Difficulties with Dedekind cuts | Real numbers and limits Math Foundations 116 | N J Wildberger
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- เผยแพร่เมื่อ 16 ก.ค. 2024
- Richard Dedekind around 1870 introduced a new way of thinking about what a real number `was'. By analyzing the case of sqrt(2), he concluded that we could associated to a real number a partition of the rational numbers into two subsets A and B, where all the elements of A were less than all the elements of B, and where A had no greatest element. Such partitions are now called Dedekind cuts, and purport to give a logical and substantial foundation for the theory of real numbers.
Does this actually work? Can we really create an arithmetic of real numbers this way? No and no. It does not really work. In this video we raise the difficult issues that believers like to avoid.
Video Content:
00:00 Intro to Dedekind's approach to "real numbers"
5:08 "Cuts" of the rationals
7:13 Principles of Mathematical Analysis
12:38 Subsets of Q
24:02 Prior theory of 'infinite sets'
32:09 Arithmetic with 'Dedekind cuts'
35:05 Unwieldy, infinite sets
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I am amazed at your acuteness in explaining the problems that mathematicians either ignored or don't want to get involved, regarding the "theory of numbers." Very good video.
Why did I watch this before doing my homework on field theory? 😭
I remember sitting in a real analysis class senior year of college after having 1 year of abstract algebra and being excited to see how the bridge from field theory to analysis would go. Then Dedekind cuts were presented to our class of 2 students and we had a lot of questions. I remember asking exactly how you constructed the dedekind cuts for pi and being disappointed that this is what it all rests on. It drove me away from analysis and caused me to view it all with an eye of skepticism.
abstract algebra is a beautiful and grounded subject. analysis is still beautiful but completely ungrounded as long as it depends on the general uncomputable reals. I believe analysis can be reformulated in such a way that most of the interesting phenomena in it is preserved or at least examined from a computable perspective.
Beautiful presentation! Thank you for being the voice of reason!
Here is another way, quite different from yours, that leads to the same view of the Dedekind cut:
The term 'irrational number' (as applied to root 2, pi, etc.) I regard as a kind of carnival huckster's trick in that it focuses one's attention on the adjective 'irrational' when our attention should be on 'number': I.e., IS root 2 a number? IS pi a number? No. Rather, there exists an algorithm to produce digits of root 2 or digits of pi For Ever, but the output from an algorithm that runs For Ever is not a number, it is (again) the output from an algorithm that runs For Ever. In other words, since root 2 and pi, etc. are not numbers, they have no place on the number line, so Dedekind need not have wasted his time trying to find a clever way to put them there. Being non-numbers, they live in a separate, non-number space of their own, a space where computer algorithms run. (And those algorithms run not 'to infinity', which is an infantile babble-phrase, but FOR EVER, which is a grown-up concept that actually works.)
@ Verschlungen Very nicely put --- I agree with you completely!
Verschlungen said "...our attention should be on 'number': I.e., IS root 2 a number? IS pi a number? No. Rather, there exists an algorithm to produce digits of root 2 or digits of pi For Ever..." But Dedekind is not defining a real number as an algorithm. Rather, he is defining a real number as a certain kind of set. After defining the real number this way, he show that they can be added, multiplied, subtracted, and divided (by non-zero real numbers) and they satisfy the usual properties for a field. In that sense, they are "numbers".
@@billh17 It is curious that you would make those remarks HERE of all places, on a channel (Insights into Mathematics) where the very argument you make has been discussed (and trounced) in other episodes in great detail. Yours is a kind of religious argument of the Math Establishment True Believer, which is no argument at all.
On a side note, people wonder "If there are so many ETs visiting, why don't they make contact?" Well, one answer to that might be that humans hold stubbornly to their superstition of 'infinity' (instead of 'for ever'), and their superstition of "pi and the golden cut" (instead of algorithms that produce certain strings for ever), and of the 'irrational numbers' generally. A species of biped simian that has come so far (its accomplishments with the bomb are not to be denied) and yet cannot yet see that those are all just superstitions would put the ET it in awkward position of not knowing whether to laugh or cry.
@@njwildberger pardon my ignorance, but there are objects that we can touch and observe with a length of sqrt(2). There are also objects that are slightly shorter and slightly longer. Why don't we attempt to unify R \ Q and Q on into a totally ordered R?
I am pretty certain that Dedekind's own book contained only that example, of the square root of 2, itself.
When we did this in my analysis class, I remember that they proved the least upper bound property of the reals before abandoning the topic of 'constructions' of the reals: But basically, they ignored this stuff even at UChicago.
Very interesting! This reminds me of some of Wittgenstein's remarks on infinity and set theory. Have you read his work in the philosophy of mathematics? I'm very curious what you think of Gödel's theorems and Wittgenstein's comments on them.
Brilliant video. I always told my students that the reals are the ugliest objects in mathematics (I had a careful look at the Rudin book -- I took it from my shelf again while I watched the video-- and that was my conclusion). Btw there was a paper at some point by the Nobel laureate Gerard 't Hooft disputing the role of the reals in physics. I can't find it any longer but at some point I am sure I had a (preprint) copy. As far as I recall, he used quantum mechanics to redefine the reals.
+kyaume21 I found this article: blogs.scientificamerican.com/critical-opalescence/does-some-deeper-level-of-physics-underlie-quantum-mechanics-an-interview-with-nobelist-gerard-e28099t-hooft/
" I’m not excluding real numbers as a good basis for a classical theory, but I’m also considering other options, such as the integers or, even better, numbers that form a finite set. "
So, he's not completely against reals in physics.
+Охтеров Егор Thanks for the link; that is an interesting discussion. There are several ways how seemingly quantum aspects can be realized in discrete classical systems.
@@kyaume21 looks like youtube had a bit of censorship. Do you still have the reference?
min 25:25 The difference between choice and algorithm, it is the key point. Yes it is.
I remember professor Franklin said in a commentary (pertaining to the infinity debate video) that the Axiom of Choice is a pure existence statement, that's right but we need an algorithm in order to be sure the elements exists , as I see.
This is great.. finally an effective and clear explanation on this.....
Those videos are great. While I sometimes wish they were a bit more advanced in content and more rigorous, it is amazing to finally see someone experienced voice out and help me structure the issues I've had with the axiom of choice throughout my maths major
Thanks. I bet a lot of mathematics students have been secretly unsure over the years over the standard dogma that is presented to them as certainties, even though it doesn't really appear to hang together logically.
Many will feel that it is probably their own fault for not understanding (I'm not smart enough..), and then eventually let their objections slide to become card-carrying members of the community.
That's pretty much my story. I have struggled to get the grades, because I struggled to get the exam answers that were convincing for ME not the marker, so I have all but given up on carrying on with maths as a research career. Some lecturers tried to appeal to the sense of aesthetics when trying to justify the ZFC, but frankly it seems a bit blemished when all of the cool results rely on Zorn's lemma. Fortunately there's always combinatorics and graphs. Those people seem to have a bit less of an inclination to have infinities all over the place
what i've been wondering however, is if the parallelism is not equivalent to axiom of choice? There is any arbitrary number of parallel lines, so it is equivalent to a choice of point we want the parallel line through. Similarly the notion of parallelism is an equivalence class in the infinite set-theoretic sense, unless we are working in an affine/projective geometry over a finite field. You did mention that perhaps a finite field for a sufficiently large prime describes our universe, and as far as I have seen there are projective geometries of any arbitrary large prime order. It's quite strange that in order for axiom of choice to not creep into the theory, you have to be equally adamant about rejecting it as the people who accept it!
@@njwildberger It's interesting that there is so much angst about reals and yet few concerns about i. What's the constructivist position on complex numbers?
In a sense, a Dedikind cut on the rational number line is analogous to trying to play a "note" between two piano keys. The rigorous structure of the piano prevents that. By definition, there is no note between two adjacent keys. To say that the "note" exists despite the rigorous structure of the piano begs the question, "well how are you going to play it on the piano without breaking the piano?"
Seems to me mathematicians in favour of Dedekind cuts and everything that follows are breaking mathematics to play their so called real number notes.
Problem #3 criticism: Your algorithm is exactly equivalent to making an infinite number of choices, so if you're trying to avoid that you don't appear to have.
It's really a bit more complicated. I tried to make the distinction between choice and algorithm and their equivalences:
1. An algorithm that selects all (finite) possible selections and terminates there is no choice analogy
2. An algorithm that selects countably many rational numbers and selects them in an order (for example Stern-Brocott tree order) A choice has to be decidable in a finite time of computation [For example the choice of x in sqrt(2) can be checked in finite amount of time: x < 0 or x^2 < 2].
3. An algorithm that selects countably many rational numbers and selects them in any order A choice (if it is in the set) has to be computable in finite time but not if it isn't. [also called recursively enumerable, the pi^2/6 choice]
4. Statements involving any choice over rational numbers. There are 2^Q choices, which is not countable! This leads to paradoxes of the axiom of choice.
Look up the infinite prisoners hat dilemma for a great illustration. The reason it leads to a problem/paradox is that we assume it is possible to create a random (martin-löb random) sequence of hats. I'm not saying it's an issue to create countably many prisoners but selecting a random hat for each one is where the problem lies. You can't write down the order of those hats in any way using an algorithm.
It is 4 that really is the issue.
Thanks for this great video...I do have a question though, if Dedekind cuts is inadequate. Do you know of a better way to construct real numbers. I am looking at Cauchy sequences and the "almost homomorphic" maps to construct real numbers from Z. Are these constructions acceptable or are there flaws in them too?
Sorry, there is no better way: the 'real numbers" are really a charade. You can tell that simply by looking at the almost complete absence of examples of how to actually do arithmetic in this domain. What is pi+e+sqrt(2)? What is their product? What is log4-cos7? These kinds of explicit questions are carefully avoided by the current orthodoxy. And the reason is: there is no real theory behind the wishful dreamings of "real number arithmetic". So Cauchy sequences, infinite decimals, Dedekind cuts, continued fractions, etc --none of them work. Naturally people will kick and scream to avoid acknowledging this, but our computers know it very well. They don't even pretend to work with real numbers!
Ok I see...Thank you for your prompt reply...So no one is trying to fix this?... But if real numbers are a charade as you say then a lot of things would lose their meaning because they are assumed to have those nice properties. Coordionates in R^n for example no? Lol interesting to see that I have been flicking through different books on Analysis and you are right they ALL talk about Dedekind cuts and give the "baby example" with Sqrt{2}. Do you know if there is a research begin done on this?
