"Daddy, we learned today that 2+3=5. Why is that?" "Well, son, here is how this goes. Number 2 partitions the whole real number set into two subsets. The one with numbers less then 2 and the one with numbers greater then 2. Same goes for number 3. So if you want to add 2 and 3 this is what you have to do. Take every number from the less then 2 subset and add it to every number from less then 3 subset. You will get a certain set of numbers. Also, add every number from the greater then 2 subset and add it to every number in the greater then 3 subset. You will also get a certain set of numbers. By the axioms of number theory it will turn out that those two sets of numbers actually define number 5. Do you understand?" "Sure, piece of cake. Is it too late for me to start training baseball?"
Ignjatovic said "Number 2 partitions the whole real number set into two subsets." It is not the "real" numbers, but rather the "rational" numbers. Ignjatovic said " By the axioms of number theory it will turn out that those two sets of numbers actually define number 5." It is not the axioms of "number theory", but rather the axioms of "set theory".
Try explaining 2+3=5 if your 'concrete model' is arithmetic as performed by a modern CPU. That is, you explain it in terms of behaviours of an interconnected bunch of microscopic transistors.
I've got q question; How can the reals be denser in irrational numbers if every irrational number has an arbitrarily close approximation in rational numbers? It is not true that the irrational have infinitely many representatives between any two rational numbers because between any two irrational numbers is an irrational number. I understand that the converse is also true. That the rational numbers can be approximated arbitrarily closely by irrational numbers, theoretically., and that between any two irrational numbers is a rational number. What is the difference in symmetry that could account for the density results? I realize that we have the concept of countability, but is there any other explanation, for example symmetry of some kind, that would justify the results?
A great explanation from MSE: "I think the most intuitive explanation I have heard is to considering writing down a rational number in decimal form. This means that either it is a repeating decimal or a terminating decimal, for example 2.373737... or 0.42, which we will write as 0.420000... Now, consider the probability of randomly writing down a number. So you have ten options every time you go to place a digit down. How likely is it that you will just "happen" to get a repeating decimal or a decimal where you only have zeros after a certain point? Very unlikely. Well those unlikely cases are the rational numbers and the "likely" ones are the irrational."
@@persistenthomology But we seem to have to assume that there is no difference between "completed" or "accomplished" infinity you speak of and the "becoming" infinity that our limit concepts deal with . However, if we only considered "becoming" then there would be no irrational numbers. Uncountability is an emergent property of completed or eternal infinity. (How do the rules for handling the two types of infinity differ?) To your specific point, we have rational numbers, 3.000..., 3.1000, 3.14000..., etc., on out to any given number of digits; whereas, the irrational has Pi. All of these numbers are said to exist, and there are countably infinitely many of them. But the collection of rational numbers does not contain Pi. On the other hand, the irrationals obviously have an infinite collection that approximates 3.000..., 3.1000..., 3.14000..., etc., to any degree, and that collection does not include any of 3.000..., 3.1---..., 3.14000..., etc. The irrational approximations within the given bounds will be uncountable. For instance, estimate 3 with error less than 0.1: 3.01...., 3.02...., etc., where ... represents a nonrepeating, nonterminating sequence of digits. We can see that the numbers in ... are uncountable given the argument for the uncountability of the reals. So, we are led to conclude that the collection of irrational approximations of a given rational number is much larger that the collection of rational approximations of a given irrational number. The argument you provide states this probabilistically but only works if we assume real numbers, i.e., completed infinite numbers, or numbers that reach their limits. Given that, it makes the density question much more intuitive than my approach.
To those wondering how much would be lost by reworking mathematics on a purely rational, logical, clear and consistent basis, with all the "constraints" such an approach entails, the answer is that *nothing would be lost at all*. Any standing result that has any real world significance, or even the possibility of ever having such significance, must be able to be improved by a more rigorous approach. If any standing mathematical result is found not to be amenable to a fortification because it absolutely relies on incoherence for its foundation, what does that say about such a result? The fear that "much of modern math would be impossible without real numbers" is unfounded; for any theorem, either it is possible to make it rigorous, or it is impossible, and if the latter it had no value in the first place. Rest assured then that every result that really is or ever could be useful cannot be harmed by being put on a more rigorous footing. The point is not that we jettison all results that contain handwaving, because making things more precise takes real work and understanding and cannot be done all at once; the point is that we make a continual effort to make foundations more rigorous. This is a crucial part of how progress is made in any intellectual investigation. You start with terms that seem clear enough, but later you encounter situations that can only be overcome by more precisely defining the initial terms. You may have to introduce finer distinctions, or adjust definitions so that they don't fall into incoherence. As you proceed to more exacting analyses and applications, you find that some terms are not carefully enough conceived to be useful at that deeper level of analysis, so they must be sharpened up. If you avoid this or worse, go the opposite direction as the mainstream has done, you end up hitting roadblocks and getting more and more removed from the pure math promise of being even potentially applicable to the real world (being homomorphic to real world situations, or at least possible ones). For example, "sqrt(2)," as an ambiguous term referring sometimes to approximations and sometimes to the incoherent notion of an endless non-repeating decimal, can be made more rigorous by defining it as "a number whose square is approximately 2" or "sufficiently close to 2 for a given purpose" and cutting any ties with the incoherent idea of a so-called number whose square is exactly 2. At first this ambiguity may not have mattered, but certainly by the time people are using that very ambiguity to justify further theory it is time to halt the proceedings and re-examine the foundations.
I find your general thesis ludicrous on its face. First, correct Mathematics need have no connection to the material world. Secondly, that having been said, as it turns out much of mathematics as currently theorized does obviously explain much of although not all of theoretical Physics. Third, I have not seen all of Professor Wildberger’s videos but his arguments against the real numbers which I have seen so far seem incredibly weak. PS: perhaps in years hence, ala Einstein’s modification of Newton’ s theory, Prof. W will after all turn out to be correct. But for the moment, his ideas are so far out of the mainstream that his theses seem not to be taught or even seriously addressed in any top tier university. or research institute. Please correct me if I’m wrong. J Wolf, PhD Math
I would say that sqrt(2) should be defined as a generator for a field extension to |Q, much the same way that sqrt(-1) is defined. In this way sqrt(2) doesn't lie on the rational number line at all, instead it is part of a more or less Lorentzian 2 dimensional rational number plane. Also the two solutions of sqrt(2) are completely symmetric this way! What these rational "approximations" of sqrt(2) should be replaced with instead are Archimedean *places* for sqrt(2). Other, Ultrametric (p-adic) *places* for sqrt(2) exist as well. This approach seem as though it could be made completely rigorous. Also, such things as cbrt(2) do not even live in a rational number plane, instead they live in a rational number space of 3 dimensions, but with only Abelian symmetries valid.
Of course, if we extend the field even more, so we get both cbrt(2) (solution in x to x^3 - 2 = 0) and a primitive cbrt(1) (solution in x to x^2 + x + 1 = 0), then we get a 6 dimensional space with non-Abelian symmetries described by Sym(3), rather than Alt(3).
By the way, the specific example of x^3 -2 = 0 is not a Galois polynomial over |Q, so it is not the best example, since its field extension has only the trivial symmetry. The polynomial equation x^3-3*x-1 is a much better example, since its field extension actually has Alt(3) as symmetries.
This is interesting. I've been dissatisfied with real numbers in mathematics many years and thought there must be another foundation that reflect our reality much better, something based purely on relations, proportions, geometry and logic. This seems to be something in that way.
Dr. Wildberger, Forgive my confusion, but I am still confused as to what exactly your objection to the real numbers are. I’ve seen various statements about strange constructions of R, but I have yet to find a problem with their definition. As an analyst and a topologist, this topic bears heavy on me. We can both agree on the metric space (Q,d): the set of rational numbers equipped with the Euclidean metric. Note, of course, that this topological space is non-Cauchy complete; a simple example is that the partial sums of the reciprocals of the squares fails to converge. What would be wrong with defining R as cl(Q), that is, the closure of Q such that the resulting set is Cauchy complete and contains all the limit points of any sequence contained in Q? Thank you for your time.
Hi Peter, There is not just one definition out there. There are several approaches to conjuring up the "real numbers", and my point is that none of them work. In this series over the last few dozen videos I have critiqued most of the major "constructions of real numbers" and shown them to be hollow. But you can test this for yourself: ask how you would demonstrate to a jury that you really know what real numbers are, that you understand the operations with them and can do arithmetic with them. Why not start on my favourite challenge: to compute "pi+e+sqrt(2)". The whole exercise immediately grinds to a halt. It is like getting a class of 8th graders to calculate, in the rational number framework, the sum "1/2+1/3+1/5" and all you get is the question repeated back to you and philosophical mumblings. I do encourage you to have a good look at this playlist and go carefully over all the "constructions of real numbers" videos.
I should also add that I most definitely do not agree with talking about "(Q,d)". There is no reasonable meaning that I can attach to the "set of all rational numbers". That is day dreaming in my view. Furthermore the metric involves taking square roots, which you cannot in general do.
@@njwildberger what if you’re wrong and reality DOES contain infinite sets. Just because there are not infinitely many particles (according to physicists) does not rule out that there exists infinitely many of something in the universe.
I have a question: How to 'convert' (transform) a real number back to an integer *WITHOUT* the usage of ceiling or floor? If x = 123.45 and we get the modulo 1 of it, say, 123.45 mod 1 I can get 0.45 and then subtract from the original x. BUT as I saw in Wikipedia and in some books, the modulo function use floor function (Knuth). I need a 'downgrade' from Real Number to Integer using pure mathematics. How can I get rid of the mantissa of a log? I'm trying to create a pure function that will count the digits of a given number without the usage of algorithm of any computer language. Example so far: n = 123.45 , log (base 10) of 123.45 + 1 = 3.09149109426795 , so, the number 123 have 3 digits + 0.091491.... how can I get rid of it? I'm watching the videos and discussions about Real Numbers, Set Theory, Logic and Infinity here , so, maybe someone can give me a hand. Thanks for reading.
There are even more simplifications in mathematics without real numbers: for example, the conditions can be simplified for the subspace theorem, "v in S => \lambda * v in S" can be eliminated, as this can be deduced from the conditions "0 in S" and "u, v in S => u + v in S".
I have some honest (and maybe stupid) questions about the central role of the rationals... how would we deal with the 'pi' we find in circles and its "real" proportions? How would we deal with 'e' and the natural logarithm or even with 'square root of two' in a square's diagonal in proportion to its side? Aren't there some "real life" applications with "real" proportions?
@DrNorman I have one question. Although rational numbers can be written and manifested as ratios of integers but are most of them not also produce repeated patterns of infinite decimals like 1/3 , 5/3 , 9/7 whose repeated patterns are obvious? There are also numbers whose repeated patterns occurs after several thousand digits and are not obvious or as you have explained in your videos that it gets harder to compute or represent numbers as they get bigger so how come rationals do not pose the same challenge as of reals. I know there is one distinction that reals cannot be represented completely but rationals can in ratio form and cannot in decimal form. PS: I am enjoying having fresh perspective on foundations of maths. It is really enlightening me in a sense that explore for solid foundations and easy to manage amd understood mathematical methods.
You should think of arithmetic with fractions as quite separate from "arithmetic with repeated infinite decimals". The latter is actually highly problematic to lay out clearly, and is notoriously avoided by educators. I know the two are conflated, but try to not confuse the clear arithmetic with fractions with the much more dubious "arithmetic" of "infinite decimals". These latter are like ghosts in mathematics.
@@njwildberger So should we totally outcast decimel expansions or we should develop a better theory about decimel expansions? After all fractional values eventually computed or approximated in decimel expansions to get results in engineering, computer science etc and when fractions get bigger does it not pose the challenge of arithmetica and computation? Can we totally ignore decimel expansions?
@@sang81 Decimal arithmetic is crucial for applied mathematics and applications, so of course it must remain. What we must rather look more carefully at are the current overblown claims that the finite decimal arithmetic of the applied mathematicians and engineers can be extended to a valid arithmetic of "infinite decimals". And it turns out that even "infinite periodic decimals" have an arithmetic which is almost nowhere spelt out completely correctly: even though in principle this is possible. Basically we have just bought into an exceedingly sloppy approach to this entire subject, propelled by massive wishful thinking.
It's also amasing sin and cos to 45 is the squire 1/root of 2, so there is a connection between pi,e, i and the sqaere root of 2. I have no idea why? :-)
+Fred Frey Sure. The average student knows the trigonometry of the 45-45-90 triangle and the 30-60-90 triangle. These are the two triangles that at least 90% of all trigonometry questions revolve around, since these are pretty well the only ones that students "know" the trigonometry of.
Here is one problem with denying irrational numbers: there are integer solutions only-that have to be expressed using square roots. Example is Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, etc. Given any 2 consecutive numbers to get the next number add those 2 numbers together. Clearly every member of a sequence is an integer. Yet to get formula for any member requires square root of 5-which you don't believe in. Similarly there are cubic equations that have only integer solutions, but formula that finds these involves square roots and cube roots. Lastly your objections that irrational numbers have infinite decimal expansion and so require more space then the universe overlooks the fact that we don't know the size of the universe (or multiverse)-so we need a system that can handle any size. Even if we knew the size of the universe-we still would need system that is bigger then the universe. Example: universe has only 3 objects-A, B, C. To list all possibilities you would need these sets: 1) empty set 2) A 3) B 4) C 5) A, B 6) A, C 7) B, C 8) A, B, C-so 8 sets in the universe of only 3 objects: more sets then its universe!
Which rational number, when multiplied with itself, yields 2? Since no such rational exist, we needed to add the set pf reals. Same as we needed to add the set of negative integers for subtracting natural numbers, as 3 - 5 is not a natural number.
