic, and not numeric, like: direction unit vectors: e_1, e_2, epsilon^2==0, etc. You can "compile" a solution that is symbolic, and the most convenient form. But you need to plugin floating point to "execute" an actual answer that can be pretty close. And some expressions that are supposed to be identical according to algebra can give wildly different answers. StdDeviation calculation for instance will soon start emitting NAN just due to constructs like: a^2-b^2 ... just being plus or minus a tiny amount can cause it to divide by zero when it should not, or flip sign...which is a catastrophe when it's part of a division. In engineering terminology: the SIGNAL is less than the NOISE in these cases. Noise gets catastrophic in the presence of feedback loops. Engineering can work in the presence of unhandled noise. Proofs must explicitly handle noise. So, people need to go to Donal Knuth books for better behaved recurrences. This is because fundamentally, floating point arithmetic isn't totally commutative, associative, etc. It's very close though.... So you can use it for engineering. But you can't really use it for proofs.
Note about actual computer number systems: - integers are typically 16, 32, or 64 bit. Mostly 32-bit for practical cases. You need 64-bit for things like file sizes, and timestamps. - doubles are 64-bits of floating point. typically this means that the integer part is only 2^53, and the other bits are sign and exponent. this fact will definitely come to you in a bug one day, probably when you are shooting json between javascript and some server-side language that expects int64. - When you write a number in decimal, it will store it in binary. - The unbounded-precision options, as noted before let you at least know WHEN it's accurate, but that works for large integers. It doesn't solve the problem of representing the fractional part at unbounded precision. - In any case, an O(2^n) solution for large n is just as incomputable as an infinite process. This is the basis of all cryptography. - In cryptography, there is no such thing as an epsilon. An answer that is off by 1 ulps will hash to a completely different value. So, what would be ideal to deal with this: - be EXPLICIT. An int64 is mod(2^64). - An arbitrarily large integer is "mod infinity"; so even with arbitrary precision, you may not be able to represent it. This is EVERYWHERE in cryptography. You can only work with a few thousand bits in practice. But note that you can know WHEN you have an exact answer. - We WRITE numbers in base10, but store them binary. Since that's not equivalent for the fractional part, perhaps writing them in floating point hex at least removes the noise contribution from input/output notation. - When doing +, -,*,/, etc.... have a number system that tracks guaranteed bounds. A noise term. - Have noise terms everywhere, like a carry. In practice, we should be making better libraries for algebraically manipulating math; and stop working on paper. When put into a machine, the work is reproduceable. Any issues with that exist become difficult to hide. Referreeing work should be a matter of determining its relevance; as it should not be able to REACH referrees until it is reproduceable. It's a big loophole that we judge correctness with people. Just like discovering things such as "2pi = tau" as breaking patterns that should match automatically; not all of the problems we will run into will invalidate prior work. But there are some cases where notation is clearly wrong. (See Johnathan Bartlett about: d^2[y]/(d[x]^2) is literally wrong. And it's easily fixed such that differentials divide algebraically like they should. It's just that higher derivatives are actually slightly more complicated with terms that are usually zero: d^2[y]/(d[x]^2) - d[y]/d[x] * d^2[x]/(d[x]^2) = actual second derivative). The mathematics is like a programming code base that has not undergone any refactoring. The advent of computing has triggered an avalanche of discoveries of notation that is wrong in some way; but was understandably done because without computers it is too verbose to handle explicitly on paper. This is what happens when you drop down from your high-level programming language to look at the actuall assembly language at the bottom turtle. (The bottom turtle of hardware is basically NAND gates - logic circuits can be NAND gates with no loops while memory is NAND gates with feedback loops. But the bottom turtle of the software is the assembly language. There isn't an assembly language for mathematics; as it's all too waffly right now.)
The thing I don’t quite understand and would like to see you address is π. It’s one thing to be able to deal with an algebraic number like the square root of 2, where you can find a finite representation (which is totally fine and quite useful). However, π, whether you like it or not, has a specific, physical definition: the ratio of any circle’s circumference to its diameter . Hence, I suspect you will have much trouble attempting to wave your hands to say it does not exist and be able to convince many others. So, how do you propose to fit π into your world of only (some) rational numbers and finite fields?
Pi is quite different from a number like 2 and as such, should be treated differently. It stands on another level. As to its approximate value, then it becomes a regular number like the rest and can be used in applied math. But as a pure mathematical object, it’s obviously more than a regular number. More work is needed to see how it can best fit in.
@Joseph Williams I agree. Probably we are just starting out learning what "pi" really is. It certainly is much more than just some other obscure "point on the number line". Getting over this intuition --where all the "real numbers" are somehow strung on a clothesline, all packed in right next to each other (sort of) with a minimal set of rationals and quadzillions of irrationals in between them: that entire intuition needs to be seriously overhauled. "Pi", whatever it is, is a hugely significant and multi-faceted mathematical object, certainly more than just one more string of digits 3.1.415... as we currently believe. Isn't exciting to realize that we are basically at the infancy of a serious analysis?
@@njwildberger What's so wrong with thinking of pi as an element of the total-ordered completion of the rational numbers? This total-ordering and completeness (no holes) is the only reason why we faithfully represent the real numbers as a continuum of points on a line... Is the notion of a completion the source of your problem?
@@schmud68 The problem is rather in the "real number" court: to explain how to compute pi+e+sqrt(2) etc. I claim this cannot be done, and that the academy is faking it. Surely mathematicians who claim there is a legitimate arithmetic with real numbers would be able to demonstrate this arithmetic explicitly and concretely.
@@WildEggmathematicscourses One can rigorously define the addition of any 2 real numbers via the addition of the equivalence classes of the Cauchy sequences of rational numbers they correspond to. Hence, rational number addition defines real number addition. One may also define the addition of any 2 real numbers via the addition of the rational numbers within the 2 LHS's of the cuts, with the remaining rational numbers for the other half. In terms of computation, as I mentioned in an earlier comment, we may only deal with arbitrarily accurate approximations of real numbers. Though, this has nothing to do with the abstract theory that defines them.
Well, you can call me a sociologist if you like. If pure mathematics was really pure then it wouldn’t be wrong or fake. If you are getting an irrational result, then it doesn’t mean that the universe or the maths is wrong, it means that you are applying the wrong math, or applying it in the wrong way in regards to how Nature actually works! To assume that the universe must be irrational, is illogical! Nature can be nothing but balanced. Prof. Wildberger is correct in his assertions that most of mathematics is just plain wrong. The reason that most of mathematics is wrong, is that it is not based on how Nature really works. However, it’s not the mathematics, but the wrong understanding of reality that is the problem. The universe is not infinite. It just produces, voids and reproduces infinitely. This, and many other false beliefs in cosmology, is the reason for irrational numbers and mathematics. True geometry and physics will produce true mathematics. We live in a polarized, electric, two-way, balanced universe, yet we apply one-way unbalanced mathematics! This must change! Just like it is difficult to visualize simultaneous events, it is also difficult to write equations that do two opposing things, simultaneously. If Prof. Wildberger has some time available, I would be happy to travel to Sydney to meet with him and explain these universal laws, natural science and philosophy in further detail. I think it is a waste of time trying to convince mathematicians that they are wrong. (Unless you really do enjoy making a career out of it!). I have experienced exactly the same problem with physicists. What has to be done in order to make any progress, is to not only explain why it is wrong, but also explain what the maths and physics of the real world actually is. Kind regards, L. Dove Arbiter - Universal Law
What does the existence of irrational numbers has to do with “balance”? As for me a circle is one of the simplest and, in a sense, “perfect” shapes. The fact that the length of its circumference divided by the length of its diameter is irrational just shows how real they are. The length of a square’s diagonal divided by the length of its side is irrational too. Both construction may exist in the real world, so how could real objects have non-real dimensions unless irrationals are as real as naturals.
@@fullfungo Excellent question. Nature reaches its perfection in carbon. Carbon is produced at the amplitude of an electric wave where a perfect cubic wavefield is produced. The ultimate forms of Nature are spheres of matter and cubic wavefields of space. These opposites are the limit that Nature can produce. Once a sphere is wound up together with its equal potential (but much larger volume) of space, however; it must, and can only, unwind and be reproduced. This basic mechanism is all there is to the mechanics of the electric wave, and to all natural forms. No reproduction of a state of motion can occur until the previous state has been voided. Therefore, in order to understand Nature, you need to understand the wave. Unfortunately, electrical engineers have not understood the correct mechanics of the electric wave, and the lack of knowledge of the exact quantum mechanics of the electric wave means that we are designing motors - coils, solenoids, armatures, transformers, etc, that do not copy the mechanics of Nature, meaning that we are not producing energy efficiently. Also this means that mathematicians, electrical engineers and physicists are describing electricity and magnetism incorrectly, using incorrect and overly complex mathematics. So, in the correct understanding of the electric wave are the entire mechanics of all creative processes, as well as the mathematics of its two opposing pressures.
I think that trying to calculate the area under a curve has to include infinite steps because the curve is supposedly composed of a continuous line of infinitesimally small curved elements going down to infinitesimal points which in reality do not exist (because everything is quantized?). So if you want to calculate the area under a "real" curve you cannot use "real" numbers or infinite steps because the real curve consists of small elements of a finite size. It may be tricky though to find out what the shape or properties of such elements would be. We would only know they are finite. So I think the solution to the problem is to find out the "real" or physical nature of the smallest elements. Maybe that's impossible and therefore we have to use some sort of abstractions. Or maybe at a fundamental level it's - as some propose - just numbers.
@Flatterman Island Please join the Algebraic Calculus One course! It is online, and currently free. And it teaches you how to do integral calculus without any hand waving about infinite processes. You can also look at the Famous Math Problems 10 lectures: th-cam.com/video/vo-ItaB28f8/w-d-xo.html is the first one.
There is no such a thing as "computationally irrational" or "rational representation." Rationality is not a property of computational algorithms or of representation strings. Rationality is the property of an abstract object we call "number." How we choose to represent it has frankly little to do with its rationality.
13:49 The truth is, a real number is a loss or gain in that which stands to that whole number called by the name of "one" in the same ratio as one magnitude stands to another of the same kind.
As usual, Our good friend Norman Wildberger is completely blind to Hilbert's synthetic approach to mathematics, and goes full-fledged for Brouwers Intuitionism. What Norman Wildberger fails to recognize in about _all_ his talks about mathematics, is that he does not understand that there are _two ways_ to understand abstract concepts. You can understand it _analytically,_ or _synthetically!_ If you understand something _analytically,_ then you _build up_ the answers. You work bottom-up. You _construct!_ Philosophically, 1. in the _analytic_ approach, you try to understand something by trying to figure out what something _is!_ You therefore _construct!_ 2. In the _synthetic_ approach, you try to understand _is not!_ you, therefore, understand in terms of _elimination!_ But the very idea, that 'what something is' is the whole story, is problematic. You can also understand something as an exclusion. To give a simple example: if I take the logical implication p -> q, or, 'if p then q', and I want to _understand what it means,_ then there are two possibilities. One is the _bottom up_ meaning. What I then mean by this statement are _three things taken together!_ These are p -> q = (p &q ) v (~p & q) v (~p & ~ q), or, in words: 'if p then q' has the same meaning as 1: p _and_ q, _or_ 2: not p _and_ q _or_ 3: not p _and_ not q Or, an example: If it rains then the roofs become wet _means_ (it rains _and_ the roof s become wet), _or_ (it does not rain, _and_ the roofs become wet), _or_ (it does not rain _and_ the roofs do not become wet) All three cases _are consistent with_ the statement: _if_ it rains, _then_ the roofs become wet, and therefore, all of them _together_ express its meaning. because, indeed, the statement says that all these three cases _can_ happen, _if_ the statement: 'if it rains, then the roofs become wet' is true. In particular, in the case it (does not rain and the roofs become wet) _can_ be true, if some helicopter sprays all the roofs with water, so that they become wet. The statement _does not exclude_ this possibility. It does not exclude _three_ possibilities. _All three possibilities it does not exclude, therefore belong to *the meaning* of the statement! In practice, this means that you can _construct_ the meaning of 'if p then q' or, (p -> q) by _building them up!_ You just _recognize_ that the implication is not just _one_ statement, but its content consists of _three_ statements! This 'in practice' is exactly the point. 'in practice' is applied mathematics. In _applied mathematics_ we _build things up!_ But in _theoretical mathematics,_ or in _synthetic mathematics_ we _exclude!_ If I want to show _the meaning of_ the statement: p -> q, and I use _the exclusion, theoretical, or pure_ approach of mathematics, then _the meaning_ of this statement is: ~(p & ~q). In other words, what I am saying is this: _the meaning of_ 'if p then q', is: NOT ('p and _not_ q)'. Or, to return to the example: 'if it rains then the roofs become wet' is the same statement as: 'it _cannot happen_ that it rains, and the roofs _will not become wet!_ This statement therefore _excludes_ the possibility, that all roofs are made _white hot,_ so that when it rains, the water falling on top of these roofs will _instantly_ become water vapor, so that the roofs will not become wet! In other words, I have understood the statement: p -> q in terms of _an exclusion!_ Let me give another example. One of the axioms of plane geometry is: 'through any two points goes one, and only one straight line'. This axiom _defines_ the concept of 'straight line' _in the form of an exclusion!_ It basically says that if we have two points, and we can draw two lines of a particular kind through these two points, then these two lines of that particular kind _are not straight lines!_ I give an example. If I have the parabola y = x ^2 - 1, then this parabola goes through the points (-1, 0) and (1, 0). But the parabola -x^2 + 1 _also goes_ through the same two points! Therefore, _the parabola is not a straight line!_ That is why the axiom: 'through two points goes one, and only one straight line' _defines_ what a straight line is, _by excluding all cases that are not straight lines!_ In the same manner, _we do not have to prove_ that, for real numbers, a + b = b + a. No, we produce a number of axioms, for the real numbers R, and _everything that does not correspond to those axions, is not a real number!_ In other words, the set of real numbers is not defined _as a construction,_ but it is defined _as an exclusion!_ This is _exactly the difference between pure and applied mathematics!_ In applied mathematics _we construct!_ In pure mathematics _we exclude!_ Therefore, Norman Wildberger has it wrong! Pure mathematicians _do not rely on a fake arithmetic!_ _Pure mathematicians DO NOT RELY ON ANY FORM OF ARITHMETIC AT ALL!_ Maybe they do not realize this, but _that_ is the difference between pure and applied mathematics! Once you understand this difference, concepts like the infinite are no longer problematic. 'Infinite' just means: _NOT FINITE!_ It is an exclusion, not a construction. If Euler proves The sum of all numbers 1/n2 up to infinity is equal to pi^2/6, he is saying that _there is no difference between the infinite sum and the outcome pi^2/6 !_ In other words_if_ you calculate the sum and whatever algorithm you use to calculate pi^2/6 up to n decimals, and this n-th decimal is not a 9, then these two calculations _will always correspond exactly_ up to the n - 1 th decimal! And Euler's proof _shows that!_
"Pure mathematicians DO NOT RELY ON ANY FORM OF ARITHMETIC AT ALL!" So, a pure mathematician cannot even add 1/2 + 1/3 + 1/5? Or, perhaps, you are overstating your claim?
While I agree there is a thing like synthetic vs analytic, I think they interact far more often than you are suggesting. In any computation there are interactions around many small interfaces. One side of each interface builds up some form of data, and the other party breaks it down. In type theory we call this introduction and elimination. There is also the more well known notion for programmers, of actual vs formal parameters. A modern approach to mathematics, pure or applied, which is to be founded at least in part on computation, will thus contain both analysis and synthesis in either case.
Also, take your pick; inclusion or exclusion, construction or elimination, applied or pure, analytic or synthetic: What is the explicit, written answer to π + e + √2, without just trivially restating the question? Clearly this kind of computation can be done for rationals such as 1/2 + 1/3 + 1/5. It can even be done for abstract things like polynomials with rational coefficients, like (x^2 + 3) + (2x^2 + x + 1). Even pure mathematicians can do this, despite your claim that they do not do any form of arithmetic at all. So, can they do it for π + e + √2? And if they cannot, then can you explain *to a sociologist* _why_ they cannot?
@@JoelSjogren0 That might be. But a computer cannot understand in either case. To be precise, a computer works with bits. But _we_ work one level higher! We work with concepts and ideas! Let me show you what I mean. What we do with computer programming, is looking at the world, and trying to capture that with 0's and 1's. To give the simplest example, how many distinctions can I make with two concepts? The answer is 4. p -> q is one of them, which is then, on bits level 1011. I can also say p & q = 1000, p v q = 1110 etc. How many of these distinctions can I make? Obviously 2^4 = 16. There are therefore 16 of such operations, and therefore 16 different distinctions I can capture in 4 bits. _That_ is the level of the computer. But I can also look at the level _we_ look. And then I must say that we use _only_ 2 concepts, and a bunch of operators, to capture 16 distinctions, and therefore 16 different 'understandings'. With three concepts I can make 8 bits, and therefore 2^8 = 256 different distinctions. Reality then consists of 256 distinctions. The computer 'sees' 8 bits, and _we_ see 2 concepts and a number of operators. In fact, we can express all distinctions by just the 'and' the 'or' and the 'not'. _Every number of distinctions_ can be expressed by a sufficient number of bits, which can then be captured in a number of different concepts and just three operators on these concepts. What a computer does, is basically capture distinctions in bits. And _we_ design computer concepts, with which we no longer interact with the world directly, but through the computer. In general, n concepts, which is the level _we_ think, become 2^n bits, which is the level of 'calculation' of the computer, and these become 2^(2^n) _distinctions_ in reality. My point is: computers work with computations. _We_ work with concepts and operations. Computers change rows of bits into other rows of bits. We _design_ these operators. The computer can change 1's in 0's, and 0's in 1's through rules. That is, basically, what a computer does. But _understanding_ does not happen on _bits_ level. It happens through concepts and operations. It happens on the p and q level and the operators. Seen from the computer, _we_ design the concepts and operators. Computers are not able to do that. To _really understand_ how big a difference this is. We make drones. They have some capacity to make their own decisions. But our most complicated drones _are no match_ for what even a simple fly, which has about 100.000 brain cells, can do to keep itself alive! Apparently, this small amount of brain cells are enough for the fly to have _simple concepts_ and _operations_ so that it can do the very complicated things it does. Including the fact that it is _the most skilled animal in flying!_
@@robharwood3538 Any pure mathematician has reached that level by moving one level higher than applied mathematics. First you learn how to calculate, and then you learn to move from the specific to the general. It takes a lot of time before you understand that 'the general' consists of constructions in which the concrete details are eliminated, so that you get a construction that can be applied to many different concrete instances. What an applied mathematician tries to do, is to solve practical problems and to find solutions. What a pure mathematician does, is asking himself if the solutions found are _the only solutions!_ To give a _very simple_ problem. Suppose I ask myself, what number, added to itself, gives the same result when I multiply this number with itself? The applied mathematician can then find the numbers 2, because 2 + 2 = 4 = 2x2. After a little thinking he finds another one, namely 0, because 0 + 0 = 0 = 0x0. The pure mathematician asks himself, are these _the only two_ solutions? He then formulates a + a = a x a = b. So, the condition for all these numbers is: 2a = a^2. Therefore a^2 - 2a = 0, or a(a - 2) = 0. The pure mathematician knows that if pxq= 0 then _either_ p = 0 or q = 0, or both. Therefore either a = 0 or a - 2 = 0. These are the only two cases. From a - 2 = 0 follows a = 2. So, indeed, 0 and 2 are the only two solutions to this problem. This can be done on far higher levels. What I think of, is that Einstein made his famous most general solution to his problem of general relativity, leading to the equation G = 8(Pi)T. The question is then: is this _the only_ solution to his problem? Some pure mathematicians later proved that, indeed, his equation _is the only solution_ to his problem. So applied mathematicians _discover_ solutions to problems. And pure mathematicians _investigate_ these solutions, and try to find out whether these solutions are _all of them,_ or whether there are more of them. That is why the work of a pure mathematician is more difficult than that of an applied mathematician. Every pure mathematician is an applied mathematician who moves to a higher level of understanding. Applied mathematicians work with constructions. Pure mathematicians take these constructions and look how far they can be extended, through _eliminating_ those factors that are only of the constructive kind. To take your example, you might wonder: what kind of formulas will be the result if the numerator is always 1? So he can try: 1/a + 1/b = (b + a)/(ab). He tries 1/a + 1/b + 1/c =(ab + ac + bc)/(abc) and then 1/a + 1/b + 1/c + 1/c =(a b c + a b d + a c d + b c d)/(a b c d) and the pattern becomes clear. In the numerator the sum of all combinations of the products one letter less than the product of all numbers in the denominators, divided by the product of all numbers in the denominators. And then comes a proof (which I do not give here) that these are _the only_ outcomes of such sums. In any case, my vision of Norman Wildberger is that he is an applied mathematician, _who thinks_ that that is _all of mathematics!_ Nevertheless, I do not underestimate him! He has made a genial discovery, for which I admire him! But _because of_ his restricted vision, he is himself not aware of what he has discovered! What I refer to, is his idea that the plane geometry of Euclid can be seen as a combined theory of linear and quadratic equations. This leads to the idea, that you can, basically, understand all algebra of linear and quadratic equations through plane geometry, by introducing _spreads_ instead of distances, for example. I do not want to go into this, here. If you are interested, you should study his book: 'Divine Proportions'.
Trouble is he is too easy on so-called rational numbers. You say the answer is 1/3? All I see there is a division you haven't done. And you can't do it, it would take forever. Down with rational numbers I say. They are fake maths.
@@jowotx He fails to understand the distinction between a number and its decimal digital string representation, which is implicitly just a sum of rational numbers with powers of 10 as denominators anyway. Also, Skewes' number is a natural number, but he could never write the digital string representation of this number in any base except for itself, so this number is also fake, I suppose. Any number that is larger than 10^(10^80) is fake, according to his argument, because no such a number can have its decimal digital expansion written, even theoretically.
@@jowotx it's all about equivalence classes and representations, one could argue for example what does "- 1" even means? What is 5+1? We say it's 6 cause that's the symbol we associate Thus we can define е+π:=ξ For example and that would get the job done. I understand that there is a concern about the foundations of the real numbers, but that's soooo XIX century like. At the end of the day mathematics is mostly a language tho
What’s the difference between 2 & pi? Both are names for numbers - 2 for a finite no of things, pi for an infinite no of things satisfying a relationship couched in mathematical language. e is another name for different infinite no of things, with a different mathematical relationship. That’s what I think, probably wrongly.
I’m sorry I cannot leave this here. 1/9 is clearly a rational number, which you state exists and has a real arithmetic, but what about the decimal representation of 1/9? You would agree that 1/9 x 9 is a possible operation, and if you were to calculate the decimal representation of 1/9, you’d get something like 0.1111111111...., so is this okay? I mean 1/9 is rational but it also has an infinite decimal representation, and if this is okay, then what happens when you want to do the same operation 1/9 x 9 on the decimal representation of 1/9, i.e. 0.1111111111... x 9 ? You would say the second arithmetic is fake, but how can the same number both exist and not exist and have a real arithmetic and a fake arithmetic at the same time? This is a contradiction, no?
@stokhosursus We defined decimal arithmetic in this video, in terms of finite decimals with an arithmetic based on integers, but with division problematic. The fraction 1/8 can be represented in this system, but the fraction 1/9 cannot. In the Babylonian base 60 system they were acutely aware of which "numbers divide" and which ones don't -- in their system 1/9 would be a valid object, that is have a proper sexagesimal expansion. Now can we extend our arithmetic with finite decimals to an arithmetic with repeating decimals? Turns out this is a very interesting challenge, and we will be talking about that in a Famous Math Problem lecture.
@@njwildberger I have no problem with you attempting to re-derive other ways to arrive at interesting mathematics results. However, there are many profound problems with merely throwing away the real numbers. I don’t see how you’re planning to address continuity, at minimum, or other larger and necessary concepts in mathematics just from an applied approach and not merely within the devil arts of differential geometry and other evil conspiracy areas of mathematics delusion that you are railing against.
As a psycho-sociologist, I found the basic problems involved in apparently only insignificant precision of long numbers to be, well, not as significant as problems of insignificant and false precision being used to validate the true as false and Vice versa. On the other hand, when I get non linear (non literal) about your speech, it makes more sense to me. As reading Principia made sense only when I realized that Newton was talking about odd reality, not even physics. As to precision in insignificance, I’m with the chemists and engineers. Hopelessly literal. Personally, I have more trouble with the reason airplanes fly is the use of imaginary numbers.... FYI, it took me a full week to accept the notion that an integral almost arbitrarily defined the convention for area under a curve. I got lost in the infinity of points and lines. Thanks for this. Strangely, it helped.
I have the answer you are seeking. It's really quite simple. In the early history of mathematics numbers were defined as the number of individual objects in a collection. In other words, this is the cardinal definitions of 'number'. But then they discovered irrational relationships such as the square root of two which appear to exist, but do not satisfy the cardinal definition of number. This is where they made their grave mistake. Instead of recognizing that irrational relationships are not cardinal numbers, they demanded that they must be. And that was the beginning of "fake numbers". In other words, 'numbers' that don't satisfy the cardinal definition of number. This was a grave error that the mathematical community actually holds up high as the greatest achievement of mathematics. When it truth it actually represents a gross mistake. What they should have done way back at the times of the ancient Greeks, was to simply recognize that irrational relationship do not satisfy the original cardinal definition for number, and then they would have realized that they have a brand new concept on their hands. But they never made that realization up to and including today. They will probably never know the true nature of irrational relationships because of this. They will never know that the source of all irrational relationships (what we've been calling irrational numbers) is self-referenced relationships. We'll probably need to wait until aliens come from some other planet and set Earthy mathematicians straight on this. Because, save for Norman Wildberger, almost no one else has recognized the problem. And even Norman doesn't appear to be aware of the actual source of this problem. The source was the day that mathematicians started treating irrational self-referenced relationships as if they could qualify as cardinal quantities. That was the mistake. A mistake that has been with the mathematical community for eons and has become so ingrained in mathematical formalism that is isn't likely that they will ever recognize their mistake or correct it. The very idea that irrational self-referenced relationships could be thought of as cardinal points on a line is a very bad idea. Yet that is precisely what all math students are being taught to believe. It was all due to a very wrong turn originally made by the ancient Greeks and never corrected. This isn't to day that irrational self-referenced relationships don't exist. They do! They just aren't cardinal quantities, and therein lies the problem for our entire mathematical formalism. Aliens would actually find human mathematics to be quite hilarious. We made an extremely stupid mistake thousands of years ago, and we haven't recognized the folly since. Norman sees that there is a problem. But does he truly understand how it came to be? I just explained the answer Norman. The problem was introduced by the ancient Greeks when they began to embrace irrational self-referenced relationships as cardinal numbers. That's the mistake right there. So we can not only point to the mistake, but we can even see when it was originally introduced into mathematics.