@@jean-francoisniglio6798 I am trying to fix it by providing an alternative -- a mathematics that does not rely on infinite processes and other dreamings. This includes Rational Trigonometry, Universal Hyperbolic Geometry, and notably the Algebraic Calculus. As it is most of the so-called "theorems" in pure mathematics are not such at all --- they are just sustained by a carefully crafted avoidance of looking at the obvious difficulties with the foundations. Look to the computer scientists and the AI people who will be moving to get maths going consistently in the computational setting --they are forced to do more correct mathematics, because their machines are required to run.
@@njwildberger The arithmetic operations on reals are possible if you allow the algorithms you demonstrated to run indefinitely. In your (counter-) argument you allow the reals to have infinite digits, but you stop the algorithms after finite time. Isn't this a logical inconsistency?
@@njwildberger Surely the real test of the exclusively rational approach would be to produce results with rationals that cannot be produced by real analysis, rather than just trying to reproduce existing results.
Your explanation is very clear, i dont agree that ZFC is a wrong foundation or that any number really exists, for me are all constructions. But your videos are excellent. Thanks a lot!
I just grasped what's potentially wrong with an infinite amount of work needed to tell whether something's true. Up till now, I have been thinking that as soon as we can ask a yes/no question about an object there must be an answer to the question, and the fact that we can't figure it out doesn't matter.
What's wrong is that some questions, like the one used to construct Russel's paradoxical set R = {x : x not in x} "is x not in x?", don't have a well-defined answer! If R is in R, then it's not, but if it's not, then it is. When defining a something, like a subset, we should be obligated to show that this situation cannot happen. If a so-called infinite amount of work is needed to tell (meaning it's in general impossible) whether an object is in a set, then how can we demonstrate that the bare question is well-defined?
At each step of this video, it is demonstrated Dedekind cuts works beautifully, the only real objection being the it assumes infinite sets. And why are infinite sets evil? Well, because I don't like them. I don't feel well around them.
Please show us this beauty explicitly by calculating pi + e + sqrt(2) for us.
So I see nothing necessary about using the Stern-Brocot construction for defining our subset of Q... Why couldn't we select our subset by constructing (geometrically) a line segment of length sqrt(2), and then imposing that upon our number line (Q), and defining our subsets on those values on each side of our lines right endpoint.
The algorithm took constant time, and we now have 2 explicitly defined sets (ie, a clear pruning of Q, when represented as a tree with branching factor of 1). And should this representation fail, what is to say there is no construction that would make the pruning algorithm finite and decidable?
Great presentation - THANK YOU for sharing
"Little one, it's a simple calculus. This universe is finite, its resources finite." - Thanos
Hi, just adding my thoughts on what a real number "is" that can be used to reason on why they "work" or not. They are sort of slippery by the very reason that they are defined as not being rationals in the first place, so no wonder it is hard to find examples like finite decimal numerals or series for real numbers (that are not rational as well). We may first define a rational number as the fraction a/b of two finite integers, where b is not allowed to be zero. A rational corresponds exactly to one point on the geometrical number line; a line in Euclidean geometry where two different points zero and one are marked out.
The definition of real numbers is most commonly a completion of the rational number points on the number line. The number line is a geometrical object, a continuum of points. All the rational numbers correspond to points on the line, where one rational number corresponds exactly to one point, but we know that some points don't correspond to rational numbers, for example the square root of two, the number e, and pi. To complete the points not corresponding to rational numbers on the number line, we introduce the irrational numbers. An irrational number is defined as all the rational number that is smaller than a certain point that is not corresponding to a rational number. The reason that we can't pick out one biggest smaller rational number is that this leads to a contradiction (then we can identify a bigger smaller number), so there exists no biggest smaller number, but all these rationals can be identified as smaller, and this is enough to define that irrational number. Sure, there are many different points defined by the same smaller rational numbers, but these points lie infinitesimally close to each other, so the interval between smaller and bigger rationals is completely filled and defined as one irrational number, and therefore all holes are filled with irrationals, and the number line is completed. If we want, we can now create the real numbers from the addition of the rational numbers to the irrationals by identifying a real rational number as all the smaller rational numbers to it, so that a real number is consistently defined from the rational numbers and irrational numbers and their corresponding points on the number line.
What we get is a number system (the real numbers) that essentially works algebraically as the rational numbers, but we are also allowed to use the square root of two, the number e, and pi. The cost is that the point precision we had in the rational numbers is replaced with a system precise down to infinitesimals. This is not a problem as long as we only use a finite number of calculations and not introduce infinitesimals, the square root of two, the number e, and pi will only correspond to one real number representation each no matter what we do with them. Working in (Euclidean) geometry, we can reach a point precise definition for these symbols though. What would a better solution than the real numbers look like, for the inclusion of the irrational points on the number line? Well, that depends on what you are working on. If the infinite decimal representation is important to you, the definition of the real numbers will guarantee that all the decimal numerals represent reals, but you will maybe struggle with some decimals (1.000... and 0.999... for example), not that this is such a big problem, since they can be viewed as the same real number from the definition of the reals; just convert to rational numbers and when a decimal represents an irrational, you deal with an irrational real number.
Set of R,Q etc., doesn't necessarily follow from a one dimensional set of number system, it can follow from a two-dimensional field/plane or a three-dimensional, or n-dimensional space should be considered. The mathematics of quantum computing involve superposition of states that involve infinity and can get rid of many paradox/ambiguity of definitions. I think.
What about just thinking of each namable irrational as being useful in these two ways:
1. As a label similar to the imaginary number i. For example, i can be thought of as a label for sqrt(-1), which isn't a "real" number, but we keep this label around in our equations to remind us that if at a later point we multiply it with itself or any other i, we can replace that multiplication with the rational number -1. We could think of irrationals in a similar sense, like sqrt(2) is just a label we keep around in our equations as a reminder that if at a later point we multiply it by itself or another sqrt(2), we can replace that multiplication with the rational number 2.
2. Also, each namable irrational could represent a finite algorithm that could take in any rational number and would return True if that number was "less than it," or False otherwise (a finite Dedekind cut algorithm). I would imagine that as long as the inputs are rationals, a finite algorithm for each namable irrational would exist, but let me know if I am wrong about this.
These two properties seem to represent the usefulness of irrationals while avoiding dealing with any infinities. I'd be interested to hear your thoughts on this.
EDIT:
I've been watching more of Norman's videos, and I've realized he's addressed both my points above in his other videos. He mentions the idea of thinking about sqrt(2) as a symbol similar to how we can think of i as a symbol for sqrt(-1) at 4:27 in this video: th-cam.com/video/cxNq-hQwvn0/w-d-xo.html. And he addresses some of the potential problems of thinking of irrational numbers as computer programs at 19:56 in this video: th-cam.com/video/EB9HZbYoD5c/w-d-xo.html
I agree with you and had the same ideas as well. I've been watching videos from Norman Wildberger for years, and I understand his points.
I work in AI and I couldn't live without log, cos, sin, exp, sqrt, decimal powers, etc. And sometimes the theoretical formulas simplifies (sqrt are squared, log are exp-ed, etc.).
More fundamentally, I think the point he is really missing is that modern maths is presented as the manipulation of symbols, not actual physical things. For example, why is an infinite amount of work impossible in our world ? Because each computation has a cost (time, ressources, ...), so an infinite sum of non zero costs diverge. But using symbols, there's no cost, there's no "friction". If computations did have friction, then surely there's an upper bound cost limit that some equations intrinsically possess, and every theorem would have to be validated a posteriori by comparing its computation cost to the limit. This is ridiculous, and shows that the cost of computation in modern maths is strictly 0. Not very small, but actual 0. And as a sum of 0 is still 0, an infinite amount of work in this framework is possible because it demonstrably doesn't take any ressource.
@@allehelgen, thanks for the reply. I find this topic very interesting to discuss. However, I'm not sure I'm on the same page as you, but maybe I can try to clarify my current thinking and see if you have any feedback.
I currently actually agree with Norman Wildberger about the idea that an infinity can't be realized. And not even just in the physical sense of work and energy, but in a logical sense as well. It seems that if you try to operate logical thinking that an infinity can be realized in completeness, you start to run into inconsistencies and paradoxes.
For example, if you believed that infinities could be completed, than Cantor's diagonal argument seems to prove that there are an uncountably infinite number of real numbers (a higher infinitiy than the countable infinitity of rationals), meaning that there are inifinitely more real numbers than there are rationals. But then you run into a paradox when you consider this thought experiment: take any two real numbers that are different (for this example lets consider two numbers between 0 and 1), and start comparing their digits from left to right until you find a place where their digits are different. If those digits are different by more than one, than you have a gap between those numbers that you could fill with an infinite number of rational digits. However, if those digits are only different by one (e.g. the last digits 0.001 and 0.002), then you look at the next digits until you find any pair of digits where the lower number's digit is not a 9 or the higher number's digit is not a zero (e.g. 0.0019997 and 0.0020000), and then you have a gap again that you could fill with an infinite number of rational numbers. And if you never find a non 9-and-0 pair (e.g. 0.00199999... and 0.00200000...), then mathematically we actually consider those two real numbers to be equal, so they actually aren't different real numbers after all. So this exercise seems to suggest that if you take any two reals that are actually different values, then you could find an infinite number of rationals between them. But this seems to contradict with the idea that there are infinitely more reals than rationals.
This paradox seems to stem from Cantor's diagonal argument, where he goes down the diagonal of the original list of numbers, changing each of the digits along that diagonal to generate a new number that is not in that list. And then claiming that since he has generated a new number not in the existing list, that there are a higher level of infinitite (uncountably infinite) real numbers. But to actually generate an actualized number, he would have to finish going through the full list of countably infinite numbers, which doesn't make sense from a logical perspective. That's like running a program that iterates over an infinite list (or in other words, enters an infinite loop) and expecting that program to eventual halt at some point as long as you run in long enough (but it won't, it's by definition a non-halting program). So in my opinion, that's where Cantor's agrument breaks down and where the paradoxes start to creep in, when we assume that an inifinity can be actualized (completed).
Nonetheless, the idea of infinity is useful, and I think the paradoxes can be avoided if we instead just think of "infinity" as a symbol to represent a non-halting program. You can run the program for as long as you want, but you just can't expect it to ever finish (which is the same as saying that the "infinity" it represents can never be realized). For example, instead of thinking of pi as a specific actualized number, we can just think of it as a program (or an alrgorithm). And you can find out as many digits of pi as you want (any finite amount) by just running the algorithm for as long as needed. But you can't expect to ever realize all the digits of pi, since the pi program by definition will never halt.
Using this idea of the real number being represented by programs (finite things, as in finite source code), I think you can represent all namable real numbers with a set of countably infinite programs. meaning that the reals could be considered as countably infinite instead Cantor's idea of uncountably infinite, and the paradoxes of considering the reals as uncountably infinite can be avoided.