@robheusd No, it definitely is not the same as introducing negative integers! This is the kind of reasoning you want to let go of. AS I discuss in the Math Foundations series, negative numbers are an effective framework -- we can say precisely what the objects are, what the operations are, and we can illustrate with a wealth of examples, not just cherry picked ones. As opposed to for example: "pi + pi = 2 pi -- see we CAN do arithmetic with real numbers!"
So, over on the real number line where I'm from, a right triangle with two sides of unit length has, by the Pythagorean theorem, a hypotenuse of length √2. What's the length of the hypotenuse in this here General Field Rational Mathematics you got goin' on?
When you are doing careful mathematics, everyday concepts do not just morph into mathematical ones, but require precise definitions. You may well think you have a very good intuitive idea of what "length" means. But in order to answer your question, you must first make clear what precise use of the term you are using when you ask "What's the length of the hypotenuse in this here General Field Rational Mathematics you got goin' on?" Here is an analogy: over where I come from, real triangles have colour. So what's the colour of the hypotenuse in this here Real Field Mathematics you got goin' on?
@@njwildberger Well, I'd say blue is the best hypotenuse color and I'd argue that all triangles should adhere to a rigid color scheme so they can be mapped to RGB computer screen color values, but I realize I'm in the minority with this opinion. More to the point, It seems you're saying that I simply can not have my dear √2 hypotenuse triangle. Can I at least have a triangle with 2 nearly equal sides, an angle of nearly 90 degrees and an almost √2 hypotenuse? Or do we all have to go back to living in tetrahedron houses?
@@sallylauper8222 I agree that blue is a nice colour for a hypotenuse, especially in Euclidean geometry. But there will be those that would prefer to consider triangles without any reference to colour, thereby avoiding prejudice and questionable definitions. So I think we can be happy with you retaining your perfect isosceles right triangles. But when we want to refer to measurements of their sides, we should stick to quadrances, which say are then 1, 1 and 2, which will surely make you happy. And happily quadrances can be computed for ANY triangles in the rational plane, so we no longer have to restrict all trig examples to those two tired 45,45,90 and 30,60,90 triangles, in all manner of contrived variations.
Sorry but the unit circle does not have an area in the usual sense. The unit square does, but the unit circle does not. However one can right down many approx-areas for it.
I would like more examples of semi regular/uniform tesselations of Universal Hyperbolic Geometry (over |Q, but also over algebraic field extensions of |Q). Especially tesselations involving polygons with vertices and edges spanning across the two main regions of the hyperbolic plane. Perhaps i could produce such examples myself someday.
So do you mean that you think sin(x) as a power series is not problematic, and using techniques you talked about early in the series, Dsin(x) =cos(x) is valid, but sin(1) as a number or a so-called randomly generated power series like f(x) = Sigma( pi(n) x^n ) where pi(n) is nth digit of pi is problematic?
That is pretty well exactly what I think. We are going to see that it is crucial to regard power series as extensions of polynomials/polynumbers, not as functions. We are going to have to worry about how to specify the coefficients of power series, so we do not fall into the same logical holes that the current theory does by assuming we can deal with an infinite amount of data. It is all somewhat delicate, and will require considerable new thinking. Thanks for the excellent comment.
You obviously have a serious bone to pick with the notion of the continuum, and hence the idea of infinity and by extension the set theoretic approach and the transfinite arithmetic. Sadly, even by moving away from the real numbers, you still won’t be able to avoid the infinities. Forcing mathematics to neatly fit into our inherent physical limitations is not the answer. Meanwhile, I’m truly surprised to see that you stomach the innately meaningless definitions of a point or a line, without much complaint!
I am not forcing mathematics to neatly fit into anything. I just want our mathematical discussion to be complete and clear, meaning that we can explicitly write down everything. This is in the same spirit that a physicist might want to restrict themselves to statements about observable quantities. The same scientific spirit needs to strengthen within mathematics.
Algebraic "irrational" extensions of Rational numbers are very natural, since we get these immediately with Vector spaces equipped with quadratic forms, and higher forms. In |Q^3 we get e.g. equilateral triangles, i.e. triangles with equal non-zero quadrance of all edges, and equal non-zero spreads of all vertices. Similarly we will get all regular polygons in |Q^n for larger n. And we will get regular polyhedra, polychora etc as well, if n is large enough. Of course we will get all those "indefinite" quadratic forms as well, besides the "positive definite" ones, which make place for geometric objects of hyperbolic geometry.
What have you in mind by a more _general_ type of mathematical field than the complex numbers (for example)? These are vectors endowed with multiplication/division as well as addition/subtraction but do you have in mind matrix representation of discrete field elements (with all rational entries)? Would complex numbers - and the field associated to them - survive in this enlarged picture as just a special case or restriction but no longer regarded as 'fundamental'?
There are many other kinds of fields. Some are related to complex numbers, others are quite different, for example finite fields, or fields of rational functions. We will be spending a good amount of time on field theory in coming videos.
@@njwildbergerAlgebraic number fields over the Rational numbers are also very worthwhile! They are somewhat analogous to the fields GF(p^n), for n > 1, though somewhat more complicated. Much better than "Complex" numbers based upon "Real" numbers, in any case.
Most likely the "Complex numbers" would be replaced with a (countably infinite dimensional) direct limit of all Algebraic number fields over the Rational number (Algebraic closure of |Q). We will get all sorts of field automorphisms over this field, both "circular" automorphisms over the cyclotomic generators (these will be much more complicated than mere "real" analogous rotations), and various sorts of "hyperbolic" automorphisms over non-cyclotomic generators.
The question of which scaling factor to use comes up for me when I am doing computations with double/float type on a computer. True 'real numbers' would require that one need not make any decision on scale, for example, 0 to 1 can represent the continuum of everything the value is trying to represent - but why stop there, why not pick 0.1, since there is infinite precision between 0 to 0.1, how about 0 to 0.01 and so on. I guess this is eased by the double/float type - where, only a fixed amount of precision can be expressed; but doing this seems to be an engineers work around to this issue of not being able to fully represent and compute with real numbers.
I hope you will, at some point, prof. Wildberger, give some thought to differential equations. These are fundamental mathematical objects in physics, and it will be very interesting for me, and for many others, I believe, to understand their correct mathematical meaning.
Sorry, I believe the proper term is acutally computable number, if you want to look it up. Essentially the set of functions N -> Z that, given an integer n, returns a rational number of increasing perscision as n increases. So pi is a computable number, but Chaitin's constant (the probability that a randomly constructed turing machine will halt) is not a computable number. One thing you may like about them is that they form a countably infinite set. :)
kratanuva725 Okay, it would be great to have a theory of `real computable numbers'. Unfortunately in the generality we currently aspire to, this is impossible. But it might be possible to have a more modest, restricted theory of computable numbers which did support an arithmetic. This would be a very major breakthrough, but I don't know if it is feasible. Certainly some people ought to think about it though.
@Josh YOu don't get rid of complex numbers, rather you just define them correctly: as expressions of the form a+bi where a,b are rational numbers, and where i is a symbol that satisfies i^2=-1.
@@njwildberger ah, makes sense! I hoping you expand a bit on Octonions. Rational numbers emerge in interesting places in the hypercomplex numbers in the realm of factorization.
@@njwildberger and keep in mind these things have important applications. Especially for quaternions, factorization tells us which rotations will give us a given orientation. Direct applications to quantum physics. In fact quantum states are rational numbers.
If you want to talk about foundations in mathematics, the first thing I would want to know is your formal definition for the natural numbers. Can you give me your formal definition of the natural numbers please? And while you're at it, please explain why real numbers do not qualify for that definition. Thank you.
One area that goes beyond rational trig where I would be curious to get educated is in the domain of quaternion algebra, but using only quaternions with rational coefficients. There is this area of Hurwitz quaternions but to me so far it’s mostly incomprehensible beyond some basic definitions. To few people seem to be working on that and those who are write extremely esoteric papers.
The more I think about this topic, I think that this is not a convincing way to talk about it. I think about mathematics as a computer without a specific and nailed-down machine language. In computer languages, there might be undefined behavior in the LANGUAGE, but when you pick an implementation to REALIZE it, then EVERYTHING gets defined whether you like it or not. Most computer security issues come from this. Mathematics is missing a virtual machine to execute upon. You can't talk about rigor unless you have a virtual machine to execute upon, where every step can be made visible. Every English language statement needs to be "compiled" down to algebra, which in turn compiles down to the assembly language of the actual mathematics machine. Ex: "C is a constant" is imprecise slang for "d[C]=0", If you write a recursive equation like: "S = 1 + (1/2)S", then you can't just argue with what happens when you both solve to find "S = -1", and find that S expands like: "S = 1 + 1/2 + 1/4 + ... + 1/(2^(n-1)) + 1/(2^n)S" (note the S on the tail-end). A virtual machine needs to be defined. Calculations that don't terminate are problematic; because you need to stop calculating one side to see if it's equal to another side. If the machine doesn't terminate, then you need to prove (in a finite number of steps!!) that the machines give the same output forever; like comparing two code functions for equal results. In computer science, this is related to "the halting problem". And most of all, you cannot argue with the result when you run the machine. You can only argue about whether the virtual machine needs to be changed to give a different result. When mathematics is being done right, we will be able to make huge databases of true statements; where a human never has to step in and object to a step that has been taken. It's crazy that math is supposedly rigorous, but defies automation. Kevin Buzzard is right; about everything.
Hi Rob, I love your comment and am deeply in agreement. I have been thinking about a video that urges us to work towards “running” mathematics in the same way that programmers run programs. Thanks for the nice input.
Many of the basic definitions are logically flawed and mathematicians know that but still use them. For example Peanos axioms define much broader concept then natural numbers namely progressions. And it should be wrong to define a concept by a definition of a much broader concept. And even Mr Tao in his Analysis ignores this and starts with Peanos 5 axioms. Operations do not have a definition. The current one just states that set of operands maps into the set of results of operations but does not state what IS an operation. Definition of relations as subsets is circular because subsets are also relations. Definition of reals as a complete field does not have a model (it is just stated it does and those "proofs" that I have seen break down at the more complex parts of proofs ie are evaded). Even Dedekind in his original work evaded the complications. He just constructed the proof that the sum of two dedekind cuts is also a dedekind cut. But when he had to do the same with multiplication and division and the power operation he just did not do it. He used the term "it is done likewise" or something to that effect. So it is not just continuum, real numbers and infinities. The sloveliness, imprecission, and I do not care attitude are all over the place. Mathematics is literally in ruins. And infinity is treated as seen fit. It is a madhouse.
Aleksandar Ignjatovic said "Definition of relations as subsets is circular because subsets are also relations." You are using "relation" in two different senses here. The relation of being a subset is not the same as what is defined when a subset of AxB of two sets A and B is called a relation. Nor is the undefined "e" (where "x e y" for sets x and y) a relation in the set-relation definition sense. The "x subset y" is defined in terms of "e". Just like "e", "subset" has all possible sets as the first argument and all possible sets as the second argument. But, there is no set of all sets. Thus, "subset" cannot be a "relation" in the set-theoretic meaning. But, even if the above were not true, the definition would not be circular. It would be fine to define "subset", then later define the set-theoretic definition of relation, and then prove that "subset" is in fact a relation under that definition.
Aleksandar Ignjatovic said " Peanos axioms define much broader concept than natural numbers namely progressions. And it should be wrong to define a concept by a definition of a much broader concept." What do you think the "concept of natural number" is? Anyway, if you don't like the idea of calling the objects of Peano axioms natural numbers, then we could just call them something else (like Peano items). The same goes with the names of "real number" or "imaginary number". If you don't like the idea of calling the objects of R or C "numbers", then we could just call them something else.
Aleksandar Ignjatovic said: He used the term "it is done likewise" This is common in mathematics. I believe Euclid did the same. Aleksandar Ignjatovic said "The sloveliness, imprecission, and I do not care attitude are all over the place. Mathematics is literally in ruins." This has always been the case. The general public idea that mathematics is precise, logical, and clear is not true. Just read any mathematical article or mathematical book or do a mathematical proof yourself, and you will see that the general public idea is not correct.
***** That very well might be somewhat true--it certainly is not uniformly applicable---but it does not prevent us from moving towards a more precise, mathematics where we DO spend more time and energy making sure that the subject hangs together logically!
***** Sorry, but all your remarks (I do not have to comment each one separately) are just rationalizations of bad ways in mathematics. Especially the one about the relations.
I'm only an amateur so it's not exactly clear to me what you're advocating here. Do you want some kind of system which gets rid of the Reals, irrational numbers and transcendental numbers like e and Pi? How would that be achievable in practice except for solving problems with algebra as opposed to numerically? As you mentioned about fractions, it quickly becomes unfeasible to represent reals as exact fractions on computers, because the numerator and denominator can become huge, and where simplifying the fraction isn't possible. They could both be huge prime numbers for example. But some programming languages do support this, for example Common Lisp implementations. But doing calculations that way is incredibly slow. And suppose we all started using fractions instead of the reals for everything and the sizes of the numerators and denominators weren't an issue in practice. Whenever our equation involved irrational or transcendental numbers, wouldn't we be forced to write the final answer as a fraction *and* an agreed name for the transcendental/irrational numbers used? If so, how is that any different than writing down the algebraic solution? Or is there a theoretical approach which gets rid of Reals, irrationals an transcendental numbers entirely? Sorry if I missed something fundamental from your lecture which addresses this. I'm just trying to understand. I agree however that exact numerical answers are pretty great. They make me warm and fuzzy inside :)
Currently modern computer systems do not use real numbers; they tradionally use floating point numbers, which are something else entirely, and increasingly these days they use full integer precision. Our computers have never played along with our real number fantasies; I am advocating facing up to this reality, and finding ways to move forward logically and carefully.