@Mystic Dreamer Thanks for the comment, which I have quite a lot of sympathy with. It is nice to remind ourselves that the ancient Greeks thought of, or at least represented general quantities geometrical, typically as line segments, or if they were being multiplied perhaps as areas. The idea of a quantity as a "point on the number line" I don't think we can fairly ascribe to the Greeks. In my next video in this series I will be discussing an aspect of ancient Greek thought re natural numbers and even there they pictured things geometrically. It is also true that the relationship between quantities, as evidenced say by the Greek theory of proportion, played much more of a role than it does today. We think of numbers as absolute objects, while they often preferred to thing of relationships between two objects (typically line segments). This way irrationality becomes incommensurability. So I think you are largely right. As for the true meaning of "irrational relationships", that may have to await a deeper understanding. See my video on "The magic and mystery of pi" at th-cam.com/video/lcIbCZR0HbU/w-d-xo.html.
@@WildEggmathematicscourses I will watch your video on Pi as I always enjoy your insight into mathematics. Are you aware that everything we call an irrational number arises from a self-referenced situation? I don't believe the mathematical community in general is aware of this.
@Uri, No I cannot support the "reals" even just as an ordered set, with now arithmetic. We have to be able to write down the objects we are talking about --- otherwise the talking is just babbling.
Uri, I'm sure you'd want at least the rationals to inject into your reals. I suppose the most conservative approach to set up mere ordinals in set theory is as transitive sets of transitive sets, but then I'd ask you which one would be the reals. And you can't do that without taking a stance on the continuum hypothesis, which says the cardinality of R is that of the smallest such ordinal :)
π+e+√2 = π+e+√2 it is indeed unfortunate that there is no simpler way to represent real numbers, but still, the mathematical object π+e+√2 can be manipulated and we can examine properties of it and formulate theorems and proceed to prove those theorems. i just don't believe real numbers need to meet the criteria you ask them to, in order to function as a consistent arithmetic. the sum π+e+√2 is in fact not, as you say, "computable" (at least not in finite time), but some of those real numbers indeed contain an infinite amount of information (in their digit sequence) so it would be silly to expect them to be subject to operations like sum or product in finite time. but, i don't think that makes arithmetic on real numbers "fake". see also: math.stackexchange.com/questions/933890/an-algorithmic-approach-to-constructing-the-real-numbers
But why do you have the right to take 1 as an object and not e? We only take numbers which have finite decimals, only numbers which can be computed? It sounds like intuitionist or constructive math. Which are fine for me, these are just different formal systems with their own truths. But I don't understand why do you complain about a sum of two real numbers (which could be stated as a sum of classes or limits, perfectly reasonable operations between objects of a theory which has infinity in it's own axioms) and not about the sum of two natural numbers, which is the same, an operation between objects of a theory
Perhaps we can reconcile both views by replacing "infinity" with "all". For instance, to sum from n to infinity, why not say sum from n to all sucessive integer numbers. That way, if you think the universe is finite and every number is finite, then you'd be happy. If you think there are a infinite number of things, 'all' still encompasses that.
Infinity is purely a human mind fiction to legalize many huge volumes of false mathematics, since integers are an endless chain of successive integers However, in practical & purely eingineering works it doesn't make much of a difference where most of mathematics is truly made by eingineers & talented carpenters by approximations that satisfy the earthy human needs, but in pure physics & ALL theoretical sciences like Logic & Philosophy or Physiology, it is definitely a disaster to human minds for sure So like for every alleged real number that is associated with fiction infinity is of course a fictional & non-existing number
@Jon Rufol, Infinity & finity are false meaningless terms in superior & truer knowledge of mathematics, they don't mean anything but a source of total confusions for the human minds, no matter if they can generate a huge volumes of false buissness mathematics for allegedly top-most genius historical & living (philosophers,logicians, physicians & especially mathematickers) The truer meaningful terms instead is existence & non-existence, & to easily understand the fictions of ALL those meaningless terms like infinity, finity, infintism, ..., etc, ask your self what is the largest FINITE natural number, & you would immediately conclude that is also INFINTE in your same terminology But in Eingineering probably earthy problem solutions by approximations, it is almost no harm to use these terms since the core issue is not at all mathematics In other words, the entire modern mathematics is infact an eingineering productions made by eingineers & scientists for purely non-mathematical purposes where people & academic proffessional mathematicians cannot strictly distinguish especially that true understanding would lead immediately to finish their huge irrelevant buissness... to truer & superior mathematics
17:26 you mention this problem with carries frequently. how bad can this wave of decimal carries become? is it naive to assume there is some limit to their effect? for example if we see enough 0's
consider this. You add a number that consists of all 9s in decimal places with 8 placed every now and then. You add another infinite decimal consisting of ones and only 2s in random places. Which decimal places in the sum have zeros? It's too easy to create multiple classes of such failing examples where you have infinite number of arbitrarily long "carry waves"
@@whistaq oh. are there irrational numbers without 0's? for example a number that contains all 9's but with 8 distributed randomly every now and then. if i google "irrational numbers without 0" then some results claim it is not possible. i have no idea myself
@@xybersurfer One of the most popular definitions of real numbers involves the so-called Axiom of Choice, which, in the case of defining a 'real' number as a decimal expansion, means that you can literally 'choose', in a completely arbitrary way, whatever digit you want at each position. So, yeah, you can have literally any pattern of digits you want, but on the other hand there doesn't have to be any pattern at all.
@@xybersurfer How about this one. Let (n) denote that you're repeating the previous number in decimal expansion n times: 0.8989(2)89(3)... you get the drift, this sequence does not repeat. Now suppose it's included beginning at an arbitrary position in a number that contains all 8s. The rest follows. If you search MathOverflow for: " Can an irrational number have a finite number of a certain digit?" you'll find the argument that I'm making :)
We can prove as well that Mathematics don't exist...or does it!? This was the result of answering the question: " How do we create new maths?" A circle don't exist (in the physical world) but it does in the human mind. But not all the thoughts in the human mind exist. A circle is different from an unicorn or a pink elephant in a person's mind. In the mind of two people two pink elephant can't be the same but a circle is the same in all minds. So a circle exist...in the mind of the humankind. But what will say about the maths when the humanking will be extinct and the universe with them. Could Mathematics exist if human kind or a mind could live or think forever? Do things exist if there are no longer any record or memory of them? The answer is no. But what about a unfinished book novel due to the demise of his author ? A book novel without his final chapter is not complete, but does not exist as a complete form. It is not a book. The same with maths. Lot's of maths has to not be discover yet therefore Mathematics is not complete. It doesn't exist. Can we discover all? We need to think transcendentally , perhaps assuming an infinite mind capable of operating in infinite universes perhaps then we could compleat the Mathematics!? But that mind would only create finite sentences (Alan Turing dictionary...) there fore will always be mathematical transcendental objects that will evade his ultimate powerful cognition. So Mathematics don't exist and what exist is only the work of the Mathematicians...or doesn't! ? Perhaps the meaning of the word "exist" doesn't exist... By the way thanks for this channel. At the moment I respectfully disagree with the author . I believe they way I learned about Real number was from Russian text book la mir, (but possibly not) In this Two thick volume the Russian scholar use the axiomatic approach to define all the property of the Real number. The axiom that bring the Real Number to life is the axiom of the existence of the extreme inferior. Any inferior bounded set of rational number has an inferior (a number smaller than any of his element but greater of any of his "minors") Then the scholat proved that the set of the Real number is unique under homeomorphism ( regardless their representation). So the Set of Real Number Exist and it is Unique. I do find problematic though the limit of theta to 0 of sin (theta)/theta = 1 I am not sure was proved or if can be proved except for geometric consideration therefore it should be considered as an axiom, anyone?? I hope I don't be taken too serious. I apologise for any mistake in the definition, I studied this long, long time ago and I misplaced my reference books. I would like to thank again the professor author of this channel for his contributions and for the great value and joy in his channel. Congratulations for the success of your channel. Finally I think I could create the App mention at 19:00 😜 I understand I don't fully understand what I am talking about. Bye
Hi Norman, love the videos and always find your perspective valuable when trying to strengthen my intuition of what mathematics really is. I feel like mathematical philosophy as a concept has been lost to the delusion of axiomatics and the esoterics of ontology, whereas mathematics to me should have a much clearer philosophical grounding to it, and your criticisms of mainstream maths have provided much food for thought. Would you agree with a construction of real analysis that explicitly deals with Cauchy sequences, and explicitly states which results are constructive algorithms, versus those that require arguments by contradiction/double negatives? There would be no canonical forms or basic comparison of sequences, so it could hardly be treated as a number system, but it seems like something which would still be worth doing, even among people such as yourself who require explicit notation of the objects being reasoned about. I can definitely see the emperor's new clothes aspect of "real" analysis, that internally consistent axioms are taken by the mainstream as reality itself. Generally people's criticism of the arguments you make amount to the obvious claim that your criticisms don't have any meaning within their axiomatic grammar, which is like defending classical physics by claiming that subatomic particles or objects with relativistic velocities don't exist! I feel like this video and its presentation of the mainstream view of mathematics is compelling as a logical argument for your point of view, but I think there is a more nuanced case to be made describing how the mainstream switches between infinite processes, convergence arguments, and 'simple' arithmetic based on what suits them, and how internally consistent technical rules are taken at face value to arrive at the intuitions they arrive at, whereas this argument you present focusses more on your side of the conflict. Maybe this is all to come but I generally feel the view from above is more productive both for individual understanding of conflict/sociology and for actually resolving the conflict as a member of the field. Thanks again for the work you put into these videos, interested to see this series unfold.
I will find this video, but my main points where agreeing that axiomatics is silly, and asking whether he would disagree with Cauchy sequences if treated formally without axiomatics. I might edit my comment so that it is clearer
I guess my comment about the view from above might also be what you are referring to. The point that I am getting at is that I don't think a mainstream mathematician would feel their side of this argument is being represented, which might be less than ideal both in terms of convincing them and in terms of a sociological discussion
@@ThePallidor Assumptions (axioms) need to made to even begin talking about anything meaningful, could you point out what is so wrong with the axioms of say, ZF? Semantic vs syntactic truths is the very essence of Godel's completeness theorem for first-order logic, which is widely accepted too. Do you have a disagreement with this theorem, if so, why? Do you disagree with the way Model theory has been built? I'm having trouble seeing what your real issue is with this foundational mathematics.
@@ThePallidor Okay, so you don't like the axiom of infinity. Now you don't even have the natural numbers, let alone the rational numbers. So one cannot talk about elementary mathematics anymore.
You are mistaken. Yes, real numbers are... abstract. We can't do arithmetic on arbitrary numbers with an infinite number of digits, so how can we claim they exist? But we can do arithmetic on finite integers, on decimal fractions with a finite number of digits, and rational numbers with finite integers in the numerator and denominator. The thing is, though, we can't set a hard limit on how big those finite numbers can be. The real number line is the only way to have a consistent system that doesn't limit them, and that doesn't exclude other numbers we can work with in other ways, like pi and the square root of two.
@John Savard With your original statement, I was hoping you were going to actually prove me wrong, by calculating pi + e + sqrt(2). Instead you just gave us the blatant pronouncement "The real number line is the only way to have a consistent system that doesn't limit them, and that doesn't exclude other numbers we can work with in other ways, like pi and the square root of two."
@Martin Suppose we want to add the "real numbers a and b". You assert something like this. Take the set A of all the rational numbers left of "a", and take the set B of all the rational numbers left of "b". Then simply form the set C of all sums x+y, where x is in A, and y is in B. Then C will be the set of all rational numbers left of "c=a+b". When you hear something like this your response could happily be: "Excellent prof! Can you demonstrate for us how to add pi + e using this fascinating technique?" Note also the pivotal use of the term "take". We do a lot of "taking" in pure mathematics.
@@WildEggmathematicscourses said: 'We do a lot of "taking" in pure mathematics.' This is a weird objection. It like saying that C++ programmers do a lot of "defining" classes. But, technically, the sum C of A and B is defined as: C = { z in Q | (Ex)(Ey)(x in A and y in B and z = x + y) }. There is no "taking".
@@billh17 I am pointing out that when it comes to set theory, we pure mathematicians often use language that suggests we are doing something, when actually we are only talking about doing something. Kind of like: "take all the fish in the sea and arbitrarily divide them into 3 non-empty subsets." You can say it easily enough, but what does it actually mean?? And if that seems a forced example... which do you think is more challenging: taking all the fish in the sea, or taking all the rational numbers less than "pi"?
@@njwildberger said: >I am pointing out that when it comes to set theory, we pure mathematicians often >use language that suggests we are doing something, when actually we are only >talking about doing something. That is because they are talking informally. In the first order language used in ZFC, there is no "doing" or even "talking about doing". There is no time, there is no change, and there is no construction of sets: all the sets already "exist" and one is just making statements about them (which may be true or may be false). And the only statements about them must only use the undefined predicate "in" and predicates defined from "in" (and, of course, constructs from first order logic). >Kind of like: "take all the fish in the sea and arbitrarily divide them into 3 >non-empty subsets." You can say it easily enough, but what does it actually >mean?? Not sure what your point is. This seems to be more an objection to your use of English as your language to do math. This is one reason why programming languages don't use English: natural languages like English are not precise and clear enough. Let's consider the word "arbitrarily" which I think is your main point. I am almost sure that "arbitrarily" cannot be define in the first order language used in ZFC together with the predicate "in". And, if it could be defined, then I would know its meaning (in ZFC) and use it to make statements about sets. In summary, I agree that your given statement is easily enough to say (in English) and that it might be hard to determine its meaning. But, that statement cannot be made in ZFC (or in C++ etc), so there is no need to seek its meaning (in ZFC). >And if that seems a forced example... which do you think is more challenging: >taking all the fish in the sea, or taking all the rational numbers less than >"pi"? I have already granted that I cannot write down in finite time on a finite piece of paper all the rational numbers less that pi. But ZFC does not claim that this can be done.
@@billh17 In the fullness of time I hope to show in the MathFoundations series why ZFC is highly dubious. Till then, please have a look at my article on Set Theory which I shall put a link to in the video description. But if you are going to defend it, why not actually use the axioms explicitly so people can see them in action. In other words, instead of saying that such and such can or cannot be "done in ZFC", explain why using reference to the actual axioms. I think if people had more exposure to these vague and even meaningless assumptions, the tide would turn rather quickly. Only by referring to them indirectly can we hope to maintain the illusion that they form a solid basis for mathematics.
Actually, Mathematica has always supported arbitrary-precision real-number arithmetic at the symbolic level but that is different from the word-size limits of floating point FPUs. So when you say "computers can't do it" there should be a distinction between FPUs and symbolic/algorithmic treatment of infinities.
@@njwildberger I have posted a perfectly innocuous and respectful reply to your question several times now and it keeps getting deleted. I'm guessing it ran afoul of youtube's algoithmn? I don't see why you would delete it.
@@njwildberger Maybe this with the link removed: Hmm. Now that I'm put on the spot, I feel a bit less confident in my answer. If true that there is no left-to-right algorithm for adding nonrepeating decimals, then it seems you are right. The exactly-precise adding of any two of those “numbers” could not even begin. A natural question is whether (A) it's provable that no L-R algorithm can exist or that (B) one might exist but is unknown. Only quick result I could come up with is here: [link removed] with “Avizienis in [2] and Wiedmer in [57] proved that an addition algorithm "from left to right" implies the redundancy of the representation:” So it looks like (A). It was unclear in the video whether you meant (A) or (B). So “arbitrary precision” in my original statement must exclude the infinite case for addition of irrational numbers. Probably same for mul, div,sub. Come to think about it, even finite, N-place precision addition for irrational numbers seems problematic now, as we are always uncertain of the carry of the N+1th digits. Yeah, it's a mess. So how do we get the exact result of e^(i*Pi)=-1 ? That result must be in a different class. I'll stick to my original beef with the oft-heard claim that “computers can't do this or that” when what is usually meant is that the built-in FPU's have limited word size, or division by zero will crash the computer. Mathematica, running on a computer, gives ComplexInfinity when you type 1/0, or can give Pi to 10,000 decimal places. It can implement all the unwarranted symbolic shortcuts that a typical Mathematician uses if he wants it to. In this regard, “computers” and mathematicians do not operate in fundamentally different compartments.
unintentionally you are showing what really happens when challenged to do arithmetics with "real" numbers: you just agree to a certain allowable error and then truncate the magical "real" number and work with either a rational number or with natural numbers and then just move the decimal point. There it is: the arithmetics of "real" numbers.
@@jgmartinezmd6867 I don't see why this is "fake". At the abstract level you can define all the arithmetic operations on the real numbers without running into paradoxes, which means imo that they are well-defined. The issue you have with real numbers is, that unlike rational numbers there isn't any "finite" way of representing the abstractly defined object. However, the reason imo why it makes sense to add obejcts like pi and sqrt(2) to the rational numbers is that there is a well-defined notion of whether these numbers are smaller or larger than any given rational number.
@@qswaefrdthzg Thanks for your reply. You say `At the abstract level you can define all the arithmetic operations on the real numbers without running into paradoxes`, i am just a hobbyist and have not studied that part, so i wont argue that statement but what bothers me of so called real numbers is that you dont really use that arithmetic, just work with truncated versions of those numbers. So we cant really construct them, neither we can use their arithmetic.. what is the use then? if it is only a logical construction unrelated to what we can find in this universe they should have named them "imaginary numbers" or something like that. i do find some paradoxes with "real numbers" when applying "natural numbers arithmetic". shouldnt they work under this arithmetic? if not, why not?
I don’t see why you accept the notion of a rational number as well as decimals, but reject that of a real. If rational 1/3 and 2/3 exist, then their corresponding decimals can’t be added as you would have 0.9999... with an infinite carry. The same as in the example with real numbers. This means that there is no “algorithm” for adding two decimal numbers. Then decimals are “fake”. As for the natural numbers, they can’t be actual numbers either since a number such as two is assigned to both a pair of doors and a pair of shoes, which cannot be equal in themself. On top of that a pair of shoes may be assigned a number 1 since a pair is a singular entity or a billion billions since it is made of atoms. You cannot decide if the numbers are “fake” solely by their real world interpretations such as “algorithms”.
@Fullfungo. Decimal numbers are finite decimal expansions, and their arithmetic reflects that of the integers. Most fractions do not have corresponding decimals, as the Babylonians realized. In their system a division was meaningful precisely when the denominator was a regular number --with factors of 2,3 and 5. Over the decimals it is more restrictive, just factors of 2 and 5. So talking about the addition of corresponding decimals to 1/3 and 2/3 is meaningless, at least with the definition of decimal number I gave. Can you try to expand that to embrace also repeating decimals? That is a very good question, and will be topic of another Famous Math Problem!
As many have pointed out in the comments, you are wrong about many things, or at least are misrepresenting them. A "decimal number" is just a representation of a real number defined by an infinite sum: x = sum( i=0,inf, a(i)/10^i ) where a(i) are integers such that |a(i)| < 10 If we have another real number y with digits b(i), then x+y is just: x+y = sum(i=1, inf, (a(i)+b(i))/10^i ) So what's the problem? It's dead easy to write an algorithm to calculate each term up to an arbitrary number of terms N, an just express it as a big fraction. It doesn't matter if the exact addition takes an infinite amount of time, the addition operator is still precisely defined!
@Kyle Brown You wrote: "It doesn't matter if the exact addition takes an infinite amount of time, the addition operator is still precisely defined!" I think not even Tinker Bell would be on board with such a far fetched claim.
@@njwildberger This far-fetched claim is literally the consensus of pure mathematicians, or is it not? EDIT: btw I just looked up your publications, you acuse me of waffle when you spend pages of text to make the point that we should teach trigonometry in terms of distance squared and sin(angle)^2 ? That's an interesting suggestion since its essentially equivalent to using distances and angles, and in some ways more intuitive, but its buried in more unreferenced statements than I could count
@@KyleDB150 I think not. You will struggle to get even a reasonable fraction of pure mathematicians to agree with this claim, in my view. Have a go if you like.
@@njwildberger Well, here is what wikipedia has to say on the "real number" page: "These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers-indeed, the realization that a better definition was needed-was one of the most important developments of 19th-century mathematics. The current standard axiomatic definition is that real numbers form the unique Dedekind-complete ordered field (R ; + ; · ;
So basically you're saying that we should do better than "true enough"? Whenever you go from pure mathematics to applied mathematics, matter's imperfect nature can be apprehended by "true enough" arithmetic, right? There's no need to be absolutely precise in applied maths. If a glass of water was designed in such a way that its volume consisted in any of those expressions you put, we could see that "true enough" arithmetic is actually good enough, a few drops more, a few drops less (a few more water molecules, a few less) make no difference to us. We should pay attention to the pure mathematics aspects then? My take on this is that numbers aren't actually just quantities. They're philosophical, transcendental entities or aspects of true reality behind this imperfect veil. When we're trying to know exactly what Pi is, we're trying to peer into an aspect of transcendental reality which we just can't comprehend in our current state of being. If we ever grasp the true meaning of a number it would be at least in the next dimension of existence, after we pass away, where our abilities to perceive reality isn't hindered by a biological machine so to say. Don't get me wrong, I think it's deffinitely worth the effort though! Would a formula in terms of natural numbers that told you exactly what is the nth decimal of Pi suffice what you're looking for? Or did I misunderstand the point you were trying to make?
@Gennady Arshad Notowidigdo Thanks!. Maybe the whole of material evolution is slowly marching towards those preposterous mathematical ideas that we tend to assume and overlook. I bet you the afterlife is closer to those conceptions than this place.
Why did you say the ability to calculate a value to an infinite number of digits is “useful”? Perhaps there is a use for mathematical approach that defines quantities in base pi instead of base 10. Then Pi would be exactly 1, pi**pi would be 10., etc. not sure the usefulness but who knows?
The challenge examples are tough mainly because we don't have easy-to-use manual tools. Expressions like "Pi to the power (e to the power {sqrt[n]} }" can be computed for specified sets of parameters (e.g., n) "close enough for government work", along with error bound. For most practical work (such as building aircraft) such computations are just as good as the "unknown" exact values.
...employ a whole machinery of symbolics... Even with symbolics, we have practical considerations. Exact values of trigonometric functions finer grained than pi/60 are beyond awkward in application. So, even symbolically only a handful of values are easy to calculate. However, we encounter nested roots which demand arbitrary precision to postpone collapsing the significant figures. In short, symbolism is not a panacea. One of my first computer experiences was manipulating identities and realizing that double precision routines spit out a string of 15 digits with only the first few places being significant and remainder being artifacts. The illusion of accuracy was shattered. The foundations of our castle rests upon the sands. Even the leaning Tower of Pisa appears vertical when viewed from a unique perspective. Appearances don't make it so. Approximations can be good enough for ordinary purposes. We tend to gloss over the limitations.
Comparing what's being done here with what we did in our many varied pure maths courses at uni, it is grossly over-simplified and wrong. What is meant by π or √2 or e or i is rigorously well defined.
@Keith Morgan Your claim would be much strengthened by sharing some of these rigorous definitions with us. Then we could make our own determination as to whether they really are rigorous or not.
@@njwildberger square root of 2 is a positive number such that, when being squared, is equal to 2. In addition, it can be proven that it is the only such number. The proof is left as an exercise to the reader.
@@njwildberger e is such a positive number that satisfies the equality e^x = d/dx e^x. It can be proven that it is the only number with these properties.
@@njwildberger pi is a ratio of the length of a circle’s circumference to the length of its diameter in the standard Euclidean plane. It can be proven that this number is unique i.e. is the same for any non-degenerate circle
@@njwildberger Or you could just take them as the limits of Cauchy sequences like any analyst would... Or perhaps dedekind cuts is more tasteful. Rather than this playtime in analysis, I am more curious on your stances on the ZF axioms, Model theory, Godel's completeness theorem and like ideas.
Why hold the assumption that something should fit in our finite universe or be computable in finite time for it to make sense? The abstraction involved in real numbers endears them to me. To the extent that “fake” means “abstract,” I don’t understand what is wrong with it. What am I missing? (PS Thank you for sharing your expertise online. Much appreciated.)
@@mehmeterciyas6844 Primarily between 24:41 to 27:44. He makes a specific concession at 27:45. Then makes a similar concession at 28:40, but shows that this second one begs the question, at 29:08. And he also raises a subtle but important issue with 'including the rationals' within the 'reals' at 29:51.
@@njwildberger 29:25 Why do you NEED to reduce the string to a single character? The reason teachers ask for that is so that they can easily check to make sure an answer is correct and for the homogeneity of the class. But whether or not you can "easily check" something doesn't affect its truth value. Why would you require some teacher's instruction to be a fundamental property of logic without any additional reasons? I think that it's perfectly fine to consider a well defined string of operators and numerical objects as a numerical object in its own right, especially since all you really need to do with numerical objects is compose them into well defined strings with operators. If you consider the reals as a vector space over the rationals, then question one is not all too dissimilar from (in Q^2) asking (1,0)+(0,1)=?. Effectively that's all your "computing challenges" amount to in my opinion.
@@SlipperyTeeth "Why do you NEED to reduce the string to a single character?" I don't have a full answer, but one particular thing that comes to mind is: in order to evaluate whether two numbers are equal or not. It's often useful to have a 'canonical form', for example. Without at least some sort of limitation on acceptable forms, then the possibilities for expressing the same value would be unlimited and essentially impossible to compare in practice. Consider the myriad ways you can express the number 0, or 1. Instead of trying to tackle that intractable problem, better to limit the number of allowable expressions to a finite number, even better if small, even better if unique.
One slight nitpick here... when you talk about electrical engineers and computer programmers, the base matters. Numbers are stored underneath all the circuits as floating point binary, not floating point decimal. This is what determines, for example, the machine epsilon. And if you're designing a system "close to the metal", or programming to a timing diagram where every clock cycle matters, in practice you absolutely need to consider things like the fact that 3/5 can be expressed as a finite decimal, but not as a finite binary expansion.