I'm not sure how well I explained this idea, but I'd be interested to hear your thoughts on it.
@@elliotwaite Thanks for the great answer ! I actually am not a fan of the real numbers at all. My research work mainly focuses on graphs and therefore discrete maths. What I don't like about real numbers is that it insidiously makes us believe that the world behave this way, with infinite precision, information and singularities. I don't believe the decimal approach is fruitful though when describing paradoxes. The decimal expansion of a real number can never be interpreted as the real number itself, because, well, as NW says these are infinite. To me, the decimal expansion is analogous to the projection in 3D of say a 4D structure : we can infer some things, but the information content is too complex to be embedded in it. Is your paradox an intrinsic characteristic of the real numbers, or does it only show the limit of the tool that approximately describe them (decimal expansion) ?
The main reason I dislike real numbers, is not theoretical, because these are based on axioms. You don't need to believe anything, just add an axiom, and then study the consequences of these axioms. It's like inventing a game, like chess, and studying the games you can play from the rules you defined. Maybe these rules will have some bad consequences (inconsistency), maybe the game will be uninteresting. I dislike real numbers mostly because of the uncomputable real numbers, which are most of the real numbers. I find these to be very unelegant, and it shows very well that the reals do not describe the real world at all.
That being said, computable reals are very important if not ubiquitous in all areas of science, and I do agree that defining them as algorithms would be so cool. As I mostly work on float64, I would love a theory of "the reals" being developed only in the float64 framework. Then you could define an algorithm that exactly describes log, exp, sqrt within 64 bits (or 32, it's arbitrary) precision. I also love the idea of some functions being only seen as manipulation of symbols that is useful, for example : if x is a float, then log(x) is just the symbolic notation that allows exp(log(x)) to be equal to x. Then, you have no problem with infinite decimals as you are only using symbol manipulation to adequately reduce expressions, simplify things, and if needed, at the end of your computation, approximate some value.
@@allehelgen, that's an interesting point about that paradox, that it may just be a limitation of the tool (decimal expansion). I'll have to think about that more. I'm still trying to develop a better understanding of what the differences are, if any, between a continuous space and a set of countably infinite distinct points. Because they seem different is the sense of continuousness vs distinctness, but when infinity is applied to distinctness, it might essentially smear it into continuousness. But maybe they are different, and it's only when we try to express the abstract idea of continuousness in a form designed for distinctness (decimal expansion) that we get the paradox.
Those other things you mentioned are interesting to. Thanks for the reply!
What do you think about Conway ' s surreal numbers
They do not allow us to escape the problems with real numbers.
Wildberger mentioned that the Dedekind cut baby example of root 2 only worked because the criteria for entry into the set A requires only two checks. My question: Would it be possible to define an arithmetic (and corresponding laws of arithmetic) of Dedekind cuts where the only real numbers allowed are the rationals and the baby algebraic irrationals? (by that, I mean root 2, root 17, and in general root q for q of type Rat).
This would at least extend the set of rationals to include some of the irrationals (and a lot at once which would be a benefit over the extension field method). Although, to get closure under some of the operations, we may need to also include complex baby algebraic irrationals among other things. I guess my real question is, Is the logic of the "cut" okay at least for the "cherry-picked" baby irrationals, or does it ultimately fail even there as well.
On can include sqrt(2) by finite means by constructing the field Q(sqrt(2)) algebraically (as couple of rationals (a,b) with the multiplication given by (a,b) (c,d) = (ac + 2bd, ad + bc)), and by "inserting the new numbers in the rationals line" via the following order: (a,b) >= (c,d) if and only if either a>=c and b>=d, or a>=c, b < d and (a-c)^2 > 2 (b-d)^2, or a < c, b >= d and (a-c)^2 < 2 (b-d)^2 (exercise: check that it is indeed an order!). Then (a,b) can be called a + b sqrt(2), we have (sqrt(2))^2 = 2 and 1.414
I'm interested in hearing the rest of this, but do you have a video somewhere where you explain exactly what you mean when you say there are only rational numbers? sqrt(2) is not rational. QED. ?
@greg55666 Sorry but your question is not well-defined. Before I can answer whether or not "sqrt(2)" is rational or not, I need to know what "sqrt(2)" is. Please do not tell me: a number starting 1.414... whose square is 2. Rather tell me exactly what this is. It is not ultimately a question about existence, but rather just about definition. As are many philosophical existence problems...
@@WildEggmathematicscourses What do you mean? I can prove to you that sqrt(2) is not rational. Surely you can too. If sqrt(2) is rational, then there exist two integers p, q such that sqrt(2) = p/q. Etc. What is not "well-defined" is your objection. sqrt(2) is the solution to the equation a^2 - 2 = 0. That solution is not rational. We give it a name, "sqrt(2)." It is perfectly well-defined, if you add it to the field of rationals, it creates a new extended field with the same natural, well-defined properties of any field, etc.
@@greg55666 Proof of existence by axiomatically stating "There exists..." is not yet sufficient, especially to be accepted to create a new extended field. Algebraic definition requires solid demonstration that your new expression really satisfies clearly defined criteria of a field.
@@santerisatama5409 Yes, and "sqrt(2)^2 - 2 = 0" is well-defined. You need to be more clear what your objection is.
Rational numbers are defined as equivalence classes of pairs of integers (a,b) with b not 0. There is no such pair of integers that will satisfy the equation a^2 - 2 = 0.
You're not explaining clearly what in the above four sentences is not "well-defined."
You and Prof. Wildberger are trying to put a burden of proof where none exists. I never made a claim that the square root of 2 "exists." What I said was that that object is not rational. I'm still not hearing if/how/why you disagree with that.
Once you acknowledge that sqrt(2) is not rational, then we can explore what we might want to do to fix this gap.
@@greg55666 I don't believe that "equivalence class" is a valid notion. I also don't believe in the AXIOM that real numbers satisfy field axioms, I consider that axiom absurd and as such unacceptable also for honest formalists who are committed to LNC or bivalent logic in general. I also don't believe that ZFC or any other similar set theory is consistent. Does anyone, really?
There's no debate over the fact that many lengths of hypotenuses etc. cannot be expressed as exact ratio of two integers.
I do think that it is possible to construct much better language for mathematical relations that are indefinite in relation to definite construction such as Wildberger's approach, language for indefinite relations that is communicable, intuitive, complementary to definite language and fruitful for further evolution of mathematics.
Real numbers and set theory are dead ends in that respect. Foundational deconstruction, intuiting, innovation and construction of something much better is of course very slow and delicate process. A nice challenge! :)
30:40
π²/6 = { a∈ℚ | ∃N∈ℕ: a < ∑(n=1,N) (1/n²)}
That's not an infinite amount of conditions.
I think his point is where your single condition is “there exists an N such that …” this is expanded into an infinite number of conditions where you are checking each natural number until you find one
Is it fair to say non-Q elements of the R set of Real Numbers is undefined, like 0 in division? When juggling non-Q numbers in the set R Real Numbers with Q numbers, one is juggling apples and oranges?
There isn’t a system of”real numbers”. So questions about them are meaningless.
35:03 is why when we used to do √3 +√6 =√9 teachers were always pissed at us. But √3/√6=√2 always pleased them.
There is another thorny question that comes to my mind.
Can we construct every irrational number using set builder notation and rational numbers, e.g. A = {a \in Q | a^2 < 2 or a
Max Percer asked: "Can we construct every irrational number using set builder notation and rational numbers..." No. There are only a countable number of set builder notation expressions, but irrational numbers are uncountable (so they won't all be expressible using set builder notation).
Even if Dedekind's construction of non-rational numbers is accepted, I don't see how it leads to the notion that these non-rational numbers are precisely equal to infinite decimals.
Thanks sir
From india
I have a question... if the real number line is not constructed properly, does it make sense to study real analysis?
If the analysis applies to the rationals, that's all you need. Irrationals exist only in the limit, and there is a rational number arbitrarily close to every one of them.
I think this boils down to the amount of time one has with a student. One has only a finite amount of time in school and even if the curriculum is changed there will still be challenges because of the limited exposure and interaction between teachers and students. But anything is possible, right (in a finite amount of time?)? There is also a wide variety of abilities and motivation of doing math. This may count for something in all this. I suppose the best math I have done has been away from the classroom and exams. Otherwise your videos have been very helpful.
Is there a higher level theory that can prove the impossibility of constructing reals with arithmetic?
The question doesn’t make sense, since the term “reals” is actually meaningless.
I am a little bit worried right now. I think in many Universities the lecture: Introduction to Real Analysis is meant to teach not only the foundations of, reals, integration differentiation, continuum, etc. But also how to think mathematically; like how to define things, prove concepts etc. If the main argument in which much of real analysis is based is kind of cracked. Would not this mean that we are learning not properly how to do mathematics and think mathematically right? At any point if I were to prove a Theorem, it means that my arguments could also have "subtle" flaws, like Dedekind's ?
The very, very sad reality is that most of pure mathematics, as currently expounded, is seriously logically flawed. Almost no-one is going to tell you this, but it is the truth. I have been a professional mathematician for more than 30 years, with dozens of learned papers etc etc and I have been thinking about the difficulties for quite a while now. It has taken me a long time to face the music myself. But at least at this channel I am trying to put together a view which aspires to a higher level of rigor.
I am an applied mathematician starting (graduate level) with the field of dynamical systems. So I found my self in the position of strengthening my knowledge on real analysis. For these I am using Rudin's book. Following what we have discussed, I guess I will just have to swallow these fact and learn the rest of the concepts... Isn't? Otherwise how should I expect to do any high quality research if I am no able to use formally and adequately the concepts from Real Analysis... especially proving the stability of a system, it seems now rather trivial... ha.
thanks
Best explaination
Problem #2 criticism: What is wrong with requiring an infinite number of operations? I would require an infinite number of operations to write out the natural numbers, but would you contend that they are somehow not non-finite in nature?
@Gennady Arshad Notowidigdo Thanks for making that distinction between infinity and completed infinity Gennady. I was wondering what AC and UHG stand for? I assume RT is rational trigonometry.
@@keithphw not sure about UGH, but AC I believe will refer to Algebraic Calculus.
@@MiroslawHorbal UHG refers to Universal Hyperbolic Geometry
In my simple understanding I would say: It is not surprising, that you can't do arithmetic with this definition, because, if you want to define irrational numbers with rational numbers and these cuts, you define them by expressing what they arent, but not what they are. Would you agree with this statement?
I'm not sure if I'm following. Could you explain, when constructing rational number tree structure, how is it acceptable to do infinitely many additions to construct those numbers, but it's not acceptable to do as many comparisons/choices to form a subset of Q?