1-Dim WildDerivative might be Df(x) = [f(x+h)-f(x-h)]/2h, where h ≈ (Planck length), and analogous integral ∫f(x)dx = h∑f(x). On the other hand, assuming lattice structure in physical World _might_ unify quantum theory and relativity under the same umbrella. But for continuous process, or describing ideal geometric figures, real numbers are indispensable.
What exactly is continuous process if any measurements under planck length and time are impossible even theoretically ? FIction ? What exactly are "ideal" geometric figures ? Fiction ? What's the point of real numbers and theory of infinite sets ? Fiction for the sake of fiction ? Ok.
pieinth3sky Mathematics basically is fiction or game, but fundamentals like geometry and arithmetics arise when our mind senses shapes of nature and steps of time. Now, if you like serious crackpottery, like L.E.J. Brouwer did, you can build your own math on those simple, finite and direct mental processes. Too bad that you would get stuck in the Stone age. Sane person would add some natural extra structure that relates different things and provides existence for the obvious, e.g. solid geometric objects are point-sets and non-parallel lines must meet.
Real numbers are not indispensible for continuous processes or ideal geometric figures. Real numbers do not exist, so how could they be indispensible for anything? Perhaps you mean that a belief in real numbers is indispensible to understand continuous processes or ideal geometric figures? I will show that claim to be false: with rational numbers we can well model continuous phenomenon, meet the needs of scientists, and develop rich theories of geometry: all without abandoning the logical clarity which is essential for true mathematics. Keep watching!
+njwildberger Hi Dr. Wildberger. Long ago I watched your video where you lecture on how the Greeks proved that the diagonal of a square is incommensurable to its sides. This is an elegant proof that continuous objects cannot be fully described in discreet terms (i.e. integers). Over the years, I have watched many of your videos, and I have not yet seen how you have defeated this proof. I respect your expertise, and I have learned a great deal from you. So it is with some discomfort that I express the following opinion. So far, I have not seen anything in your videos that addresses the real problem, that being that rational numbers cannot express every actual continuous magnitude. What I have seen appears to me to be more mathematical sophistry than a solution. For example, regarding the diagonal of a square you prefer to express it as a "quadrant". Why? Because the quadrant is expresable as a rational number. All this shows is that you can square an incommensurable magnitude to get a commensurable one. But this was never the point, and therefore solves nothing. The point was always about how to express the original magnitude, not "what is the square of this magnitude we cannot express". According to you, real numbers (e.g. irrational numbers) do not exist. But the magnitudes that real numbers represent do, and these magnitudes cannot be expressed by rational numbers, as proven by the Greeks. You have yet to show me where the Greeks were wrong. I have watched your videos, and there isn't a single case where you express the actual magnitude. You square it (not the same thing) or you express it implicitly (again, not the same thing, and in science we want to know how things relate explicitly, implicit expression is of very limited use), and so I find many of your methods detrimental for many reasons. I still think you are a pioneer, and I believe your methods will find a place as an addition to mathematical techniques, but not as a replacement to fix a system that isn't broken if properly understood. Just one of your thousands of online students, Mark Botirius
How do you APPROXIMATE rational numbers: For example, if I have the number 0.53476853456566 then the simplest way to approximate this number is to IGNORE the remaining digits: 0.53476853 Very simple.! Now I have (233)(65)(18) / (234)(44)(19). This number is EXACT! It is very tempting to use this representation instead. I would love to use this representation in my computer. However I need to ensure that that during the calculation, there will not be an overflow. Although the coefficients are huge, the number itself is reasonable. It is hard to predict in advance where it will crash. This situation does not happen with real numbers because the Floating Point Unit simply ignores the remaining bits. But with the rationals it is not as simple. I could prepare my numbers in some kind of series where the arithmetic unit does not overflow? I am clueless?
Concatenating non-number "-" meaning negative with an aspatio-temporal unique number may rob you of the relationship between a number and it's representative numeral.
I gave a thumbs-up to this video but found it shocking. IMO, the reals and complex numbers are well-defined and solve many real-world problems. I have no doubt that numbers such as e, pi, square roots, converging sequences, etc. exist and are very useful. I also see axiomatic set theory as providing or, at the very least, a good attempt to provide a solid foundation to mathematics. I do agree that a lot of math is far removed from realty and suffers from vague, arbitrary definitions without concrete examples. And some of it is stupid such as analitical extensions that "prove" that the sum of some positive numbers is equal to minus 1/12. NJ is obviously an accomplished mathematician and I look forward to seeing more of his videos.
Tip for differential equations... A likely future direction will be away from exact solutions, and more emphasis on asymptotics, rigorous approximation, and extracting information from PDE's using ideas from information theory combined with asymptotics. For some lectures on such directions I highly, highly, recommend Carl Bender's PIRSA lecture series. Physicist do appreciate the work of pure mathematicians on exact solutions, but typically they are important only for toy models, the real world of physics never deals with exact solutions to DE's because they just do not exist in any practical sense.
AJ AJ but what exactly are useful digits of pi ? I dont see any practical purpose in knowing the 1e12 th digit of pi - beside sports - similar to knowing the next bigger prime... is it even right to ask for the 'value' of pi ?
AJ AJ Well said. Bolzano's genuine "foundational" work on infinite sets is explicitly metaphysical. Moreover it is anti-physical in sense that it expels time and motion from mathematics. All further "development" of "pure mathematics" like "Dedekind-infinite sets" ,"Cantor's diagonal argument" or "Goedel's incompleteness theorem" or AC looks like bootstrapping of metaphysical objects by appeal to even more metaphysical objects.
AJ AJ Please tell me, is the following a process or a number: S = 1/2 + 1/4 + 1/8 + 1/16 + · · · = 1. If you call it process, then I argue that it will permeate and contaminate the whole edifice of rational arithmetics. If you call it number, then I argue that every other finite limit is also a number and we can do arithmetics just like we do with rationals. For me process and its limit value are just two sides of the same coin.
Another thing about diagonals of the unit square... What the non-existence of a number whose square is 2 is telling us about the diagonal lengths is simply that they are incommensurable with the side lengths. There is no way to directly measure the length of a diagonal using the sides as measuring sticks. It can be done approximately but not exactly. *That* is the fundamental reality. Creating a fantasy number called "the square root of 2" doesn't circumvent that fundamental incommensurability. It just covers it up with a fake hack. Hence having to do without that fantasy number doesn't create the incommensurability anew, it merely exposes what was there all along. Doing without real numbers will seem to create new problems, but that is only because real numbers gave the illusion of solving those problems. Don't shoot the messenger, so to speak. Rational mathematics means seeing what is really there, so that real progress can be made by addressing real problems. It's like the brutally honest person who tells you your weak points, which you had learned to ignore but that are weighing you down nonetheless. Ignoring is blissful for a time, but ultimately not helpful. And if you keep doing it you end up trapped in the mud of all those contradictions and denials, unable to move forward. Taking an honest look at the hard reality is so painful that it can feel like the very act of looking is what's creating the problems, but that is of course not the case. And once you realize that, you can start to come to grips with these challenges and truly take the bull by the horns.
You can talk of a collection of underestimates for the length of the diagonal, and a collection of overestimates. An overestimate will be better if it is smaller in magnitude, and an underestimate better if it is greater in magnitude. The 'real number sqrt(2)' then corresponds to the 'theoretical best approximation possible', or rather the asymptotic limit of a process of continually refining the approximations so as to be closer than any given rational. That, of course, is the point of the Dedekind cut approach.
There always tis question if mathematics is found or invented and if the universe can be described by pure mathematics. Dr. Wildberger have sure convinced me about the mathematical problems with real numbers, infinities, intiniticimals.... ;-) Eulers equation suggest that there is some relationships we dont understand e^pi*i+1=0
@@HDitzzDH Yes, Thanks. It was 5 years ago. I've taken Math A since then :-) I still feel that Dr. Wildberger fundamental right about the problems with real numbers :-) Happy New Year.
Do you agree that the notion of real number is directly connected to the notion of infinite divisibility of length? ie: if we do away with real number, then we have to do away with infinite divisibility of length...
No I do not agree, since first of all `length' is a problematic notion, and not necessarily involved in discussion of the nature of the number line, and secondly because the continuum as the rational number line has inherent in it the possibility of division of segments or intervals into arbitrarily large numbers. If we want to divide [0,1] into 10^10 pieces, we can do that. I am not sure however if that is what you mean by `infinite divisibility'. It is a great exercise to ask, before asking a question: do I know precisely what all the words that I am using here mean?
njwildberger "No I do no agree, since first of all `length' is a problematic notion, and not necessarily involved in discussion of the nature of the number line, and secondly because the continuum as the rational number line has inherent in it the possibility of division of segments or intervals into arbitrarily large numbers. " I was referring geometric length in Euclidean geometry, not the number line rational or otherwise. I see the purpose of real numbers as giving a quantifier to euclidean length... Do you think the notion of length needs to removed from Euclidean geometry?
otakurocklee Somewhat surprisingly, length is NOT a part of Euclidean geometry, in the following sense: Euclid in his book The Elements did not introduce a notion of length of line segments. This is a modern attribution to Euclid. Why did Euclid not introduce such a concept? My guess is because he well understood the difficulties with having a prior number system in which lengths can have values.
Complex analysis definitely goes in the direction of algebraic calculus and topology always goes in the combinatorial direction. Professors talk and youtubers work. I've started thinking about definitions of a manifold.
Totally agree almost on everything, but perhaps I'm even more radical. I think there shouldn't be rational or natural numbers, just numbers. Those numbers can have certain conditions such as the limitations which natural numbers have in comparison to rational numbers. Of course you can still use the name natural number, but the mathematics is all around just the definition of a number. Then the logical steps/results which evolve from this definition would be mathematics.
Finally a scholar who dares. Why are we always trying to make all shapes into squares. Though they may be rectangles, triangles, curves etc. We're always asking what is the root, as if all things should revert to square. One is slave to the standard one sets. And the givens you define today, if not controlled, will invariably be your frustration tomorrow. There is a fundamental reason why a rectangle has unequal sides & that reason gives a unique meaning to what it is to be a rectangle. Same with a circle, its a unique object, a type, complete with its own geometry & math. Seems to me that there is a math-by-type logic embedded in the construct of each type of object. And unifying by ignorantly setting single standards, generalizes over the separate & unique. We got no clue what the common basic building block(s) of physical objects are (if there is only one) yet we claim its nature Type-by-type, with respect given to uniqueness may be the way to go. And if approximation is our fate, let it be with at least sensible.
To the person who asked how to find the length of a diagonal of a unit square, my first question is, "How thick are the sides?" Remember they cannot be infinitesimal in size, since we're now rejecting such fantasies. When you have a definite thickness to the sides, length becomes either approximate (no real numbers needed) or reliant on what exactly you're measuring. It's no longer obvious that the length of a diagonal is incommensurable with that of the legs.
Furthermore, it would have to be the case that the picture in physics of fields existing on a manifold is not the way to go. Some people work on theories of quantum gravity where our spacetime is hoped to emerge as an approximation of some discrete structure. In this picture, `squares' would only be an approximation with no fundamental place in physics.
Disclaimer! I am NOT a mathematician! Root 2 is often shown as the most easily provable "real" number. There are many videos showing proofs by contradiction all over TH-cam channels. We want to assume that the diagonal of a square of side-length 1 has a length, and it's easy to show that it ought to be some value, namely "root 2", and the units of measurement don't matter.... But aren't we "measuring" sidelengths according to the typical Cartesian co-ordinate system, aligned to a right-angled pair of axes, generally referred to as x and y, or horizontal and vertical? Well, almost any measure of length along a diagonal in this reference frame will lead to some root, some irrational or "real" number... Ok. Accepting this conclusion, let's look at the intersection of the hypotenuse with either of the legs: two lines meeting at a 45° angle. Now, imagine measuring the length of the two sides which intersect. And we take more and more measurements of a finer and finer scale, to try to refine the measurement of the hypotenuse. The side-length reads 1 unit but to finer abd finer precision. Fine. But the measurement of the hypotenuse will never intersect perfectly with the end of the side-length!! Even with an "infinite" number of repetitions, a diagonal "unit" can never equal a vertical "unit" in a Cartesian set-up, unless you rotate one to match the other. And even if you did, you will never come to a fine enough measurement to make one expressable in terms of the other. Now, mathematically we assume that a geometric "line" has no width, and is a "one-dimensional" object. If so, how could a diagonal line ever hope to intersect with a straight line? One would be completely and utterly unable to conceive or interact with the other
Unless a line has some width, even speaking of "Intersection" is meaningless, it seems to me. I can accept the (for want of a better term) "Absolute length" of the Euclidean point of view of a line segment which has no dependance on any underlying grid. But as soon as you combine the Cartesian Plane with 1-dimensional, abstract, lines or line segments, how can they possibly intersect, or for that matter, even interact at all? The applied math POV to choose some level of precision makes sense to me, but to speak of 2 1d objects "touching"... How?
Jeremy Nasmith I think sqrt(2) is easily proved to be not rational. But that doesn’t prove it’s “real” unless this is the definition of “real” in which case that is an axiom, not a proof. As far as I can tell this video is questioning the coherence of the definition of real numbers.