Great philosophical discussion. I have a few questions (certainly no offense meant!): 1st: Why do we need an algorithm? What if we simply accept the fact that maths s not algorithmic? What if it was only symbolic? Why would we care about a different answer to "pi+e+root(2)" than its symbolic representation? "What do we actually get?" Well, the number itself of course! Which can be symbolic, why not? "What is pi^2/6?" Well, I can give you an approximation, if you want an approximation, but asking "what is it?" to me is nonsensical, because the answer simply is "pi^2/6". 2nd: You explained the problem for addition concerning infinite decimals. I agree there. Why though does addition fail when it comes to equivalence classes of Cauchy sequences, for example? We know that it works for rationals, so everything should work fine by adding the n-th term to the n-th term and then considering the resulting Cauchy sequence. (Of course, we need to make sure that it is independent of the representative, but that has been proven.) 3rd: Infinite number of tasks. Why would you say that doing an infinite number of tasks is even required? Why isn't talking about doing an inifinite number of tasks sufficient? I feel like it all comes back to my first question, because to me maths is not at all algorithmic (and certainly doesn't have to be algorithmic to be real). I would wager that there could be a different, certainly less negatively connotated word than "fake" for pure mathematics, because if we allow for this definition of fakeness, we end up discussing reality in general - in that case nothing at all would be real except (going by Descartes) something which can think itself to be I.
@Indikativ How are you going to react when your Grade 5 daughter comes home from school to announce that her maths class is now going completely symbolic as you suggest? She is thrilled, because now the answer to 34 x 69 is ... 34 x 69. The answer to 1445 + 87 x 7 is 1445 + 87 x 7. She and her friends are now even getting the hang of powers, which was overly complicated before ... now 7 ^ 5 is just ... 7 ^ 5. She can even do really really big arithmetic: 4234 ^ 19 is ... 4234 ^ 19 !! Evidently studies have shown that kids do much better with this more agreeable approach to mathematics.
@@njwildberger The problems you have described in the video only arise when dealing with irrational numbers. For rational numbers we can prove everything you doubt easily. So my approach of making maths purely symbolic is supposed to be a solution to only the problems you described. Calculation with everything else should obviously still be taught. Also, we can certainly talk about approximations and computation using approximations, as long as we are aware of the fact that it is no more than that, an approximation.
Hi Prof Wildberger, I have some thoughts on this subject. I, like you, despise the notion of actual infinites, contradictory as they are. However, it seems to me that this talk of "numbers" misses the point. I find it very strange that you accept the existence of rational numbers and yet reject transcendental numbers. A finitist should conceive of the *process* "divide two by 3" completely differently to a natural number or integer. The finitist should make this distinction because dividing two by three is an unbounded procedure; it never finishes. The idea that it can simultaneously represent a *value* which we might plot, is a rather unusual one. The constructivists, especially Jan Brouwer said we should place more emphasis on the actual computation of the resulting values. This was long before computer science. Nowadays, mainstream mathematics now labels practically everything a "number" of some kind. It is clear to me that there are really two things: *values* and *procedures* . In the theory of computation we have *images* and *machines* . An image is what you would expect: a finite euclidean space filled with boolean values. The description of any finite object is an image. For example, to back up your computer, you make a system image and save it to a hard drive. The image is the complete description of the system at some moment in time. An image does not have the dimension of time. It has only spatial dimensions. A *machine* , on the other hand, is a computational model. It consists of three parts: The current state (an image), how the next state is computed (some rules), that there is a next state (a clock). A machine has agency (can make things happen). When we run a machine (simulate its behaviour), its current state changes. It is clear to me that division of one natural number by another is a procedure, not a value. We encode each numbre as as image in the radix representation. This allows us to define a machine which computes a division of one number by another. We simply encode them as two images using the radix representation, and then modify the initial state of the machine to include the two images as appropriate. However, division is not a bounded procedure; that is, it is not guaranteed to finish in finite time. Instead, if ever we have a remainder of '0', then we can return the radix representation of a natural number as the result. Alternatively, we could modify our machine to have a fixed level of accuracy, and always return a truncated result after so many digits. Pure division, though, is not a bounded procedure, and will not always produce a final result. This is in contrast to addition, which always produces a result in finite time. Surds, trigonometric functions, infinite summations, limits, etc. are all unbounded procedures. To my mind, they are exactly the same sort of thing as division; unbounded procedures, liable to stop and return some finite answer, but equally capable of continuing indefinitely. It should be noted that the special case where we return a finite answer is artificially enforced. In its more natural form the machine would continue to print '0's but we have specified that if ever we would reach that state, instead notify us that no more "useful" digits will be calculated. A similar case arises with repeating digits. From my finitist, computationalist perspective, most other people seem very confused. Yet I don't suffer any confusion at all. I am delighted to see you bravely challenging the existence of values with actually infinite resolution. This is the main point of confusion nowadays. Typically, by "number" people mean *value*; But to call a procedure like sin(0.734) a value is horribly wrong! Instead, we must combine unbounded procedures with some resolution bound. Then we can calculate a value of finite resolution. That's how trigonometery and the laws of physics are applied to the real world; You take initial values as inputs to the appropriate procedure and then combine the procedure with the lowest resolution bound of your initial values to find an answer. For example, we might measure two values to be 0.76m and 0.433m. This means between 0.755m and 0.765m, and between 0.425m and 0.435m since we have only measured to those levels of resolution. Before we can calculate an answer from our procedure (let's say sqrt(0.76^2 + 0.433^2) ) we have to know what level of resolution our answer should be, otherwise we might mislead others with an answer which is too precise, or not precise enough. Even worse, we might get stuck in a loop! In this example we should calculate our answer to two decimal digits of resolution, because the lowest resolution measurement value had just two decimal digits of resolution. As we can see, no actual infinities are involved anywhere. Just unbounded analytic expressions which can be applied an any resolution. I'd love to speak with you and hear your thoughts on these issues.
You should check out his Math Foundations series, and also his Rational Trigonometry series. In MF, he starts out as dead-simple as possible to form a basis for numbers, and by the time he builds up to the rational numbers, you'll see why the rational number 2/3 does not need to be seen as a process. It is just a finite written expression of a finite number of symbols, and the 'division' aspect of it can be interpreted geometrically, if need be (though it doesn't necessarily *need* to be), and it is completely finite, not an infinite process, or it can be dealt with entirely algebraically, which is another very useful perspective to have on it. In his Rational Trig, you will be introduced to how powerfully simple this approach to rational numbers can be, and how it can be used to break the illusionary dependence on 'real' numbers from a topic such as trig (in other courses he uses the same/similar techniques to break the dependence also; it's just particularly impressive when it's done with trig, because trig is so messed up because of this dependence, and at first it doesn't seem like it would be possible to break it, but it is). In fact, maybe start with his Rational Trig first, just to get a feel for where he's going with things, and then go into Math Foundations to get into the nitty gritty of how it can actually be accomplished. Just a note of caution: It takes time to lay out the whole groundwork and the arguments, it's not something that is (yet) packaged up, even as an introduction, into a few short videos. Maybe one day it will get to that kind of polished state, but it's not yet. On the other hand, each video tends to be interesting in its own right, so it's not like he'll leave you hanging for multiple videos without any tangible/useful ideas. I'm just saying it may take 2, 3, 4, or more videos to really start to see where he's going with things. On the other hand, since you're already a finitist you might instantly pick up on it right away. Just giving you a heads up in case it takes longer than 1 or 2 videos. He also has a bunch of other (math-related) topics in his two channels, e.g. Math History, Projective Geometry, Famous Math Problems, Hyperbolic Geometry, etc., so sometimes it's refreshing to take a break from one topic for a bit to check out something else. That's how I tend to watch his stuff, anyway. Cheers! 😊
@@robharwood3538 Hi! Thanks for this. I will follow your reccommendation : ) I want to quickly respond to your comment about rationals though. I completely agree with you when you say "[a rational] is just a finite written expression of a finite number of symbols, and the 'division' aspect of it can be interpreted geometrically, if need be... ...or it can be dealt with entirely algebraically, which is another very useful perspective to have on it." This is true, and is why I, as a finitist completely accept rationals but also surds, trig functions, etc. Because, all procedures have a finite description. All machines are finite in size, even if they will grow when simulated. As a result, not just division but all mathematical procedures are legitimate (in my current understanding). The problem is not that they "don't exist" or something, the problem is that people are confused because they don't understand these "numbers" as procedures. If we understood them this way, things would be a lot more clear. I will watch his rational trig series, and see if there is in fact no geometrical or algebraic interpretation. For me, that would be a symptom of some kind of problem. Let's see... Thanks, God bless.
Unsurprising truth be told, I am highly sceptical of your position. However, I want to know more about the critiques you mention here that are on your playlist. Could you recommend me any set of videos from the playlist I would need to watch to have a better grasp of your claims? If so, I would thank you, and pardon if my English is rather sloppy. Sincerely, an amateur mathematician.
For me, the most important problem is about the properties in real number arithmetic bc, not having an algorithm to add irrational numbers term by term seems imposible by the fact that is imposible to know all the decimals of such one irrational number with an algorithm. There is no periodicity, and no periodicity means infinite calculations to know every single decimal of an irrational (It is not equivalent to say that any particular decimal of an irrational could not be deduced by not doing the computional algorithm through the point. Them also could be deduced by some advanced sketch)
@@isaacstamper7798 "I am a communist, so no" - If you actually survive in real society, then you cannot be a communist. It's probably only your fantasy. You have to necessarily behave as if you accept money, as if ploretarian are NOT the primery movers of history, as if you DON'T struggle to overthrow bourgeois, as if private property IS real, as if people have a nature that cannot be changed and as if self-interest IS this nature. Even if you were an intellectual or politician, you would still have to behave according these things which you believe to be fiction or mystifications of capitalist ideology. It must be cool to believe in a law of history that manifests only if whole world believes in it (only when the proletariat realized that "all they ahve to lose are their chains" - which they never did "realize").
So are the election results in the United States right now, but that won't change anything. It is literally impossible to do arithmetic with irrational numbers, so it's hard to see how it isn't fake when nobody in the universe can do it.
I find this ridiculous. Mr. Wildberger is confusing value with representation. Representing the base of the natural logarithm with the letter 'e' is no less valid than representing the sum of 1 and 2 with the symbol '3'. Both numbers exist and have definite values. Both can be represented in different ways. When we write '12.3' there's nothing canonical or necessary about that notation. In fact it's just shorthand for the longer expression "1x10^1+2x10^0+3x10^-1". The two expressions represent the same value, the same point on the number line. Neither is any more the "real" value than the other.
There's no inherent contradiction in computing a derivative. There's no necessity to do an infinite set of calculations. Taking a limit isn't implying an infinite set of calculations. It's a way of saying something like "If we could do an infinite number of calculations, the resulting value would be..." To put it another way, we don't need to do an infinite number of calculations because we have another way to find the value.
As a high school mathematics teacher I find this whole thing to be part of the distressing and dysfunctional belief that mathematics is a set of problems with correct answers. The standard curriculum starts out on the wrong foot by reducing the whole subject to a matter of calculating the value of an expression, and it never recovers. Normally you have to get past the first year or two of university mathematics to discover that that's not at all what mathematics is.
What Mr. Wildberger is showing isn't that arithmetic is fake. He's showing that computers are bad at math.
Well said.
Do you have some sort of notion of information theory?
If a computer in principal cannot perform the math, then it's not math.
As a math professor myself I agree with you fully.
@@keystoneperspectives OK, so I'm being a bit cute when I say computers are bad at math. I'm not sure what you mean by "the math". Do you mean computing the result of an arithmetic operation? Math is so much more than that, but even if we limit ourselves to arithmetic calculator, a computer can only represent an infinitesimally small proportion of all integers, and an even smaller proportion of real numbers. One way to show this is as follows: in a spreadsheet cell type "=(2^50+1)-(2^50)". That's obviously 1 but on my PC the result is 0. Given the set of all possible arithmetical computations, which can be shown by a simple diagonal construction to be uncountably infinite, a computer can only perform an infinitesimally small number of them.
Most of this is flat out wrong. For example:
- Operations on the real numbers is well defined via Cauchy sequences. The proofs are not that hard.
- Limits, integration and such don't require you to do infinite tasks. It's always just about being closer than a given epsilon in FINITE steps.
Also, what's the obsession with writing out numbers in decimal about?
Consider the standard view that a derivative describes the tangent at a point. With the limit definition, every single case considered deals with the secant across some interval. Are we justified to say that the limit describes a point, even though we know we cannot bring that interval to zero?
@@keystoneperspectives That is exactly the problem we had to overcome while making the concept of limits. We can never bring it to 0, we can only bound it arbitrarily close to 0 in a neighbourhood close to it, which is the next best thing.
@@aviralsood8141 Are you sure the concept of limits did not create a new contradiction as fallacy/sophism? Derivatives can and should be described as varying coefficients of polynomial function without limits and any geometrical interpretations, pure algebraicaly. With limits all very not simple where in one expression we have three and more dimentional/variable, and sophism of limits becomes self-evident.
@@dmitryezhov8352 Not everything is a polynomial man. And please tell me if you found a contradiction in the concept of limits.
@@aviralsood8141 If we can never bring the denominator to zero, why do we think that the two points defining the secant can be brought together to produce the tangent?
19:25
You’re just flat out wrong. Given an algorithm for the nth digit of some number (rational or irrational) A and an algorithm for the nth digit of some number B- it’s actually quite trivial to create an algorithm for the nth digit of A+B.
Let’s call the output of our algorithm for A given the input n A(n) and similarly for B.
Then here is the algorithm S(n) for generating the nth digit of the sum of A and B.
Take A(n) + B(n).
If your result is greater than 9, subtract 10. Otherwise keep it. We’ll call this number s.
Now take the sum of A(n+1) and B(n+1). This result will fall into three cases-
Case 1- it’s greater than 9. Then add 1 to s or if s is 9, make s=0, and your done. You have the nth digit of A+B.
Case #2 Your result is less than 9 and you’re done. s is the nth digit of A+B.
Case #3 Your result is 9. So you move on to summing A(n+2) and B(n+2). This result will once again fall into three cases and you follow the same steps previously used only in the case of it being 9, you move on to the sum of A(n+3) and B(n+3). And so you continue this process until you reach a non-9 result, and you now have the nth digit of the sum of A and B.
Now, I’ll acknowledge that it is possible this sum generates an infinite number of 9s at some point- so there may be some specific point where you’re summing algorithm won’t terminate. But if you’d argue that means you don’t truly have an algorithm for this- then I’d counter that the same reasoning can be applied to the decimal system in general when trying to calculate the sum of 1/3 and 2/3 (represented as decimals) and so you have to acknowledge that regular old decimal addition of the rationals ALSO is “fake” because there’s “no algorithm” for computing the sum of any two rational numbers.
Incidentally- this also means when you have the algorithms for the decimal expansion of any set of irrational numbers, you can generate an algorithm for their sum- which basically answers most of the “challenges” posed later in the video. For the multiplication and exponential problems- it’s still fairly trivial to develop general algorithms for those results although you have to go quite a bit further to the right in order to make sure your results aren’t being changed by carrying digits.
I didn't watch the whole video. I stopped at 20:11.
Several things here:
- Real numbers are usually defined together with the arithmetic operations on them. So arithmetic works in this case by defintion.
- Your statement is basically "Because there is no algorithm which can give you any n-th digit of a sum of all two real numbers after finite time , arithmetics is ill defined".
But it is quite a basic thing in math to show existence (that is logical consistency) of an object without actually providing an algorithm to construct it (which can be impossible)
e.g.
-- existence of a smallest sigma algebra
-- existence of uncomputable numbers
-- A functional analysis proof of the existence of continuous but nowhere differentiable functions.
etc.
But even if you'd demand an algorithm for the sake of practicability or whatever the following Algorithm (which stops after finite time) is completely sufficient:
Input: a,b, e >0
Output: c s.t. |c-(a+b)|
Thanks for saving me 20 minutes and 11 seconds
"- Real numbers are usually defined together with the arithmetic operations on them. So arithmetic works in this case by defintion. "
do you realize that this allows for anything? mouse + cheese = elephant by definition in my new mathematical system, how useful is it?
"quite a basic thing in math to show existence (that is logical consistency) .."
that is the heart of the topic, those are not the same thing. You can have logical consistency of whatever joke-based math. that in no way helps us model the real world.
"I don't see why you'd need an algorithm which can give you the correct digit after finite time since this digits are just representations anyway."
sorry, what? isn't it the purpose of math operations to obtains precise results? this thing ("real" numbers) are at best an useful engineering solution to math problems, it is not rigorous math.
@@jgmartinezmd6867
- Real numbers are useful as you know if you look at all kinds of sciences. My point is that arithmetics on real numbers are by definition well-defined. You cannot think of numbers without the arithmetic operations on them. Therefore it is a fruitless task to check if arithmetics works on real numbers.. it does.
- As I understood he makes a mathematical point here.. I assumed he really thinks that arithmetics is ill defined (in a mathematical sense) on real numbers. This is not true.
- It is not necessarily the point of math to provide an algorithm to calculate stuff. Often the task is too show properties of some objects which can be useful in real life applications. The arguments used in showing these properties can be quite abstract. Showing something by constructing it is one proof strategy but by far not the only one. But in this case you actually have an algorithn to provide arbitrary precise results. Note that getting arbitrary precise results is not equivalten to getting the n-th digit right. Consider the following toy example to get the point:
a=1/3=0.333...
b=2/3=0.666...
obviously a+b=1.000...
numerically you could e.g. take the truncated version of a und b up to the n-th digit and add them. that means:
a_0= 0; a_1=0.3 ; a_2=0.33 ... a_n=0.333...333 (up to the n-th digit)
a_0= 0; a_1=0.6 ; a_2=0.66 ... a_n=0.666...666(up to the n-th digit)
therefore a_n+b_n = 0.999...999 (up to the n-th digit)
As you see you never get any digits right but you get arbitrarly close to 1
Note that digits are just a representation of the number. It's not problematic that you are not getting a digit right as long as the results gets arbitrarly close
Hi Dr. Wildberger, This video really fascinated me, because I have been having similar thoughts without ever coming across other peoples views such as yours. I do not want to speak for you, but I believe in a fakeness that possibly goes way beyond what you are suggesting. I believe the whole real number line is completely misunderstood through a deep social conditioning of our understanding of reality. The 'order' of numbers on the real line is fake, it is an assumption of our reality that we believe in because we are deeply socially conditioned to do so. I actually believe that every number is infinitely dimensional (even natural numbers), and they live in a chaos space where there is no order or structure whatsoever. The apparent 'order' of the numbers on the real line is a contrivance of our reality that goes onto create our reality as we experience it. Put another way, the order we believe numbers sit in, create the type of reality we experience. If we believed that the numbers sit in a different order say, 1,2,3,4,6,5,7,8,9, this belief would create an entirely new reality for us all. I believe the fakeness goes way beyond this. For example the equals sign is the biggest fakeness of all, nothing is ever equal to something else that looks different, i.e. 1+2 is not the same as 3, they are fundamentally different things. I think we need to abandon 'equals', and replace it with 'transforms into'. I am also with you on connecting, sociology, psychology, and pure mathematics, again I take this to the extreme, because I believe that when we peel off all the layers of our ego, we will see all of truth, and this is not just a belief, as I have had mystical experiences where I have experienced exactly this. I would love to talk more with you, if this is of interest to you, I have also put up a few videos on these sort of ideas in my TH-cam Channel. Thank you so much for the wonderful videos you have published.
Gödel is a very great thinker. It’s not apparent untold it’s apparent
Not a mathematician here, merely an amateur, but I do have a thought I'd like to share.
Aren't numbers merely labels we apply to abstract objects? Some of these objects are more agreable to our understanding and lend themselves to simpler, easier to manipulate labelling schemes, like the integers.
From there, computation really amounts to finding the tidiest, shortest, cleanest label for an object. Computing 1/2 + 1/3 is not asking "What is this?' but rather "Which is a more agreable label for this object." '2' is a tidier label than '6 - 4', but both really represent the same thing.
By "tidier" I really mean "more manageable for our human brains".
Our most widely accepted/used labelling system for these awkward 'real numbers' does produce rather unwieldy labels for them but do all system not have the same kind of limitations?
For instance, when it comes to large integers, we do rightly resort to 'computational' notation as a more practical label (ex: the googolplex), not to mention how complicated some basic comparisons like > become with these larger numbers.
Again, we seem to have unwieldy labels for objects that are further removed from our human scales.
In that sense that number systems are human inventions to make foreign objects manageable by our human brains, aren't we bound to hit a hard limit of complexity/unwieldyness whatever the system we come up with?
As always your videos and the insights you kindly share with us are invaluable. Thank you so much for that!
The real numbers minus the rational numbers arguably don't even exist. Yes, including the irrational numbers, as a construct such as sqrt(2) is meaningless in nature. That they are impossible is easily proven: if there were an algorithm that can compute an infinite number, then either it must require an infinite amount of time, or an infinite amount of computing power to do so. Neither apply to anything that exists in nature., especially not physical objects with very much bounded dimensions. The very best you'd get is an approximation that terminates after n digits.
If they don't actually exist in nature, they must only exist in the minds of people. Abstract objects don't exist. What is the locus of abstract objects? None, except the minds of people. Those numbers are not merely "more manageable" for our brains, they are a product of them wholesale. So what is the relevance of -- as you put it -- creating symbols to denote mathematical objects us humans have come up, except the study of said human constructs?
You can have real, irrational, complex etc. numbers no problem. As long you accept they are fictional.
@@ekszentrik To be fair, Prof. Wildberger is enthusiastically willing to work with *algebraic* extensions to the rationals (or any other basic field), such as a symbolic sqrt(2), defined as a number with the property that when you square it, the result is 2. Similarly, he has no problem with the complex/imaginary numbers like i, or the i, j, and k of quaternions, etc. But the *underlying* field he'll use will be the rationals, rather than the 'reals'. And he won't mess around with trying to approximate the symbolic sqrt(2) unless/until it's needed for some practical approximate purpose at the end of whatever calculation it arises/remains in.
,
@@ekszentrik Interesting point, with which I'd be tempted to agree. But I fail to follow your meaning when you say that sqrt(2) is fictional. If you're correct wouldn't it follow that the right triangle with sides 1 is also fictional? If it is so for other reasons than simple physical imprecision in building one, doesn't it in turn render the number 1 fictional as well? Ad absurdum it feels like 'fictionality' as in having its locus in the human mind is a rather weak concept, if not entirely circular and devoid of any epistemological significance. What do you think?
@@ekszentrik Ok so your concept of existence is that something exists, if it is part of the "real" world right?
How do you know that the real world even "exists"? What does it mean to be fictional and what does it mean to "calculate" something?
Mathematicians have long pondered about these questions, and the best way to solve it is to not talk about "existence", but rather to formulate mathematics as a language (i.e. a set of rules how to but together signs on a peace of paper, where the grammatical rules how one is allowed to put these letters together are called the "axioms". Then a mathematical proof for a "Statement" (which is also, by definition, just a sequence of letters) is defined to be a Sequence of letters satisfying the grammatical rules such that the Statement is written at the end of this (finite) sequence.
In this framework one can for example take the axiom System of ZFC (Zermelo Frenkel Choice), and then one can "talk about" mostly everything that is commonly done in pure mathematics and one can for example prove (in the sense I described above) that there exists something that we call the "real numbers" and it is unique up to unique isomorphism (when one makes this precise).
In principle, for any mathematical proof (in the classical sense) to be "correct" (for example in ZFC) it should be possible to translate this proof in the language described above, so that when a computer parses this finite string, it satisfies all grammatical rules required in ZFC.
Of course ZFC could have contradictions, maybe because it talks about infinitely big sets. However then the theory is not "proken" but rather any statement can be proven in the above sense and the problem is that the theory will just become uniteresting.
I'm not an expert in logic, but if you want a reference for this look at "Combinatorial Set Theory" by "Lorenz Halbeisen"
When you finish a math major but you never understood what an equivalence class was lol
O.o
Don't add decimals, but rather add two cauchy sequences (with limit in real number) component-wise
Sequences don't exist to him. This is the problem. *sigh*
@Jukke jukke Sorry I am not too adept at adding Cauchy sequences in their entirety. Perhaps you could illustrate the technique by adding the Cauchy sequences for pi and e together?
@@njwildberger Let a and b be two Cauchy sequences of rational numbers.
Define a + b = { (i, s) | i in N and s = a(i) + b(i) }.
This can be shown to be a Cauchy sequence of rational numbers.
Then, pi + e = { x | (Ea)(Eb)( a in pi and b in e and x is an equivalent Cauchy sequence of rational numbers to a + b) }
@@billh17 Your definition is inconsistent. If a and b are Cauchy sequences given by some functions a(i) and b(i), why have you switched to set notation for their sum? You also concluded that "pi + e = { x | (Ea)(Eb)( a in pi and b in e and x is an equivalent Cauchy sequence of rational numbers to a + b) }". Hence I suppose you would suggest to a primary school student that "1/2 + 1/3 = { x | (Ea)(Eb)( a in 1/2 and b in /13 and x is an equivalent Cauchy sequence of rational numbers to a + b) }" in the "arithmetic of real numbers". Would such an answer get full marks? At what point are you going to admit that this is all just talking??
@@njwildberger functions are sets of ordered pairs. This is a completely consistent definition. Why would you bring up primary school? Should all of mathematics be simple enough to explain to an 8 year old? That's absurd
When I started in primary school, I used to ask questions about arithmetic , like how can 3x4 be the same as 4x3.
My teacher had never of the commutative law. When I got to secondary school and began studying calculus , there wasn't time to ask any more questions, exams had to be got ready.
@Thomas Kember Nothing like a busy schedule to keep the unwanted questions at bay. I wonder how many analysis or calculus classes around the world are going to start on Monday morning with the prof saying: "There has been some interesting discussion coming from Wildberger's claim that our real number arithmetic is fake. Let's spend the hour actually investigating that claim, and see how it stands up." If this happens in your course, please let us know!
Wow, it looks like you were unlucky to get a bad teacher. The best and worst thing about teachers is that they form our attitude toward their subject. They make us experience it only from their point of view. If you don’t like a subject, your teacher failed to inspire you.
It is a good question. Because when we get to infinite ordinals, or "infinite positive integers", the multiplication is no longer commutative.
look up peano axioms
That's because you never took a basic course on abstract algebra.