@MikSu, We are not doing infinitely many additions. We cannot construct "all" of the rational number tree, only a given tree up to a certain level, as I have done here. However we can extend what we have done, to go further, if we have additional resources (time, space, energy etc). We never have to debate about whether or not someone can "do" an "infinite number of actions": until someone comes along and does something like this, it has no meaning. Like "jumping to Andromeda"--it's just pointless to have a scientific discussion in which this plays a central role.
@@njwildberger Oh, and I just realized that if you choose any rational number, you can construct a tree that contains that number in finite time. But you need infinite time to construct a tree that contains enough rational numbers that define a dedkind cut of an irrational number uniquely.
@@njwildberger And thank you for a quick reply! Since you answered I want to ask, if you have a video or something that explains exactly, what you think the rationals are. If you don't think they are an infinite set, what object are they? And how do you describe that object mathematically? If it's only that tree model, it really raises some questions. How many levels deep we go? Is there a largest rational, say a. Is there then no rational a+1? If the tree is only a finite constructions, how do we know we have constructed all the rationals we need?
@@miksurankaviita My answer would be along the lines of what most computers would understand: a rational number is an expression of a given type, perhaps a list [m,n] where m and n are integers, representing what we usually write as a/b, subject to a routine to determine equality ie [m,n]=[r,s] precisely when ms-nr=0, and then with specified rules of addition, multiplication etc. At no point is the computer going to be interested in determining "all" the rational numbers. This is clearly a senseless quest, as our ability to handle really large numbers diminishes with their size, and eventually ceases altogether. We don't need to know about "all the fish in the universe" to go fishing.
@@njwildberger Ok, so you follow the constructions that are usually made after getting naturals from PA. But then the question is, what object your natural numbers form. Do you have an equivalent of PA for your system? And if you say that the naturals are a finite collection, then what determines when to stop the sequence? What lets us say that there is some a that is a natural number, but a+1 is not?
How about this definition: x is a real number iff x is not rational, but can be expressed in an equation with rational numbers. So e.g. V2 (yes, the baby example^^) would be a real number, because it's the solution of the equation x² = 2 and because it can't be a rational number.
Then I would just postulate things like: If x < y and y < z then x < z and go from there to build all arithmetic with reals. So I would basically not construct reals, but postulate them, at least that seems more honest. How about that? Could that work?
Your point about using an equation written in rationals to define algebraic real numbers is exactly what a field extension is.
Ah, ok, but what about non-algebraic real numbers? To my knowledge they can also be written down as an equation like Pi = ... or e = ....
Not polynomial equations certainly. They are transcendental numbers. Idk but perhaps there are ingenious ways to write them down.
Right, I'd only define a real number by being not rational and by being expressed with an (polynomial or non-polynomial) equation. If these two assumptions are true then it would be a real number. That's it. Then I would just pretend that a, b and c are real numbers and I would move on to axiomize things like: a < b and b < c then a < c and addition and so on.
Maybe there is also another very interesting question: What would happen if tomorrow we all agree with NW and cut real numbers and only use rationals. What would happen practically and theoreteically if we wouldn't have Pi, e, V2 and all the others? Would things collapse, would things get overly complicated? Why - in detail - do we need reals so badly? Because if there is no good reason then occham's razor tells us what to do....
@gennady: I'd say that it's a real number if it makes any equation "x = a" true and if it is not a rational. E.g. "x = V2" is an equation and we know that x cannot be a rational number, so the real number r for V2 is the number that makes true: "r = V2". Well, of course "x = V-1" would be also a real number and therefore there were no complex numbers^^. Oh well....
Again, I am not a mathematican, not even close, so bear that in mind.
Does anybody make the argument that numbers don't exist because you can't go to a store and buy one?
I can go to the store to buy six apples or I could buy a plastic toy molded into the shape of a six, but I cannot go to the store and buy a six.
This seems identical to the argument that infinity doesn't exist because you can never count to it.
It's really a bit more complicated. I tried to make the distinction between algorithm and choice of the cuts and their equivalences:
1. An algorithm that selects all (finite) possible selections and terminates there is no choice analogy
2. An algorithm that selects countably many rational numbers and selects them in an order (for example Stern-Brocott tree order) A choice has to be decidable in a finite time of computation [For example the choice of x in sqrt(2) can be checked in finite amount of time: x < 0 or x^2 < 2].
3. An algorithm that selects countably many rational numbers and selects them in any order A choice (if it is in the set) has to be computable in finite time but not if it isn't. [also called recursively enumerable, the pi^2/6 choice]
4. Statements involving any choice over rational numbers. There are 2^Q choices, which is not countable! This leads to paradoxes of the axiom of choice.
Look up the infinite prisoners hat dilemma for a great illustration. The reason it leads to a problem/paradox is that we assume it is possible to create a random (martin-löb random) sequence of hats. I'm not saying it's an issue to create countably many prisoners but selecting a random hat for each one is where the problem lies. You can't write down the order of those hats in any way using an algorithm.
It is 4 that really is the issue. I can't possibly select an arbitrary cut (that's the equivalent of selecting a real number uniformly at random, it's not possible).
bvoq said "It is 4 that really is the issue. I can't possibly select an arbitrary cut (that's the
equivalent of selecting a real number uniformly at random, it's not possible)." Bringing in the
axiom of choice is a red herring. The axiom of choice is not needed to define the set of real
numbers as Dedekind cuts. Let Q be the rational numbers and P(Q) be the set of all subsets of
rational numbers (i.e., the power set of Q). Then the real numbers as Dedekind cuts is defined to be
the set { A in P(Q) | A != the empty set and A != Q and (if x in Q and y in Q and x < y and y in A,
then x in A) and (if x in A, then (Ey)( y in A and y > x)) }. It is not required to be able to
explicitly specify each element of each Dedekind cut. It is only required to specify when a subset
of Q is a Dedekind cut.
Really enjoying your videos. As someone who stopped studying mathematics in favor of computer science, this really resonates with me. During my studies, I found mathematics to be really "esoteric" in its use of real numbers, infinite sets and infinite sequences. I used to think I'm just too dumb to get it, but perhaps I was the only person in the room being completely honest with myself.
Your thoughts about viewing mathematics from a more computational perspective is a very pragmatic and sensible approach in my opinion. There is absolutely no practical use for supposed real numbers or infinite sequences when you cannot map them to anything in the real world. I think they are a convenient abstraction over very large numbers and operations performed on them. Contrary to you, I also think that rational numbers aren't really numbers. I believe both the rational numbers and the real numbers to be "intermediate forms of numbers", or more precisely, algorithms operating on natural numbers, which I believe to be the only numbers that actually exist in the real world. If we believe that our concrete universe cannot be made up of magical waves, but must be constructed with some sort of smallest building block, then really the count of all of these building blocks in our universe must be the biggest natural numbers to REALLY exist.
In that sense, I think rational numbers are helper constructs for when we cannot perform arithmetic with such fantastically large numbers, but want to "zoom out" a little. If we regard a lenght as 1 meter, then that is just an abstraction over an "unknown, but very large amount of smallest building blocks, whose number is impractically large to do arithmetic with". Since half a meter obviously exists and is a practical thing to consider, we have constructed the division operation (which means building ratios of natural numbers: the very large number of smallest building blocks contained in a meter divided by 2). I see the "division" as a binary operation (algorithm) that is performed on two natural numbers.
In that sense, I don't believe that rational division is an actual thing, but just a helpful abstraction. Integer division (with remainder) is the only applicable division in the real world. The square root of any number is just an iterative algorithm performed on natural numbers because we cannot count the length of a diagonal in terms of its smallest, discrete building blocks. Since we cannot possibly run such an algorithm infinitely long (limited time and space in our universe), we should think of the square root as an algorithm that does not terminate (WHILE-program). If we want a practical number out of such an algorithm, we'd need to have the algorithm terminate at some arbitrary point and yield an arbitrarily accurate result (FOR-program).
Doing arithmetic with real numbers should not be seen as proper arithmetic. It's more like combining different WHILE-programs for so long that, at some point, you can hopefully reduce the resulting algorithm to an actual (natural) number again. Similar to how sqrt(2) - which is an algorithm - times sqrt(2) - another algorithm - is 2 - an actual number - again. This also applies to rational numbers in my opinion.
I haven't really fleshed this out properly, but wanted to share my thoughts while I had the time to do so. Maybe you find some obvious flaws!
If you should ever read this, then please know that your work is important and that you are a very precious individual for inquiring deeper than others. You represent what is greatest in us humans: insatiable curiosity. Thank you.
This is presented in such a strange way. It is as if it is a refutation of the construction of the real numbers when it is not. What you are describing are essentially just the numbers which can be computed by an algorithm (or a Turing machine). This is a fine way to view numbers, as you will almost never see a number which is not computable. The real numbers however are different, there are uncomputable real numbers. There are only countably many turing machines, and thus countably many computable numbers, but there are uncountably many real numbers, so almost all real numbers are uncomputable.
The real numbers can be constructed, assuming you have the powerset axiom. The powerset axiom seems to be the axiom of ZF which you have a problem with. It says that given a set X, the class {Y subset of X} is a set. This is the set of all subsets, where a subset of X is a set Y such that x in Y => x in X. If we accept that the set of all subsets of a set X, called the powerset of X, does indeed exist, then Dedekind cuts do indeed yield a construction of the real numbers, i.e. a complete ordered field containing Q. If the powerset exists, then there is no problem taking arbitrary subsets, since you know they exist, and then you just pick out the subsets which satisfy the properties of a Dedekind cut (using the axiom schema of specification). This can be proven to exist from the axioms of ZF (you don't even need to use the axiom of choice). Without the powerset axiom, I don't think you can even construct uncountable sets.
This could have been presented under the light of: "what happens if we remove the powerset axiom". Instead, it is presented as: "the real numbers don't exist". If you do not accept the powerset axiom, and use ZF - powerset, then I don't think you can even construct the product of two sets, i.e. X x Y = {(x,y) | x in X, y in Y}. The standard way to construct this set is to say that (x,y) is really just {x,{y}} which can be constructed in ZF - powerset, and then X x Y is the subset of the powerset of (X union powerset(Y)) which are of the form {x,{y}} (this is not super precise, the exact definitions can be found in Jech, Set Theory). You would need to replace the powerset axiom with some other axiom along the lines of some "computable" powerset axiom, which allows you to construct the set of all computable subsets. If you take this axiom, then I am pretty sure that if you define the reals as all computable Dedekind cuts, it is probably still a complete ordered field, but only under these axioms. The point is that they are probably still complete, i.e. have the least upper bound property, since you cannot ask for the LUB of a non-computable subset (since those no longer exist).