'Completed' infinite sets e.g. N, Q, R, C are heuristics, one does not need to actually take a position on the truthiness or ontological status of 'actual infinity' versus potential infinity. Also the use of 'completed' infinite sets is valuable especially for pedagogical reasons. In a discussion about things that are imaginary, which are numbers, can you honestly put limits on the imagination, e.g. claiming that infinite sets are past the set of what-is-possible-to-think-of-with-the-imagination? It seems to me that you are saying that a completed infinity is a 'lie of the imagination'. But i could be misinterpreting your position. If I was pressed to take a position, without thinking too deeply into it, the concept of infinity does not seem incoherent. It seems to be another order or level of imagination. unless you are claiming that we must always tie mathematics to generalizations about real objects (sets of trees, sets of sheep, etc) , and since no one has experienced infinity, from whence comes this generalization of large sets come from. But this is like an extreme empiricist position which seems unnecessary. Since when were mathematicians bothered by reality, they are philosophical by nature and daring in their ability to imagine. From an empirical point of view it seems that infinity is taking the idea of 'and so on' and bracketing it. In most instances of the use of infinity, it can be substituted for 'arbitrary large' or 'arbitrarily small' or arbitrarily close expressions. To approximate irrational numbers we use functions that wait for the call of how much precision you want. What is π? What is √2, they are functions that have a certain type of arbitrary precision when requested ( so far i have avoided talk about infinity). "The natural numbers are infinite" - that just means that given any number you think is the biggest because you grew tired of counting, i can find a number bigger than it. "The prime numbers are infinite" - that just means that given any prime number, no matter how big, I can give you a procedure to find a bigger prime or has a prime factor that is bigger than your prime. But how do you account for ordinals 0,1,2, ... w, w+1, ... , then i guess it would correspond to multiple levels of bracketing , like a nested function (makes me think of C programming, the variables are deallocated when the function is returned). There does seem to be a need for completed infinities in proofs by contradiction , e.g. cantor's real number uncontability theorem. There is a lot to think about here. Maybe infinity is a lie after all, an imagination lie. but its a useful lie because we can pack a lot of information into this lie. As a society we realized that god does not exist (maybe speaking for myself here), and now infinity does not exist. What's left? only the 'and so on' I hope I didn't strawman your position. its a little hard to pinpoint your views because you haven't said straightforwardly that 'infinity is incoherent' or 'infinity is a lie' or 'infinity is a nonsense word' (because it doesnt refer to any coherent concept). Maybe the best analogy for infinity is as a do-while loop procedure, infinity( ), that never terminates. I don't have to actually call this function,i.e., i never have to run infinity ( ) , but i know it's there.
I don't understand this, but I think that this has connections wit you subject of the last weeks: Ideas and Explorations : Brouwer's Road to Intuitionism authors Kuiper, Johannes John Carel source Utrecht University Repository (2004) full text [Full text] document type Dissertation discipline Wijsbegeerte abstract This dissertation is about the initial period of Brouwer's role in the foundational debate in mathematics, which took place during the first decades of the twentieth century. His intuitionistic and constructivistic attitude was a reaction to logicism (Russell, Couturat) and to Hilbert's formalism. Brouwer's own dissertation (1907) is a first introduction to his intuitionism, which was the third movement in the foundational debate. This intuitionism reached maturity from 1918 onwards, but one of my aims is to show that there are demonstrable traces of this new development of mathematics as early as 1907 and even before, viz. in his personal notes, which are composed of his numerous ideas in the field of mathematics and philosophy. To mention some important ones: 1. Mathematics is entirely independent of language. Mathematics is created by the individual mind (Brouwer certainly is a solipsist) and the role of language is limited to that of communication a mathematical content to others and is also useful for one's own memory. 2. The ur-intuition of the 'move of time', that is, the experience that two events are not coinciding, is the most fundamental basis of all mathematics. A separate space intuition (Kant) is not needed. 3. A strict constructivism. Only that what is constructed by the individual mind counts as a mathematical object. 4. Logic only describes the structure of the language of mathemqatics. Hence logic comes after mathematics, instead of being its basis. 5. An axiomatic foundation is rejected by Brouwer. Axioms only serve the purpose of describing concisely the properties of a mathematical construction. These five items have far-reaching consequences for Brouwer's mathematical building. To mention the most relevant ones: - The only possible cardinalities for sets are: finite, denumerably infinite, denumerably infinite unfinished and the continuum. - The continuum is not composed of points (Aristotle already said so), but is given to us in its entirety in the ur-intuition. It can be turned into an everywhere dense measurable continuum by constructing a rational scale on it. - Cantor's second number class does not exist as a finished totality for Brouwer, since there is no conceivable closure for the elements of this class. - The continuum problem is a trivial one: Every well-defined subset of the continuum is finite, denumerably infinite, or has the cardinality of the continuum. Finally, in my dissertation the sixth chapter is devoted to Brouwer's view on the application of mathematics to the human evironment and on his outlook on man and on human society in general (chapter 2 of Brouwer's dissertation). His opinion about humanity turns out to be a pessimistic one: All man's effort, when applying mathematics to the surrounding world, is aimed at a domination over his environment and over his fellow men. keywords constructive second number class, intuitionism, ur-intuition of mathematics, possible cardinalities, logic and mathematics, solipsism, continuum, objectivity, apriori, actual infinite
If length is truly quantized then it makes no sense to talk about irrationals anyway - to me the question becomes whether we want our pure mathematics to capture the real mathematical universe, or talk about going further at the expense of seemingly useless accuracy. I think this overhaul has a lot to do with reality vs a "super reality" accuracy with our math... the irony being the latter is so error ridden.
There really has to be a systematic study on identities. There is so much 'heavenly assistance' in proving your UHG laws. The identities need to be numbered and named systematically somehow. Perhaps we can start with (a+b)^2 = a^2 + b^2 + 2ab and name it (1) and then the binomial theorem 1.1, as its generalization.
Yes we do need a much more systematic and thorough understanding of identities. It's a subject that has only really come into view in the last few decades, with computers helping us find identities much more efficiently.
Wouldn't it be great if there was a mathematical manipulative that helped the young guide themselves towards the foundations: (your)multi-set theory, chromogeometry, UHG/rattrig. Maybe puzzles can be very useful here. Think about all the video games that have been made and the mathematics that have been used to make them... But maybe we need the finite fields at the heart and not the rational numbers. You know that the finite fields are key, we all wish you would stand on them. Can you solve a rubiks cube? Have you used a rubiks cube to study the theories you have developed? I think you might see that this will help you guide us, we need a lot of help. Maybe if you showed us how a rubiks cube can model the continuum very nicely using chromogeometry, rat trig, UHG, multi-set theory... You have really brought us out of the darkness... but keep in mind what motivated you to rat trig. What motivates me towards the foundations of mathematics is a HONEST conversation with students.
@Jo Reven, Complex numbers are fine, as long as you define them properly, meaning as combinations a+bi where a and b are rational numbers. Similarly the true octonions are built from rational numbers. The fictional realm of "real numbers" ought to play no role in careful algebra, where we aspire to making computations on the page, completely precisely and correctly.
Yes, yes. "Dihedrons and their conjugates" -> looking at Bott Periodity from a Information Theoretic (Computer Science like perspective) more work needed on Math Itself edgeoforder.org/mathitself.html
Dear Norman. Since you are going to talk about axioms of real numbers, could you please have following commentary at the very end of your next MF120 video: " Now we know that with the least upper bound property every positive real number has a square root, that is again a real number. But, of course, that is not the case with _my_ _precious_ _rational_ _numbers_ [now you should smile like Gollum]. It is delusional, wishful thinking to even ask what is the square root of a number. There is no such calculation! It makes no sense! It makes no sense!! ... "
Hi Dr. Wildberger,I've watched all of your math history videos, and I've gotten through many of your math foundations videos. I have learned a lot from you, and I respect people who are brave enough to challenge the status quo. It is therefore with great hesitation that I must disagree with your position on real numbers. I posted a video reply here: th-cam.com/video/qiZP5sakM9M/w-d-xo.html. In all of the videos I've watched I have been hoping that you would address the elephant in the room with regards to integers. You haven't. I wonder if it is because this elephant represents the fatal flaw in your argument to do away with irrational numbers entirely, or if it is because you haven't considered this difficulty seriously.The problem with your position is that the ontology of existence is continuous. The proof of this lies in the fact that discreet entities (those entities that can be successfully described using integers) arise from continuous entities. However the reverse is not true, you can never have a continuous entity arise from the discreet. For example, the intersection of two continuous lines is a discreet entity called a point, which can be described with an integer. However, no matter how many points you have you can never create a continuous line, the best you can do is have a row of points placed side by side. Even with an "infinite" number of points, you will never have a continuous object, because this would constitute a contradiction.This is the deep reason why there exists geometrical relationships that are physically impossible to describe with integers. The relationship of a circle (a continuous entity) to the line that bisects it (another continuous entity) is an example. Another example is the relationship to the line that makes up the side of a square (lines are continuous entities) to the line that runs diagonal through a square (also made up of lines, which are continuous). In these cases, it is logically impossible to describe these relationships with integers because integers can only fully describe that which is not continuous. Because the discreet is in reality a sub-category of the continuous, and integers can only be used to describe the discreet, you are left with a system that is strictly limited to this sub-category. When faced with a relationship that falls outside of the limitations of your system you simply claim that anything outside your limited system does not exist. For example, when graphing the square root function, you draw circles anywhere an integer cannot describe the continuous, claiming that the point does not exist. The reality is that the point lies in a place on the continuum that cannot be described in terms of a unit. Not only does this magnitude exist, it MUST exist, or the curve wouldn't be continuos in the first place.This is a defining characteristic of the continuous. A CONTINUOUS ENTITY IS THAT WHICH HAS MAGNITUDES THAT CANNOT BE DESCRIBED WITH INTEGERS. IF EVERY MAGNITUDE COULD BE DESCRIBED WITH INTEGERS, THEN THERE IS NO REASON TO ASSUME THAT IT IS CONTINUOUS. This is also the deep meaning behind the Greek's proof regarding the Pythagorean theorem. The Greeks proved simply that integers can only describe part of continuous objects, and therefore any system based on integers (including yours) will be likewise limited. You have tried to get around this either by simply claiming that anything outside your limited integer-based system does not exist (and it certainly does) or you have employed techniques that mask the problem (such as "quadrants"). A quadrant only demonstrates that you can square a magnitude that is not describable with integers to find one that is. It still does nothing to express the true magnitude of the original incommensurable side itself. So how do we express the incommensurable with honesty? Certainly not by trying to deny its existence or mask it through some operation such as squaring it. Conventional mathematics has developed the use of non-integer based symbols to express that which cannot be expressed with integers or any system based on integers (such as yours).What you are attempting to do is worse than what you are trying to "fix". You are attempting to sell a contradiction. I write this with deep respect. I have learned a lot from you, and it gives me no pleasure to disagree. Respectfully, Mark Botirius
+Mark Botirius Hi Mark, Thanks for the thoughtful and extensive comment. I suppose my main reply is that: existence is usually not the prime issue. The prime issue is usually always definition. The difficulty with the continuous is that it is highly challenging to even define what we mean by it. There is also something of that difficulty in defining the discrete, almost always in terms of natural numbers or something close to them. In my view, this is the essential reason why the discrete trumps the continuous when we inquire into the foundations of mathematics_ it is just a lot simpler defining arithmetic with natural numbers than it is defining arithmetic with rational numbers. And that is orders of magnitude easier than defining arithmetic with "real numbers". So I would ask you to consider your comment in the following light: which of the terms that you are using has a prior accepted definition? Without such prior definitions, I am afraid the argument drifts towards the philosophical.
+Streets, blocks, experimental music Hello, You asked, " If you have an exact theory of geometry (with all fundamental definitions), can you refer to that?" The basis for my comments are ancient and go all the way back to the Greeks. This controversy is not new. It has existed for millennia. Dr. Wildberger has simply brought it back into view, which is good, because with the invention of analytical geometry and the Cartesian plane, we have forgotten some very fundamental truths established by Aristotle long ago. Like Dr. Wildberger, the Pythagoreans insisted that everything was number. Later, Aristotle corrected the Pythagorean view by correctly asserting that everything was not number, there was a distinction between number and magnitude. For Aristotle, number referred to the discreet only, whereas magnitude referred to the continuous. According to Aristotle, continuous objects are not made up of points. By definition they can't be, because that would be a contradiction. Integers (i.e. rational numbers), being a discreet entity (by definition) can only refer to some point on a continuous object. However, because the object is continuous, there will always exist some magnitude that lies outside of any two points, The magnitude certainly exists, however it is simply not possible to express it in terms of a unit (i.e. a number) In other words, the discreet is contained WITHIN the continuous, NOT the other way around. Real numbers are simply the method employed by modern mathematics to account for what lies outside the realm of integers. Of course, real numbers do not make sense in terms of rational numbers. How could they? Real numbers are trying to express the continuous to human beings that can really know only the discreet. But then again, they aren't trying to make sense in terms of rational numbers, hence the reason why we place them in a category by themselves. In other words, in this very ancient debate, it appears to me that Dr. Wildberger has taken the side of the Pythagoreans, and I have chosen the side of Aristotle. Which side are you on? Mark Botirius
+Streets, blocks, experimental music Hello, Thanks for the reply. I want you to know that this is a very important discussion. Unfortunately, I have a report to write for my immunology class and I have to study my organic chemistry at the moment (I am a microbiology student), so I will not have the time to give as thorough of a reply as I would like, and I may need to put our conversation on hold for a week or two while I complete my assignments. If it is ok with you, hopefully we can continue the discussion a few weeks from now when my load is not as heavy. I did not want to leave your questions unanswered, however, so I hope this very brief reply will be sufficient until a later date when I have more time. Regarding Aristotle, I don't know if you will be able to find the information you seek on the internet, so I wanted to share with you where I get my information from. For example, on page 342 of Thomas Little Heath's book "A History of Greek Mathematics" we learn that Aristotle held that the continuous "could not be made up of indivisible parts; the continuous is that which the boundary or limit between the two consecutive parts, where they touch, is one and the same, and which, is kept together, which is not possible if the extremities are two and not one." Heath not only informs us of Aristotle's mathematics, but as the name of the book implies, he walks through the mathematics of the Greek's from what is considered to be the first mathematician/philosopher, Thales. I've owned a copy of this book for quite some time (recommended by Dr. Wildberger in his math history videos as well), maybe you can find one at your local library. In addition, I used to own pdf's of Aristotle's works on my Ipad, but a while back, they mysteriously disappeared, and I haven't replaced them. I intend to buy paper copies when I am not so financially distressed in the future. But, of course, most of my information came directly from the works of Aristotle himself. Lastly, I want to point out that I agree with your points regarding the theoretical world we imagine in our thought experiments and the actual real world around us (that they should agree). My interest, and the reason that I care about this topic so deeply, is that my concern is that we understand the actual world, and that our mathematics reflect the actual world. However, this is a very deep and complex issue. If we are able to continue our discussion at a later date, I will state my case as to why I hold that, existence qua existence is continuous. This ultimately will bring the discussion into the realm of physics, and unfortunately I am no physicist (I am no mathematician either), but I will do my best. I look forward to continuing our discussion at a later date. Until then. Mark Botirius
+Mark Botirius Norman addressed all this numerous consideration in this series. That's why MF is so lengthy, too lengthy for comfortable comprehension to my taste. You didn't watch key videos or didn't listen/followed his arguments and references. >My interest, and the reason that I care about this topic so deeply, is that my concern is that we understand the actual world, and that our mathematics reflect the actual world., this is a very deep and complex issue. It has been formalized by Turing, modern mathematicians just ignore him and stick with mysteries of platonic objects.