Hi Dr. Wildberger: I would offer one minor suggestion. Of the three academic disciplines of psychology, sociology, and anthropology the one, it seems to me, that you should be appealing to is anthropology not sociology. These three fields all ask similar questions, often, but they have very different methodologies. A psychologist researcher might try to conduct a controlled experiment to study your issue, a sociologist might try to go back through historical records and analyze data or maybe take a poll. An anthropologist will actually get in there and study mathematicians “in the field” and I suspect that is what you are after.
However, such a study would be conducted without any judgement regarding the “correct” resolution of the issue. Their work may yield a lot of information about how and why the culture of mathematicians dismisses outlying ideas, or how TH-cam gives minority perspectives a stronger voice. Their work will not address “Why can’t’ most mathematicians see the obvious”. That is, anthropologists (or “sociologists” as you call for) will never assist you to shine light on this interesting issue. That burden will always reside with you and the merits/usefulness of your alternative ideas.
@Xyl... That is a good point --- I would love to include anthropologists and indeed psychologists in this investigation. But I am a bit more hopeful than you about getting reasonable social/cultural/psychological answers to the question "Why can’t’ most mathematicians see the obvious?”. And indeed there is another potentially interested cohort that I would like to get thinking about these issues --- the pure mathematicians themselves! I bet a lot of them would have some pretty interesting takes on all of this; once we get them over the hurdle of actually considering the possibility that they have been seriously duped, and are currently doing the same to the next generation.
@@njwildberger It's not that they don't *see* what you're saying-- they see it fine, they just *disagree* with you about it. As to _why_ they disagree, almost certainly because you haven't actually made a good argument. Your whole point seems to be that everything should be computable because they better corresponds to the real world, but pure mathematicians generally aren't concerned with modeling the real world anyway, so your point isn't really _relevant_ to their work.
The thing that you don't seem to understand is that _all_ mathematical objects are purely abstract ideas. *None* of them "exist" in the usual sense of the word. That's as true for the rationals as it is for the reals. They "exist" in the same sense that magic "exists" in the world of _Harry Potter._ That is, they're just _ideas,_ and so, as long as they're logically consistent, they're all equally valid.
You can say that the real numbers don't "exist" _in the constructable universe_ and that's fine, but it's the same kind of statement as, "There's no such thing as magic in the world of _Batman._" _True,_ but it does nothing to invalidate the statement, "In the world of _Harry Potter_ magic exists."
It's all about what universe(s) of discourse we're working in. Just because _your_ preferred universe of discourse doesn't have real numbers doesn't justify claiming that there isn't another, equally valid, universe of discourse that does.
@@Lucky10279 it sounded like he argues that they aren't logically consistent. For example, the addition of some transcendentals can't be evaluated. Or, that they are treated in analysis like discreet numbers and ongoing processes interchangeably. In this 39 minute video he does not simply claim they don't exist- he gives several examples of their apparent logical incoherence. If you can clarify those examples we all might learn something.
@@Robinsonero Well I didn't watch the whole video, as I was already familiar with argument from an article of his. If you tell me a specific example I'd be happy to try to clarify it though.
@@Lucky10279 25:55
evaluate pi + e + root2
I think the ironic thing about this lecture is that it presupposes that common arithmetic is “more real” simply because it’s the more common human experience. IMO, the universe doesn’t care enough to count things, implying that counting is a human abstraction itself.
19:45 : is the example correct? How can you tell these 2 numbers have infinite sequence of digit? Only by their actual algebraic definition - as the roots of an equation for instance. And it is starting from this definition, the only thing conveying all the information about them, that you can hope to create the right algorithm for calculating (any approximation of ) their sum. That there's no universal algorithm to calculate the sum of 2 real numbers doesn't invalidate the definition of the real numbers. It's a solution space like the complex numbers. You can deny this - but at the expense of staying stuck in trivia.
I have a doubt if the concept of infinity is non sensical then math should consist of a finite number of theorems , but according to godel incompleteness theorems math is incomplete, in other words no matter how many theorems we prove, there will be true statements which cannot be proved by our current system, and if math is complete and finite then it's inconsistent. So how to deal with this?
Hi Norman,
Thanks for making a clear presentation outlining your view.
I conceed that with your proposed questions, the best I could do probably arrive at re-expressing them as infinite sums etc.
However, I think your notion of "definition" is much stronger than what most pure mathematians require a definition to be.
Never in pure mathematics has it been the case that a definition must be some kind of algorithm from which we obtain explicit objects.
Definitions need not provide explicit representations or "answers" or algorithms for allowing humans to digest them.
The brilliance of the foundation of R uses definitions which don't give explicit representations of objects time and time again.
You're absolutely right, barely any operations on real numbers give explicit answers but that doesn't mean they are not valid objects.
Your qualm with infinite sums I think you already know is not quite right and highlights exactly the point I'm making here. It's defined as the limit of partial sums. Who said you have to evaluate explicitly all these partial sums? Or know explicitly what the limit is for that matter.
Limits are defined also in a way which does not prescribe an algorithm or explicit represention of them but they are nonetheless well defined.
Even if some limits will never be obtainable by humans in explicit form, they are valid objects nonetheless.
I think what needs to be assessed here is the fact that explicit representations or "answers" of things is subjective.
Here is a (degenerate) example of my point:
1+2=3 means:
"the unique natural number 1+2 is equal to 3"
As opposed to:
"1+2 evaluates to 3"
This latter interpretation makes no sense, it's like saying that 1+2 represents something that "is to be done". This is incorrect. The sum has already "happened" and it so happens to be equal to 3.
1+2 has an "explicit representation" or "answer" of 3.
Apologies for rambling, I may edit my response to make the message clearer.
Thanks,
David
@ David Alexander Are you really confident that this is an approach to mathematics that is coherent, or that would be approved by the great mathematicians from days gone by, or that it actually is well-defined? Isn't there something solid for your Grade 3 student to learn that 6 + 7 = 13, and not some range of possible wafflings? Are you happy that if "real numbers" include natural numbers and integers, then over the real numbers all arithmetical questions posed by primary school teachers become trivial by your own standards? So for example 6 + 7 is equal to 6 + 7, because that is a "valid object". Full marks?
@@WildEggmathematicscourses it hardly seems like the approval of greats from days gone by makes for a good criterion of soundness. Pythagoreans were excellent geometricians but would have scoffed at the work of Galois for relying upon symbolic notation rather than construction, and yet his own work is powerful enough to demonstrate the limitations of their craft in a completely rigorous fashion. We can't blame them for it, but the greats of old are not so blessed as we with the benefit of hindsight
@@WildEggmathematicscourses you don't need the approval of past greats, we have living great experts today that are perfectly capable to evaluate any approach. Mathematics done by Gauss, Newton etc is not the only valid mathematics. This sort of reasoning is appeal to authority fallacy, it exposes a deficiency in your argument. Yes it is perfectly adequate to teach children that 6 + 7 is precisely, identically, unmistakably 6 + 7. However we also need to teach that 13 = 6 + 7. Although we only teach transitivity at university level, it is extremely practical to teach children the next step: 6 + 7 = 13. Don't forget school is not meant to teach only one side (theoretical or practical). You can finish a higher level mathematics course without doing a single computation (if you choose to not define demonstrations as computations), but you can't finish an applied mathematics course without doing computations, they are the whole point of these courses. So is the point of school, both teach you foundations and applications.
'So for example 6 + 7 is equal to 6 + 7, because that is a "valid object". ' - in an ideal world, children would leave primary school knowing this.
@@njwildberger You are entirely misreading that statement. That last statement was but a response to your last questions: "Are you happy that if 'real numbers' include natural numbers and integers, then over the real numbers all arithmetical questions posed by primary school teachers become trivial by your own standards? So for example 6 + 7 is equal to 6 + 7, because that is a 'valid object'. Full marks?". It was not meant in any way to justify real number arithmetic. I used to teach arithmetic to children (Kumon system) and one the main issues I've seen some children develop is failing to understand expressions as "surrogates" for numbers. This fault diminished their intuition. I fully expect the success case for primary mathematical education to include the notion that valid (closed) expressions and numbers are identical.
@@renato360a I think it is very important to distinguish between the arithmetical objects themselves and expressions which may evaluate to them.
Speaking as a mathematician, it seems to me that while sociologists may be able to give an account of how mathematicians come to 'pretend that all is well', they can't actually fix the math foundational problem. Maybe it can't be, anyhow! - but otherwise, few sociologists are mathematicians by training or inclination. So it's up to the mathematical community to clean out its own stables - and Hercules the Sociologist, can take the week off!
At a personal level, MJW, I'm glad to see you looking a whole lot better. Your apparent absence for a while had me worried! Whatever it is you're doing - keep right on doing it!
It’s funny how people have such affinity to what they’ve learned. Still, Norman is 100% correct from a pure mathematical sense. He has shown most trig and calc can be done in a better way, built on a stronger foundation. I believe the hope is that future minds can build on that. However, the current limitations must first be recognized before they can be overcome.
All he has shown is an alternative way which takes a few definitions and changes them to not rely so heavily on real numbers. I like those methods but still the fact remains that real numbers are part of a majority of mathematical topics and there has been no one who has come up with a 'better' way.
@@aviralsood8141 In this instance i would argue all he has done is incorrectly used the term arithmetic. rather than actually talk about binary operators on a field.
Dimensional Gauge Symmetry; my TOE explains the most with the least. It is God's combinatorial infinity. We know they are all stealing from me, and now God does too.
*Still, Norman is 100% correct from a pure mathematical sense.*
Except he isn't.
@@dsm5d723 so what does gauge symmetry have to with any of this? Interestingly, if one considers the reals as ill-defined, not a single Lie group survives, and not a single gauge theory survives.
Can we consider a real number X to be an algorithm that emits rational numbers that approximate X up to any finite level of accuracy that we input to the algo? Note this is different from asking for an algo that emits a certain number of correct decimal places, since it is immune to the waves-of-carries problem.
It is straightforward to define such algorithms for pi and e, in fact they are quite compact. We can then easily generate an algorithm meeting the same requirement for pi + e.
The algorithms are themselves finite artifacts, and we only ever ask for a finite (but arbitrary) level of precision.
I think there is another problem in the "program that returns i-th digit" formulation: with finite amount of memory and deterministic processor we will always end up with a sequence that starts to repeat itself starting from some position.
Thus, we will be only able to get a rational numbers, which can be easily added.
@Edward Newell, Please have a very careful read of your first sentence. Can you see that it has an essentially self-referential nature? What if your algorithm seems to be outputing numbers that are close to the outputs of my algorithm. How do we conclude they are in fact the same? In any case trying to define "real numbers" via algorithms is tempting. But no-one has been able to make it work. Maybe for a more limited notion there is some hope. Part of the problem: you first have to wrestle with the general definition of "algorithm"including again the issue about when two algorithms are "the same". And defining the arithmetical operations is fraught with difficulty.
That’s basically a definition using a Cauchy sequence
Imagine two different algorithms that both claim to produce pi+e. How would you be able to tell they are the same if all you can ever calculate is an approximation? Likewise for whether neither one, either one or both are correct?
@@nazlfrag algorithms don’t claim anything. You have either proven that it indeed does, that it does not or you cannot yet make any definite claim.
The calculation of the sum of two numbers with infinite digits is defined although there is no algorithm to calculate all its digits. Mathematics is based on well-defined concepts rather than computable algorithms. It seems that your philosophical position is intuistism, which believes that mathematics must be intuitive and rejects all well-defined mathematical concepts when there is no algorithm to calculate the concepts. This position is too restrictive to do Maths.
Any time you introduce any kind of infinity into an expression, you've entered a whole different kind of math. Why should having an infinite number of non-repeating digits (even describing a finite number) be any less problematic than infinities of any other kind?
There are certain objects which are only represented infinitely in certain bases, like 1/3 in base ten. Also, 1/3 in decimal notation is a repeating decimal, so the problem of being cot computable doesn't exist. It's clear that 0.3333333.... plus 0.3333333... is 0.6666666. But transcendental numbers cannot be represented this way.
All notions of "infinite sets" are equally problematic.
Assuming you're correct, what solution do you suggest?
making it to the end of the video, I understand what the problem is: you're hung up on Zeno's paradox. Yes, it's that "trivial" of a problem. You see, in theoretical thinking we don't need time or space. Mathematics was never about the real world. In a theoretical world everything goes, at least everything the mind can conceive of. It only depends on your ability to create or choose axioms. We can readily have the result of an infinite number of operations. Admit to yourself that you CAN imagine it, and you'll find yourself to be able to. I claim to be able to solve the paradox. Proof?
I didn't define my axioms first. Here they go (although this is all still very informal): 1. Peano arithmetic. 2. First-order logic (sorry, not trying to be aggressive, but is this "jargon" to you as you point out at the end? I mean, if you want to have an "adult" conversation about math you can't escape some rigor). [My personal axioms:] 3. (In our hypothetical reality) we can hold an infinite number of characters. 4. (In our hypothetical reality) we can perform an infinite number of tasks. [In other words, there is no time or space, they're unnecessary concepts]
Proof: I can map Achilles' steps and the tortoise steps to infinite sequences of numbers. Using my axioms I can take the sums of these sequences, then I can verify they're identical.
Now, this only solves the paradox in my make-believe theoretical world. Zeno's paradox obviously has applications to both theoretical worlds and the physical world. If you want to solve it physically you need to go beyond mathematics, into physics or philosophy or something. What I set out to do here is make an analogy between the paradox and the process of taking infinite sums, and show that it is easily doable if you have the diligence to build the appropriate foundations.
As a token of good will, I will attempt to solve your computing challenges. The primary answer is this: your questions are ill-defined. If I just repeat the left-hand side expression you won't accept the answer, whereas the answer is valid and perfectly sound. You didn't specify what format you want your answer to be in. You only said in a hand-waving manner that all the obvious answers are unsatisfactory. So, in the absence of an explanation of what would be considered satisfactory, we can only make assumptions. I'll assume you don't want to see symbols that represent numbers "beyond" rational or infinities (sums). I'll further assume that if I can compute any digit, then I can claim I know the precise answer. Now go ahead and give a decimal place for any of these questions and I'll give you a digit. Mind the fact that, for your own convenience, farther digits will require a long time to compute. Even if this process extrapolates the age of the universe, it's in principle always doable and that's all that matters.
But I'll grant you this. There are points of contention in mathematics. There are open questions. There are questionable axioms. Sets, which are to mathematics as the quarks are to Physics, are not defined in a way that makes everybody happy. But everything above these matters, everything that stems from these definitions is unquestionably sound, including real numbers.
thanks for the insight :)
Video Content
00:00 Introduction
02:36 Number systems
06:41 What system do scientists generally use?
10:09 What system do pure mathematicians use?
15:27 Arithmetic with 'real numbers'
24:43 Computing challenges for analysts
The key reason we programmers like "floating point" is because of time complexity. The floating point math, while inaccurate, will always run at the same speed, due to the truncation of lesser significant figures. This is rather important for certain applications that repeatedly do the same calculations (realtime systems), as you need the speed to be predicable.
However, consider a rational number: as you add, sub, mul and div, the complexity of the output rapidly increases, i.e. both the numerator and denominator constantly get bigger. Eventually (albeit often quickly) this saturates your integer word size (usually 64 bit), meaning you have to use multiple words, which takes CPU time (you can only process 1 word at a time). This is highly problematic because you want calculations in realtime systems (such as video games, flight computers, etc) to run at a predictable speed, not slow down pseudo arbitrarily depending on input device data (well a lot of games do this anyway, so imagine how much worse it would be if the hardware calculations themselves could slow down in this way (well actually they can, due to CPU caching and branch prediction, but this isn't a problem with arithmetic)).
Real arithmetic is fake because you don't actually do arithmetic? You just say what you did algebraically (if even that). Rather useless from a practical standpoint. Really just glorified philosophy. My problem with "real" number theory is it will get you stuck in "analysis paralysis" if you try to use it in any practical application. Things work much better when you just say "infinity is NaN (not a number)", and strangely enough, this doesn't seem to limit our capability in any meaningful way.
Is the square root of 2 a number? If so it is not a natural number, not an integer, and not a rational number. Or equivalently if you construct a square 1 unit of distance on each side and apply the Pythagorean Theorem then does the length of the diagonal exist or not or does the Pythagorean Theorem not work in that case - or any other case of a square for that matter. If you construct a rectangle with sides of length 3 and 4 then the diagonal is 5 and magically the Pythagorean Theorem works in cases like this.
Norman Wildberger is of the opinion that the metric concepts of length and angle are very problematic. In their place he uses the quasi metric concepts of quadrance and spread which are much more well behaved. Intuitively quadrance is the same as length squared and spread is the same as [sine angle] squared. Although it is better to think of length as square root of quadrance and of angle as arcus sine of square root of spread, which is generally ill defined, but works sometimes, inside the rational numbers.
That said, as an *abstract symbolic* concept, there is a sense in which a square root of 2 exists (just like a square root of -1 exists), but this is a point on an axis dimensionally independent from the rational axis, it is not a metric quantity anymore than imaginary i is a metric quantity.
There are algebraic extensions of rational numbers, of far higher dimensionality than the usual nonsense about "the complex numbers being an algebraically complete but metrically 2 dimensional extension of the real numbers". So forget about only using 2 dimensions for expressing field algebraic solutions to polynomial equations!
As a mathematician who now works with computer science, I understand your point. Basically, for you something should only be considered to exist if it is computable. And yes, in the computable universe, Real numbers do not exist. Heck, any infinity set does not exist. But the way you formulate your ideas - an attack to one of the pillars of mainstream mathematics - makes mathematicians on the internet complete disregard your video as a random internet madman, although it does make it more viral. Try to reformulate it as a movement for a new kind of mathematics that is closer to the computable world and the real world by avoiding infinity axioms and only dealing with objects which are computable or can be sufficiently approximated by computable things.
@GNavarro This is not an isolated video, although maybe you have just arrived here just now. I have more than 600 videos across a wide range of mathematics. The Math Foundations series presents a consistent critique of mainstream pure mathematics, but also lays out many new directions. The Algebraic Calculus One course is running now on openlearning, you can join for free, and start learning Calculus in the right way. Please have a look at the other playlists too. And mathematicians do not consider me a random internet madman, many know about what I am doing full well. They just don't like it, because it undermines the status quo upon which many people's life's work rests. But there is more at stake --- the education of a new generation of maths students!
@@njwildberger Norman, I love you because you never stop giving. You have added great value to my life. Thank you! I purchased your book, a fine product, excellent work my Brother.
Not all mathematicians ignore his work. He does an excellent job of explaining some very interesting ideas in an accessible way. Occasionally going into what a lot of us would disagree with. Still he offers a valuable service here that even a platonic can appreciate.
I live in leafy Surrey in England. It is autumn and many leaves have fallen. There are a finute number of leaves although nobody can tell me how many leaves have fallen at a particular instant. Or even if an odd or even number of leaves have fallen. This does not mean that the number of leaves does not exist. Similarly we cannot specify all of the digits of pi + e but I can give you any number of the digits that you want or better still, need. 2⅔ + 3⅗ is a perfectly good number. Just as 2+3 can be used in place of 5 in any computation.
Does this mean, that root 2 doesn’t exist?
It means rather that root 2 is not well defined. So its existence is a moot point.
@@njwildbergerThough we can treat two conjugate square roots of 2 in a very similar vein to the way we treat two conjugate square roots of -1. Purely algebraic field extensions (of rational numbers or prime fields) that is, polynumbers i think you call them.
Adjoining such square roots of 2 (or of other positive numbers) gives us a kind of relativistic/hyperbolic geometry over the rationals, rather than the elliptic geometry we get by adjoining square roots of -1 or of -3 (or to a lesser extent of other negative numbers).
This view of geometry as algebraic extensions of rational numbers or similar well defined base fields opens up the possibility of seriously studying 3 dimensional, 4 dimensional, 6 dimensional, 5 dimensional etc commutative field geometry (with conjugate roots in a sense determined by the Galois group of the defining polynomial of the Galois field extension) in a sense the possibility of which is generally masked by the illusion of complex numbers being a "complete" 2 dimensional field extension of the real numbers (both number systems being little more than illusions), and all higher dimensional algebraic geometries (over real numbers) demanding either zero divisors or non commutativity etc.
Stirring the ant nest again...
Eh, it's more of just putting food the ant nest, except it just tastes bad.
You seem to impose the condition that it all has to be algorithmic and computable. The current consensus in the (most of) mathematical community is the opposite. This is a difference of opinions, not facts. To say it differently - yes, the operations are not finitary or computable, but for the way pure mathematicians use them this is not a problem. I consider the use of the word 'fake' in this context to be misleading.
Regarding your question about concrete values of something like e + pi. Before I comment on that, I offer a counter question - what is the circumference of a circle of radius one? Or it's area? Is it a number? If yes, tell me which, concretely, as a finitary, number-like expression. If it is not, please explain why it is unreasonable to have such a number. Or maybe it is the case that circles do not really exist, but only their rational approximations? I did not watch your videos on foundations of math, and I assume that somewhere there might be an answer, but I most likely won't go through those videos anytime soon.
Now coming back to your question. There is no concrete object in the form of a 'palpable' rational-like finite expression and you know that. But what does it prove? It only proves the point that these things are not in general finite or computable - which is something that no one really disputes (but yes, the audience might find it strange and surprising). Say you have a motorbike, I have car and you send a bunch of sociologists to ask me how many wheels my vehicle has - of course the answer is 4, but that only proves that I have a car (and I never claimed otherwise), not that having a car is somehow wrong. The problem is that the way you set up the question makes it seem that any answer other than 2 is just bad. Yes, a friend of mine who thought I had a motorbike may be surprised by me having a car, but it does not make your point that having cars is bad stronger. The way mathematics is set up does not really need the objects it speaks of to be finitary or computable. Engineers need that, but not mathematicians.
The problem you have with all this seems to be mainly philosophical - you just don't think that this is what mathematics should be, I guess because it is too distant from reality as we know it. I guess a lot of philosophical problems of this type come from the fact that mathematics actually does not work with _any_ objects - it is purely syntactical and the meaning/interpretation of its theorems and objects it speaks of is just in our heads. No actual construction of objects takes place, the only thing that happens is theorem proving (symbol manipulation). Sadly, this is something I have basically never seen explained properly, in a common sense fashion. I would really like to elaborate on this here, but this post would be just too long. I definitely agree with you that the foundations are usually not taught in enough detail.
By the way, you say that rationals are ok. Fine, but what actually _is_ a rational number? If 2/3 and 4/6 are the same number, then neither of them actually _is_ the number, but they are both a representation of something. What actually _is_ this object they represent? Is it finite? Can you exhibit it fully? You can certainly write a computer program which tells you that two strings like 2/3 and 4/6 represent the same number, and you can also do arithmetic by a program. But do you actually want to postulate that only computable 'objects' are fine? Even then - what _are_ these objects anyway? Are they actually just the programs (finite strings of symbols) themselves or something like a collection of pairs of the form (input,output)?
PS: I am actually a fan of your lectures, you have a very nice no-nonsense teaching style.
@Leonard Szekeres First of all you have a famous last name--- George Szekeres was a friend of mine at UNSW and one of the best mathematicians in Australia during his day. Are you perhaps related? Let me answer some of your questions: a circle of radius 1 in the usual affine plane does not have a well-defined circumference, nor a well-defined area. Archimedes knew this already more than 2000 years ago. What we can do is find approximate circumferences, and approximate areas, and that is exactly the kind of question Archimedes set out to answer. Modern pure mathematicians are uncomfortable following Archimedes --they want exact answers, even if they have to impose them artificially. As for pi+e+sqrt(2), I take it that your best answer is: there is no concrete form for it. Fine, thus we can all agree this is a fake arithmetic. Additional extensive discussion about what mathematics should or should not be is irrelevant. You can't add the basic numbers in your system, therefore it is a fake system. I know it is hard to swallow that you have been misled, perhaps for many years. Why not have a go at the MathFoundations series?
@@njwildberger No, I am fairly certain I am not related to him, even distantly.
What I take issue with is your use of word 'fake'. One can also say the following "In your mathematics even the most basic geometrical objects like circles do not have some very basic attributes such as area or circumference. Fine, thus we can all agree this is a fake mathematics."
Of course, you can object to the above and say "Well, if a circle of radius one has a circumference, then _what_ is it? Please give me a concrete answer". To which I can reply "Well if a circle of radius one exists, then _what_ is it? Concretely please". And you can tell me that it consists of all rational points with distance 1 from 0. But that's a description of the object, not the object itself. You can say that this description is at least computable, but there are many programs which describe it - so which one _is_ the circle? Or all of them? That's a pretty large object, and how is that exhibited concretely?
"Additional extensive discussion about what mathematics should or should not be is irrelevant." - your entire argument is based on something that you think mathematics _should_ be (you say that arithmetical operations _should_ be computable) and at the same time you say that any discussion about what mathematics should or shouldn't be is irrelevant? Come on...
" What we can do is find approximate circumferences, and approximate areas..." - but what is it that you are approximating? By using smaller and smaller angles and line elements you can get a number which is arbitrarily close to X, but what is X? If X does not exist then you cannot speak about approximating it. Of course, you can get by without having an actual object which you are approximating, but then you end up with a Cauchy sequence. And then it is convenient to at least pretend we have them and we can do operations with them, even if we do not believe they actually 'exist' in the real world (neither do circles with infinitely many points).
If I have a finite resolution to work in I can give you exact values for circumference and area, and at no point will I be invoking pi or any other irrational in my calculations.
"We're disturbing the priest, won't you please come to our feast
Do we mind disturbing the priest, not at all, not at all, not in the least" -- Black Sabbath
For the final example shown between 24:42 and 31:43, each term in that symbolic statement and the sum can be expressed as finite numbers if you change the base accordingly. The least common multiple of the denominators 2, 3, and 5 is 30, so in base 30 the problem becomes 0.F + 0.A + 0.6 = 1.1. Left in base 10, the sum is simplified to 31/30 rather than leaving it as 1/2 + 1/3 + 1/5. Hypothetically, the other problems could be shown in finite form if expressed in base pi, base e, base sqrt(2) and so forth, but then that may create problems with expressing whole numbers from the decimal system in finite form, depending on the problem. Logarithms can be used to convert between bases as needed, though this would reintroduce those errors of finite calculations.
@Dorian, No-one has seriously suggested a number system with base pi or base e. With sqrt(2) we can't make such a sweeping statement, because algebraic approaches go someways in this direction. Once again the confusion arises with our terminology, of mixing up sqrt(2) as a magnitude somewhere between 1.4 and 1.5, and as a novel symbol involved in a quadratic extension field of the rationals. Please watch the Famous Math Problems 20, 21 and 22 videos (the latter is still on its way..)