This is interesting and all, but I ultimately don't think it is really worthwhile, unless you are a set theorist who is interested in this kind of stuff. Not using the powerset axiom just makes mathematics harder to use and doesn't make it any better of a tool. In real life, we will pretty much never see anything that can be represented by an uncomputable number, but that doesn't mean that we shouldn't include them in our theory, especially when it makes it much easier.
Even if Dedekind's construction of non-rational numbers is accepted, I don't see how it leads to the notion that these non-rational numbers are precisely equal to infinite decimals. Edit: Ah, Problem 7.
I also appreciate your efforts to highlight all these difficulties. I did by far not expect the swampy fundaments of a Dedekind cut based theory of the reals
When I looked at Stern-Brocot tree I got the idea that there are as much numbers between 0 an 1 as between 1 and Infinity
Good insight! That is indeed true! It's one of the properties of the Rationals, that for any two rationals, you can divide the 'space' between them in exactly the same way that you can divide the space between 0 and 1. The Stern-Brocot tree is indeed a really useful tool to help illustrate this intuitively. 👍
You may also be interested in the closely related topics of Farey Sequences and Ford Circles. Prof. Wildberger has at least a few different videos where he either explores these things directly, or uses them to supplement the discussion of a related math problem or topic. Personally, I have gained a lot of insight from his lectures on these (SB trees and Ford circles in particular), and by playing around with the ideas on my own as well. Cheers!
@@robharwood3538 I am following these lectures on a daily basis although I don't fully understand all what is being presented but what I am after is the plinks of ideas such as this one
In problem 4, the real number 1 + 1/4 + 1/9 + ... can be expressed as a Dedekind cut with its criterion being a finite condition.
Let f(n) be the partial sum up to the nth term. Let A = { a in Q | a < f(n) for some natural number n}. The condition "a < f(n) for some natural number n" is a finite condition.
It is easy to see that f(n) is in A for every natural number n. In particular 1 is in A. Thus, A is non-empty. A is not all of Q since it is easy to show that f(n) is bounded above by 3, which implies 4 is not in A. Thus, part 1 of definition of Dedekind cut is satisfied.
Part 2 of definition of Dedekind cut is clear.
Part 3 of definition of Dedekind cut is easy to show.
Some may object to my introduction of the function f(n) above. This would lead to a discussion of whether recursive definitions of functions is ok (but this is more a question of logic).
Actually, "The condition 'a < f(n) for some natural number n' is a finite condition." is not strictly a true statement. The problem is the phrase "for some natural number n". The set of natural numbers is infinite, so there isn't necessarily a way in finite time of deciding whether or not such a natural number n exists. This is why finitists have a problem with first order logic when applied to infinite domains: it isn't always possible in an algorithmic way to decide whether a statement is true or false. Quantifying over an infinite set, in particular, may lead to undecidable statements. Formalists, such as myself, don't have a problem with that. We simply accept that some statements might indeed be undecidable, but many such statements ARE decidable and we can do very useful things with them. (Such as develop the real number system.)
The condition "a < f(n) for some natural number n" is not really a finite condition, since this does not provide a systematic way to check in a bounded time if this condition is satisfied. To illustrate that, you can think about how you would check that 1.644935 does not satisfy the condition (Even checking for 3 requires the "think outside the box" in some sense).
@@josephmathmusic It is a finite condition since it consists of a finite string of letters. First, in ZFC, there is no "time". Second, in the set builder notation, there is no need to prove or check the statement "(En)(a < f(n))". The statement needs only to be a well-formed statement in order to invoke the axiom that justifies using set builder notation. Third, set builder notation need not be used at all: just use the axiom itself to justify that such a set exists.
If the square root of two is not a rational number and since there are no irrationals, which number is it?
There is no such thing as "square root of two".
Hey, thanks for all these great videos, i'm studying at Macquarie and they complement my material very well. I am not quite sure I agree or understand how you treat infinity as an object. For example you say that it is a problem that an infinite amount of work/operations is required. Could you elaborate why you believe this to be a problem? As i see it, infinity should be treated as a concept, and therefore should not be treated as,or likened to the physical world but rather an extension of it. Thanks again.
MrGatward Nice to hear from you. You use the word ìnfinity as if it already has a well-defined existence. I am skeptical. I like to see examples explicitly laid out, and as a mathematics student you should also be oriented to this direction. If someone wants to convince you that they understand what an infinite set is, let them show you an example.
njwildberger Thanks for the responce. Could this situation not be likened to the situation with a klein bottle? Of course i cannot see an example of a klein bottle but i can still understand what it is. What is it that's different between these two situations?
@@njwildberger Some people use the example of the set of natural numbers?
32:02 that was a direct point to teachers 😂
I shall cover the curriculum, whether or not the students the message got!
😂😂
gonna take real variables next term. Doing some review right now. Got stuck on the idea of an upper bound of a set of sets(in rudin analysis). Ok, now I know these sets could be mapped into real numbers and a set of sets can be considered as a set of real numbers. WOW, this surely helps a lot. Thanks for bringing the whole idea of a cut in the beginning
It seems polytopes and geometry makes a pretty strong case.
Are you familiar with the textbook "Real Mathematical Analysis" by Charles Chapman Pugh? There is a somewhat rigorous treatment of Dedekind cuts in that book, although I'm sure you would disagree with just about everything he says given your ideology.
Still waiting for the - one would think - inevitable discussion of logic and axiomatic set theory.
ANSIcode We will get to all of that.
Hehehe
-Kurt
@@njwildberger How the actual fuck 'A does NOT have a 'biggest' element"???????
Dear NJ,
You really had me when explaining Dedekind Cuts require an infinite amount of steps to form a set. We cannot simply claim to have completed a supertask.
But in the next frame of Problem 4, you really, really had me... laughing. "Dogma". Spot on.
Isn't it time for another debate - "Do the real numbers exist?".
This would be much more interesting than the usual "Does God (which god?) exist?" nonsense.
the problem I have with the Dedekind cut is since we know the rationals are countable, that means their subsets are also countable and that would entail that real numers pertaining to these subsets would have to be also countable, and we know they are not...
Indeed. Consider that if you accept the illusion that infinite decimal expansions represent any real number using a unique sequence, then the imaginary set of real numbers must be countable because now they all have names.
A set is countable if and only if all its members can be systematically named.
Euler played a huge role in this mess with his biggest blunder S = Lim S:
www.linkedin.com/pulse/eulers-worst-definition-lim-john-gabriel
That's incorrect. With the Dedekind cut, each real number is indeed defined by a particular countable subset of the rationals. But this means we select the reals from the power set of the rationals. Since the cardinality of the power set of any S of is necessarily greater than the cardinality of S, it follows that the cardinality of possible subsets of the rational is *not* countable. Nothing said so far precludes the set of reals - being defined by such subsets - from being uncountable under the Dedekind definition.
confuses does it exist with can you do arithmatic
Chatlin showed (if you believe him) that most mathematical statements are true for incidental reasons. I wonder if this idea is connected to difficulties with dedekind cuts.
Thanks again for a great lecture! I'm really quite interested in the topics you've presented in the last couple of videos and to be honest I've also actually tried to think what the number line is. (It may be that I've got some inspiration from you.) I can't wait for the next video!
How I think about numbers and what they are is not really clear. The number line as far as I'm concerned doesn't exist. We like to put it that way, as a nice straight line with dots but what can it really be? Can it even be anything or everything just the imagination of man? I've heard from another professor that he thinks about numbers actually being bunched together like in a jelly. That's just one way of thinking about how to order numbers. Numbers on the numberline are said to be zero dimensional points. Why are they zero dimensional points? Another invention by mankind. What does "zero dimensional" even mean? It has no place in our world. And if you think about numbers like that, like some concrete things, that's wrong too. Or at least I think so. I could say like... Thoughts don't take space either. Are thoughts zero dimensional?
The numberline is an invention of man and if you think about all the points on the line being zero dimensional points wouldn't that be the same to say that they take no space - in essential they are not there. All the infinitude of numbers would then sink to a singularity - a point that didn't even exist in the first place. Also! I would like to point that maybe there really is some kind of magic behind numbers. Like you, Norman, say that we don't live in a fairy tale world... Maybe we do? Just like dark matter. We don't know what it is, cannot see it or so, but yet it affects our Universe. (I'm not saying "dark matter is magic" ooohhhh, rather just threw some kinda metaphor in the air. I like to speak presenting metaphors...)
Also thumbs up and respects to anyone who read through this block of text. I salute and bow to you. You gave your time to listen my thoughts.
I appreciate your perspective and your efforts to discredit infinite sets and therefore the real numbers.
With that said what I believe is a knock down argument comes from homotopy type theory. Although HoTT may not be the next best thing like many of its proponents suggest it is still a foundation for mathematics.Within HoTT we have that no sense of the law of excluded middle, the axiom of choice is not a axiom - it's a theorem and we can construct the real numbers within HoTT. To discredit the existence of the real numbers you seem to need to discredit what ever allows HoTT to construct the real numbers.
I would bet HoTT does not construct the reals
I enjoy your lectures a great deal! Today, I literally lost my lunch to watching one of them (no regrets though!) :) What is sometimes missing is the links to the previous and the next one in the series of lectures. Or a least a link to the playlist where they are collected.
+malilsisbro I suggest you click on the Playlist you are interested in. There you can see all the videos in their (hopefully) correct order, and view them in sequence if you like.
+njwildberger Thanks. I've sort of figured it out already. I'm taking a linear algebra class now (offline), and I'm also following your lectures on linear algebra. Though the list is in reverse order! :)
I really want someone to construct the reals in a convincing way like the rationals can be constructed. Until then I will continue to think that we can only get our favorite irrationals through various representations of those numbers (outside of set theory or using some set algorithm). I will continue to think that they do not come magically through "the set of all reals".
+Euquila Or you could just jettison all unclear thinking in this direction by forgetting about "irrational numbers" the same way most scientists eventually stopped thinking about "ghosts". Whenever an old fashioned "irrational number" appears, just replace it with a convenient rational/ floating point approximation.
Yeah it's very odd that "set theory" and "dedekind cuts" are considered to be the the fundamental building blocks for analysis, but as soon as we are donw with them we can completely forget everything we did and it doesn't affect how we do analysis. But what should be done? Seems like throwing out analysis is like throwing the baby out with the bathwater. So do we try to have new foundations for analysis based on arithmetic? How exactly would that be different? Is it possible that we would just talk about reals like pi in a different way, say as an algorithm that can give you as many decimal places as you want? Is there a complaint here other than philosophical/ontological or do you think that entire big branches of math are actually useless and unapplicable?
Dedekind cuts are defined in the context of set theory. It is disingenuous to claim it "doesn't work" when what you're really saying is that you don't like set theory as a foundation.