+pieinth3sky Hello, >You didn't watch key videos or didn't listen/followed his arguments and references On the contrary, I made it through 45 of his Math Foundations videos (I've participated by leaving comments on several of them). I simply didn't have the time to watch all of them. There are hundreds! However, I've seen several videos make the same claims as the one that you bought to my attention. Claims for example, that there exists magnitudes where a circle and it's diameter do not actually intersect because we are unable to express the relationship in terms of rational numbers. The video you brought to my attention, much later in the series than I had the time to wade through, did not make any new claims, and therefore my counter arguments were not affected. Thanks for bringing it to my attention however. > just ignore him and stick with mysteries of platonic objects. Thanks for the advice. Have a great day! Mark Botirius
I fail to see any reason to believe that a noticable number of mathematicians will stop using real numbers any time soon. Furthermore, I am still under the impression that the "problems" with real numbers touched upon here are philosophical and should be discussed in the context of set theory.
I am hoping that the realization that the theory of real numbers is logically flawed, as I have at length described in the last dozen or so videos, will provide a strong inducement for mathematicians to start to re-examine their assumptions.
njwildberger Something can be logically flawed and still be useful in practice. If real numbers offer a utility and ease of use that can't be found in the alternatives mathematicians will continue to use them, flawed or not.
Here is how Dedekind proves strictly mathematically that there are infinite systems 66. Theorem. There exist infinite systems. Proof My own realm of thoughts, i. e., the totality S of all things, which can be objects of my thought, is infinite. For if s signifies an element of S, then is the thought s0, that s can be object of my thought, itself an element of S. If we regard this as transform _(s) of the element s then has the transformation _ of S, thus determined, the property that the transform S0 is part of S; and S0 is certainly proper part of S, because there are elements in S (e. g., my own ego) which are different from such thought s0 and therefore are not contained in S0. Finally it is clear that if a, b are different elements of S, their transforms a0, b0 are also different, that therefore the transformation _ is a distinct (similar) transformation (26). Hence S is infinite, which was to be proved. The point is in the first sentence. He actually said that infinite sets exist because he is able to think of infinite number of things. He even mentioned his ego in this very scientific proof. As opposed to him, Cantor stated that actually infinite sets exist because God is actually infinite. I never thought of God as a set but there you are.
njwildberger If you don't like Dedekind's cuts, go and read (Super boring): Edmund Landau - Foundations of Analysis. If you don't like Cauchy sequences, go and read (Super interesting): Terence Tao - Analysis 1.
Juho x If you have read them, please report to us: do they address the concrete issues that I have raised in my recent videos? But I will be talking about those books in some future video discussing more absent theories of `real numbers'.
njwildberger Our axioms origin how our mind understand nature. Existence surpasses human computability (e.g. two lines _really_ meet, whether you like it or not.)
Construction by description only is a main tenet of postmodern left philosophy, i.e,, woke and cancel culture. Is there a causal connection between the idea of real numbers being real and postmodern leftist political philosophy? Is real number theory an overreach of Kantian/Aristotelian Reason? And the procedural processes of logic?
@@njwildberger Prof. Wildberger, I never dreamed I would engage in discourse with you! The next step should be a 3way talk with philosophy prof. Stephen R.C. Hicks. Thank you!
Constructive Mathematics has made beautiful progress recently & Intuitionistic Type Theory has had some elegant software, go downoad COQ and Isabelle people!
I completely understand that the rationals are more beautiful than the "real" numbers. Yet, it is unavoidable that we deal with irrationals like the square root of 2 or pi. Before, you said the best way of going about these numbers were to bound them, like Archimedes did. Though doesn't this take away from the exactness we are looking for?
In applied mathematics, we will always need approximate values of sqrt(2) or pi. That is different from postulating that these are exact numbers belonging to some distant `real number field' whose definitions elude us but that we insist form the basis of most mathematical thought. As we progress through this series, your dependence on `real numbers' like `sqrt(2)' and `pi' will naturally diminish.
True, though if we want to study circles without their circumferences or areas, then we simply aren't doing them justice. Similarly with squares and their diagonals.
Swiftclaw123 As we will see, the quadrance of a line segment (the sum of the squares of the vector coordinates) is a more than adequate replacement for its length. It is more accurate, more algebraic, and one can compute with it in a finite precise way, actually over a general field. It also extends to more general bilinear forms. So we should not be too quick in assuming that the transcendental notions that are currently used are necessarily the best ones: often they demonstrably are not!
No, no, no Norman.. "Boo" to matrix algebras. Please look in to geometric (multivector) algebra. If you value elegance and naturalness in mathematics matrices are abominable. All the matrix algebras can be reexpressed far more "natively" with the Spin groups embedded in geometric algebra. See David Hestenes' work, and the papers from the Cambridge GA Research Group.
While I think GCH algebra (a much better term than "geometric algebra", please consider it) has lots of potential, I see it as a more advanced tool. To establish it simply and logically in an elementary setting--without making key assumptions, is not so easy!
You keep saying that the real number system/theory is logically flawed. But in what way, can this be shown by a proof, logical or mathematical? Is it possible to show a contradiction in the system, or is "logically flawed" just an opinion that it might be shown logically inconsistent in the future, but we are not really sure which is right at the moment? I certainly see both advantages and disadvantages with real numbers as compared to rationals. One disadvantage is the not too exact correspondence to the non-dimensional points in the line, where rationals are exact and irrationals are, well, not so exact--in that there are many different points corresponding to single irrational real numbers due to the completeness property (pairing up all the points on the line with irrationals where ther are no corresponding rationals). Maybe the completeness property is what is bothering you most, the lack of precision in the reals compared to the rationals? If that is the case, I don't see a solution in other number systems, because they work in similar fashion to the reals, but maybe one could construct a better, "more complete" number system that could represent all points on a line with separate numbers. Until then, maybe a hyperrational system might give good enoughapproximations to do math with, extending the rationals with infinitesimals and infinites. That way you would get perfect precision AND infinity in math!
MisterrLi As several people have pointed out in comments of prior videos, it is unreasonable to ask someone who is critical of a subtely flawed theory to come up with a contradiction. There are other, much more common, ways in which logical weaknesses manifests themselves: in a lack of explicit examples and computations, in arguments which border on the philosophical, in confusions about terminology. If I point out weaknesses in your theory of real numbers, you have an obligation to defend your position and answer the charges, don't you think? In modern mathematics, contradictions are finessed away as semi-amusing paradoxes. For example, in a recent video, I have shown that with a Cauchy sequence point of view, all real numbers are in fact the same. Is this a contradiction?
njwildberger "Is this a contradiction?" If you really did show that the Cauchy sequences all lead to the same number (or something in that direction), that would certainly qualify as a contradiction, debunking the real number system as we know it. Why not publish that? If, on the other hand, it is all about which axioms or definitions to pick in the first place, it is less convincing. Is ZFC ok? Not really sure what you mean by "logical weakness" if the system can't be shown to be false in some way. Well, I do agree that the real number theory could be explained much clearer. This is something I enjoy doing, visualize math concept using images and animations to make them easier to grasp. With the addition of interactivity and textual info of course.
MisterrLi Actually I did publish that result! It was published on this TH-cam channel, on Nov 23, 2014, in the video MF114: Real numbers as Cauchy sequences don't work! with link th-cam.com/video/3cI7sFr707s/w-d-xo.html.
is this guy a mathematician or a sociologist? sometimes i get mad and yell at cantor until i put on some schoenberg and reconsider the radical subject. anyway, he comes off a little schizoid in the videos with the constant video cuts, and some of his comments were insightful whereas others were strange. perhaps some of this will occur in progression but there are 'famous potholes' that have gotten in the way of some of the more radical shifts he suggests. the psychology of mathematics is perhaps going to become more adept at its methods, and theological mathematics is a much stronger social force than many pseudorationals expect (prosperity gospel is a booming business and number theory portal for the masses). philosophy will probably expand into a nightmare. high science will be safe from major inconsistencies. the economics will likely mitigate anti-traditional approaches up to the point of futility. "alternate" opinions will not have access to mathematics, but may bubble up 'math-isms' which have huge importance in fields not conventionally labelled mathematics.
Mr. Wildberger, you "hate" the concept of real numbers and "love" the rational numbers. Thinking about that, it seems to me that what you hate is infinity, because when the numerators amd denumerators of real numbers become larger and larger you naturally end up having real numbers??
+Juha Immonen Let's say that I am crusading against imprecision. I have no problem with infinity, in the same way I have no problem with ghosts. If someone is able to define either of these terms, then we can have a discussion about whether or not such things exist. But first we must clarify what it is we are talking about.
njwildberger Thank you for the reply! One question: If we have the right triangle for ex. with sides 3, 4 and 5, do you find it meaningfull to speak about the lengths of all sides? If we then have a more general right triangle with two integer or rational side lengths, would you say that we might have a third side which has no length at all ? :-)
Tiny nitpick: you pronounce David Hestenes’s last name a bit differently than he pronounces it himself. You can hear him say it right at the beginning of this lecture, th-cam.com/video/ItGlUbFBFfc/w-d-xo.html
First of all I want to say I'm a long time fan of your videos and to be quite honest I've really enjoyed and learnt alot from them, even and also from the lectures in which you present this problem of real numbers. But I want to say something that could refresh your way of thinking and teaching. I think you are too much focused on this fixation of yours that real numbers must be eradicated. And by being too narrowly focused on this one problem makes your vision of other problems more blurry. I'm definitely not saying you are blind to other problems in mathematics but I think priorities must be made clear before presenting anything at all. I think TH-cam is a great platform to learn any subject, like mathematics, and I would so much like to see more videos from you that really teach mathematics. Videos such as this one I think should be a side project. You can't just state a number 2 to students before teaching that 1 + 1 makes 2. In a simple sense. Like someone said earlier, this seems more like philosophy almost. You have a great teaching style and would be really happy to REALLY learn mathematics. Teach us the way, master!
Sir ,, I really admire and feel satisfied to see some one who is an anomaly in this closed , no room for discussion kind of world of Mathematics. Let me ask you something . Have you been interested in Indian Spiritual and Mystical Traditions ever ?
A radical remedy is needed if we want to brush irrational numbers under the carpet: The notion of _distance_ is not defined in a plane, since it's pure 1-dimensional concept. More 2-dimensional is to talk about _area_. This works since square of a rational number is again rational, and taking square roots is not even defined! Now we can actually construct everything in *direct* and *finite* fashion, so no need of fictitious infinities or _argumentum_ _ad_ _absurdum_ or wishful thinking in our proofs! (This was all parody)
njwildberger So was I {:?) . I am sorry Juho, think of infinity as being stuck in an obsessional compulsive loop, or, think of it as mathematical delusion. Ok, they are not infinite because we die before we get to the largest number.
njwildberger Why should I be converted? With analysis we (all) can do difficult-convoluted-geometrical-calculations and get the perfect answer within ε, with ε as small as we wish. It's not an ugly approximation -- it's pure mathematics! But if you don't know what the notion _limit_ means, only God can help you. Altought in _continious_ math everything goes _smooth_ (pun int.) I'm willing to belive that physical world is discrete: energy in quantized (_quanta_), charge is quantized (_elementary_ _charge_), matter is quantized (_quark_), so it's likely that time and space are quantized too. Maybe that fundamental scale (Planck's lenght/time) is the realm where "discrete calculus" is needed. But changing the notion of metric with every dimension for just to salve the rational numbers... well, that's just bullshit...
Juho x Pure mathematics based on a lie? Just because it is simple? What happened to the quest for the truth? But who cares for the truth if belief in God makes ones everyday life easier (and also I will live on after death, tralalala...) In one of analysis books the author gives the usual definition of an ordered pair (a,b)=: {{a},{a,b}}. He calls this definition "startling" (of course it is totally senseless) but then he praises it because it is easy to use it to prove basic properties of ordered pairs. What kind of mathematical thinking is that?!?!?! Just because a lie makes life easier does not make it excusable.