@@njwildberger Angles are calculated mod 2pi which is the same as having a 2pi based nomber system though...
30 years programming I have never used floating point outside of school. Counting numbers and addition work trillions upon trillions of times without error. Thus when you say this is not used, I am quite perplexed. I guess you have heard of floating point and somehow extrapolated that onto all of computer programming?
I agree. For geometry / statistics / simulation we often use float (maybe double), but for counting things we use int (maybe uint64 or occasionally even bigger) unless we are using a really crippling programming language that forces us to use float. He could have been more careful to draw this distinction.
Once the integer numbers get big enough in some calculation, I think you have little choice but to go to floating point.
@@WildEggmathematicscourses I have yet to imagine how this could be correct. It feels similar to arguing that everyone MUST wear life jackets 24/7 on the basis that some people might not know how to swim and then arguing that no one knows how to swim using the universal life jacket rule as evidence.
I still ask myself, how do we know there is foundation of mathematics? And if it is the foundation then how can we reach it with our modern Mathematics that is built on it? Maybe Mathematics is like Physics, maybe in this universe it exists in that way and knowing beyond the universe is impossible
To be able to calculate, or compute, the challenge expressions 1-4 at 24:44 in the video, we would have to define some precision
first. The results, as reals, may be expressed as intervals of decimals, with a very high precision (to the 10th decimal digit), like this:
1) 7.2740880444 < π+e+√2 < 7.2740880445
2) 12.0770079567 < π×e×√2 < 12.0770079568
3) 1.6344452924 < π÷(e÷√2) < 1.6344452925
4) 110.8924304703 < π^(e^√2) < 110.8924304704
Perhaps we would like a perfect expression for each answer, but not by just repeating the expression
in each challenge. Sometimes you are lucky, as in the expression 8 that results is the number 1, but most often you will probably end up with an irrational number and get, expressed as a decimal, a non-repeating string of digits.
If we would like exact decimal expressions for irrational numbers like π, e and √2, we can't expect
these to be expressed in their entirety, since by definition these expressions are infinite in size,
but maybe the interval expressions are sufficient, since they can be extended to a precision of 100
digits, 1 000 digits, 1 000 000 digits or, in fact, any finite number you like (or has the time to compute).
If we actually need exact expressions for irrational numbers, is this even possible in the real
numbers? If we look at the definition of an irrational real number, we can see it is defined down to
infinitesimal size, which means that if the constant has non-zero infinitesimal addition it can't be exactly
matched by a real number to differentiate it from another constant that has no infinitesimal addition. This calls for another number system that is able to express infinitesimal additions, but this extension doesn't solve the whole problem, since a constant may differ with an infinitesimal to the infinitesimals in that system and so on.
Surely, sometimes you are lucky enough to have an expression that entirely defines the irrational number,
so maybe such an expression (an infinite series perhaps) can be used instead of a decimal interval if that is necessary in the context. If precision is really important, you could even work in an precision extended number system and then go back to reals (or rationals maybe) and in the process all non-zero infinitesimals vanishes.
So, are we satisfied? It depends on what we really want, it also depends on how precise we want the constants to be defined and so on. There is simply
no perfect number system that does all that, but we can at least pick one that is optimal for our needs, and when infinitesimal precision isn't necessary real number intervals will almost always do.
You say you are happy with applied mathematics making approximations. Approximations to what?
It’s better to think of running algorithms up to a certain point, and then terminating and having an approximate solution. That’s what applied mathematicians do. Pure mathematicians pretend that they can run algorithms to infinity to get exact solutions which they can’t write down. So they have an elaborate language of talking about all these objects which are not actually obtainable.
@@njwildberger An approximate solution to what?
Dr. Wildberger is an inspiration, and so many have benefited from his teachings
@@brendawilliams8062 And yet he can't answer a simple question.
@@chrimony you are so wrong. He knows the answers to so very,very many things. Light years above many others as qualified in mathematics
I think that how mathematics is framed has sociological (and economic) implications to how humans develop technologically. Therefore, questioning mathematical assumptions ( such as ability to perform infinite tasks) is sociologically relevant. I thank Prof. Wildberger for bringing this question into my attention. May his efforts continue to be a blessing to all of humanity like the mathematicians of the old.
mathematics is independent of society. even if no one was around to view it. mathematical processes would still occur. just like physics!
I get your point. But the sociological relevance of questioning is not always positive. Take flat-earth, vaccination and global warming, for example. Every child and teenager has to question these concepts as they learn about them. But the questioning of these concepts openly in society is doing nothing but harm. The same way, this prof. does not have valid claims and for any reason (that I won't judge here) failed to check himself. All this is doing is confusing the public, essentially it's disinformation.
@@Kav107 No it is not. Mathematics describes real physical processes. How are the processes transformed into human knowledge? Through complex social processes. Ignoring the observer would be poor science.
@Ben Catechi mathematics doesn't describe -just- real physical processes. It also describes impossible physical processes and predicts physical processes in which are un-testable. In fact it's logic is self contained. Concepts of unity, zero and infinity remain constant at any point in the universe and bypasses society or viewer. The simple fact is, it doesn't matter what society you live in..here there or anywhere. Or if there was no society at all. The concept of 1+1=2 is just that.
@@Kav107 No. Those concepts refer to actual physical processes. The problem is you think that a physical process as newtonian mechanics, when really a physical process can be a lot more complicated than that, including human society itself. Ideas can be objectively correct, but the form those ideas take is highly subjective. I'm not arguing that there is no such thing as objective truth. I'm saying that your insistence that human mathematical and logical systems, capture that objective truth without being filtered through society only puts your knowledge farther away from actual objective truth.
Any engineer would ask but how do you explain that the values we calculate using these perhaps slightly flaky procedures yield accurate results. Its worse for the physicists who sum known divergent series that miraculously agree with measurements to many decimal places.
The answer is that you don't. Arithmetic with real numbers is literally impossible. What you do is convert numbers like pi and e into rational approximations at the 11th hour before doing the actual calculation.
Of course, a divergent series doesn't produce a specific value. What has been been done instead is to introduce extra operations, such as weights, that select only some terms within a filter, which amounts to admitting that outside that range the mathematical model is not a good fit. There are other similar "regularization" techniques, not all of them being logical. One misadventure comes from bosonic string theory, where one of Ramanjuan's sums has been used to replace the sum of all natural numbers by -1/12, and thus the spacetime manifold's dimension was "proved" to be D=26, whereas the replacement wasn't necessary in the first place since the unmodified model produced D=2 without applying any trickery. (The original hope was that the model ought to "prove" that D=4 and thereby validate the theory.)
In the replies to this comment, I will present an extensive breakdown of the video point by point and a conclusive analysis. Please bear with me since this is an extremely long breakdown.
0:32 - 0:37 Contention in the community? I have no idea of what you are talking about. Arithmetic is not at all controversial right now in the mathematical community.
0:46 - 0:48 I assume you will define what "fake arithmetic" is later, correct?
2:41 - 2:51 This is not really correct. To begin with, the phrase "number system" is completely meaningless and ambiguous as you have presented it. Also, arithmetic operators are defined axiomatically, thus it is inaccurate to present operators and their axioms separately. To further worsen your argument, in mathematics, we often perform arithmetic with objects that, generally, are not considered numbers, such as sets, vectors, matrices, higher-order arrays, and other types of formal abstract structures. Many of them have no absolutely no relationship to the real numbers. A more accurate presentation of the topic that is fit even for non-mathematicians, such as sociologists, for example, would be to say that an arithmetic underlies (1) a collection of objects (2) a collection of rules that prescribe how these objects behave.
3:37 - 3:52 No, this is completely incorrect. Firstly, there is no such a thing as "the set of decimal numbers," and pretending that such a set exists, when you are presenting this to an audience which you yourself acknowledged is not necessarily mathematically inclined, is very dishonest. All this does is create misconceptions and confusion for the sociologists and anger the mathematicians and logicians. The set should merely be presented as the set of real numbers, and then just provide some examples, such as the numbers π or the golden ratio φ, numbers a non-mathematician may be familiar with, and this will give them an idea of what you mean without being misleading.
Secondly, those numbers are not contained in the set of rational numbers. To the contrary: the set of rational numbers is a subset of the set of real numbers.
5:26 - 5:34 Once again, this is misleading, because as I stated earlier, mathematical operations are defined based on certain properties, not the other way around.
6:20 - 6:41 You need to be careful with how you are wording what you say, because first you said these properties need to be proven, but then you stated that you need very precise definitions, not acknowledging that operations are themselves defined by properties. The problem is that you are starting this discussion without even having introduced the terminology properly to sociologists, and you have not explained to them what is the distinction between an axiom, a definition, and a theorem, and how those are all related.
9:01 - 9:23 I am not sure if this claim is supposed to be general about arithmetic, or this is only specific to floating-point systems and other similar things in applications. However, *just in case,* I need to clarify that this claim is only true precisely in an applied mathematics context. In general, though, this is not the case. Mathematicians generally work with exact, precise arithmetic, not approximations, unless they are studying approximation theory.
9:58 - 10:09 I am glad you acknowledge this, because a very popular misconception among non-mathematicians is that this is not the case.
10:45 - 10:54 Presenting this as the key point is somewhat misleading, though I understand what you were trying to get at. The set of real numbers are defined via limits, be it indirectly with Dedekind cuts, or directly as equivalence relations of Cauchy sequences. In either case, these numbers that are expressible as an infinite string in decimal digital representation are just the limit of a sequence, and it so happens that this limit has at least one decimal representation.
10:55 - 11:09 I seriously hope you are not going to present this video with the premise that infinity is a problematic concept. Such a premise is fundamentally flawed.
11:45 - 12:00 This is completely and beyond inaccurate. π & e as numbers are much older than what you are claiming them to be. The irrationality of numbers such as e & π was proven centuries ago, but in addition to that, you are neglecting to mention that *algebraic* irrational numbers, such as sqrt(2) and φ, were known to exist and to be rational by the Greek millennia ago. Constructible numbers in general include those irrational numbers, called quadratic irrational numbers, and they were of great importance to the Greek. I am no sociologist, but I do know also that in other ancient cultures, some of the metallic ratios were considered important as well for at least some applications in architecture and the visual arts. As for proving that their decimal digit representation required an infinite string, yes, this did occur later - though not that much later, but the existence itself of these numbers was known for a very long time. So saying that it was only now that we had to acknowledge that other types of numbers besides the irrational numbers exist is just false.
17:56 - 18:00 Challenging? Yes. Problematic? No.
18:05 - 18:14 This is false. Others in the comments section have already given examples of this, so I myself will not bother having to repeat what they said, but you should know you can also do a quick search on Google and see for yourself that such algorithms do exist, and applied mathematicians will tell you this. Many of these algorithms have already been implemented in computing, though to a limited degree, obviously. Stating there is no algorithm is inaccurate, and this is stemming from the fundamental misconception that there is only one valid sequence of steps to add any two numbers expressed in decimal notation with a finite string. Actually, you need not appeal to decimal strings to add real numbers at all.
19:19 - 19:24 Sure, for computable numbers, this is true. I fail to understand how this is problematic. At best, all this implies is that string representation is limited, and so are computers, which is a moot point, because computers are already limited anyway by virtue of physics. You can never accomplish with a computer all that a human can accomplish because that is just the nature of computers: they are different from humans. However, you need not know what the decimal string representation of a number is to be able to work with the number in the relevant context and understand it. Also, you already discussed earlier how, in applications, it is strictly necessary to have a flexibility for truncations and for approximations, so this is not a problem. These numbers are still definable, and from the definition alone, you can always make some amount of progress. Also, it should be noted that even the formal definitions of computable functions from any given computing model appeal to a concept of theoretical lack of limits about a given thing.
19:25 - 19:30 No, this is not true. There are plenty of real transcendental numbers which are computable numbers. For example, this is true of π and e. π + e is a computable number, as are π & e themselves. More impressively, πe & π/e are also computable numbers. Not only are they computable, they are efficiently computable, because there exist fast-converging sequences of partial sums of a function-generated sequence with which you can calculate these. Computing them is easier than proving their rationality or lack thereof.
19:31 - 19:39 Yes, you can. Your statement has been thoroughly debunked. The mathematical literature on the computational analysis and computer science, as well as the existence of irrational numbers that are also computable numbers is almost a century old and is quite enormous. I have nothing else to add here because, as I said, the literature is overwhelmingly large and comprehensive, and it does far more than just addressing the false claims you are presenting here.
19:44 - 19:51 What? Nonsense. Because uncomputable numbers exist and you fail to understand how they work, this means real number arithmetic is fake? That is just now how it works. An operation need not be defined algorithmically. In fact, they are not defined algorithmically. Some functions are computable, and some are not, but what makes a function satisfy the definition of a function has nothing to do with computability. All a funcion is, it is a subset of the Cartesian product of two sets, such that for every element of the first factor, there exists at most one element of the second factor which the former forms a pair that is an element of the subset.
19:53 - 19:56 They are properly defined. They are the supremum of a set of numbers and the infimum of its complement, they are the limit of a set of Cauchy sequences. The limit operator, the supremum operator, and infimum operator, as well as the sets they act on, are all properly defined. Cauchy sequences are well-defined. Therefore, the numbers are properly defined. There is nothing problematic here. I fail to understand what the problem is. Uncomputable numbers exist? What exactly is the problem with the existence of uncomputable numbers? Mathematics is a collection of formal theories, not a prescription or description of how the physical world operates.
19:57 - 20:03 Your presentation of real analysis as it is understood by mathematicians today is completely inaccurate, as is your presentation of the understanding that we have of set theory and group theory. I cannot tell if this is deliberate dishonesty, or deliberate ignorance and misunderstanding of the literature, but regardless, the fact that you are presenting these obviously incorrect claims to an audience you yourself acknowledged may not have the sufficient education to be able to discern as true or false for themselves in order for to present constructive criticism and logical refutations, is completely abhorrent.
There is no denying that some of your other work is respectable, and being allowed to present skeptical viewpoints is necessary for the community to work, but when you actively try to mislead an unknowing audience that is seeking to be educated on a topic for the sake of getting a validation card, that is when the conversation stops being about skepticism and it starts being about your ethics. This is not a critical argument you are presenting about mathematics, you are merely trying to fill an agenda disguising it as a presentation on mathematics you happened to construct very poorly.
20:20 - 20:28 Yes. Sometimes, some things are too complicated to explain properly. You want to tell you how the Big Bang works? Well, I cannot explain that to you if you lack certain prerequisite knowledge to understand the explanation you are asking for. That is just how life works. You cannot run without first learning how to walk. This is supposed to be a flow of the educational system? The education system is completely broken, I agree with this, but not for this reason. This is also by no means a flaw with real analysis, this is ultimately just a mere inconvenience inherent in the way learning fundamentally works. I cannot teach you about computable numbers if you are still trying to learn how to add numbers, so of course I would have to tell you "that is for a more advanced course."
20:28 - 20:38 Such as?
20:38 - 20:41 You cannot seriously say that given the hundreds of thousands of number theoretic proofs that exist, not only on the theorems of arithmetic themselves, but even on the effective finite axiomatization of arithmetic or impossibility thereof. This is just more dishonesty. If you could at least bother to give examples, then that would somewhat help your case.
21:33 - 21:38 What does that actually mean? If by "evaluate," you mean "to compute the complete decimal digit string representation of the number," then sure, it is true that we cannot write such a number in a complete decimal digit string representation. However, this is an entirely arbitrary, unnecessarily restrictive, unhelpful, and misguided definition of the verb "to evaluate." Such a definition would also imply that numbers such as 10^(10^100) and TREE(3) are not actually numbers that exist, because it is impossible to write the complete decimal digit string representation on a paper. This is particularly true for uncomputably large numbers, such as the busy beaver numbers and the Rayo numbers. You may as well deny the existence of any natural number larger than 10^(10^80).
Also, this claim seems to point to a fundamental conflation between the symbolic representation of a number and the number itself. The symbolic machinery you so despise is employed to represent *every* number, because numbers are fundamentally abstract conceptualizations not present in the physical world, so a physical actualization of the pure concept of the number is impossible. When I say every number, I really mean *every* number. The symbol "2" is just that: a symbol for the number. Other languages have other symbols for it as well. The symbol should not be confused for the number it represents. In this regard, if your logical deductions are consistent, then using the symbol "2" in Kindergarden arithmetic is no less problematic than using the symbol "log(3)" to represent the irrational numerical value it represents. In fact, why should you care about decimal digital string representations when computers operate in binary digital string representations? We should be representing the number as 10, not 2, at least if we want to maintain logical consistency with what your comments seem to imply.
By any sane and reasonable definition of the verb "to evaluate," these symbolical representations are already trivially evaluated, because we already know what abstract object to identify the symbolic representation with. The equality relation, when written as part of a symbolically-written equation of two expressions, merely identifies a representation of a number with another representation. The number itself is already known from recognizing the value represented by either string of symbols, or from using definitions and axioms to simplify the expressions via identification with other expressions: other string-symbolic representations of the same number, or object, more generally, since this also applies for matrices and vectors, for examples. This process of identifying an expression written with a string of symbols representing an object with another expression whose corresponding number value it represents is already known and recognized, is what mathematicians call an evaluation. It has nothing to do with computing a string of digits in any partocular base, be it decimal or binary.
Mathematicians do not think of digital strings as being numbers. These digital strings are merely how we choose to represent them because they are a very historically convenient way of actually keeping track of tallies and counts. You need to stop thinking as if numbers had to be fundamentally representable this way to be valid. Plenty of non-mathematicians, and especially younger students, think of the decimal representation as the number itself. So if you tell them 0.(9) = 1, they lose their mind, because they are incapable of conceptualizing the idea that a number can be represented by a decimal digital string in two different ways or more, because they think of each representation as a number, not as a representation. This would be akin to thinking that 2/4 and 2/1 are different rational numbers because the integers being divided are different. Strangely enough, they also may find themselves thinking that 1 and 1.0 are different numbers. This is a mistake that we need to address.
22:15 - 22:18 It is interesting that you mention γ in your list of examples, because it actually demonstrates why your previous claims are silly. Did you know that it is not known whether γ is rational or not? It has been proven that if γ were to be rational, then it would have an astronomically large denominator. Writing the number in quotient form would be completely unfeasible. It also would have a decimal digital string representation of infinite length. This would be an example of a rational number that, according to your argumentation, could not exist, it would be a "fake" number from a "fake" arithmetic. Also, independently of whether γ itself is rational or not, such rational numbers do exist, you even mentioned them earlier.
23:26 - 23:48 Wait, so you ARE arguing that these rational numbers are "fake" as you would call them because they cannot be written? This seems to contradict statements you made earlier in your presentation. Anyhow, I am glad you are at least now choosing to stay logically consistent, although you had to go quite late into the video to make the decision.
24:05 - 24:09 This infinite sum is an example of a computable real number, so it is inaccurate to say you cannot calculate it. Anyhow, yes, that symbol is the number's "name," whatever that means. Numbers do not have names. We assign them names because we humans need a way to refer to them, but the names depend on the language we choose to communicate with. As far as the conceptual abstract object that we call number itself is concerned, though, it has no name, and it does not care if it as a name or not. It just is what it is: an abstract object that has the value of a quantity.
24:39 - 24:42 To this moment, you have failed to successfully explain why it does not work. All you did was claim there is no algorithm to compute these numbers, which is false, and then proceeded to say that the entire system is ill-defined without any proof of this, and then began to proliferate arguments that are based on flawed premises and misunderstandings of concepts and of the current paradigm. A lot of finitist ideas are based on misconceptions and a lack of understanding of concepts as well, but here, as I stated earlier, I get the impression that you maybe do understand the concepts, and just pretend not to for the sake of an agenda. I still have not been given sufficient reason to forgive the amount of dishonesty in the video, so pardon me for continuing to repeatedly call you out on your intentions and your dishonest presentation of the concepts to an unknowing audience. Anyhow, this was all a rant just for to explain that, no, you have not proven the system does not work. All you did was present a few claims that can be easily shown to be false, and decided to move on from there.
25:14 - 25:21 So you are telling sociologists to prevent us from being dishonest in the exact same way to decided to be dishonest throughout most of the video by completely dodging the question and failing to present a proof of concept? Yes, I approve of that, although it makes you seem like a hypocrite.
25:35 - 25:39 Hearing you say this is quite hilarious, when in reality, the entire premise of finitism is reliant on philosophical obfuscation and not any sort of operational formal theory of logic.
26:03 - 26:09 π + e + sqrt(2) = 7.274088044421933522624619578841863460524088368452013469118591958
What? You wanted more digits? You never said how many digits you wanted the answer to have. This is what you get for asking a loaded dishonest question and not giving any room to explain why it is a silly question. You wanted an answer? I gave an answer, and you *have* to accept it, because you said you would not allow for any jargon or philosophical arguments about the validity of the question or the answers
If you adding/multiplying real numbers then you are an engineer. That also means you know how to calculate the margin of error in the bottom line so that the bridges would not fail due to rounding errors. If you are a pure mathematician you don’t really need to deal with such busy work.
I think strictly speaking engineers use rational approximations of reals, since reals can't be computed. Maybe some are computing pi exactly, but it might be a long while before those bridges get built.
@@Robinsonero Irrational numbers do not fit in finite number of bits but that is no reason for not building bridges, skyscrapers or spaceships. We can calculate the upper bounds in our calculations and we always use a safety factor appropriate for the application in engineering.
@@EnginAtik I agree, we can rely on rational numbers, including rational approximations of pi, phi or e whenever we need to compute or manipulate numbers (a process known as math). Of course the pure complete transcendental notion of these numbers can't ever be obtained by mathematical operations, and they can't be used mathematically either. We can philosophize about the nature of transcendental values but we must always retreat to the rational numbers whenever we want to do actual math.
You ended with the sociological question "how did we get here in the first place?" and I say we got here because the Industrial Revolution needed bodies and the sensibilities of the Industrial Revolution valued the Engineer above all else. Whether consciously devised or simply iteratively emperically driven, it was accepted that the gross human capacity for rote memorization vastly outweighed gross human capacity for abstract thought. In the absence of omniscience required to know which 6 year olds were the best "investments" of effort, mathematics were framed with infinitums and inherently unanswerable impossibilities to discourage people from asking "Why?" and encouraging them to just move onto their next task on the educational assembly line; if the output was good enough for government work, it was deemed sufficient and it never mattered to them that they were discouraging people of seeker deeper understanding by presenting them the illusion of results commensurate with a "Higher Power."
Prof. Wildberger: The problem is that you are removing the real numbers from the number line without considering what's left in their place. This leaves your math to be far less powerful than infinity-based math, which is why I suspect that it hasn't become mainstream. When you remove the real numbers from the number line, what you are left with are continua in their place. When you do not complete infinite summations, what you are left with is a potentially infinite process. Only by incorporating continua and potentially infinite processes into your philosophy can finite math compete with infinite math. If you want to see what I'm talking about, check out this video on a finite perspective on the dartboard paradox: th-cam.com/video/LQmwZZNUMNA/w-d-xo.html&t
I'd love to discuss potentially-infinite math with you if you're interested.
@@ThePallidor By incorporating "potentially infinite process", I am not suggesting that we complete them. I'm saying that "potentially infinite processes" have value in themselves even if they cannot be completed. Pi and e may not be numbers, but they're still mathematical entities which can be finitely manipulated. Rejecting them leaves the resulting mathematics with a big void.
@Gennady Arshad Notowidigdo I view these 'potentially infinite processes' as computer programs which do not terminate. If we interrupt it, then we produce a rational number. This is what you are talking about in how it has value in applied math. I agree, but I am not talking about this. I am talking about studying the programs themselves and seeing how the programs relate. Although these programs cannot complete their output, they can often be written with finite lines of code and manipulated with a finite number of operations. I see no issue with rigour in this pursuit. The issue is only when we believe that these programs can be executed in completion.
While I do believe that it is extremely hard to make the establishment change to an entirely finite mathematics (I'm fighting this fight too and I feel the pain - I've made many TH-cam videos in this area and I struggle to get views), I think Norman's primary problem is that his version of finite mathematics is lacking. Specifically, it's lacking because his mathematics is based on points (which are inherently linked with infinity) when it should be based on continua.
@@ThePallidor Yes pi and e are processes, let's call them algorithms. But who says an algorithm is not a valid object? Who says there is no value in seeing how these algorithms relate? The problem is not in working with the algorithms, the problem is in assuming that we can work with the output of the algorithms.
@@ThePallidor With a few lines of code, I can write a program to spit out the terms corresponding to a Cauchy sequence, one by one. The *execution of the program* is dynamic and can never be completed. It is meaningless to talk about the complete output of this program.
But the program itself (the few lines of code) is static, finite, and a valid mathematical object. It's just not a number. Numbers are not the only valid mathematical objects.
On points: Points are 0-dimensional. What's your point?
On pure math: Math that is in principle beyond the realm of any possible computer is not math, it's fiction. But it is a stretch to say that math needs to be tied to some application. There are well defined finite numbers that have no connection to anything physical. Are these "nothing"?
@@ThePallidor "The problem comes when they are presented as numbers"
Agreed.
"Points are dots of definite but negligible width"
Are you saying that a point has non-zero width?
I'm not sure I understand what the problem is--I watched the video and it seems like "lazy evaluation" is a solution; I've always thought of real numbers as "recipes" for cooking up decimals, rather than decimal objects in and of themselves. If you combine two different recipes using a procedure which results in a third recipe, then that seems like a satisfactory way to combine recipes.
For instance, we'll take the case of pi + e. For simplicity, I'll leave out the sqrt(2), but it will be clear how to re-incorporate it once I have explained the procedure. "Pi" and "e" are recipes for cooking up decimals--the real numbers are the recipes, not the decimals. We start cooking: First, we add the 3 + 2 = 5. Then, we mix in the 0.1 + 0.7 = 0.8, sift out the 5 from 5.8, and set it aside: we are now confident that 5 is the ones digit in pi + e. We then carry on in this fashion to obtain more and more digits of the decimal number--that is, we have obtained a recipe for calculating the real number which is result of "+(e,pi) = e + pi" (which we should not expect to have a better symbolic identity, because lists of symbols in any alphabet are a countable set, and we know that the cardinality of the real numbers is uncountable). We took two recipes and combined them to cook up a third recipe--a perfectly reasonable addition of two real numbers.