Mihai Garba said "Also if the "cut" is on Q, how can this definition transcends Q and becomes R?"
The definition of the real numbers is taking place in the framework of sets with the axioms of ZFC taken to be true. For the axioms of ZFC, see the link:
en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
Thus, a cut is just some particular set in ZFC. The rational numbers in Q are also sets. Everything in ZFC is a set. The set of cuts is defined to be the set of real numbers. A cut is a real number. There is no "becoming" since those sets exist before we concentrate our attention on them and declare them to be real numbers.
Assuming that the set Q of rational numbers has already been defined in ZFC, then we do the following to define the real numbers. First, consider the power set of Q, which exists by the axiom of power set in ZFC. Let P denote the power set of Q. As an examples of an element of P, we have {-3, 1/3, 10/7} is in P. Note that all cuts are elements of P.
The next step is to take a particular subset of P and that subset will be R. The first restriction is that we don't want Q to be a cut. Thus, Q is not in R, even thoughtQ is in P.
Another restriction is that we don't want the empty set { } to be a cut. Thus, { } is not in R, even though { } is in P.
The final two conditions on being a cut further restricts what can and cannot be in R. This is allowed by the axiom schema of specification of ZFC.
The above only gives what sets (the "cuts") are to be considered as real numbers. There still remains the task to define addition and subtraction and prove that the ordinary rules hold (like commutativity and associativity).
So the power sets exit by the power set axiom?
On rare occasions (happened a couple of times) some kid of my friends would ask me advice on mathematics career. I tell them to go to the applied mathematics and actually do something useful and meaningful. What artificial load of BS mathematics has become.
I have known for some time now that power sets exist by the power set axiom. If the axiom was missing I would wonder if there was such a thing as the power set of Q. Good job we have axioms to tell us what exists and what does not.Now that I have understood the full importance of ZFC I might get to the next step and try to figure out what the hell is PI to the power of e if both have infinite number of decimals. But I guess there is somewhere or other artificial meaningless answer to that question.
Aleksandar Ignjatovic said "Good job we have axioms to tell us what exists and what does not."
I assume that your statement is being sarcastic. But, I don't understand why you think this.
Are you against Euclid's axioms like:
1. "To draw a straight line from any point to any point."
4. "That all right angles are equal to one another."
Also, are you against Euclid's axioms like:
1. Things that are equal to the same thing are also equal to one another
2. If equals are added to equals, then the wholes are equal
Mathematics is a deductive system and it needs axioms to have something to work from.
*****
It's about how to read definitions used to state axioms.
There are generally two ways to interpret definitions like "A line is a length without breadth".
First, you can spot that a book is named "Elements of Geometry"(Constructions of Measures of Earth) and conclude that it's the book of science and engineering.
Therefore you have to read "A line is a length without breadth" with Thales' razor in mind like "As long as breadths of my lines are less than a length of my shortest line segments about 1000 times my constructions are accurate up to about 0.1%"
Second, you can read the book like Holy Scripture and take every word literally.
Therefore you have to read this definition like Pythagorean style existential statement "Breadthless lengths exist!". And what if such objects can't be found anywhere in real world ? It's too bad for real world, screw real world, let's declare that real world is flawed. Also let's ban beans, because I hate beans.
Proper deductive systems are grounded by constructive axioms/rules of inference and have known limits.
Arbitrary deductive systems are meaningless.
*****
I am not against axioms in general I am against axioms that proclaim the existance of something. Axioms should be used to define properties of already existing objects. I am against Frenkel axiom that states "Set of natural numbers is an actual infinite object" (this way he proclaims existance of actual infinity by an axiom) or Bourbaki axiom that similarly states "There is one actual infinite set". The way I see things in maths is that the theory of actual infinities is arbitrary and far from logically sound (otherwise I would not have anything against actual infinities per se) so that mathematicians should stick to potential infinities only.
As for sarcasm, here is why I am sarcastic (nothing to do with you personaly). Mathematical objects have been formed as models of reality. Like Euclidean space and all geometrical objects, vectors, tensors, matrices etc. Some objects like groups and fields etc have been formed inside maths for various concrete reasons, like to study the structural similarities of maths theories etc. Also infinities are only part of mathematical model and not of real world around us, So infinities should not be taken too siriously. Even physicists use models to study physical world, like Hawking had said - We do not know any reality outside a certain mathematical model. But some mathematicians have become so arrogant that to them mathematical model (including infinities) have become the real world and the real world is just a poor finite approximation of the mathematical world (which turns maths into some kind of religion). Infinities are real only inside a mathematical model but seems like many are forgetting that.
I mean the reflections are there.
just curious. have you tried defining real numbers yourself?
He believes (and personally, I have come to agree with him) that the so-called 'Real' numbers are not a functional/working concept. They are actually a fictional concept that no one can actually produce a meaningful example of in reality. The closest thing we can get in *practical* terms is simply rational *approximations* to something like sqrt(2) or e or pi. So, instead of pretending the 'Reals' are actually *real,* he just sticks with the *actually* real Rational numbers themselves.
But instead of settling only for approximations, since he is focused on 'Pure Mathematics', he has been working on developing Pure Mathematics using the Rational numbers themselves, rather than so-called 'Real' numbers. Examples are that he is currently well into developing his Algebraic Calculus, which does not use 'Real' numbers, but only Rational numbers. Also, he has previously developed his Rational Trigonometry for doing all of Trigonometry using only Rational numbers.
(He _has_ also explored things like using special matrices to develop a concrete, algebraic way of using Infinitesimals, as well. And it even works over *finite* fields, which is really quite amazing, IMHO.)
@@robharwood3538 are you saying that he is essentially developing a better way to describe Real numbers, by only referring to Rational numbers?
@@xybersurfer He is saying he is stupid
Problem with Criticism #5: You construct the sum by taking the set of all sums from elements in each black box.
I'm not a mathematician, so this might be stupid, but isn't finding a real number using Dedeking cut equivalent as finding the edge of a fractal? Also if the "cut" is on Q, how can this definition transcends Q and becomes R?
When playing the game of logical refutation of Real Numbers, shouldn't one compute also the possibility that the other player counters with his (perhaps) strongest move (instead of noob baby move), namely Conway's theory of Surreal numbers, which claims that "we obtain a theory at once simpler and more extensive than Dedekind's theory of the real numbers just by defining numbers as the strengths of positions in certain games." I hope that fellow fan of fantastic game of Go gives his best move to counter also the number theory (including a definition of reals) that was developed as mathematical analysis of Go end game. The game could be said to have the characteristic about winning more area of logically oriented mental space than the opponent. Or should this game be left for the Google AI to play? ;)
Great stuff. My students will sit an exam (not set by me) in which the real numbers will, no doubt, be mentioned several times. Irrational numbers and trigonometric ratios will also feature prominently. I feel obliged to teach these concepts even though I know they are dubious at best. If I was to teach them what I have learned from your videos, it would, of course, conflict with their previous "understanding". It makes me feel kind of guilty for "playing along" but do I really have any other choice given my position?
If you read Rudin's book more carefully, you will find his field axioms useless and ridiculous. He never ever actually defined what "addition" means. The axiom of addition did not "define" addition. It is basically assuming that everything in Q can be added by default, and yet no true meaning can be assigned to this animal called addition. Then Rudin went ahead to define some of the properties of the undefined notion of addition. And he called these properties of the undefined notion of addition "axiom of addition".
Then he said :"The field axioms clearly hold in Q, the set of all rational numbers, if
addition and multiplication have their CUSTOMARY meaning. Thus Q is a
field. " This is like saying, you define addition by yourself using your own customary meaning of addition. Then whatever meaning you see fit to define addition, apply my axiom of addition to that thing you call addition, then everything should be OK. LOL
Can you add a pig to an elephant? Why can you add 1 to 1 and 1 to 2 etc? Can you add a complex number to a real number? Can you add 1 to the biggest number in the world? What is the real meaning and limitations of addition? If addition is undefined, we can add everything, or can we?
Also, you might be interested to know that I may have gotten a math major to be skeptical about so called real number arithmetic.
As I was waiting for class to begin, a student of mine (a math major named Norm) saw me reading my copy of Divine Proportions. We got to talking and since he's a senior I felt it appropriate to ask him: "How do you define real numbers?" He answered quickly and confidently ... "As Dedekind cuts". I asked him if he had ever proved the laws of arithmetic for real nos defined that way. He said "No, but of course it's been done. Laws like the distributive and associative properties are done essentially the same way with cuts as they are with rationals". My reaction was !#&?!%*! Keep in mind this kid has been through two semesters of real analysis already (getting A's in both). After mentioning a few quotes and examples from your YT lectures, he conceded "I am suddenly very curious to read that book you got there?"
Well, I've done my proselytizing for the semester.
Well done. We need to try to engage more people, especially young people, to consider a broader possible mathematical canvas.
Read Landau!
Well, just because Rudin's definition's of a set are lacking doesn't mean they are senseless in axiomatic set theory, which is only defined in means of first-order-logic combined with a lot of syntactic sugar. I have a book where one has to proof as an exercise that
c in {a} implies c = a for example. Or that is actually a set. Or that the class of all x for which x=x is not a set. So pretty rigorous.
So, when we only use first order logic, one already has an algorithmic approach, right? The rules of arithmetic would then follow by the rules for logic, which are very similar.
I just started, so I don't know the answer yet. Hope u respond :-)
Any cut for a rational number r, called r*={p in Q| p
We don't need the "theory" of Dedekind cuts to construct the rational numbers ... from the rational numbers.
@@WildEggmathematicscourses I was replying specifically to the erroneous comment in the video that there is "only one example" of a cut in Rudin, nothing more. Also, considering that any ordered field contains an isomorphic copy of the rationals, yes, your construction of some other ordered field would have to have the rational numbers being constructed as well.
@@WildEggmathematicscourses Sorry, maybe I misunderstood this comment... I guess I kinda get the point he's making in the vid, I don't know enough about philosophy of math stuff really to tell what sort of objects "should" exist in "mathematical reality". If you subscribe to the idea that, informally stated by an amateur (apologies), infinite amounts of work aren't mathematically honest in some way, why would that be? I guess that's really the idea I've been trying to wrap my head around after watching these lectures.
Number theory is a vast and fascinating field of mathematics, sometimes called "higher arithmetic," consisting of the study of the properties of whole numbers.
The theory is applied to computational and theoretical physics. The great difficulty in proving relatively simple results in number theory prompted no less an authority than Gauss to remark that "it is just this which gives the higher arithmetic that magical charm which has made it the theoretical science of the greatest mathematicians, not to mention its inexhaustible wealth, wherein it so greatly surpasses other parts of mathematics." Gauss, often known as the "prince of mathematics," called mathematics the "queen of the sciences" and considered number theory the "queen of mathematics" (Beiler 1966, Goldman 1997).
arxiv.org/abs/math/0010298
I hope you know what term "queen" means in a mouth of German of first half of XIX century.