Aleksandar Ignjatovic said "the usual definition of an ordered pair (a,b)=: {{a},{a,b}} ... (of course it is totally senseless)" You don't accept that finite sets exist? If you do accept that finite sets exists, why would {{3},{3,7}} be totally senseless?
You won’t readily find a willingness for debate on all these ideas. What you are proposing requires nothing short of a radical reformulation of modern mathematics. Instead of constantly banging your head against the wall and talking and complaining about it till you’re blue in the face, start working on your own comprehensive treaties on these subjects and show people by example what could be done in all those areas that you had pointed out. Then it would be up to the mathematics community to decide which approach it preferes to follow. However, if your aim is to recruit converts to do the work for you, you’d be seriously swimming against the tide.
@NothingMaster, Swimming against the tide is no problem, often the view is better. If you check out the Algebraic Calculus One course, or the Universal Hyperbolic Geometry or Wild Trig series, you will see that I am doing just what you propose--showing by example that there is a simpler and better mathematics. But at the same time I can also pleasantly point out that the current form of pure mathematics is logically very weak, and explain exactly how. As for whether or not pure mathematicians currently can appreciate these concerns, that is much less important than providing a road for future mathematicians to explore. Forward!
Thank you for the MF* videos. Your "new math" reminds me something. Galois re-defines the essence of symmetry thanks to group action over sets* (and then over fields). So groups becames the bricks of Mathematics. Now also of physics, from General relativity to Supersymmetry. Algebra instead of Geometry. The central role of the rational and finite fields in your approach reminds me this (old) "new language". Boole vision about algebraic logic. The Boolean Algebra is one of my favorite (old) "new language". One of the rivers that flows from it is the concept of partially ordered set, and then the "Universal Algebra" (thanks to Whitehead and then G. Birkhoff). Algebra instead of Logic. The central role in your "new math" of "algebraic structures that can be defined by identities" (this is the definition of varieties in Universal Algebra) reminds me this (old) "new language". One of the most strange and beautifull theory that cames out in this context is the "Pointless Topology", in which we focus on lattice instead on neighborhood. Poset instead of Topology. This reminds me your work on Rational Trigonometry: quadrance and spread instead of length and angle. In rational trigonometry we can regain length and angles but also in pointless topology we can define "points". This is a difficult message to universalize. Also the central role of Algebraic Geometry in your Rational Trigonometry (as your articles points out) is in the direction of "Algebra instead something". Working on the (e.g. qradrance and spread), in an affine or projective context, is one standard strategy of translation between other languages to Algebra. Other powerfull "translation tools" are Linear Algebra and Computability**. You cite them. Stanley (article of 1989, book: Topics in Algebraic Combinatorics, 2013) works with Linear Algebra to prove that the Boolean lattice is Sperner. Algebra instead Combinatorics. I think that your "new math" lives in the present, not in the future. The previous examples deal all with the question: can we overcame Analysis? All these theories wants to subtract something to Analysis. All the previous theories (and Analysis) lives together, and we can learn from them. *Is not exactly a group action over sets as we now define it in 2014. ** The "hash table revolution" in Computer Science give us functions as "first-class" object. Is Object-Oriented programming "the way" to overcame floating point? I don't think so. Big integer algebra (also thanks to DFT) is mature?
"Daddy, we learned today that 2+3=5. Why is that?"
"Well, son, here is how this goes. Number 2 partitions the whole real number set into two subsets. The one with numbers less then 2 and the one with numbers greater then 2. Same goes for number 3. So if you want to add 2 and 3 this is what you have to do. Take every number from the less then 2 subset and add it to every number from less then 3 subset. You will get a certain set of numbers. Also, add every number from the greater then 2 subset and add it to every number in the greater then 3 subset. You will also get a certain set of numbers. By the axioms of number theory it will turn out that those two sets of numbers actually define number 5. Do you understand?"
"Sure, piece of cake. Is it too late for me to start training baseball?"
Ignjatovic said "Number 2 partitions the whole real number set into two subsets." It is not the "real" numbers, but rather the "rational" numbers.
Ignjatovic said " By the axioms of number theory it will turn out that those two sets of numbers actually define number 5." It is not the axioms of "number theory", but rather the axioms of "set theory".
***** Indeed but it does not make it less ridiculos.
Try explaining 2+3=5 if your 'concrete model' is arithmetic as performed by a modern CPU. That is, you explain it in terms of behaviours of an interconnected bunch of microscopic transistors.
I've got q question; How can the reals be denser in irrational numbers if every irrational number has an arbitrarily close approximation in rational numbers? It is not true that the irrational have infinitely many representatives between any two rational numbers because between any two irrational numbers is an irrational number. I understand that the converse is also true. That the rational numbers can be approximated arbitrarily closely by irrational numbers, theoretically., and that between any two irrational numbers is a rational number. What is the difference in symmetry that could account for the density results? I realize that we have the concept of countability, but is there any other explanation, for example symmetry of some kind, that would justify the results?
A great explanation from MSE:
"I think the most intuitive explanation I have heard is to considering writing down a rational number in decimal form. This means that either it is a repeating decimal or a terminating decimal, for example 2.373737... or 0.42, which we will write as 0.420000... Now, consider the probability of randomly writing down a number. So you have ten options every time you go to place a digit down. How likely is it that you will just "happen" to get a repeating decimal or a decimal where you only have zeros after a certain point? Very unlikely. Well those unlikely cases are the rational numbers and the "likely" ones are the irrational."
@@persistenthomology But we seem to have to assume that there is no difference between "completed" or "accomplished" infinity you speak of and the "becoming" infinity that our limit concepts deal with . However, if we only considered "becoming" then there would be no irrational numbers. Uncountability is an emergent property of completed or eternal infinity. (How do the rules for handling the two types of infinity differ?)
To your specific point, we have rational numbers, 3.000..., 3.1000, 3.14000..., etc., on out to any given number of digits; whereas, the irrational has Pi. All of these numbers are said to exist, and there are countably infinitely many of them. But the collection of rational numbers does not contain Pi.
On the other hand, the irrationals obviously have an infinite collection that approximates 3.000..., 3.1000..., 3.14000..., etc., to any degree, and that collection does not include any of 3.000..., 3.1---..., 3.14000..., etc. The irrational approximations within the given bounds will be uncountable. For instance, estimate 3 with error less than 0.1: 3.01...., 3.02...., etc., where ... represents a nonrepeating, nonterminating sequence of digits. We can see that the numbers in ... are uncountable given the argument for the uncountability of the reals.
So, we are led to conclude that the collection of irrational approximations of a given rational number is much larger that the collection of rational approximations of a given irrational number.
The argument you provide states this probabilistically but only works if we assume real numbers, i.e., completed infinite numbers, or numbers that reach their limits. Given that, it makes the density question much more intuitive than my approach.
To those wondering how much would be lost by reworking mathematics on a purely rational, logical, clear and consistent basis, with all the "constraints" such an approach entails, the answer is that *nothing would be lost at all*. Any standing result that has any real world significance, or even the possibility of ever having such significance, must be able to be improved by a more rigorous approach.
If any standing mathematical result is found not to be amenable to a fortification because it absolutely relies on incoherence for its foundation, what does that say about such a result?
The fear that "much of modern math would be impossible without real numbers" is unfounded; for any theorem, either it is possible to make it rigorous, or it is impossible, and if the latter it had no value in the first place. Rest assured then that every result that really is or ever could be useful cannot be harmed by being put on a more rigorous footing.
The point is not that we jettison all results that contain handwaving, because making things more precise takes real work and understanding and cannot be done all at once; the point is that we make a continual effort to make foundations more rigorous.
This is a crucial part of how progress is made in any intellectual investigation. You start with terms that seem clear enough, but later you encounter situations that can only be overcome by more precisely defining the initial terms. You may have to introduce finer distinctions, or adjust definitions so that they don't fall into incoherence. As you proceed to more exacting analyses and applications, you find that some terms are not carefully enough conceived to be useful at that deeper level of analysis, so they must be sharpened up.
If you avoid this or worse, go the opposite direction as the mainstream has done, you end up hitting roadblocks and getting more and more removed from the pure math promise of being even potentially applicable to the real world (being homomorphic to real world situations, or at least possible ones).
For example, "sqrt(2)," as an ambiguous term referring sometimes to approximations and sometimes to the incoherent notion of an endless non-repeating decimal, can be made more rigorous by defining it as "a number whose square is approximately 2" or "sufficiently close to 2 for a given purpose" and cutting any ties with the incoherent idea of a so-called number whose square is exactly 2.
At first this ambiguity may not have mattered, but certainly by the time people are using that very ambiguity to justify further theory it is time to halt the proceedings and re-examine the foundations.
Amanojack A This is an excellent comment, thank you very much!
I find your general thesis ludicrous on its face. First, correct Mathematics need have no connection to the material world. Secondly, that having been said, as it turns out much of mathematics as currently theorized does obviously explain much of although not all of theoretical Physics. Third, I have not seen all of Professor Wildberger’s videos but his arguments against the real numbers which I have seen so far seem incredibly weak. PS: perhaps in years hence, ala Einstein’s modification of Newton’ s theory, Prof. W will after all turn out to be correct. But for the moment, his ideas are so far out of the mainstream that his theses seem not to be taught or even seriously addressed in any top tier university. or research institute. Please correct me if I’m wrong. J Wolf, PhD Math
I would say that sqrt(2) should be defined as a generator for a field extension to |Q, much the same way that sqrt(-1) is defined. In this way sqrt(2) doesn't lie on the rational number line at all, instead it is part of a more or less Lorentzian 2 dimensional rational number plane. Also the two solutions of sqrt(2) are completely symmetric this way! What these rational "approximations" of sqrt(2) should be replaced with instead are Archimedean *places* for sqrt(2). Other, Ultrametric (p-adic) *places* for sqrt(2) exist as well. This approach seem as though it could be made completely rigorous.
Also, such things as cbrt(2) do not even live in a rational number plane, instead they live in a rational number space of 3 dimensions, but with only Abelian symmetries valid.
Of course, if we extend the field even more, so we get both cbrt(2) (solution in x to x^3 - 2 = 0) and a primitive cbrt(1) (solution in x to x^2 + x + 1 = 0), then we get a 6 dimensional space with non-Abelian symmetries described by Sym(3), rather than Alt(3).
By the way, the specific example of x^3 -2 = 0 is not a Galois polynomial over |Q, so it is not the best example, since its field extension has only the trivial symmetry.
The polynomial equation x^3-3*x-1 is a much better example, since its field extension actually has Alt(3) as symmetries.
This is interesting. I've been dissatisfied with real numbers in mathematics many years and thought there must be another foundation that reflect our reality much better, something based purely on relations, proportions, geometry and logic. This seems to be something in that way.
J6k_O6i74fw Take a look at the youtube video entitled: Dynamic Maths. The Fifth Arithmetical Operation. The Wonders of Number
@@redpillmath You answered someone "J6k_O6i74fw" -- who is that? I can only see one answer to my comment, and it is yours.
Dr. Wildberger,
Forgive my confusion, but I am still confused as to what exactly your objection to the real numbers are. I’ve seen various statements about strange constructions of R, but I have yet to find a problem with their definition. As an analyst and a topologist, this topic bears heavy on me. We can both agree on the metric space (Q,d): the set of rational numbers equipped with the Euclidean metric. Note, of course, that this topological space is non-Cauchy complete; a simple example is that the partial sums of the reciprocals of the squares fails to converge. What would be wrong with defining R as cl(Q), that is, the closure of Q such that the resulting set is Cauchy complete and contains all the limit points of any sequence contained in Q?
Thank you for your time.
Hi Peter, There is not just one definition out there. There are several approaches to conjuring up the "real numbers", and my point is that none of them work. In this series over the last few dozen videos I have critiqued most of the major "constructions of real numbers" and shown them to be hollow. But you can test this for yourself: ask how you would demonstrate to a jury that you really know what real numbers are, that you understand the operations with them and can do arithmetic with them. Why not start on my favourite challenge: to compute "pi+e+sqrt(2)". The whole exercise immediately grinds to a halt. It is like getting a class of 8th graders to calculate, in the rational number framework, the sum "1/2+1/3+1/5" and all you get is the question repeated back to you and philosophical mumblings.
I do encourage you to have a good look at this playlist and go carefully over all the "constructions of real numbers" videos.
I should also add that I most definitely do not agree with talking about "(Q,d)". There is no reasonable meaning that I can attach to the "set of all rational numbers". That is day dreaming in my view. Furthermore the metric involves taking square roots, which you cannot in general do.
@@njwildberger what if you’re wrong and reality DOES contain infinite sets. Just because there are not infinitely many particles (according to physicists) does not rule out that there exists infinitely many of something in the universe.
@@njwildberger i think you are permanently handicapping yourself by being so closed minded.
@@mlevy2429 What if I am wrong and there are enlightened leprechauns? Or hierarchies of transcendent angels? Or just turtles all the way down? Yawn.
I have a question: How to 'convert' (transform) a real number back to an integer *WITHOUT* the usage of ceiling or floor? If x = 123.45 and we get the modulo 1 of it, say, 123.45 mod 1 I can get 0.45 and then subtract from the original x. BUT as I saw in Wikipedia and in some books, the modulo function use floor function (Knuth). I need a 'downgrade' from Real Number to Integer using pure mathematics. How can I get rid of the mantissa of a log? I'm trying to create a pure function that will count the digits of a given number without the usage of algorithm of any computer language. Example so far: n = 123.45 , log (base 10) of 123.45 + 1 = 3.09149109426795 , so, the number 123 have 3 digits + 0.091491.... how can I get rid of it? I'm watching the videos and discussions about Real Numbers, Set Theory, Logic and Infinity here , so, maybe someone can give me a hand. Thanks for reading.