I think the (much) more interesting problem which isn't really talked about that much in this video is that determining if two real numbers are equal, in general, is an undecidable problem (which is why we need to provide proofs that two numbers are equal--because it is a small miracle when it happens!). If you have two recipes for decimals, then you can compare them digit by digit. If they ever differ, then you can say if one is different from the other (moreover, you can say which one is bigger). However, if they really are the same, then your procedure doesn't halt. That is, you can never purely computationally prove that two real numbers are equal in finite time. Instead, you need metacomputational methods, including proofs and abstract reasoning.
@Alexander Sanchez Trying to get a theory of "real numbers" using recipes/algorithms/computer programs is tempting and it is a reasonable thing to try to do. Sadly no-one has done it yet. It appears to be much easier to just suppose that such a thing could be done, and then move on to more "advanced and interesting topics". I will be touching on the computational aspects of a much more restricted and tractable problem --- how to do arithmetic with repeating decimals. Turns out this has not been worked out carefully either, as far as I know.
It is absolutely remarkable how many important foundational issues have been abandoned to scrabble after the ever-diminishing returns of yet more sophisticated, narrow and incomprehensible investigations.
@@njwildberger I appreciate your thoughtful response, professor! I'll be interested to see what you have to present; it seems like one potential problem is the fact that you can get repeating 9's and never get anywhere. One way I know that actually fixes this issue is to use "bijective numeration" instead of the standard alphabet of symbols--that at least makes it so that you're always guaranteed to make some progress. However, there seems to be a difference between the question of the existence of an algorithm and the practicality of such an algorithm. I wonder: what are your goals for a satisfactory arithmetic with repeating decimals? Take care!
There’s no problem with algorithmic addition of Cauchy sequences of rationals to arbitrary specified precision, as long as we can query the Cauchy sequences to arbitrary precision (an epsilon/2 argument). This is unrelated to the ambiguity in the decimal representation of real numbers.
That would only be an approximate answer, not a true, exact answer, in the pure mathematics sense. 1/2 + 1/3 + 1/5 = 31/30. That's the complete, true, exact answer. It can be done for rational numbers. What's the complete, true, exact answer to π + e + √2, using whatever definition of real numbers you please? That's his point. You can't do the same thing with so-called 'real' numbers that you can do with rationals. Especially when you need to actually write down the answer, rather than just talk about the 'answer' as if it were already written down. For example, what is the explicit, written down "algorithmic addition of Cauchy sequences of rationals to arbitrary specified precision" answer to that calculation? Now suppose the 'arbitrary precision' we're specifying is 'exact precision', for usage in pure mathematics, rather than an approximation; can you actually exhibit such an explicit, written down answer?
@@robharwood3538 Thanks Rob, very well put.
@Rob Harwood "Especially when you need to actually write down the answer, rather than just talk about the 'answer' as if it were already written down."
{Sorry for my English, I speak Spanish.}
This comment strikes me... We are not carefully distinguishing the Experience from the "talk about the Experience". This is very common in our present way of relating to our observations, ideas, speech... We pretend, and many many times we don't know we are pretending.
We confuse, take our beliefs for granted... instead of checking them.
We finally must learn to listen, deep listening to do Math & Science. Otherwise we sell "pescado podrido" (rotten fish, idiom) to our students.
Thanks for this video, and all the material. Thanks Prof. Wildberger.
Diego Vallejo / Prof. Cálculo / Facultad Ingeniería UNLP/Arg
@@matematica6pi Thanks Diego: I love the line "Otherwise we sell "pescado podrido" (rotten fish, idiom) to our students."
Could you explain why computational short cuts are to be expected? For example, in a free magma there are no non-trivial identities. Is it fruitful to assume that all expressions are equal to a "simpler" expression? It seems that mathematics is in the identities, not the actual computation.
(btw I kind of agree with you - I'm just playing devil's advocate)
Here is how the defense might respond: That two unknowns should be equal does not determine what either of them is; it only determines that they are not diverse according to quantity.
Math isn’t physics and thus isn’t bound to this reality Norman…..I believe thou protest too much!😉😎🎓
The definition of a Dedekind cut seems to be circular since every number is defined but all the numbers it is not, put into two infinite sets of those numbers larger and smaller respectively. Is this a paradox or am I missing something?
It is a common technique/principle which is not bad in and of itself.
Imagine you have two approaches C and D to a single ostensible phenomenon P.
Here P is "real numbers" and C is some vague conception of them, which nonetheless is meaningful to you personally.
D on the other hand is the axiomatic notion of a Dedekind cut, which does not reveal the purpose of its being, but which is hopefully more strict, less vague, more communicable.
Now let's agree that sqrt(2), pi and e ought to be examples of the phenomenon P. You are assumed to have some understanding of this within C.
What you can attempt to do is you characterize these examples by abstract statements involving C. Then you reinterpret these as statements involving D instead of C. If you are lucky, they are still true when reinterpreted in D, and if you are very lucky, they characterize certain things uniquely (which we should call sqrt(2), pi and e) even according to approach D.
In this way of analogy, knowledge is transferred from C to D!
(Furthermore, if a new approach E, still to P, is discovered later, you should reinterpret the abstract statements above as statements involving E instead, to test this approach and hopefully learn what sqrt(2), pi and e must mean according to E.)
Not sure if that makes sense the way I have put it. I learned this in category theory, which offers more insights into abstraction than set theory, but which does not provide a good (powerful) foundation for real numbers either, as far as I can tell. A simple example of an "abstract statement" (as used above) in category theory is the "universal property of the product", which you can look up if you want. This "abstract statement" allows you to decide what the analogue of the cartesian product should be in various niches of mathematics where set theory is not the primary object of study.
It is a bit weird but if you plot y=x²-2 then if the x axis consists of only rational numbers then the graph does not cut the x axis for any value. i.e. the x axis has infinitely many gaps. The presentation seems to ignore these points.
@@joecotter6803Yes this is a necessary consequence of working over the rational numbers rather than over the (Archimedean) metric algebraic "real" numbers. This problem can be somewhat "solved" in a certain sense however, without admitting transcendental numbers of any kind, by just providing "real places" (or "complex places") for various algebraic numbers.
If we still insist on working over just the rational numbers, then we get counter intuitive results about under which circumstances various conics intersect, or whether certain points or lines are inside or outside a geometric figure. Iirc i have mentioned this to prof. Wildberger in the yt comments of some of his (highly promising!) lectures about geometry, especially the universal hyperbolic geometry and rational trigonometry lectures.
So, professor Wildberger, you want to know what pi+e+sqrt(2) is? Well, pi=3.1416±0.0001; e=2.7183±0.0001; sqrt(2)=1.4142±0.0001, therefore pi+e+sqrt(2)=7.2741±0.0003. One may never represent an irrational number with a nice fraction, but we can get the error as small as we want, given time to compute. I have been teached about real analysis and IEEE-754 computer floating point arithmetic system, physics &c., in my time in university. It amuses me, that more than often, we have to break an integral formula to finite sums to give the task to computer i.e program it. But these integral formulas have been already achieved by (Riemann) summing infinitely. But even if it amuses me, I certainly wouldn't want the equations start with summation symbol, capital sigma. The curve of the integral sign, the dx,dV,dxdy, they talk to me. They pleasure my eye.
Hello, just to recap are you saying that addition for real numbers doesn't work because it will take infinite time to finish ? then what about computing natural numbers that takes a long time to compute ? Like Ackermann function which deals basically with natural numbers. For example it is estimated that A(4, 2) will take roughly 2^(65536) steps to compute. That's why I'm interested of your insight for those (natural) numbers
Doesn't one have the same problem with fractions? 6/3 evaluates to 2. But if you apply the division algorithm to 1/3 it never terminates. So we simply keep the symbol "1/3" and claim that's the real number.
So if one can claim that 1 can be divided into 3 equal parts, why cannot be claimed that a unit circle has a circumference?
@otraguardia Our use of fractions, or rational numbers, to seemingly avoid division is interesting and warrants discussion. When we write 1/3 we are not so much dividing 1 by 3 as introducing an entirely new symbol that stands for that operation in our minds, but is logically separate from it. Is this yet another example of empty manipulation with symbols? No, because we define the arithmetic with fractions consistently in terms of the symbols we introduce. Thus for example we define a/b+c/d = (ad+bc) / bd . However one problem, or rather challenge, with this is to deal with the multiplicity of forms of a fraction ie 1/3 = 2/6 = (-5) / (-15) etc. More generally we declare a notion of equality a/b = c/d precisely when ad-bc=0. But that does mean that we have to check that our operations and statements are well-defined --- that is independent of the forms used in the definition. Doing this for the addition above is an important exercise that I bet 90% of math majors have never done. So yes, our arithmetic with fractions certainly warrants some criticism. But it is able to respond!
Note how different this is from the "real number" symbolics. With the "real numbers" there is no set restriction to the number of new symbols invoked. And crucially there is no clearly defined test for equality. How do we tell whether my algorithm / app for an infinite decimal is the same as your algorithm /app? If we run both to a certain extent and discover some difference in outputs, then we can confidently declare the "real numbers" different. But how to tell if they are the same if they actually are? This turns out to be a theoretical impossibility in general! Yet another point where analysts instinctively know: the less said the better.
@@njwildberger, I see where you draw the difference. Thanks for the detailed response!
Do you think the p-adic numbers are more logical than the reals, or do you have a problem with their infinite string representations as well?
From watching most of his videos on this topic, I think it's safe to say that he considers all forms of the 'reals', i.e. all the forms that have been proposed so far, to be deficient in one way or another. Usually it boils down to the necessity to actually 'finish' an 'infinite' computation in one way or another. (Although that is not his only objection. Some definitions of 'reals' have other problems as well, for example, and in particular, anything involving the 'Axiom of Choice' or an equivalent.)
@@robharwood3538 said "Some definitions of 'reals' have other problems as well, for example, and in particular, anything involving the 'Axiom of Choice' or an equivalent.)" I don't know of any definition of the "reals" that use the axiom of choice. Do you have an example?
@@billh17 The ZFC axioms. I guess you're right in the sense that they don't *actually* 'define' what a set is, and so -- since reals are supposedly founded on set theory -- they don't actually define what a real number is, either. But that's just another problem regarding the fakeness of real number arithmetic then, right? 😉
@@robharwood3538 answered "The ZFC axioms." Ok, but I was assuming that we were in the framework of ZFC. The standard definitions of "reals" don't depend upon the axiom of choice.
@@robharwood3538 said "they don't actually define what a real number is, either" But this video itself says "real" numbers are defined in ZFC by Dedekind cuts or by sets of equivalent Cauchy sequences.
@Account Name; Actually the p-adics are much more tractable than the "real numbers". Their arithmetic is much cleaner and actually more interesting. Have a look at th-cam.com/video/XXRwlo_MHnI/w-d-xo.html where I call them "reversimals" for reasons that are closely related to the superiority of their arithmetic. But you still have to be careful!!
Computers use floating point because real number arithmetic is impossible to actually implement. Yup. You can say that the approximations are good enough. But in some uses, if something is exactly 0.0, or exactly 1.0 it determines what TYPE it is (ie: of a multivector, you calculate the type: scalar, vector, bivector, trivector... based on what coefficients are 0.0). Two unit vectors are parallel if their dot product squared is 1. They are perpendicular if their cross product squared is 1.0. So, if you do a calculation and get 0.000000001 for a vector part, you can't decide whether it's supposed to be a vector. In floating point, you CANT use the "==" operator reliably. Of course, floating point is excellent for engineering approximations. But you cant use it for proofs, in general.
But I would also say something that NJW might object to.... You can't really implement exact FRACTIONAL arithmetic either; though you can know when you DO have an exact answer - which will be the case in most practical situations. If have a sequence that multiplies a whole bunch of rational numbers; you reach a point where you may not have enough digits for the numerator and the denominator in practice. If you don't run out of SPACE, calculations get so slow that you run out of TIME. Anything that has O(2^n) running time is not calculable in practice when n actually gets large. An example with Fourier transforms: As long as the best known algorithm is O(n^2), you can't really do anything useful with it in practice. An O(n lg[n]) algorithm was discovered for it, so suddenly it's useful in practice. In fact, these RUNNING times, and RUNNING space are MORE relevant than numerical accuracy for doing engineering work.
When we go back to symbols, you can almost think of "pi" as an irreduceable "unit"; as is "e", etc. ("i" is a unit, but it's an ambiguous unit, as it doesn't specify the plane or orientation that it's in; which may or may not matter, depending on what you are doing; it is a thing that squares to -1, without identifying which thing it is. Clifford Alegebra (ie: Geometric Alegebra) fixes this.) Other units are algebraic, and not numeric, like: direction unit vectors: e_1, e_2, epsilon^2==0, etc. You can "compile" a solution that is symbolic, and the most convenient form. But you need to plugin floating point to "execute" an actual answer that can be pretty close. And some expressions that are supposed to be identical according to algebra can give wildly different answers. StdDeviation calculation for instance will soon start emitting NAN just due to constructs like: a^2-b^2 ... just being plus or minus a tiny amount can cause it to divide by zero when it should not, or flip sign...which is a catastrophe when it's part of a division. In engineering terminology: the SIGNAL is less than the NOISE in these cases. Noise gets catastrophic in the presence of feedback loops.
Engineering can work in the presence of unhandled noise. Proofs must explicitly handle noise.
So, people need to go to Donal Knuth books for better behaved recurrences. This is because fundamentally, floating point arithmetic isn't totally commutative, associative, etc. It's very close though.... So you can use it for engineering. But you can't really use it for proofs.
Note about actual computer number systems:
- integers are typically 16, 32, or 64 bit. Mostly 32-bit for practical cases. You need 64-bit for things like file sizes, and timestamps.
- doubles are 64-bits of floating point. typically this means that the integer part is only 2^53, and the other bits are sign and exponent. this fact will definitely come to you in a bug one day, probably when you are shooting json between javascript and some server-side language that expects int64.
- When you write a number in decimal, it will store it in binary.
- The unbounded-precision options, as noted before let you at least know WHEN it's accurate, but that works for large integers. It doesn't solve the problem of representing the fractional part at unbounded precision.
- In any case, an O(2^n) solution for large n is just as incomputable as an infinite process. This is the basis of all cryptography.
- In cryptography, there is no such thing as an epsilon. An answer that is off by 1 ulps will hash to a completely different value.
So, what would be ideal to deal with this:
- be EXPLICIT. An int64 is mod(2^64).
- An arbitrarily large integer is "mod infinity"; so even with arbitrary precision, you may not be able to represent it. This is EVERYWHERE in cryptography. You can only work with a few thousand bits in practice. But note that you can know WHEN you have an exact answer.
- We WRITE numbers in base10, but store them binary. Since that's not equivalent for the fractional part, perhaps writing them in floating point hex at least removes the noise contribution from input/output notation.
- When doing +, -,*,/, etc.... have a number system that tracks guaranteed bounds. A noise term.
- Have noise terms everywhere, like a carry.
In practice, we should be making better libraries for algebraically manipulating math; and stop working on paper. When put into a machine, the work is reproduceable. Any issues with that exist become difficult to hide. Referreeing work should be a matter of determining its relevance; as it should not be able to REACH referrees until it is reproduceable. It's a big loophole that we judge correctness with people.
Just like discovering things such as "2pi = tau" as breaking patterns that should match automatically; not all of the problems we will run into will invalidate prior work. But there are some cases where notation is clearly wrong. (See Johnathan Bartlett about: d^2[y]/(d[x]^2) is literally wrong. And it's easily fixed such that differentials divide algebraically like they should. It's just that higher derivatives are actually slightly more complicated with terms that are usually zero: d^2[y]/(d[x]^2) - d[y]/d[x] * d^2[x]/(d[x]^2) = actual second derivative).
The mathematics is like a programming code base that has not undergone any refactoring. The advent of computing has triggered an avalanche of discoveries of notation that is wrong in some way; but was understandably done because without computers it is too verbose to handle explicitly on paper. This is what happens when you drop down from your high-level programming language to look at the actuall assembly language at the bottom turtle. (The bottom turtle of hardware is basically NAND gates - logic circuits can be NAND gates with no loops while memory is NAND gates with feedback loops. But the bottom turtle of the software is the assembly language. There isn't an assembly language for mathematics; as it's all too waffly right now.)
The thing I don’t quite understand and would like to see you address is π. It’s one thing to be able to deal with an algebraic number like the square root of 2, where you can find a finite representation (which is totally fine and quite useful). However, π, whether you like it or not, has a specific, physical definition: the ratio of any circle’s circumference to its diameter . Hence, I suspect you will have much trouble attempting to wave your hands to say it does not exist and be able to convince many others. So, how do you propose to fit π into your world of only (some) rational numbers and finite fields?
Pi is quite different from a number like 2 and as such, should be treated differently. It stands on another level. As to its approximate value, then it becomes a regular number like the rest and can be used in applied math. But as a pure mathematical object, it’s obviously more than a regular number. More work is needed to see how it can best fit in.
@Joseph Williams I agree. Probably we are just starting out learning what "pi" really is. It certainly is much more than just some other obscure "point on the number line". Getting over this intuition --where all the "real numbers" are somehow strung on a clothesline, all packed in right next to each other (sort of) with a minimal set of rationals and quadzillions of irrationals in between them: that entire intuition needs to be seriously overhauled. "Pi", whatever it is, is a hugely significant and multi-faceted mathematical object, certainly more than just one more string of digits 3.1.415... as we currently believe. Isn't exciting to realize that we are basically at the infancy of a serious analysis?
@@njwildberger What's so wrong with thinking of pi as an element of the total-ordered completion of the rational numbers? This total-ordering and completeness (no holes) is the only reason why we faithfully represent the real numbers as a continuum of points on a line... Is the notion of a completion the source of your problem?
@@schmud68 The problem is rather in the "real number" court: to explain how to compute pi+e+sqrt(2) etc. I claim this cannot be done, and that the academy is faking it. Surely mathematicians who claim there is a legitimate arithmetic with real numbers would be able to demonstrate this arithmetic explicitly and concretely.
@@WildEggmathematicscourses One can rigorously define the addition of any 2 real numbers via the addition of the equivalence classes of the Cauchy sequences of rational numbers they correspond to. Hence, rational number addition defines real number addition. One may also define the addition of any 2 real numbers via the addition of the rational numbers within the 2 LHS's of the cuts, with the remaining rational numbers for the other half. In terms of computation, as I mentioned in an earlier comment, we may only deal with arbitrarily accurate approximations of real numbers. Though, this has nothing to do with the abstract theory that defines them.
@@schmud68 sorry about my somewhat snarky response, I have since deleted it. Let’s keep the discussion spirited but respectful, thanks.
Well, you can call me a sociologist if you like.
If pure mathematics was really pure then it wouldn’t be wrong or fake. If you are getting an irrational result, then it doesn’t mean that the universe or the maths is wrong, it means that you are applying the wrong math, or applying it in the wrong way in regards to how Nature actually works!
To assume that the universe must be irrational, is illogical! Nature can be nothing but balanced.
Prof. Wildberger is correct in his assertions that most of mathematics is just plain wrong. The reason that most of mathematics is wrong, is that it is not based on how Nature really works.
However, it’s not the mathematics, but the wrong understanding of reality that is the problem.
The universe is not infinite. It just produces, voids and reproduces infinitely. This, and many other false beliefs in cosmology, is the reason for irrational numbers and mathematics.
True geometry and physics will produce true mathematics. We live in a polarized, electric, two-way, balanced universe, yet we apply one-way unbalanced mathematics! This must change! Just like it is difficult to visualize simultaneous events, it is also difficult to write equations that do two opposing things, simultaneously.
If Prof. Wildberger has some time available, I would be happy to travel to Sydney to meet with him and explain these universal laws, natural science and philosophy in further detail. I think it is a waste of time trying to convince mathematicians that they are wrong. (Unless you really do enjoy making a career out of it!). I have experienced exactly the same problem with physicists. What has to be done in order to make any progress, is to not only explain why it is wrong, but also explain what the maths and physics of the real world actually is.
Kind regards,
L. Dove
Arbiter - Universal Law
What does the existence of irrational numbers has to do with “balance”? As for me a circle is one of the simplest and, in a sense, “perfect” shapes. The fact that the length of its circumference divided by the length of its diameter is irrational just shows how real they are. The length of a square’s diagonal divided by the length of its side is irrational too. Both construction may exist in the real world, so how could real objects have non-real dimensions unless irrationals are as real as naturals.
@@fullfungo Excellent question.
Nature reaches its perfection in carbon. Carbon is produced at the amplitude of an electric wave where a perfect cubic wavefield is produced.
The ultimate forms of Nature are spheres of matter and cubic wavefields of space. These opposites are the limit that Nature can produce. Once a sphere is wound up together with its equal potential (but much larger volume) of space, however; it must, and can only, unwind and be reproduced.
This basic mechanism is all there is to the mechanics of the electric wave, and to all natural forms.
No reproduction of a state of motion can occur until the previous state has been voided.
Therefore, in order to understand Nature, you need to understand the wave. Unfortunately, electrical engineers have not understood the correct mechanics of the electric wave, and the lack of knowledge of the exact quantum mechanics of the electric wave means that we are designing motors - coils, solenoids, armatures, transformers, etc, that do not copy the mechanics of Nature, meaning that we are not producing energy efficiently. Also this means that mathematicians, electrical engineers and physicists are describing electricity and magnetism incorrectly, using incorrect and overly complex mathematics.
So, in the correct understanding of the electric wave are the entire mechanics of all creative processes, as well as the mathematics of its two opposing pressures.
Producing videos like this is a consequence of going down the wrong rabbit hole.
Videos like this are filling in the rabid [not misspelled] hole that has vacuated mathematics for more than 100 years.
I think that trying to calculate the area under a curve has to include infinite steps because the curve is supposedly composed of a continuous line of infinitesimally small curved elements going down to infinitesimal points which in reality do not exist (because everything is quantized?). So if you want to calculate the area under a "real" curve you cannot use "real" numbers or infinite steps because the real curve consists of small elements of a finite size. It may be tricky though to find out what the shape or properties of such elements would be. We would only know they are finite. So I think the solution to the problem is to find out the "real" or physical nature of the smallest elements. Maybe that's impossible and therefore we have to use some sort of abstractions. Or maybe at a fundamental level it's - as some propose - just numbers.
@Flatterman Island Please join the Algebraic Calculus One course! It is online, and currently free. And it teaches you how to do integral calculus without any hand waving about infinite processes. You can also look at the Famous Math Problems 10 lectures: th-cam.com/video/vo-ItaB28f8/w-d-xo.html is the first one.
@@WildEggmathematicscourses thanks, I will take a look into it soon and hopefully learn something new :-)
Sir, Is there any certain answer of this question, why we needed mathematics.??
Could you argue that something like sqrt(2) expressed in this form is the rational "representation" of something that is Computationally irrational?
There is no such a thing as "computationally irrational" or "rational representation." Rationality is not a property of computational algorithms or of representation strings. Rationality is the property of an abstract object we call "number." How we choose to represent it has frankly little to do with its rationality.
No and we don't need to do it. Irrational numbers are fine and don't need fixing.
13:49 The truth is, a real number is a loss or gain in that which stands to that whole number called by the name of "one" in the same ratio as one magnitude stands to another of the same kind.
As usual, Our good friend Norman Wildberger is completely blind to Hilbert's synthetic approach to mathematics, and goes full-fledged for Brouwers Intuitionism.
What Norman Wildberger fails to recognize in about _all_ his talks about mathematics, is that he does not understand that there are _two ways_ to understand abstract concepts.
You can understand it _analytically,_ or _synthetically!_
If you understand something _analytically,_ then you _build up_ the answers. You work bottom-up. You _construct!_
Philosophically,
1. in the _analytic_ approach, you try to understand something by trying to figure out what something _is!_ You therefore _construct!_
2. In the _synthetic_ approach, you try to understand _is not!_ you, therefore, understand in terms of _elimination!_
But the very idea, that 'what something is' is the whole story, is problematic. You can also understand something as an exclusion.
To give a simple example: if I take the logical implication
p -> q, or, 'if p then q',
and I want to _understand what it means,_ then there are two possibilities.
One is the _bottom up_ meaning. What I then mean by this statement are _three things taken together!_ These are
p -> q = (p &q ) v (~p & q) v (~p & ~ q), or, in words:
'if p then q' has the same meaning as
1: p _and_ q, _or_
2: not p _and_ q _or_
3: not p _and_ not q
Or, an example: If it rains then the roofs become wet _means_
(it rains _and_ the roof s become wet), _or_ (it does not rain, _and_ the roofs become wet), _or_ (it does not rain _and_ the roofs do not become wet)
All three cases _are consistent with_ the statement: _if_ it rains, _then_ the roofs become wet, and therefore, all of them _together_ express its meaning.
because, indeed, the statement says that all these three cases _can_ happen, _if_ the statement: 'if it rains, then the roofs become wet' is true.
In particular, in the case it (does not rain and the roofs become wet) _can_ be true, if some helicopter sprays all the roofs with water, so that they become wet. The statement _does not exclude_ this possibility. It does not exclude _three_ possibilities. _All three possibilities it does not exclude, therefore belong to *the meaning* of the statement!
In practice, this means that you can _construct_ the meaning of 'if p then q' or, (p -> q) by _building them up!_ You just _recognize_ that the implication is not just _one_ statement, but its content consists of _three_ statements!
This 'in practice' is exactly the point. 'in practice' is applied mathematics. In _applied mathematics_ we _build things up!_
But in _theoretical mathematics,_ or in _synthetic mathematics_ we _exclude!_
If I want to show _the meaning of_ the statement: p -> q, and I use _the exclusion, theoretical, or pure_ approach of mathematics, then _the meaning_ of this statement is:
~(p & ~q). In other words, what I am saying is this:
_the meaning of_ 'if p then q', is: NOT ('p and _not_ q)'.
Or, to return to the example: 'if it rains then the roofs become wet' is the same statement as: 'it _cannot happen_ that it rains, and the roofs _will not become wet!_ This statement therefore _excludes_ the possibility, that all roofs are made _white hot,_ so that when it rains, the water falling on top of these roofs will _instantly_ become water vapor, so that the roofs will not become wet!
In other words, I have understood the statement: p -> q in terms of _an exclusion!_
Let me give another example. One of the axioms of plane geometry is:
'through any two points goes one, and only one straight line'.