I don't know, but I hope it is allegorical like I intended in this comment
Women were forbidden to rule and teach in Germany.
Queen never was a ruler.
Thanks! Yes I knew women were excluded and still are for many reasons denied equal opportunities in many academic fields yet Emmy Noether still made incredible contributions to 19th century male dominated german mathematics. Many today still think it takes an individual with great powers to overcome obstacles but it's more likely that the cumulative effect of many discrete contributions can accomplish the most difficult challenges.
I would recommend anyone watching this hoping to gain insight into analysis to watch Francis Su's videos for a more careful treatment of the development of analysis. Indeed, Francis Su covers the arithmetization of the Dedekind cuts. Also, most books on analysis assume naive set theory rather than drown the reader in all the set details. Most sets you imagine coming up from the real numbers will not let you down! This is the "intuitive" power of naive set theory. If you have more of an interest, read an introduction to set theory.
This takes care of a couple of Wildberger's criticisms. However, there is a better argument that he is implicitly making, and you can decide whether to accept it or not:
Dr. Wildberger is intentionally omitting a typical distinction in mathematics: that of a set definition being delineable versus a definition being constructive. So long as you accept that a set can be defined in such a manner as to know what kinds of things are in it (without the formality of a construction to prove this), then analysis sits on a firm foundation.
@Christopher Campbell: Perhaps the test of a mathematical theory is not whether or not its proponents claim that it sits on firm foundations. Perhaps it has more to do with making representative computations, exhibiting concrete examples of claims, laying out definitions and theorems for all to see, and actually answering questions that critics ask?
Here are three direct challenges for you and any others who would like to maintain the status quo re "real numbers":
1. What is pi+e+sqrt(2)? (i.e. can you actually do arithmetic with "real numbers"?)
2. How to factor x^6-x-3 into linear and quadratic factors? (i.e. can you actually do algebra with "real numbers"?)
3. What are the angles in the triangle with vertices [0,0], [3,1] and [5,2]? (i.e. can you actually do geometry with "real numbers"?)
Sure, OK so long as you wish to test mathematical theory in such a way, I cannot give you a valid answer to 1-3. However, I do not accept that making representative computations is a necessary condition for a rigorous theory to have.
Also, from what I have studied in analysis, the theory has certainly made concrete examples of its claims (as long as we're not talking about requiring finitude for a concrete example) and its claims have been scrutinized and improved upon for hundreds of years, so I believe the field has been open to criticism.
1. Any real number is a cut. Cuts are closed under addition and multiplication and follow all the regular arithmetic rules, so long as you aren't required to construct the things with some algorithm (which I do not believe should be a requirement). Furthermore, pi and e have been given a treatment from these first principles. So we define cuts first, then develop the "infinite" series which characterize the transcendentals (each uniquely!). Since all real numbers are cuts, and we know what e and pi are, it makes sense to add pi + e + sqrt(2).
2. I think this follows from 1 but I'm not sure what you're getting at? I would think the number of factors follows from complex analysis and the arithmetization of cuts.
3. From my perspective, geometry is a special case of the development of the S and C functions on the reals, where C_1(x):=1, S_1(x):=x, S_n(x) := Integral_0^x C_n(t) dt, and C_n+1(x) := 1 - Integral_0^x S_n(t) dt.
The appropriate geometric notions can be derived from taking the limit of S_n(x) above.
Mihai Garba "Also if the "cut" is on Q, how can this definition transcends Q and becomes R?"
It is true that some cuts are on Q, but not all cuts. The square root of two lies in a hole of the rationals. You have an old proof that no rational number is exactly equal to the square root of two. So, to include it in the numbers we have to construct something like the real numbers. Then it can be used in algebra together with the other numbers (Naturals, Rationals etc), that are also transformed to being Dedekind cuts. There are many different ways to create R, but they all have their pluses and minuses. Dedekind cuts are not symmetrical för example, a property found in other methods.
If instead 'a' was a non-computable (and of course in standard mathematics most numbers are non-computable) I would find the idea of a Dedekind cut particularly unsettling. The very existence of such a cut seems to contradict the idea that it's non-computable by allowing you to produce the term by term decimal expansion. It seems like an algorithm to me, but presumably we are supposed to consider it a meta-mathematical/meta-physical algorithm.
Is random function allowed in compter programming? If so Axiom of Choice is not needed but random function instead.
The problem is that with algorithm method you are sort of uncovering decimals of a concrete number but with random/choice method you do not know where you are headed.
Aleksandar Ignjatovic A program which generated true random numbers would not be an algorithm. So call 'random' numbers generated by programs are pseudo random. You must assume the axiom of choice (or its equivalent Zorn's lemma) to get Reals. I think I'm right is saying that Norman (along with a significant minority of mathematicians) rejects the axiom of choice and as result doesn't believe in Reals. It would be interesting to see how far you could go building a number system without invoking the axiom of choice - could you build a logical and coherent system from the computable numbers for example?
Zantorc
Thanks for the answer.
To say "one believes in Real numbers" or does not, is not the right way to put it. Maths are not a thing to believe or not. It is rather a qustion of can what a math theory claims be checked in practise. 2+2=4 not because it can be proved in mathematics but because when you use it in calculations you always get a right result. So differential and integral calculus work in practise too. And they do even without the artificial set of numbers (real numbers) with which it is impossible even to define basic operations. To the minority of us it pretty clear that matematicians have failed to put analysis on firm logical grounds. It does not matter if we are minority. I have come to believe (maybe wrongly) that within the present mathematical model (euclidean geometry as the model of the real three dimensional world) it is impossible to create a logical theory of analysis. This is where the model fails, so to speak. In mathematics, a theory should be of inpeccable logical structure or it fails, it is as simple as that.
Mattematicians have successfully forgotten that the real strength in maths is that it is a set of methods for practical use. Without that mathematics would be worthless. Actual infinities, my rear end. And also Cantors numerology of the infinite.
Aleksandar Ignjatovic "Mathematicians have successfully forgotten that the real strength in maths is that it is a set of methods for practical use. Without that mathematics would be worthless."
One of the main themes of the book 'A Mathematician's Apology' by G.H.Hardy is the beauty that mathematics possesses, which Hardy compares to painting and poetry. For Hardy, the most beautiful mathematics was that which had no practical applications in the outside world (pure mathematics) and, in particular, his own special field of number theory. Hardy contends that if useful knowledge is defined as knowledge which is likely to contribute to the material comfort of mankind in the near future (if not right now), so that mere intellectual satisfaction is irrelevant, then the great bulk of higher mathematics is useless. He justifies the pursuit of pure mathematics with the argument that its very "uselessness" on the whole meant that it could not be misused to cause harm. On the other hand, Hardy denigrates much of the applied mathematics as either being "trivial", "ugly", or "dull", and contrasts it with "real mathematics", which is how he ranks the higher, pure mathematics.
I agree with him, even though his particular examples turned out to be wrong in the long term, given the application of number theory to cryptography, he was right at the time he wrote it in 1940. It follows that we differ on this point.
Zantorc
Ok, we do differ on that point. But then. how do you know what is correct and what is not (or true and false if you want) and what is corrct logic of working with stuff that cannot be checked in reality. I am not disputing the beauty of mathematics or even the interest of pursuit of useless knowledge, but if you lose touch with reality, mathematics becomes more and more arbitrary and even messy - at some point it became all right to use circular definitions, misuse the word infinity and the word limit, lose application of logic and what not. Even to accept what cannot be defined ("a to the power of b" or "a times b" in the infinite number of decimals world) etc. There is oriental phrase "building neither on Earth nor in the Sky". Thats exactly where maths is today.
I do not claim I am right. I am posting a lot on these topics as people arround me are completely disintersted in all this so this is a good chance that I discuss this all with the people that are interested.
We can model the same ideas in a much simpler setting. If we take take 3 apples (in order) and make a cut in the middle we get the apple nr.2 if we take 4 apples and cut the set in the middle we get no apple, instead we have two sets {1,2} and {3,4}, but we cannot point to the middle-most apple.
What is x, if 1
"What is x, if 1
I think the theory of extension fields should be introduced to the 10th grade in replacement of the real number system chapter. This makes math more like what it is. Prof, we need you to change the curriculum!
We will be talking about that important topic shortly in this series! As for changing the curriculum--I'm trying, but it might take a bit of time :)
njwildberger Do you want to change the curriculum? READ THIS and you will be closer to make it happen.
You always affirm with total conviction to be sure about your claims. For example, in your description of this video, you say:
"Does this actually work? Can we really create an arithmetic of real numbers this way? No and no. It does not really work."
Prof. Wildberger, I ensure you that if you write a paper in which you rigorously show that every single claim you make in your videos (instead of just making the videos), you will reach your goal of changing something. If you don't do this, you will always feel sure about your intuitions and claims, but you will not convinced the majority of mathematicians with your typical assertions like when you said at 27:05 ...
"I cannot remember ever reading a book where there was a significant second example haha, ok? there is always only this one example. Please open your local analysis text and check if this actually holds or not...A theory with one example...kind of dubious isn't it?"
...as if that were a rigorous way to convince a professional mathematician. And let me tell you...not even with the elaborated examples that you present to point at difficulties will serve to change things for good. You should do what you want people to do...Do things rigorously.
Keep in mind that the opposite side, no matter how wrong they are, don't actually have to do anything to keep things the same way. Unfortunately, the burden is on you. Write a paper and submit it to be peer reviewed.
So, from the definition at 7:30, or page 3, all the elements of R are elements of Q? Doesn't that mean that R is a Subset of Q? I thought R is supposed to be bigger than Q.
I still can't understand why there needs to be a distinction between choice and algorithm. If I was attempting to make some kind of infinite object (set, sequence, or whatever it may be) I wouldn't see any problem with allowing its elements to be arbitrary or possibly random. If I was attempting to do some calculation however, I would just restrict myself to objects whose elements could be determined by algorithm, because (most likely) those would be the only kinds of objects i would be interested in.
Objects whose elements are random or determined by "choice" might be problematic for calculation, but in my view, surely their mere existence shouldn't be so troublesome as to invalidate a theory.
quotevg The question is one of definition, not existence. What do we mean by an infinite process, set or function?? Are we talking about something produced by an infinite number of independent choices, necessarily occupying an infinite amount of time and required an infinite amount of energy (i.e. completely impossible in the known universe) or are we talking about a finite algorithm that produces such a process, set or function?
There is a huge difference. And if we are not going to specify what camp we are in, then we are not being precise.