Hello i read that Completeness of the real numbers axiom is sufficient to proof everything we need for real numbers is this not not true??
There are even more simplifications in mathematics without real numbers: for example, the conditions can be simplified for the subspace theorem, "v in S => \lambda * v in S" can be eliminated, as this can be deduced from the conditions "0 in S" and "u, v in S => u + v in S".
I have some honest (and maybe stupid) questions about the central role of the rationals... how would we deal with the 'pi' we find in circles and its "real" proportions? How would we deal with 'e' and the natural logarithm or even with 'square root of two' in a square's diagonal in proportion to its side? Aren't there some "real life" applications with "real" proportions?
@DrNorman I have one question. Although rational numbers can be written and manifested as ratios of integers but are most of them not also produce repeated patterns of infinite decimals like 1/3 , 5/3 , 9/7 whose repeated patterns are obvious? There are also numbers whose repeated patterns occurs after several thousand digits and are not obvious or as you have explained in your videos that it gets harder to compute or represent numbers as they get bigger so how come rationals do not pose the same challenge as of reals.
I know there is one distinction that reals cannot be represented completely but rationals can in ratio form and cannot in decimal form.
PS: I am enjoying having fresh perspective on foundations of maths. It is really enlightening me in a sense that explore for solid foundations and easy to manage amd understood mathematical methods.
You should think of arithmetic with fractions as quite separate from "arithmetic with repeated infinite decimals". The latter is actually highly problematic to lay out clearly, and is notoriously avoided by educators. I know the two are conflated, but try to not confuse the clear arithmetic with fractions with the much more dubious "arithmetic" of "infinite decimals". These latter are like ghosts in mathematics.
@@njwildberger So should we totally outcast decimel expansions or we should develop a better theory about decimel expansions? After all fractional values eventually computed or approximated in decimel expansions to get results in engineering, computer science etc and when fractions get bigger does it not pose the challenge of arithmetica and computation? Can we totally ignore decimel expansions?
@@sang81 Decimal arithmetic is crucial for applied mathematics and applications, so of course it must remain. What we must rather look more carefully at are the current overblown claims that the finite decimal arithmetic of the applied mathematicians and engineers can be extended to a valid arithmetic of "infinite decimals". And it turns out that even "infinite periodic decimals" have an arithmetic which is almost nowhere spelt out completely correctly: even though in principle this is possible. Basically we have just bought into an exceedingly sloppy approach to this entire subject, propelled by massive wishful thinking.
It's also amasing sin and cos to 45 is the squire 1/root of 2, so there is a connection between pi,e, i and the sqaere root of 2. I have no idea why? :-)
Could you please elaborate on the point that trig only "adequately deals with" two triangles?
+Fred Frey Sure. The average student knows the trigonometry of the 45-45-90 triangle and the 30-60-90 triangle. These are the two triangles that at least 90% of all trigonometry questions revolve around, since these are pretty well the only ones that students "know" the trigonometry of.
Here is one problem with denying irrational numbers: there are integer solutions only-that have to be expressed using square roots.
Example is Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, etc. Given any 2 consecutive numbers to get the next number add those 2 numbers together. Clearly every member of a sequence is an integer. Yet to get formula for any member requires square root of 5-which you don't believe in.
Similarly there are cubic equations that have only integer solutions, but formula that finds these involves square roots and cube roots.
Lastly your objections that irrational numbers have infinite decimal expansion and so require more space then the universe overlooks the fact that we don't know the size of the universe (or multiverse)-so we need a system that can handle any size. Even if we knew the size of the universe-we still would need system that is bigger then the universe. Example: universe has only 3 objects-A, B, C. To list all possibilities you would need these sets: 1) empty set 2) A 3) B 4) C 5) A, B 6) A, C 7) B, C 8) A, B, C-so 8 sets in the universe of only 3 objects: more sets then its universe!
Mark Igolnikov I would suggest getting to work on figuring it out then. Try not using set theory.
Which rational number, when multiplied with itself, yields 2? Since no such rational exist, we needed to add the set pf reals. Same as we needed to add the set of negative integers for subtracting natural numbers, as 3 - 5 is not a natural number.
@robheusd No, it definitely is not the same as introducing negative integers! This is the kind of reasoning you want to let go of. AS I discuss in the Math Foundations series, negative numbers are an effective framework -- we can say precisely what the objects are, what the operations are, and we can illustrate with a wealth of examples, not just cherry picked ones. As opposed to for example: "pi + pi = 2 pi -- see we CAN do arithmetic with real numbers!"
Exciting stuff!
So, over on the real number line where I'm from, a right triangle with two sides of unit length has, by the Pythagorean theorem, a hypotenuse of length √2. What's the length of the hypotenuse in this here General Field Rational Mathematics you got goin' on?
When you are doing careful mathematics, everyday concepts do not just morph into mathematical ones, but require precise definitions. You may well think you have a very good intuitive idea of what "length" means. But in order to answer your question, you must first make clear what precise use of the term you are using when you ask "What's the length of the hypotenuse in this here General Field Rational Mathematics you got goin' on?" Here is an analogy: over where I come from, real triangles have colour. So what's the colour of the hypotenuse in this here Real Field Mathematics you got goin' on?
@@njwildberger Well, I'd say blue is the best hypotenuse color and I'd argue that all triangles should adhere to a rigid color scheme so they can be mapped to RGB computer screen color values, but I realize I'm in the minority with this opinion. More to the point, It seems you're saying that I simply can not have my dear √2 hypotenuse triangle. Can I at least have a triangle with 2 nearly equal sides, an angle of nearly 90 degrees and an almost √2 hypotenuse? Or do we all have to go back to living in tetrahedron houses?
@@sallylauper8222 I agree that blue is a nice colour for a hypotenuse, especially in Euclidean geometry. But there will be those that would prefer to consider triangles without any reference to colour, thereby avoiding prejudice and questionable definitions. So I think we can be happy with you retaining your perfect isosceles right triangles. But when we want to refer to measurements of their sides, we should stick to quadrances, which say are then 1, 1 and 2, which will surely make you happy. And happily quadrances can be computed for ANY triangles in the rational plane, so we no longer have to restrict all trig examples to those two tired 45,45,90 and 30,60,90 triangles, in all manner of contrived variations.
@@njwildberger Ok, yes, the Q1, Q1, Q2 triangle makes me happy.
How do you calculate de area of a circle using only rationals?
Sorry but the unit circle does not have an area in the usual sense. The unit square does, but the unit circle does not. However one can right down many approx-areas for it.
I would like more examples of semi regular/uniform tesselations of Universal Hyperbolic Geometry (over |Q, but also over algebraic field extensions of |Q).
Especially tesselations involving polygons with vertices and edges spanning across the two main regions of the hyperbolic plane.
Perhaps i could produce such examples myself someday.
I am an undergraduate who needs a curriculum for proper finite math. What should I read?
We want a WildCalculus lectures ? your approach for the subject is very deep .Thats gonna help us :)
Yes, something like WildCalculus is on its way, but it will probably be a few years yet!
njwildberger Rather see algebraic topology continued to 3-manifolds :)
So do you mean that you think sin(x) as a power series is not problematic, and using techniques you talked about early in the series, Dsin(x) =cos(x) is valid, but sin(1) as a number or a so-called randomly generated power series like f(x) = Sigma( pi(n) x^n ) where pi(n) is nth digit of pi is problematic?
That is pretty well exactly what I think. We are going to see that it is crucial to regard power series as extensions of polynomials/polynumbers, not as functions. We are going to have to worry about how to specify the coefficients of power series, so we do not fall into the same logical holes that the current theory does by assuming we can deal with an infinite amount of data.
It is all somewhat delicate, and will require considerable new thinking. Thanks for the excellent comment.
You obviously have a serious bone to pick with the notion of the continuum, and hence the idea of infinity and by extension the set theoretic approach and the transfinite arithmetic. Sadly, even by moving away from the real numbers, you still won’t be able to avoid the infinities. Forcing mathematics to neatly fit into our inherent physical limitations is not the answer. Meanwhile, I’m truly surprised to see that you stomach the innately meaningless definitions of a point or a line, without much complaint!
I am not forcing mathematics to neatly fit into anything. I just want our mathematical discussion to be complete and clear, meaning that we can explicitly write down everything. This is in the same spirit that a physicist might want to restrict themselves to statements about observable quantities. The same scientific spirit needs to strengthen within mathematics.
Algebraic "irrational" extensions of Rational numbers are very natural, since we get these immediately with Vector spaces equipped with quadratic forms, and higher forms. In |Q^3 we get e.g. equilateral triangles, i.e. triangles with equal non-zero quadrance of all edges, and equal non-zero spreads of all vertices.
Similarly we will get all regular polygons in |Q^n for larger n. And we will get regular polyhedra, polychora etc as well, if n is large enough.
Of course we will get all those "indefinite" quadratic forms as well, besides the "positive definite" ones, which make place for geometric objects of hyperbolic geometry.
What have you in mind by a more _general_ type of mathematical field than the complex numbers (for example)? These are vectors endowed with multiplication/division as well as addition/subtraction but do you have in mind matrix representation of discrete field elements (with all rational entries)? Would complex numbers - and the field associated to them - survive in this enlarged picture as just a special case or restriction but no longer regarded as 'fundamental'?
There are many other kinds of fields. Some are related to complex numbers, others are quite different, for example finite fields, or fields of rational functions. We will be spending a good amount of time on field theory in coming videos.
@@njwildbergerAlgebraic number fields over the Rational numbers are also very worthwhile! They are somewhat analogous to the fields GF(p^n), for n > 1, though somewhat more complicated. Much better than "Complex" numbers based upon "Real" numbers, in any case.
Most likely the "Complex numbers" would be replaced with a (countably infinite dimensional) direct limit of all Algebraic number fields over the Rational number (Algebraic closure of |Q).
We will get all sorts of field automorphisms over this field, both "circular" automorphisms over the cyclotomic generators (these will be much more complicated than mere "real" analogous rotations), and various sorts of "hyperbolic" automorphisms over non-cyclotomic generators.
The question of which scaling factor to use comes up for me when I am doing computations with double/float type on a computer. True 'real numbers' would require that one need not make any decision on scale, for example, 0 to 1 can represent the continuum of everything the value is trying to represent - but why stop there, why not pick 0.1, since there is infinite precision between 0 to 0.1, how about 0 to 0.01 and so on. I guess this is eased by the double/float type - where, only a fixed amount of precision can be expressed; but doing this seems to be an engineers work around to this issue of not being able to fully represent and compute with real numbers.
I hope you will, at some point, prof. Wildberger, give some thought to differential equations. These are fundamental mathematical objects in physics, and it will be very interesting for me, and for many others, I believe, to understand their correct mathematical meaning.
Are an advocate of the computational numbers as a replacement for the reals then?
Hmm... what are the computational numbers??
Sorry, I believe the proper term is acutally computable number, if you want to look it up. Essentially the set of functions N -> Z that, given an integer n, returns a rational number of increasing perscision as n increases. So pi is a computable number, but Chaitin's constant (the probability that a randomly constructed turing machine will halt) is not a computable number. One thing you may like about them is that they form a countably infinite set. :)
kratanuva725 Okay, it would be great to have a theory of `real computable numbers'. Unfortunately in the generality we currently aspire to, this is impossible. But it might be possible to have a more modest, restricted theory of computable numbers which did support an arithmetic. This would be a very major breakthrough, but I don't know if it is feasible. Certainly some people ought to think about it though.
How do you get rid of complex numbers?
@Josh YOu don't get rid of complex numbers, rather you just define them correctly: as expressions of the form a+bi where a,b are rational numbers, and where i is a symbol that satisfies i^2=-1.
@@njwildberger ah, makes sense! I hoping you expand a bit on Octonions. Rational numbers emerge in interesting places in the hypercomplex numbers in the realm of factorization.
@@njwildberger and keep in mind these things have important applications. Especially for quaternions, factorization tells us which rotations will give us a given orientation. Direct applications to quantum physics. In fact quantum states are rational numbers.
If you want to talk about foundations in mathematics, the first thing I would want to know is your formal definition for the natural numbers. Can you give me your formal definition of the natural numbers please? And while you're at it, please explain why real numbers do not qualify for that definition. Thank you.
these videos have become my evenings
I was wondering if anything like the Wiki Maths project came into being after 5 years?
What about, for example, the squareroot of 2, or pi - these are obiously not rational numbers - how shold we relate to these things?
One can view them as functions. The have all the characteristics. They are not values.
One area that goes beyond rational trig where I would be curious to get educated is in the domain of quaternion algebra, but using only quaternions with rational coefficients. There is this area of Hurwitz quaternions but to me so far it’s mostly incomprehensible beyond some basic definitions. To few people seem to be working on that and those who are write extremely esoteric papers.
The more I think about this topic, I think that this is not a convincing way to talk about it. I think about mathematics as a computer without a specific and nailed-down machine language. In computer languages, there might be undefined behavior in the LANGUAGE, but when you pick an implementation to REALIZE it, then EVERYTHING gets defined whether you like it or not. Most computer security issues come from this. Mathematics is missing a virtual machine to execute upon. You can't talk about rigor unless you have a virtual machine to execute upon, where every step can be made visible.