This axiom _defines_ the concept of 'straight line' _in the form of an exclusion!_
It basically says that if we have two points, and we can draw two lines of a particular kind through these two points, then these two lines of that particular kind _are not straight lines!_
I give an example. If I have the parabola y = x ^2 - 1, then this parabola goes through the points (-1, 0) and (1, 0). But the parabola -x^2 + 1 _also goes_ through the same two points!
Therefore, _the parabola is not a straight line!_
That is why the axiom: 'through two points goes one, and only one straight line' _defines_ what a straight line is, _by excluding all cases that are not straight lines!_
In the same manner, _we do not have to prove_ that, for real numbers, a + b = b + a.
No, we produce a number of axioms, for the real numbers R, and _everything that does not correspond to those axions, is not a real number!_ In other words, the set of real numbers is not defined _as a construction,_ but it is defined _as an exclusion!_
This is _exactly the difference between pure and applied mathematics!_ In applied mathematics _we construct!_ In pure mathematics _we exclude!_
Therefore, Norman Wildberger has it wrong! Pure mathematicians _do not rely on a fake arithmetic!_
_Pure mathematicians DO NOT RELY ON ANY FORM OF ARITHMETIC AT ALL!_
Maybe they do not realize this, but _that_ is the difference between pure and applied mathematics!
Once you understand this difference, concepts like the infinite are no longer problematic. 'Infinite' just means: _NOT FINITE!_ It is an exclusion, not a construction.
If Euler proves The sum of all numbers 1/n2 up to infinity is equal to pi^2/6, he is saying that _there is no difference between the infinite sum and the outcome pi^2/6 !_ In other words_if_ you calculate the sum and whatever algorithm you use to calculate pi^2/6 up to n decimals, and this n-th decimal is not a 9, then these two calculations _will always correspond exactly_ up to the n - 1 th decimal! And Euler's proof _shows that!_
"Pure mathematicians DO NOT RELY ON ANY FORM OF ARITHMETIC AT ALL!"
So, a pure mathematician cannot even add 1/2 + 1/3 + 1/5? Or, perhaps, you are overstating your claim?
While I agree there is a thing like synthetic vs analytic, I think they interact far more often than you are suggesting. In any computation there are interactions around many small interfaces. One side of each interface builds up some form of data, and the other party breaks it down. In type theory we call this introduction and elimination. There is also the more well known notion for programmers, of actual vs formal parameters.
A modern approach to mathematics, pure or applied, which is to be founded at least in part on computation, will thus contain both analysis and synthesis in either case.
Also, take your pick; inclusion or exclusion, construction or elimination, applied or pure, analytic or synthetic: What is the explicit, written answer to π + e + √2, without just trivially restating the question? Clearly this kind of computation can be done for rationals such as 1/2 + 1/3 + 1/5. It can even be done for abstract things like polynomials with rational coefficients, like (x^2 + 3) + (2x^2 + x + 1). Even pure mathematicians can do this, despite your claim that they do not do any form of arithmetic at all. So, can they do it for π + e + √2?
And if they cannot, then can you explain *to a sociologist* _why_ they cannot?
@@JoelSjogren0
That might be. But a computer cannot understand in either case.
To be precise, a computer works with bits. But _we_ work one level higher!
We work with concepts and ideas!
Let me show you what I mean.
What we do with computer programming, is looking at the world, and trying to capture that with 0's and 1's.
To give the simplest example, how many distinctions can I make with two concepts? The answer is 4.
p -> q is one of them, which is then, on bits level 1011. I can also say p & q = 1000, p v q = 1110 etc.
How many of these distinctions can I make? Obviously 2^4 = 16. There are therefore 16 of such operations, and therefore 16 different distinctions I can capture in 4 bits. _That_ is the level of the computer. But I can also look at the level _we_ look. And then I must say that we use _only_ 2 concepts, and a bunch of operators, to capture 16 distinctions, and therefore 16 different 'understandings'.
With three concepts I can make 8 bits, and therefore 2^8 = 256 different distinctions. Reality then consists of 256 distinctions. The computer 'sees' 8 bits, and _we_ see 2 concepts and a number of operators. In fact, we can express all distinctions by just the 'and' the 'or' and the 'not'. _Every number of distinctions_ can be expressed by a sufficient number of bits, which can then be captured in a number of different concepts and just three operators on these concepts.
What a computer does, is basically capture distinctions in bits. And _we_ design computer concepts, with which we no longer interact with the world directly, but through the computer.
In general, n concepts, which is the level _we_ think, become 2^n bits, which is the level of 'calculation' of the computer, and these become 2^(2^n) _distinctions_ in reality.
My point is: computers work with computations. _We_ work with concepts and operations. Computers change rows of bits into other rows of bits. We _design_ these operators. The computer can change 1's in 0's, and 0's in 1's through rules. That is, basically, what a computer does. But _understanding_ does not happen on _bits_ level. It happens through concepts and operations. It happens on the p and q level and the operators. Seen from the computer, _we_ design the concepts and operators. Computers are not able to do that.
To _really understand_ how big a difference this is. We make drones. They have some capacity to make their own decisions. But our most complicated drones _are no match_ for what even a simple fly, which has about 100.000 brain cells, can do to keep itself alive! Apparently, this small amount of brain cells are enough for the fly to have _simple concepts_ and _operations_ so that it can do the very complicated things it does. Including the fact that it is _the most skilled animal in flying!_
@@robharwood3538 Any pure mathematician has reached that level by moving one level higher than applied mathematics. First you learn how to calculate, and then you learn to move from the specific to the general. It takes a lot of time before you understand that 'the general' consists of constructions in which the concrete details are eliminated, so that you get a construction that can be applied to many different concrete instances.
What an applied mathematician tries to do, is to solve practical problems and to find solutions. What a pure mathematician does, is asking himself if the solutions found are _the only solutions!_
To give a _very simple_ problem. Suppose I ask myself, what number, added to itself, gives the same result when I multiply this number with itself?
The applied mathematician can then find the numbers 2, because 2 + 2 = 4 = 2x2. After a little thinking he finds another one, namely 0, because 0 + 0 = 0 = 0x0.
The pure mathematician asks himself, are these _the only two_ solutions?
He then formulates a + a = a x a = b. So, the condition for all these numbers is: 2a = a^2. Therefore a^2 - 2a = 0, or a(a - 2) = 0. The pure mathematician knows that if pxq= 0 then _either_ p = 0 or q = 0, or both. Therefore either a = 0 or a - 2 = 0. These are the only two cases. From a - 2 = 0 follows a = 2. So, indeed, 0 and 2 are the only two solutions to this problem.
This can be done on far higher levels. What I think of, is that Einstein made his famous most general solution to his problem of general relativity, leading to the equation G = 8(Pi)T. The question is then: is this _the only_ solution to his problem? Some pure mathematicians later proved that, indeed, his equation _is the only solution_ to his problem.
So applied mathematicians _discover_ solutions to problems. And pure mathematicians _investigate_ these solutions, and try to find out whether these solutions are _all of them,_ or whether there are more of them. That is why the work of a pure mathematician is more difficult than that of an applied mathematician.
Every pure mathematician is an applied mathematician who moves to a higher level of understanding. Applied mathematicians work with constructions. Pure mathematicians take these constructions and look how far they can be extended, through _eliminating_ those factors that are only of the constructive kind.
To take your example, you might wonder: what kind of formulas will be the result if the numerator is always 1?
So he can try: 1/a + 1/b = (b + a)/(ab). He tries 1/a + 1/b + 1/c =(ab + ac + bc)/(abc) and then 1/a + 1/b + 1/c + 1/c =(a b c + a b d + a c d + b c d)/(a b c d) and the pattern becomes clear. In the numerator the sum of all combinations of the products one letter less than the product of all numbers in the denominators, divided by the product of all numbers in the denominators. And then comes a proof (which I do not give here) that these are _the only_ outcomes of such sums.
In any case, my vision of Norman Wildberger is that he is an applied mathematician, _who thinks_ that that is _all of mathematics!_
Nevertheless, I do not underestimate him! He has made a genial discovery, for which I admire him! But _because of_ his restricted vision, he is himself not aware of what he has discovered!
What I refer to, is his idea that the plane geometry of Euclid can be seen as a combined theory of linear and quadratic equations. This leads to the idea, that you can, basically, understand all algebra of linear and quadratic equations through plane geometry, by introducing _spreads_ instead of distances, for example.
I do not want to go into this, here. If you are interested, you should study his book: 'Divine Proportions'.
I'm loving this video, it's hilarious.
If you did this for a bet, you deserve to win.
Unfortunately he does this for living
Unfortunately, most of what the video states is wrong. But you are correct: the video is hilarious.
Trouble is he is too easy on so-called rational numbers.
You say the answer is 1/3? All I see there is a division you haven't done. And you can't do it, it would take forever. Down with rational numbers I say. They are fake maths.
@@jowotx He fails to understand the distinction between a number and its decimal digital string representation, which is implicitly just a sum of rational numbers with powers of 10 as denominators anyway. Also, Skewes' number is a natural number, but he could never write the digital string representation of this number in any base except for itself, so this number is also fake, I suppose. Any number that is larger than 10^(10^80) is fake, according to his argument, because no such a number can have its decimal digital expansion written, even theoretically.
@@jowotx it's all about equivalence classes and representations, one could argue for example what does "- 1" even means?
What is 5+1?
We say it's 6 cause that's the symbol we associate
Thus we can define е+π:=ξ
For example and that would get the job done. I understand that there is a concern about the foundations of the real numbers, but that's soooo XIX century like.
At the end of the day mathematics is mostly a language tho
Sir, Can we replace existing arithmetic by another kind of foundation with completely different approach..?
What’s the difference between 2 & pi? Both are names for numbers - 2 for a finite no of things, pi for an infinite no of things satisfying a relationship couched in mathematical language. e is another name for different infinite no of things, with a different mathematical relationship. That’s what I think, probably wrongly.
Are you saying that a computer can't tell you the n^th digit of e + pi?
@KaiokenComplete More precisely there is no algorithm that will give the n-th digit of e+pi.
@@njwildberger equally, the maximum positive value for a 32-bit signed binary integer is 2,147,483,647. that doesn't mean 2,147,483,648 doesnt exist
@@Kav107 Try to explain that to your Commodore 64.
@@njwildberger However, there is an algorithm for computing (effectively) the nth hexadecimal digit of pi, so it's not obvious.
I’m sorry I cannot leave this here. 1/9 is clearly a rational number, which you state exists and has a real arithmetic, but what about the decimal representation of 1/9? You would agree that 1/9 x 9 is a possible operation, and if you were to calculate the decimal representation of 1/9, you’d get something like 0.1111111111...., so is this okay? I mean 1/9 is rational but it also has an infinite decimal representation, and if this is okay, then what happens when you want to do the same operation 1/9 x 9 on the decimal representation of 1/9, i.e. 0.1111111111... x 9 ? You would say the second arithmetic is fake, but how can the same number both exist and not exist and have a real arithmetic and a fake arithmetic at the same time? This is a contradiction, no?
@stokhosursus We defined decimal arithmetic in this video, in terms of finite decimals with an arithmetic based on integers, but with division problematic. The fraction 1/8 can be represented in this system, but the fraction 1/9 cannot. In the Babylonian base 60 system they were acutely aware of which "numbers divide" and which ones don't -- in their system 1/9 would be a valid object, that is have a proper sexagesimal expansion. Now can we extend our arithmetic with finite decimals to an arithmetic with repeating decimals? Turns out this is a very interesting challenge, and we will be talking about that in a Famous Math Problem lecture.
@@njwildberger I have no problem with you attempting to re-derive other ways to arrive at interesting mathematics results. However, there are many profound problems with merely throwing away the real numbers. I don’t see how you’re planning to address continuity, at minimum, or other larger and necessary concepts in mathematics just from an applied approach and not merely within the devil arts of differential geometry and other evil conspiracy areas of mathematics delusion that you are railing against.
As a psycho-sociologist, I found the basic problems involved in apparently only insignificant precision of long numbers to be, well, not as significant as problems of insignificant and false precision being used to validate the true as false and Vice versa. On the other hand, when I get non linear (non literal) about your speech, it makes more sense to me. As reading Principia made sense only when I realized that Newton was talking about odd reality, not even physics. As to precision in insignificance, I’m with the chemists and engineers. Hopelessly literal. Personally, I have more trouble with the reason airplanes fly is the use of imaginary numbers.... FYI, it took me a full week to accept the notion that an integral almost arbitrarily defined the convention for area under a curve. I got lost in the infinity of points and lines. Thanks for this. Strangely, it helped.
One question: does this have anything to do with Gödel Theorem?
@@ThePallidor Thank you. I wish to understand more on the subject. Ive always find so interesting his works on logic.
@@ThePallidor the liar paradox? Is it the same paradox exposed by Russel in set theory?
No, it has nothing to do with any of Gödel’s theorems.
It literally has nothing to do with the Gödel theorems.
@@angelmendez-rivera351 Thank you. What then Gödel theorem exposed? Seems to me it has something to do with arítmetics fails.
I have the answer you are seeking. It's really quite simple.
In the early history of mathematics numbers were defined as the number of individual objects in a collection. In other words, this is the cardinal definitions of 'number'.
But then they discovered irrational relationships such as the square root of two which appear to exist, but do not satisfy the cardinal definition of number. This is where they made their grave mistake.
Instead of recognizing that irrational relationships are not cardinal numbers, they demanded that they must be. And that was the beginning of "fake numbers". In other words, 'numbers' that don't satisfy the cardinal definition of number.
This was a grave error that the mathematical community actually holds up high as the greatest achievement of mathematics. When it truth it actually represents a gross mistake.
What they should have done way back at the times of the ancient Greeks, was to simply recognize that irrational relationship do not satisfy the original cardinal definition for number, and then they would have realized that they have a brand new concept on their hands. But they never made that realization up to and including today.
They will probably never know the true nature of irrational relationships because of this. They will never know that the source of all irrational relationships (what we've been calling irrational numbers) is self-referenced relationships.
We'll probably need to wait until aliens come from some other planet and set Earthy mathematicians straight on this. Because, save for Norman Wildberger, almost no one else has recognized the problem. And even Norman doesn't appear to be aware of the actual source of this problem.
The source was the day that mathematicians started treating irrational self-referenced relationships as if they could qualify as cardinal quantities. That was the mistake. A mistake that has been with the mathematical community for eons and has become so ingrained in mathematical formalism that is isn't likely that they will ever recognize their mistake or correct it.
The very idea that irrational self-referenced relationships could be thought of as cardinal points on a line is a very bad idea. Yet that is precisely what all math students are being taught to believe.
It was all due to a very wrong turn originally made by the ancient Greeks and never corrected.
This isn't to day that irrational self-referenced relationships don't exist. They do! They just aren't cardinal quantities, and therein lies the problem for our entire mathematical formalism. Aliens would actually find human mathematics to be quite hilarious. We made an extremely stupid mistake thousands of years ago, and we haven't recognized the folly since.
Norman sees that there is a problem. But does he truly understand how it came to be? I just explained the answer Norman. The problem was introduced by the ancient Greeks when they began to embrace irrational self-referenced relationships as cardinal numbers. That's the mistake right there. So we can not only point to the mistake, but we can even see when it was originally introduced into mathematics.
@Mystic Dreamer Thanks for the comment, which I have quite a lot of sympathy with. It is nice to remind ourselves that the ancient Greeks thought of, or at least represented general quantities geometrical, typically as line segments, or if they were being multiplied perhaps as areas. The idea of a quantity as a "point on the number line" I don't think we can fairly ascribe to the Greeks. In my next video in this series I will be discussing an aspect of ancient Greek thought re natural numbers and even there they pictured things geometrically. It is also true that the relationship between quantities, as evidenced say by the Greek theory of proportion, played much more of a role than it does today. We think of numbers as absolute objects, while they often preferred to thing of relationships between two objects (typically line segments). This way irrationality becomes incommensurability. So I think you are largely right. As for the true meaning of "irrational relationships", that may have to await a deeper understanding. See my video on "The magic and mystery of pi" at th-cam.com/video/lcIbCZR0HbU/w-d-xo.html.
@@WildEggmathematicscourses I will watch your video on Pi as I always enjoy your insight into mathematics.
Are you aware that everything we call an irrational number arises from a self-referenced situation?
I don't believe the mathematical community in general is aware of this.
Hello professor Wildberger, would you object to using the reals solely as an order, with no arithmetic? This is more computable
@Uri, No I cannot support the "reals" even just as an ordered set, with now arithmetic. We have to be able to write down the objects we are talking about --- otherwise the talking is just babbling.
Uri, I'm sure you'd want at least the rationals to inject into your reals. I suppose the most conservative approach to set up mere ordinals in set theory is as transitive sets of transitive sets, but then I'd ask you which one would be the reals. And you can't do that without taking a stance on the continuum hypothesis, which says the cardinality of R is that of the smallest such ordinal :)
π+e+√2 = π+e+√2
it is indeed unfortunate that there is no simpler way to represent real numbers, but still, the mathematical object π+e+√2 can be manipulated and we can examine properties of it and formulate theorems and proceed to prove those theorems.
i just don't believe real numbers need to meet the criteria you ask them to, in order to function as a consistent arithmetic.
the sum π+e+√2 is in fact not, as you say, "computable" (at least not in finite time), but some of those real numbers indeed contain an infinite amount of information (in their digit sequence) so it would be silly to expect them to be subject to operations like sum or product in finite time.
but, i don't think that makes arithmetic on real numbers "fake".
see also: math.stackexchange.com/questions/933890/an-algorithmic-approach-to-constructing-the-real-numbers
But why do you have the right to take 1 as an object and not e? We only take numbers which have finite decimals, only numbers which can be computed? It sounds like intuitionist or constructive math. Which are fine for me, these are just different formal systems with their own truths. But I don't understand why do you complain about a sum of two real numbers (which could be stated as a sum of classes or limits, perfectly reasonable operations between objects of a theory which has infinity in it's own axioms) and not about the sum of two natural numbers, which is the same, an operation between objects of a theory
@@ThePallidor what's an approximate number? Why 1 isn't one?
Perhaps we can reconcile both views by replacing "infinity" with "all". For instance, to sum from n to infinity, why not say sum from n to all sucessive integer numbers. That way, if you think the universe is finite and every number is finite, then you'd be happy. If you think there are a infinite number of things, 'all' still encompasses that.
Infinity is purely a human mind fiction to legalize many huge volumes of false mathematics, since integers are an endless chain of successive integers
However, in practical & purely eingineering works it doesn't make much of a difference where most of mathematics is truly made by eingineers & talented carpenters by approximations that satisfy the earthy human needs, but in pure physics & ALL theoretical sciences like Logic & Philosophy or Physiology, it is definitely a disaster to human minds for sure
So like for every alleged real number that is associated with fiction infinity is of course a fictional & non-existing number
@Jon Rufol, Infinity & finity are false meaningless terms in superior & truer knowledge of mathematics, they don't mean anything but a source of total confusions for the human minds, no matter if they can generate a huge volumes of false buissness mathematics for allegedly top-most genius historical & living (philosophers,logicians, physicians & especially mathematickers)
The truer meaningful terms instead is existence & non-existence, & to easily understand the fictions of ALL those meaningless terms like infinity, finity, infintism, ..., etc, ask your self what is the largest FINITE natural number, & you would immediately conclude that is also INFINTE in your same terminology
But in Eingineering probably earthy problem solutions by approximations, it is almost no harm to use these terms since the core issue is not at all mathematics
In other words, the entire modern mathematics is infact an eingineering productions made by eingineers & scientists for purely non-mathematical purposes where people & academic proffessional mathematicians cannot strictly distinguish especially that true understanding would lead immediately to finish their huge irrelevant buissness... to truer & superior mathematics
17:26 you mention this problem with carries frequently. how bad can this wave of decimal carries become? is it naive to assume there is some limit to their effect? for example if we see enough 0's
consider this. You add a number that consists of all 9s in decimal places with 8 placed every now and then. You add another infinite decimal consisting of ones and only 2s in random places. Which decimal places in the sum have zeros? It's too easy to create multiple classes of such failing examples where you have infinite number of arbitrarily long "carry waves"
@@whistaq oh. are there irrational numbers without 0's? for example a number that contains all 9's but with 8 distributed randomly every now and then. if i google "irrational numbers without 0" then some results claim it is not possible. i have no idea myself
@@xybersurfer One of the most popular definitions of real numbers involves the so-called Axiom of Choice, which, in the case of defining a 'real' number as a decimal expansion, means that you can literally 'choose', in a completely arbitrary way, whatever digit you want at each position. So, yeah, you can have literally any pattern of digits you want, but on the other hand there doesn't have to be any pattern at all.
@@xybersurfer You'd have to take for granted my ability to construct a non repeating infinite sequence using unly two symbols :)
@@xybersurfer How about this one. Let (n) denote that you're repeating the previous number in decimal expansion n times:
0.8989(2)89(3)... you get the drift, this sequence does not repeat. Now suppose it's included beginning at an arbitrary position in a number that contains all 8s. The rest follows. If you search MathOverflow for: " Can an irrational number have a finite number of a certain digit?" you'll find the argument that I'm making :)
We can prove as well that Mathematics don't exist...or does it!?
This was the result of answering the question:
" How do we create new maths?"
A circle don't exist (in the physical world) but it does in the human mind. But not all the thoughts in the human mind exist.
A circle is different from an unicorn or a pink elephant in a person's mind. In the mind of two people two pink elephant can't be the same but a circle is the same in all minds. So a circle exist...in the mind of the humankind.
But what will say about the maths when the humanking will be extinct and the universe with them. Could Mathematics exist if human kind or a mind could live or think forever?
Do things exist if there are no longer any record or memory of them?
The answer is no.
But what about a unfinished book novel due to the demise of his author ?
A book novel without his final chapter is not complete, but does not exist as a complete form. It is not a book. The same with maths.
Lot's of maths has to not be discover yet therefore Mathematics is not complete. It doesn't exist.
Can we discover all?
We need to think transcendentally , perhaps assuming an infinite mind capable of operating in infinite universes perhaps then we could compleat the Mathematics!?
But that mind would only create finite sentences (Alan Turing dictionary...) there fore will always be mathematical transcendental objects that will evade his ultimate powerful cognition.
So Mathematics don't exist and what exist is only the work of the Mathematicians...or doesn't! ?
Perhaps the meaning of the word "exist" doesn't exist...
By the way thanks for this channel.
At the moment I respectfully disagree with the author .
I believe they way I learned about Real number was from Russian text book la mir, (but possibly not)
In this Two thick volume the Russian scholar use the axiomatic approach to define all the property of the Real number. The axiom that bring the Real Number to life is the axiom of the existence of the extreme inferior.
Any inferior bounded set of rational number has an inferior (a number smaller than any of his element but greater of any of his "minors")
Then the scholat proved that the set of the Real number is unique under homeomorphism ( regardless their representation).
So the Set of Real Number Exist and it is Unique.
I do find problematic though
the limit of theta to 0 of sin (theta)/theta = 1
I am not sure was proved or if can be proved except for geometric consideration therefore it should be considered as an axiom, anyone??
I hope I don't be taken too serious.
I apologise for any mistake in the definition, I studied this long, long time ago and I misplaced my reference books.
I would like to thank again the professor author of this channel for his contributions and for the great value and joy in his channel.
Congratulations for the success of your channel.
Finally I think I could create the App mention at 19:00 😜
I understand I don't fully understand what I am talking about.
Bye
Isn't the problem due to the existence of non-contructable quantities, such as what are demonstrated by the Banach-Tarski paradox?
Hi Norman, love the videos and always find your perspective valuable when trying to strengthen my intuition of what mathematics really is. I feel like mathematical philosophy as a concept has been lost to the delusion of axiomatics and the esoterics of ontology, whereas mathematics to me should have a much clearer philosophical grounding to it, and your criticisms of mainstream maths have provided much food for thought.
Would you agree with a construction of real analysis that explicitly deals with Cauchy sequences, and explicitly states which results are constructive algorithms, versus those that require arguments by contradiction/double negatives? There would be no canonical forms or basic comparison of sequences, so it could hardly be treated as a number system, but it seems like something which would still be worth doing, even among people such as yourself who require explicit notation of the objects being reasoned about.
I can definitely see the emperor's new clothes aspect of "real" analysis, that internally consistent axioms are taken by the mainstream as reality itself. Generally people's criticism of the arguments you make amount to the obvious claim that your criticisms don't have any meaning within their axiomatic grammar, which is like defending classical physics by claiming that subatomic particles or objects with relativistic velocities don't exist!
I feel like this video and its presentation of the mainstream view of mathematics is compelling as a logical argument for your point of view, but I think there is a more nuanced case to be made describing how the mainstream switches between infinite processes, convergence arguments, and 'simple' arithmetic based on what suits them, and how internally consistent technical rules are taken at face value to arrive at the intuitions they arrive at, whereas this argument you present focusses more on your side of the conflict. Maybe this is all to come but I generally feel the view from above is more productive both for individual understanding of conflict/sociology and for actually resolving the conflict as a member of the field.
Thanks again for the work you put into these videos, interested to see this series unfold.
I will find this video, but my main points where agreeing that axiomatics is silly, and asking whether he would disagree with Cauchy sequences if treated formally without axiomatics. I might edit my comment so that it is clearer
I guess my comment about the view from above might also be what you are referring to. The point that I am getting at is that I don't think a mainstream mathematician would feel their side of this argument is being represented, which might be less than ideal both in terms of convincing them and in terms of a sociological discussion
@@ThePallidor Assumptions (axioms) need to made to even begin talking about anything meaningful, could you point out what is so wrong with the axioms of say, ZF? Semantic vs syntactic truths is the very essence of Godel's completeness theorem for first-order logic, which is widely accepted too. Do you have a disagreement with this theorem, if so, why? Do you disagree with the way Model theory has been built? I'm having trouble seeing what your real issue is with this foundational mathematics.
@@ThePallidor Okay, so you don't like the axiom of infinity. Now you don't even have the natural numbers, let alone the rational numbers. So one cannot talk about elementary mathematics anymore.
@@ThePallidor What are your qualms with the other axioms? Why do you think they're broken?
You are mistaken. Yes, real numbers are... abstract. We can't do arithmetic on arbitrary numbers with an infinite number of digits, so how can we claim they exist? But we can do arithmetic on finite integers, on decimal fractions with a finite number of digits, and rational numbers with finite integers in the numerator and denominator. The thing is, though, we can't set a hard limit on how big those finite numbers can be. The real number line is the only way to have a consistent system that doesn't limit them, and that doesn't exclude other numbers we can work with in other ways, like pi and the square root of two.