I encourage you to also take a look at a book titled "Analysis 1" by Terence Tao. He gives an alternative construction of the real numbers without using Dedekind Cuts. He also goes very in depth into set theory, and into the paradoxes of set theory in the main text. It is a truly unique book. I think it would address your concerns as you've laid them out in this video.
As it so happens, he has made all his course materials available free online. Lecture notes similar to his book, but not exactly the same as his book (but pretty close) are free to download and read here (the set theory seems to be missing from these notes though): www.math.ucla.edu/~tao/resource/general/131ah.1.03w/
The treatment in Analysis 1 by Terence Tao is guilty of the usual errors. It does not suffice to create a viable theory of "real numbers". Unfortunately there are essentially no modern analysis texts that deal adequately with this issue. And why is that? Because the "theory" of "real numbers" is a mirage.
Well, no more than the mirage of sqrt(-1), yet it helps us solve any algebraic equation and gives us fractal geometry that describes nature. And I'd say that perfect unit squares are also a mirage since you can't make a perfect geometry in the real world, yet we use pythagorean's theorem to build houses. In this way, we use derivatives to build car engines and integrals to build swimming pools. Even if it's not accurate to real life, it is useful. Now the question of whether it exists in the "Platonic math universe" doesn't really matter to me.
Especially if your main issue is with set theory and the idea of a subset of Q.
For instance, if infinity is unreasonable because it's larger than the known physical universe, then aren't triangles also equally unreasonable since there isn't one triangle in the known universe? I'm not seeing how Greek geometry get's a pass but real numbers do not. I also don't understand how sqrt(2) is a problem but sqrt(-1) is not. And in all seriousness, philosophically speaking, there can be strong cases to be made that our own sense-perception isn't even accurate to begin with, so who knows if the physical universe is even a legitimate comparison, and by the same token, who knows if our own thoughts aren't complete nonsense without empirical evidence to back them up. Math walks the metaphysical tightrope between rationalism and empiricism and I think it matters less how one defines the pure math concepts, and more what they do.
Because in the end nothing is surely true metaphysically, except for maybe hard skepticism.
Its a bit like arguing that the old Norse Gods really do exist because they explain so well the thunder and lightning. In fact all those houses, car engines and swimming pools you discuss were built with floating point arithmetic. The engineers were able to use that arithmetic effectively precisely because they were able to circumvent the "infinite arithmetic" that the analysts dream about.
A set isn't well defined, true, but why is that a problem? Geometry is built on something called a point that isn't well defined. Euclid says: Definition 1 . A point is that which has no part (reminds me of a set is a collection of objects). If each point contributes zero to a one-meter length segment, then wouldn't the entire segment have length zero? You get this strange questions when things aren't well defined. Same with Set Theory. That is why they are axioms thought because you start from them. Why is that a problem?
@Luis Munoz It is a problem because in mathematics we want everything to be clearly defined. If we start right off the bat with the attitude that unclear definitions are somehow OK, then the entire subject is compromised. The point is that if one approach requires or at least involves unclear definitions, then we have to find a better way of dealing with that subject. Please have a look at Rational Trigonometry, and of course with more of the videos in this playlist. You will hopefully get a better appreciation for the crucial importance of clear and precise definitions.
@Wild Egg mathematics courses Thank you for reply. I am taking for granted that all definitions bottom out and are in fact circular. Euclid calls these primitive terms - terms which at the outset of a deductive system are introduced and explained but never precisely defined. I imagine you accept that axioms are a necessity to bottom out the system, you just want the axioms to have something about them, perhaps that they are simple to understand (type A in the lectures). I don't see the fundamental reason to prefer that abstract criteria over picking an axiom that is convenient i.e, produces results.
The thing is, depending on which axioms you use (probably ZF), sets are well defined, the author of this video just refused to give a definition of a set.
And all these supposedly real numbers that cannot be generated algorithmically. That is so far out. That is extrapolation gone wild :-)
I suppose the question is whether or not we believe that anything can be truly arbitrary. Doesn't the axiom of choice imply that an algorithm always inherently exists in choice, even if we can't describe it?
So basically three German guys wanted to fill up the gaps in rational number line. So they walked along the number line and every time they found a gap they proclaimed - By the sacred power of the Holy institution of the Allmighty Axiomatic Definitions we proclaim there is an irrational number here.
Everything else is just a rationalization.
Correct!
To clarify, your (sole?) objection with the construction of R via Dedekind cuts is due to our inability to denumerate every subset of Q?
you troublemaker ;)
How much do you expect a book on mathematical analysis to devote to philosophy and set theory?
A debate about the actual infinity is, in a sense, a debate about the existence of God, the analogy being literal when Cantor first presented his ideas.
I find most, if not all, of your objections to the actua infinity ludicous, but that is just me. What is wrong with Cantor's point iof view that any object that falls within the scope of a consistent axiom system is real in the mathematical sense whether or not it has any connection to the physical world or is explicitly constructible?
e+pi? Easy. Add decimal approximations to get a decimal approximation. Since 0 is the lower limit to the accuracy of the approximation, the sum is a well defined real: i.e. define addition on R to be the continuous extension of its definition on the dense set of rationals. What is the objection? I accept the existence of a musical note between middle C and its neighboring C#, whether or not I ever actually hear it.
I find nothing more troublesome about the axiom of infinity than I do about the empty set axiom. One could just as easily regress centuries and come up with philosophical objections to the existence of the empty set and thus deny the existence of the number 0. One could argue that there is no such thing as a perfectly straight line and thus reject geometry. Nothing exists.
35:00 Rudin’s book proves that the axioms of a field hold for Dedekind Cuts. Saying that no book discusses it is a complete straw man. It’s like you didn’t even read the book you were talking about.
Sorry, but have you read this proof? Do you accept that it is valid?
@@njwildberger I’ve been working on the exercises in the book, and I believe the proof is valid. He’s able to show that the Dedekind cuts behave like a complete, ordered field, so I don’t think it’s much of a problem to base our calculations on them. It might be hard to conceptualize something like their sum, but I don’t think it’s much different from how we think of vector spaces being made of an infinite collection of linear combinations.
Assuming you buy ZFC, you can construct R as Dedekind cuts, the completion of Q under the usual metric, and probably other ways I don't know about... If you don't think that ZFC is a consistent set of axioms, and in particular you don't believe that there is a model for it, then much of modern math doesn't hold... and if that is the case you could make a million other videos criticizing almost everything in modern math by just going back to the set theory behind it, like you do here. What does it add to the discussion than just saying that you don't think ZFC describes something real or that it is consistent?
Yeah, he should just scream into the void that he hates ZFC and be done with it. Goedel's theorem basically says that whatever system of logic this flat earther likes is not going to be any better.
We _could_ spend all of our efforts cutting off the various 'twigs' and 'branches' of the old 'tree', and by the time we die we will not be hardly anywhere closer to making any significant progress.
Or, we could, as the title of this series of lectures ("Math Foundations") suggests, cut directly at the 'roots' of the entire 'tree'. It may take some extra time and effort -- at least initially, compared with cutting off a single 'twig' or two (or even some large 'branches') -- but once we cut the root all the way through, we have essentially cut _all_ the 'branches' off in one fell swoop.
Of course, it's not *all* just about 'cutting down'. Wildberger has spent considerably more energy into 'planting' and 'raising up' alternative 'trees' to the standard 'old tree'.
"What does it add to the discussion than just saying that you don't think ZFC describes something real or that it is consistent?"
To the discussion, it adds a hefty 'chop' at the 'root' of the 'tree'. ZFC is considered by most to be 'the foundation' of modern mathematics. But it's not a good foundation. To establish (a) new foundation(s), it's beneficial to explain to folks *why* new foundations are needed in the first place. One good way to do this is to show the logical difficulties and problems with the old foundations.
Why not clip off all the little twigs first? Because it's a far more efficient use of our time to just chop right through the roots, and cut the whole tree down at once.
@@robharwood3538 I dunno. Pissy comments by people who clearly haven't a good logical foundation for their irritation are not worth serious response. They are defending an ignorant point of view in the name of not having to question their prior reasoning. Okay, I say, keep your ignorance right there on the floor with the pisspot. Makes little difference either way.
300 comments!
Without the abstractions of pi and e (just to make an example), it would be very difficult or even impossible to establish the amazing Eulers's identity e^(i times pi)+1=0. And this is true for an endless amount of interesting and useful statements and theorems in the majority of branches of mathematics. The point of view of Dr. Wildberger is perfectly understandable but wanting to re-found analysis on only the rationals and obtain all its results, theorems, etc. making use only of finite approximate methods sounds like a chimera.
Dr. Wildberger doesn't believe in SQRT(2) for example, so what does this mean? that he doesn't believe in the existence of the hypotenuse of a right triangle with both catheti equal 1?. Here we would have a perfectly defined object in geometry but that wouldn't exist in Wildberger's analysis which sounds pretty absurd. We must assume the existence of radicals, and in general irrationals (at least a countable set of them) etc.: playing the math game without them would be extremely cumbersome.
I respect profoundly his thoughts but I consider he is going too far with his extreme constructivistic approach to analysis. Nevertheless his videos are very interesting and I am a fan of them and don't miss any.
Illogic Math he only believes that the quandrance of the hypotenuse is 2 he doesn’t assume anything about distances.
Euler's identity is a reflection of the relationship some infinite processes that you can define (e and pi) have.
What I can't buy in the view that it takes an infinite amount of work and that this is making a definition illegal is the mixing of me (or any physical calculator) with the definition. This way it is very hard to define anything including infinite or even very big (most natural numbers are very big) parts. Since the natural numbers, the rationals, and the reals are all infinite in number, will they then disappear as math concepts if they can't be counted in a finite time? Also, you can divide a finite number endlessly many times in theory, but only with an infinite amount of work. So how divisible is then a finite number?
I use to solve these questions with a realization, that IF we could do an infinite amount of work, THEN we could count all the natural numbers and do the other choices and algorithms, and we can keep the concept of infinity and use it where we like, but with some caution of course. There are a lot of traps in using the concept of infinity, I agree on that, and you have to be aware not to cause any errors by making false assumptions about infinity in math. My view seems to be more positive than in the video, I see infinity as a very useful mathematical concept that will find more and more use in the future. See infinitesimal calculus based on hyperreals and non-standard probability for examples.
Have you watched the earlier videos in this series?
njwildberger I have, nice lectures. I am also a teacher, so I know a little about what is a good way of teaching compared to a bad way, and you definitely know what you're doing. I'm very interested in the paradoxes of math, reading and spending time here on youtube trying to figure them out.
I can recommend this video on some of the intriguing paradoxes of infinity:
th-cam.com/video/dDl7g_2x74Q/w-d-xo.html