@John Savard With your original statement, I was hoping you were going to actually prove me wrong, by calculating pi + e + sqrt(2). Instead you just gave us the blatant pronouncement "The real number line is the only way to have a consistent system that doesn't limit them, and that doesn't exclude other numbers we can work with in other ways, like pi and the square root of two."
How do you usually define the arithmetic operations with Dedekind cuts? 🤔
@Martin Suppose we want to add the "real numbers a and b". You assert something like this. Take the set A of all the rational numbers left of "a", and take the set B of all the rational numbers left of "b". Then simply form the set C of all sums x+y, where x is in A, and y is in B. Then C will be the set of all rational numbers left of "c=a+b".
When you hear something like this your response could happily be: "Excellent prof! Can you demonstrate for us how to add pi + e using this fascinating technique?"
Note also the pivotal use of the term "take". We do a lot of "taking" in pure mathematics.
@@WildEggmathematicscourses said: 'We do a lot of "taking" in pure mathematics.' This is a weird objection. It like saying that C++ programmers do a lot of "defining" classes. But, technically, the sum C of A and B is defined as: C = { z in Q | (Ex)(Ey)(x in A and y in B and z = x + y) }. There is no "taking".
@@billh17 I am pointing out that when it comes to set theory, we pure mathematicians often use language that suggests we are doing something, when actually we are only talking about doing something. Kind of like: "take all the fish in the sea and arbitrarily divide them into 3 non-empty subsets." You can say it easily enough, but what does it actually mean?? And if that seems a forced example... which do you think is more challenging: taking all the fish in the sea, or taking all the rational numbers less than "pi"?
@@njwildberger said: >I am pointing out that when it comes to set theory, we pure mathematicians often
>use language that suggests we are doing something, when actually we are only
>talking about doing something.
That is because they are talking informally. In the first order language used in ZFC,
there is no "doing" or even "talking about doing". There is no time, there is no change,
and there is no construction of sets: all the sets already "exist" and one is just
making statements about them (which may be true or may be false).
And the only statements about them must only use the undefined predicate "in" and
predicates defined from "in" (and, of course, constructs from first order logic).
>Kind of like: "take all the fish in the sea and arbitrarily divide them into 3
>non-empty subsets." You can say it easily enough, but what does it actually
>mean??
Not sure what your point is. This seems to be more an objection to your use of English
as your language to do math. This is one reason why programming languages don't use
English: natural languages like English are not precise and clear enough.
Let's consider the word "arbitrarily" which I think is your main point. I am almost
sure that "arbitrarily" cannot be define in the first order language used in ZFC
together with the predicate "in". And, if it could be defined, then I would know its
meaning (in ZFC) and use it to make statements about sets.
In summary, I agree that your given statement is easily enough to say (in English)
and that it might be hard to determine its meaning. But, that statement cannot be
made in ZFC (or in C++ etc), so there is no need to seek its meaning (in ZFC).
>And if that seems a forced example... which do you think is more challenging:
>taking all the fish in the sea, or taking all the rational numbers less than
>"pi"?
I have already granted that I cannot write down in finite time on a finite piece
of paper all the rational numbers less that pi. But ZFC does not claim that this
can be done.
@@billh17 In the fullness of time I hope to show in the MathFoundations series why ZFC is highly dubious. Till then, please have a look at my article on Set Theory which I shall put a link to in the video description. But if you are going to defend it, why not actually use the axioms explicitly so people can see them in action. In other words, instead of saying that such and such can or cannot be "done in ZFC", explain why using reference to the actual axioms. I think if people had more exposure to these vague and even meaningless assumptions, the tide would turn rather quickly. Only by referring to them indirectly can we hope to maintain the illusion that they form a solid basis for mathematics.
Actually, Mathematica has always supported arbitrary-precision real-number arithmetic at the symbolic level but that is different from the word-size limits of floating point FPUs. So when you say "computers can't do it" there should be a distinction between FPUs and symbolic/algorithmic treatment of infinities.
Excellent. Can you please use this arbitrary-precision real-number arithmetic to compute Pi+e+sqrt(2) for us?
@@njwildberger I have posted a perfectly innocuous and respectful reply to your question several times now and it keeps getting deleted. I'm guessing it ran afoul of youtube's algoithmn? I don't see why you would delete it.
@@njwildberger Maybe this with the link removed:
Hmm. Now that I'm put on the spot, I feel a bit less confident in my answer. If true that there is no left-to-right algorithm for adding nonrepeating decimals, then it seems you are right. The exactly-precise adding of any two of those “numbers” could not even begin. A natural question is whether (A) it's provable that no L-R algorithm can exist or that (B) one might exist but is unknown. Only quick result I could come up with is here: [link removed] with “Avizienis in [2] and Wiedmer in [57] proved that an addition algorithm "from left to right" implies the redundancy of the representation:” So it looks like (A). It was unclear in the video whether you meant (A) or (B).
So “arbitrary precision” in my original statement must exclude the infinite case for addition of irrational numbers. Probably same for mul, div,sub. Come to think about it, even finite, N-place precision addition for irrational numbers seems problematic now, as we are always uncertain of the carry of the N+1th digits. Yeah, it's a mess. So how do we get the exact result of e^(i*Pi)=-1 ? That result must be in a different class.
I'll stick to my original beef with the oft-heard claim that “computers can't do this or that” when what is usually meant is that the built-in FPU's have limited word size, or division by zero will crash the computer. Mathematica, running on a computer, gives ComplexInfinity when you type 1/0, or can give Pi to 10,000 decimal places. It can implement all the unwarranted symbolic shortcuts that a typical Mathematician uses if he wants it to. In this regard, “computers” and mathematicians do not operate in fundamentally different compartments.
The simple answer to your challenge is, tell me how many digits do you want and I'll give you an algorithm to compute them.
You have now calculated a rational number, not an irrational one.
unintentionally you are showing what really happens when challenged to do arithmetics with "real" numbers: you just agree to a certain allowable error and then truncate the magical "real" number and work with either a rational number or with natural numbers and then just move the decimal point. There it is: the arithmetics of "real" numbers.
@@jgmartinezmd6867 I don't see why this is "fake". At the abstract level you can define all the arithmetic operations on the real numbers without running into paradoxes, which means imo that they are well-defined.
The issue you have with real numbers is, that unlike rational numbers there isn't any "finite" way of representing the abstractly defined object. However, the reason imo why it makes sense to add obejcts like pi and sqrt(2) to the rational numbers is that there is a well-defined notion of whether these numbers are smaller or larger than any given rational number.
@@qswaefrdthzg Thanks for your reply. You say `At the abstract level you can define all the arithmetic operations on the real numbers without running into paradoxes`, i am just a hobbyist and have not studied that part, so i wont argue that statement but what bothers me of so called real numbers is that you dont really use that arithmetic, just work with truncated versions of those numbers. So we cant really construct them, neither we can use their arithmetic.. what is the use then? if it is only a logical construction unrelated to what we can find in this universe they should have named them "imaginary numbers" or something like that.
i do find some paradoxes with "real numbers" when applying "natural numbers arithmetic". shouldnt they work under this arithmetic? if not, why not?
I don’t see why you accept the notion of a rational number as well as decimals, but reject that of a real. If rational 1/3 and 2/3 exist, then their corresponding decimals can’t be added as you would have 0.9999... with an infinite carry. The same as in the example with real numbers. This means that there is no “algorithm” for adding two decimal numbers. Then decimals are “fake”.
As for the natural numbers, they can’t be actual numbers either since a number such as two is assigned to both a pair of doors and a pair of shoes, which cannot be equal in themself. On top of that a pair of shoes may be assigned a number 1 since a pair is a singular entity or a billion billions since it is made of atoms. You cannot decide if the numbers are “fake” solely by their real world interpretations such as “algorithms”.
@Fullfungo. Decimal numbers are finite decimal expansions, and their arithmetic reflects that of the integers. Most fractions do not have corresponding decimals, as the Babylonians realized. In their system a division was meaningful precisely when the denominator was a regular number --with factors of 2,3 and 5. Over the decimals it is more restrictive, just factors of 2 and 5. So talking about the addition of corresponding decimals to 1/3 and 2/3 is meaningless, at least with the definition of decimal number I gave. Can you try to expand that to embrace also repeating decimals? That is a very good question, and will be topic of another Famous Math Problem!
If we limit our math to only finitely representatable quantities, we get problems with completeness and computability.
To the contrary. This limitation excludes anything uncomputable from our mathematics.
As many have pointed out in the comments, you are wrong about many things, or at least are misrepresenting them.
A "decimal number" is just a representation of a real number defined by an infinite sum:
x = sum( i=0,inf, a(i)/10^i )
where a(i) are integers such that |a(i)| < 10
If we have another real number y with digits b(i), then x+y is just:
x+y = sum(i=1, inf, (a(i)+b(i))/10^i )
So what's the problem? It's dead easy to write an algorithm to calculate each term up to an arbitrary number of terms N, an just express it as a big fraction.
It doesn't matter if the exact addition takes an infinite amount of time, the addition operator is still precisely defined!
@Kyle Brown You wrote: "It doesn't matter if the exact addition takes an infinite amount of time, the addition operator is still precisely defined!" I think not even Tinker Bell would be on board with such a far fetched claim.
@@njwildberger This far-fetched claim is literally the consensus of pure mathematicians, or is it not?
EDIT: btw I just looked up your publications, you acuse me of waffle when you spend pages of text to make the point that we should teach trigonometry in terms of distance squared and sin(angle)^2 ? That's an interesting suggestion since its essentially equivalent to using distances and angles, and in some ways more intuitive, but its buried in more unreferenced statements than I could count
@@KyleDB150 I think not. You will struggle to get even a reasonable fraction of pure mathematicians to agree with this claim, in my view. Have a go if you like.
@@njwildberger Well, here is what wikipedia has to say on the "real number" page:
"These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers-indeed, the realization that a better definition was needed-was one of the most important developments of 19th-century mathematics. The current standard axiomatic definition is that real numbers form the unique Dedekind-complete ordered field (R ; + ; · ;
So basically you're saying that we should do better than "true enough"?
Whenever you go from pure mathematics to applied mathematics, matter's imperfect nature can be apprehended by "true enough" arithmetic, right? There's no need to be absolutely precise in applied maths. If a glass of water was designed in such a way that its volume consisted in any of those expressions you put, we could see that "true enough" arithmetic is actually good enough, a few drops more, a few drops less (a few more water molecules, a few less) make no difference to us.
We should pay attention to the pure mathematics aspects then?
My take on this is that numbers aren't actually just quantities. They're philosophical, transcendental entities or aspects of true reality behind this imperfect veil. When we're trying to know exactly what Pi is, we're trying to peer into an aspect of transcendental reality which we just can't comprehend in our current state of being. If we ever grasp the true meaning of a number it would be at least in the next dimension of existence, after we pass away, where our abilities to perceive reality isn't hindered by a biological machine so to say. Don't get me wrong, I think it's deffinitely worth the effort though!
Would a formula in terms of natural numbers that told you exactly what is the nth decimal of Pi suffice what you're looking for? Or did I misunderstand the point you were trying to make?
@Gennady Arshad Notowidigdo Thanks!. Maybe the whole of material evolution is slowly marching towards those preposterous mathematical ideas that we tend to assume and overlook. I bet you the afterlife is closer to those conceptions than this place.
Why did you say the ability to calculate a value to an infinite number of digits is “useful”? Perhaps there is a use for mathematical approach that defines quantities in base pi instead of base 10. Then Pi would be exactly 1, pi**pi would be 10., etc. not sure the usefulness but who knows?
The challenge examples are tough mainly because we don't have easy-to-use manual tools. Expressions like "Pi to the power (e to the power {sqrt[n]} }" can be computed for specified sets of parameters (e.g., n) "close enough for government work", along with error bound. For most practical work (such as building aircraft) such computations are just as good as the "unknown" exact values.
...employ a whole machinery of symbolics... Even with symbolics, we have practical considerations. Exact values of trigonometric functions finer grained than pi/60 are beyond awkward in application. So, even symbolically only a handful of values are easy to calculate. However, we encounter nested roots which demand arbitrary precision to postpone collapsing the significant figures. In short, symbolism is not a panacea. One of my first computer experiences was manipulating identities and realizing that double precision routines spit out a string of 15 digits with only the first few places being significant and remainder being artifacts. The illusion of accuracy was shattered. The foundations of our castle rests upon the sands. Even the leaning Tower of Pisa appears vertical when viewed from a unique perspective. Appearances don't make it so. Approximations can be good enough for ordinary purposes. We tend to gloss over the limitations.
zero is not natural
Then why does it feel so right
It is
@@dionysianapollomarx There are N and N0 sets.
Comparing what's being done here with what we did in our many varied pure maths courses at uni, it is grossly over-simplified and wrong. What is meant by π or √2 or e or i is rigorously well defined.
@Keith Morgan Your claim would be much strengthened by sharing some of these rigorous definitions with us. Then we could make our own determination as to whether they really are rigorous or not.
@@njwildberger square root of 2 is a positive number such that, when being squared, is equal to 2. In addition, it can be proven that it is the only such number. The proof is left as an exercise to the reader.
@@njwildberger e is such a positive number that satisfies the equality e^x = d/dx e^x. It can be proven that it is the only number with these properties.
@@njwildberger pi is a ratio of the length of a circle’s circumference to the length of its diameter in the standard Euclidean plane. It can be proven that this number is unique i.e. is the same for any non-degenerate circle
@@njwildberger Or you could just take them as the limits of Cauchy sequences like any analyst would... Or perhaps dedekind cuts is more tasteful. Rather than this playtime in analysis, I am more curious on your stances on the ZF axioms, Model theory, Godel's completeness theorem and like ideas.
Why hold the assumption that something should fit in our finite universe or be computable in finite time for it to make sense? The abstraction involved in real numbers endears them to me. To the extent that “fake” means “abstract,” I don’t understand what is wrong with it. What am I missing? (PS Thank you for sharing your expertise online. Much appreciated.)
@tantzer How about trying your hand at the arithmetical challenges that I explicitly set?
@@njwildberger timestamp please
@@mehmeterciyas6844 Primarily between 24:41 to 27:44. He makes a specific concession at 27:45. Then makes a similar concession at 28:40, but shows that this second one begs the question, at 29:08. And he also raises a subtle but important issue with 'including the rationals' within the 'reals' at 29:51.
@@njwildberger 29:25 Why do you NEED to reduce the string to a single character? The reason teachers ask for that is so that they can easily check to make sure an answer is correct and for the homogeneity of the class. But whether or not you can "easily check" something doesn't affect its truth value. Why would you require some teacher's instruction to be a fundamental property of logic without any additional reasons?
I think that it's perfectly fine to consider a well defined string of operators and numerical objects as a numerical object in its own right, especially since all you really need to do with numerical objects is compose them into well defined strings with operators.
If you consider the reals as a vector space over the rationals, then question one is not all too dissimilar from (in Q^2) asking (1,0)+(0,1)=?. Effectively that's all your "computing challenges" amount to in my opinion.
@@SlipperyTeeth "Why do you NEED to reduce the string to a single character?"
I don't have a full answer, but one particular thing that comes to mind is: in order to evaluate whether two numbers are equal or not. It's often useful to have a 'canonical form', for example. Without at least some sort of limitation on acceptable forms, then the possibilities for expressing the same value would be unlimited and essentially impossible to compare in practice. Consider the myriad ways you can express the number 0, or 1. Instead of trying to tackle that intractable problem, better to limit the number of allowable expressions to a finite number, even better if small, even better if unique.
One slight nitpick here... when you talk about electrical engineers and computer programmers, the base matters. Numbers are stored underneath all the circuits as floating point binary, not floating point decimal.
This is what determines, for example, the machine epsilon. And if you're designing a system "close to the metal", or programming to a timing diagram where every clock cycle matters, in practice you absolutely need to consider things like the fact that 3/5 can be expressed as a finite decimal, but not as a finite binary expansion.
I wasted my time to watch an irrational argument against irrational numbers
oh wise man, what is your counterargument?
Yes, please enlighten us with your superior reasoning.
Great philosophical discussion. I have a few questions (certainly no offense meant!):
1st: Why do we need an algorithm? What if we simply accept the fact that maths s not algorithmic? What if it was only symbolic? Why would we care about a different answer to "pi+e+root(2)" than its symbolic representation? "What do we actually get?" Well, the number itself of course! Which can be symbolic, why not? "What is pi^2/6?" Well, I can give you an approximation, if you want an approximation, but asking "what is it?" to me is nonsensical, because the answer simply is "pi^2/6".
2nd: You explained the problem for addition concerning infinite decimals. I agree there. Why though does addition fail when it comes to equivalence classes of Cauchy sequences, for example? We know that it works for rationals, so everything should work fine by adding the n-th term to the n-th term and then considering the resulting Cauchy sequence. (Of course, we need to make sure that it is independent of the representative, but that has been proven.)
3rd: Infinite number of tasks. Why would you say that doing an infinite number of tasks is even required? Why isn't talking about doing an inifinite number of tasks sufficient? I feel like it all comes back to my first question, because to me maths is not at all algorithmic (and certainly doesn't have to be algorithmic to be real). I would wager that there could be a different, certainly less negatively connotated word than "fake" for pure mathematics, because if we allow for this definition of fakeness, we end up discussing reality in general - in that case nothing at all would be real except (going by Descartes) something which can think itself to be I.
@Indikativ How are you going to react when your Grade 5 daughter comes home from school to announce that her maths class is now going completely symbolic as you suggest? She is thrilled, because now the answer to 34 x 69 is ... 34 x 69. The answer to 1445 + 87 x 7 is 1445 + 87 x 7. She and her friends are now even getting the hang of powers, which was overly complicated before ... now 7 ^ 5 is just ... 7 ^ 5. She can even do really really big arithmetic: 4234 ^ 19 is ... 4234 ^ 19 !! Evidently studies have shown that kids do much better with this more agreeable approach to mathematics.
@@njwildberger The problems you have described in the video only arise when dealing with irrational numbers. For rational numbers we can prove everything you doubt easily. So my approach of making maths purely symbolic is supposed to be a solution to only the problems you described. Calculation with everything else should obviously still be taught. Also, we can certainly talk about approximations and computation using approximations, as long as we are aware of the fact that it is no more than that, an approximation.
Hi Prof Wildberger, I have some thoughts on this subject.
I, like you, despise the notion of actual infinites, contradictory as they are. However, it seems to me that this talk of "numbers" misses the point. I find it very strange that you accept the existence of rational numbers and yet reject transcendental numbers. A finitist should conceive of the *process* "divide two by 3" completely differently to a natural number or integer. The finitist should make this distinction because dividing two by three is an unbounded procedure; it never finishes. The idea that it can simultaneously represent a *value* which we might plot, is a rather unusual one.
The constructivists, especially Jan Brouwer said we should place more emphasis on the actual computation of the resulting values. This was long before computer science. Nowadays, mainstream mathematics now labels practically everything a "number" of some kind. It is clear to me that there are really two things: *values* and *procedures* . In the theory of computation we have *images* and *machines* . An image is what you would expect: a finite euclidean space filled with boolean values. The description of any finite object is an image. For example, to back up your computer, you make a system image and save it to a hard drive. The image is the complete description of the system at some moment in time. An image does not have the dimension of time. It has only spatial dimensions.
A *machine* , on the other hand, is a computational model. It consists of three parts: The current state (an image), how the next state is computed (some rules), that there is a next state (a clock). A machine has agency (can make things happen). When we run a machine (simulate its behaviour), its current state changes.
It is clear to me that division of one natural number by another is a procedure, not a value. We encode each numbre as as image in the radix representation. This allows us to define a machine which computes a division of one number by another. We simply encode them as two images using the radix representation, and then modify the initial state of the machine to include the two images as appropriate. However, division is not a bounded procedure; that is, it is not guaranteed to finish in finite time. Instead, if ever we have a remainder of '0', then we can return the radix representation of a natural number as the result. Alternatively, we could modify our machine to have a fixed level of accuracy, and always return a truncated result after so many digits. Pure division, though, is not a bounded procedure, and will not always produce a final result. This is in contrast to addition, which always produces a result in finite time.
Surds, trigonometric functions, infinite summations, limits, etc. are all unbounded procedures. To my mind, they are exactly the same sort of thing as division; unbounded procedures, liable to stop and return some finite answer, but equally capable of continuing indefinitely. It should be noted that the special case where we return a finite answer is artificially enforced. In its more natural form the machine would continue to print '0's but we have specified that if ever we would reach that state, instead notify us that no more "useful" digits will be calculated. A similar case arises with repeating digits.
From my finitist, computationalist perspective, most other people seem very confused. Yet I don't suffer any confusion at all. I am delighted to see you bravely challenging the existence of values with actually infinite resolution. This is the main point of confusion nowadays. Typically, by "number" people mean *value*; But to call a procedure like sin(0.734) a value is horribly wrong! Instead, we must combine unbounded procedures with some resolution bound. Then we can calculate a value of finite resolution. That's how trigonometery and the laws of physics are applied to the real world; You take initial values as inputs to the appropriate procedure and then combine the procedure with the lowest resolution bound of your initial values to find an answer. For example, we might measure two values to be 0.76m and 0.433m. This means between 0.755m and 0.765m, and between 0.425m and 0.435m since we have only measured to those levels of resolution. Before we can calculate an answer from our procedure (let's say sqrt(0.76^2 + 0.433^2) ) we have to know what level of resolution our answer should be, otherwise we might mislead others with an answer which is too precise, or not precise enough. Even worse, we might get stuck in a loop! In this example we should calculate our answer to two decimal digits of resolution, because the lowest resolution measurement value had just two decimal digits of resolution. As we can see, no actual infinities are involved anywhere. Just unbounded analytic expressions which can be applied an any resolution.
I'd love to speak with you and hear your thoughts on these issues.
You should check out his Math Foundations series, and also his Rational Trigonometry series. In MF, he starts out as dead-simple as possible to form a basis for numbers, and by the time he builds up to the rational numbers, you'll see why the rational number 2/3 does not need to be seen as a process. It is just a finite written expression of a finite number of symbols, and the 'division' aspect of it can be interpreted geometrically, if need be (though it doesn't necessarily *need* to be), and it is completely finite, not an infinite process, or it can be dealt with entirely algebraically, which is another very useful perspective to have on it.
In his Rational Trig, you will be introduced to how powerfully simple this approach to rational numbers can be, and how it can be used to break the illusionary dependence on 'real' numbers from a topic such as trig (in other courses he uses the same/similar techniques to break the dependence also; it's just particularly impressive when it's done with trig, because trig is so messed up because of this dependence, and at first it doesn't seem like it would be possible to break it, but it is).
In fact, maybe start with his Rational Trig first, just to get a feel for where he's going with things, and then go into Math Foundations to get into the nitty gritty of how it can actually be accomplished.
Just a note of caution: It takes time to lay out the whole groundwork and the arguments, it's not something that is (yet) packaged up, even as an introduction, into a few short videos. Maybe one day it will get to that kind of polished state, but it's not yet.
On the other hand, each video tends to be interesting in its own right, so it's not like he'll leave you hanging for multiple videos without any tangible/useful ideas. I'm just saying it may take 2, 3, 4, or more videos to really start to see where he's going with things. On the other hand, since you're already a finitist you might instantly pick up on it right away. Just giving you a heads up in case it takes longer than 1 or 2 videos.
He also has a bunch of other (math-related) topics in his two channels, e.g. Math History, Projective Geometry, Famous Math Problems, Hyperbolic Geometry, etc., so sometimes it's refreshing to take a break from one topic for a bit to check out something else. That's how I tend to watch his stuff, anyway. Cheers! 😊
You'll especially like Famous Math Problem #9!
@@robharwood3538 Hi! Thanks for this. I will follow your reccommendation : )
I want to quickly respond to your comment about rationals though. I completely agree with you when you say "[a rational] is just a finite written expression of a finite number of symbols, and the 'division' aspect of it can be interpreted geometrically, if need be... ...or it can be dealt with entirely algebraically, which is another very useful perspective to have on it." This is true, and is why I, as a finitist completely accept rationals but also surds, trig functions, etc. Because, all procedures have a finite description. All machines are finite in size, even if they will grow when simulated. As a result, not just division but all mathematical procedures are legitimate (in my current understanding). The problem is not that they "don't exist" or something, the problem is that people are confused because they don't understand these "numbers" as procedures. If we understood them this way, things would be a lot more clear.
I will watch his rational trig series, and see if there is in fact no geometrical or algebraic interpretation. For me, that would be a symptom of some kind of problem. Let's see...
Thanks, God bless.
I always think many infinite convergent series should not use equal sign. it should use approach sign.
Sir, how much do you appreciate ,David Hilbert for his contribution to the foundation of mathematics..?
Unsurprising truth be told, I am highly sceptical of your position. However, I want to know more about the critiques you mention here that are on your playlist. Could you recommend me any set of videos from the playlist I would need to watch to have a better grasp of your claims? If so, I would thank you, and pardon if my English is rather sloppy. Sincerely, an amateur mathematician.
For me, the most important problem is about the properties in real number arithmetic bc, not having an algorithm to add irrational numbers term by term seems imposible by the fact that is imposible to know all the decimals of such one irrational number with an algorithm.
There is no periodicity, and no periodicity means infinite calculations to know every single decimal of an irrational (It is not equivalent to say that any particular decimal of an irrational could not be deduced by not doing the computional algorithm through the point. Them also could be deduced by some advanced sketch)
Norman, I would consider you a 'reactionary' mathematician (in the political sense).
Based.
Read Moldbug a bit too much? :)
@@Contra1828 I am a communist, so no
@@isaacstamper7798 "I am a communist, so no" - If you actually survive in real society, then you cannot be a communist. It's probably only your fantasy. You have to necessarily behave as if you accept money, as if ploretarian are NOT the primery movers of history, as if you DON'T struggle to overthrow bourgeois, as if private property IS real, as if people have a nature that cannot be changed and as if self-interest IS this nature. Even if you were an intellectual or politician, you would still have to behave according these things which you believe to be fiction or mystifications of capitalist ideology. It must be cool to believe in a law of history that manifests only if whole world believes in it (only when the proletariat realized that "all they ahve to lose are their chains" - which they never did "realize").
The claim that pure mathematics is based on fake arithmetic is disputed.
So are the election results in the United States right now, but that won't change anything. It is literally impossible to do arithmetic with irrational numbers, so it's hard to see how it isn't fake when nobody in the universe can do it.