Every English language statement needs to be "compiled" down to algebra, which in turn compiles down to the assembly language of the actual mathematics machine. Ex:
"C is a constant" is imprecise slang for "d[C]=0",
If you write a recursive equation like: "S = 1 + (1/2)S", then you can't just argue with what happens when you both solve to find "S = -1", and find that S expands like: "S = 1 + 1/2 + 1/4 + ... + 1/(2^(n-1)) + 1/(2^n)S" (note the S on the tail-end).
A virtual machine needs to be defined. Calculations that don't terminate are problematic; because you need to stop calculating one side to see if it's equal to another side. If the machine doesn't terminate, then you need to prove (in a finite number of steps!!) that the machines give the same output forever; like comparing two code functions for equal results. In computer science, this is related to "the halting problem". And most of all, you cannot argue with the result when you run the machine. You can only argue about whether the virtual machine needs to be changed to give a different result.
When mathematics is being done right, we will be able to make huge databases of true statements; where a human never has to step in and object to a step that has been taken. It's crazy that math is supposedly rigorous, but defies automation. Kevin Buzzard is right; about everything.
Hi Rob, I love your comment and am deeply in agreement. I have been thinking about a video that urges us to work towards “running” mathematics in the same way that programmers run programs. Thanks for the nice input.
Many of the basic definitions are logically flawed and mathematicians know that but still use them. For example Peanos axioms define much broader concept then natural numbers namely progressions. And it should be wrong to define a concept by a definition of a much broader concept. And even Mr Tao in his Analysis ignores this and starts with Peanos 5 axioms.
Operations do not have a definition. The current one just states that set of operands maps into the set of results of operations but does not state what IS an operation.
Definition of relations as subsets is circular because subsets are also relations.
Definition of reals as a complete field does not have a model (it is just stated it does and those "proofs" that I have seen break down at the more complex parts of proofs ie are evaded).
Even Dedekind in his original work evaded the complications. He just constructed the proof that the sum of two dedekind cuts is also a dedekind cut. But when he had to do the same with multiplication and division and the power operation he just did not do it. He used the term "it is done likewise" or something to that effect.
So it is not just continuum, real numbers and infinities. The sloveliness, imprecission, and I do not care attitude are all over the place. Mathematics is literally in ruins. And infinity is treated as seen fit. It is a madhouse.
Aleksandar Ignjatovic said "Definition of relations as subsets is circular because subsets are also relations."
You are using "relation" in two different senses here. The relation of being a subset is not the same as what is defined when a subset of AxB of two sets A and B is called a relation.
Nor is the undefined "e" (where "x e y" for sets x and y) a relation in the set-relation definition sense.
The "x subset y" is defined in terms of "e". Just like "e", "subset" has all possible sets as the first argument and all possible sets as the second argument. But, there is no set of all sets. Thus, "subset" cannot be a "relation" in the set-theoretic meaning.
But, even if the above were not true, the definition would not be circular. It would be fine to define "subset", then later define the set-theoretic definition of relation, and then prove that "subset" is in fact a relation under that definition.
Aleksandar Ignjatovic said " Peanos axioms define much broader concept than natural numbers namely progressions. And it should be wrong to define a concept by a definition of a much broader concept."
What do you think the "concept of natural number" is?
Anyway, if you don't like the idea of calling the objects of Peano axioms natural numbers, then we could just call them something else (like Peano items). The same goes with the names of "real number" or "imaginary number". If you don't like the idea of calling the objects of R or C "numbers", then we could just call them something else.
Aleksandar Ignjatovic said: He used the term "it is done likewise"
This is common in mathematics. I believe Euclid did the same.
Aleksandar Ignjatovic said "The sloveliness, imprecission, and I do not care attitude are all over the place. Mathematics is literally in ruins."
This has always been the case. The general public idea that mathematics is precise, logical, and clear is not true. Just read any mathematical article or mathematical book or do a mathematical proof yourself, and you will see that the general public idea is not correct.
***** That very well might be somewhat true--it certainly is not uniformly applicable---but it does not prevent us from moving towards a more precise, mathematics where we DO spend more time and energy making sure that the subject hangs together logically!
***** Sorry, but all your remarks (I do not have to comment each one separately) are just rationalizations of bad ways in mathematics. Especially the one about the relations.
I'm only an amateur so it's not exactly clear to me what you're advocating here. Do you want some kind of system which gets rid of the Reals, irrational numbers and transcendental numbers like e and Pi? How would that be achievable in practice except for solving problems with algebra as opposed to numerically?
As you mentioned about fractions, it quickly becomes unfeasible to represent reals as exact fractions on computers, because the numerator and denominator can become huge, and where simplifying the fraction isn't possible. They could both be huge prime numbers for example. But some programming languages do support this, for example Common Lisp implementations. But doing calculations that way is incredibly slow.
And suppose we all started using fractions instead of the reals for everything and the sizes of the numerators and denominators weren't an issue in practice. Whenever our equation involved irrational or transcendental numbers, wouldn't we be forced to write the final answer as a fraction *and* an agreed name for the transcendental/irrational numbers used? If so, how is that any different than writing down the algebraic solution? Or is there a theoretical approach which gets rid of Reals, irrationals an transcendental numbers entirely?
Sorry if I missed something fundamental from your lecture which addresses this. I'm just trying to understand. I agree however that exact numerical answers are pretty great. They make me warm and fuzzy inside :)
Currently modern computer systems do not use real numbers; they tradionally use floating point numbers, which are something else entirely, and increasingly these days they use full integer precision. Our computers have never played along with our real number fantasies; I am advocating facing up to this reality, and finding ways to move forward logically and carefully.
It's not a question of getting rid of the reals, so much as acknowledging that they never properly were defined in the first place.
1-Dim WildDerivative might be Df(x) = [f(x+h)-f(x-h)]/2h, where h ≈ (Planck length), and analogous integral ∫f(x)dx = h∑f(x). On the other hand, assuming lattice structure in physical World _might_ unify quantum theory and relativity under the same umbrella.
But for continuous process, or describing ideal geometric figures, real numbers are indispensable.
What exactly is continuous process if any measurements under planck length and time are impossible even theoretically ? FIction ?
What exactly are "ideal" geometric figures ? Fiction ?
What's the point of real numbers and theory of infinite sets ? Fiction for the sake of fiction ?
Ok.
pieinth3sky
Mathematics basically is fiction or game, but fundamentals like geometry and arithmetics arise when our mind senses shapes of nature and steps of time. Now, if you like serious crackpottery, like L.E.J. Brouwer did, you can build your own math on those simple, finite and direct mental processes. Too bad that you would get stuck in the Stone age. Sane person would add some natural extra structure that relates different things and provides existence for the obvious, e.g. solid geometric objects are point-sets and non-parallel lines must meet.
Real numbers are not indispensible for continuous processes or ideal geometric figures. Real numbers do not exist, so how could they be indispensible for anything? Perhaps you mean that a belief in real numbers is indispensible to understand continuous processes or ideal geometric figures? I will show that claim to be false: with rational numbers we can well model continuous phenomenon, meet the needs of scientists, and develop rich theories of geometry: all without abandoning the logical clarity which is essential for true mathematics.
Keep watching!
***** There are aspects of religion in the modern mathematical belief system.
+njwildberger
Hi Dr. Wildberger. Long ago I watched your video where you lecture on how the Greeks proved that the diagonal of a square is incommensurable to its sides. This is an elegant proof that continuous objects cannot be fully described in discreet terms (i.e. integers). Over the years, I have watched many of your videos, and I have not yet seen how you have defeated this proof. I respect your expertise, and I have learned a great deal from you. So it is with some discomfort that I express the following opinion.
So far, I have not seen anything in your videos that addresses the real problem, that being that rational numbers cannot express every actual continuous magnitude. What I have seen appears to me to be more mathematical sophistry than a solution. For example, regarding the diagonal of a square you prefer to express it as a "quadrant". Why? Because the quadrant is expresable as a rational number. All this shows is that you can square an incommensurable magnitude to get a commensurable one. But this was never the point, and therefore solves nothing. The point was always about how to express the original magnitude, not "what is the square of this magnitude we cannot express".
According to you, real numbers (e.g. irrational numbers) do not exist. But the magnitudes that real numbers represent do, and these magnitudes cannot be expressed by rational numbers, as proven by the Greeks. You have yet to show me where the Greeks were wrong. I have watched your videos, and there isn't a single case where you express the actual magnitude. You square it (not the same thing) or you express it implicitly (again, not the same thing, and in science we want to know how things relate explicitly, implicit expression is of very limited use), and so I find many of your methods detrimental for many reasons.
I still think you are a pioneer, and I believe your methods will find a place as an addition to mathematical techniques, but not as a replacement to fix a system that isn't broken if properly understood.
Just one of your thousands of online students,
Mark Botirius
How do you APPROXIMATE rational numbers:
For example, if I have the number 0.53476853456566 then the simplest way to approximate this number is to IGNORE the remaining digits: 0.53476853 Very simple.!
Now I have (233)(65)(18) / (234)(44)(19). This number is EXACT! It is very tempting to use this representation instead. I would love to use this representation in my computer. However I need to ensure that that during the calculation, there will not be an overflow. Although the coefficients are huge, the number itself is reasonable. It is hard to predict in advance where it will crash. This situation does not happen with real numbers because the Floating Point Unit simply ignores the remaining bits. But with the rationals it is not as simple.
I could prepare my numbers in some kind of series where the arithmetic unit does not overflow? I am clueless?
Concatenating non-number "-" meaning negative with an aspatio-temporal unique number may rob you of the relationship between a number and it's representative numeral.
I gave a thumbs-up to this video but found it shocking. IMO, the reals and complex numbers are well-defined and solve many real-world problems. I have no doubt that numbers such as e, pi, square roots, converging sequences, etc. exist and are very useful. I also see axiomatic set theory as providing or, at the very least, a good attempt to provide a solid foundation to mathematics. I do agree that a lot of math is far removed from realty and suffers from vague, arbitrary definitions without concrete examples. And some of it is stupid such as analitical extensions that "prove" that the sum of some positive numbers is equal to minus 1/12. NJ is obviously an accomplished mathematician and I look forward to seeing more of his videos.
Tip for differential equations... A likely future direction will be away from exact solutions, and more emphasis on asymptotics, rigorous approximation, and extracting information from PDE's using ideas from information theory combined with asymptotics. For some lectures on such directions I highly, highly, recommend Carl Bender's PIRSA lecture series. Physicist do appreciate the work of pure mathematicians on exact solutions, but typically they are important only for toy models, the real world of physics never deals with exact solutions to DE's because they just do not exist in any practical sense.
So in this system of mathematics, square root of 2 doesn't exist, because it can be proven that it is not rational?
How can we describe objects like pi and e without real numbers?
AJ AJ This is a very good comment, thanks! Viewers should read it carefully.
AJ AJ
but what exactly are useful digits of pi ? I dont see any practical purpose in knowing the 1e12 th digit of pi - beside sports - similar to knowing the next bigger prime...
is it even right to ask for the 'value' of pi ?
AJ AJ
Well said.
Bolzano's genuine "foundational" work on infinite sets is explicitly metaphysical.
Moreover it is anti-physical in sense that it expels time and motion from mathematics.
All further "development" of "pure mathematics" like "Dedekind-infinite sets" ,"Cantor's diagonal argument" or "Goedel's incompleteness theorem" or AC looks like bootstrapping of metaphysical objects by appeal to even more metaphysical objects.
AJ AJ
Please tell me, is the following a process or a number: S = 1/2 + 1/4 + 1/8 + 1/16 + · · · = 1.
If you call it process, then I argue that it will permeate and contaminate the whole edifice of rational arithmetics.
If you call it number, then I argue that every other finite limit is also a number and we can do arithmetics just like we do with rationals.
For me process and its limit value are just two sides of the same coin.
You can't describe them WITH real numbers!
Another thing about diagonals of the unit square...
What the non-existence of a number whose square is 2 is telling us about the diagonal lengths is simply that they are incommensurable with the side lengths. There is no way to directly measure the length of a diagonal using the sides as measuring sticks. It can be done approximately but not exactly. *That* is the fundamental reality.
Creating a fantasy number called "the square root of 2" doesn't circumvent that fundamental incommensurability. It just covers it up with a fake hack. Hence having to do without that fantasy number doesn't create the incommensurability anew, it merely exposes what was there all along.
Doing without real numbers will seem to create new problems, but that is only because real numbers gave the illusion of solving those problems. Don't shoot the messenger, so to speak. Rational mathematics means seeing what is really there, so that real progress can be made by addressing real problems.
It's like the brutally honest person who tells you your weak points, which you had learned to ignore but that are weighing you down nonetheless. Ignoring is blissful for a time, but ultimately not helpful. And if you keep doing it you end up trapped in the mud of all those contradictions and denials, unable to move forward. Taking an honest look at the hard reality is so painful that it can feel like the very act of looking is what's creating the problems, but that is of course not the case. And once you realize that, you can start to come to grips with these challenges and truly take the bull by the horns.
You can talk of a collection of underestimates for the length of the diagonal, and a collection of overestimates. An overestimate will be better if it is smaller in magnitude, and an underestimate better if it is greater in magnitude. The 'real number sqrt(2)' then corresponds to the 'theoretical best approximation possible', or rather the asymptotic limit of a process of continually refining the approximations so as to be closer than any given rational. That, of course, is the point of the Dedekind cut approach.
Wow, you are more of a constructivist than any other other mathmatician I know.
So much of this topic is studied & has clever answers under the domain of transcendental number theory.
There always tis question if mathematics is found or invented and if the universe can be described by pure mathematics.
Dr. Wildberger have sure convinced me about the mathematical problems with real numbers, infinities, intiniticimals.... ;-)
Eulers equation suggest that there is some relationships we dont understand e^pi*i+1=0
We certainly do understand Euler's equation, see for example the entire field of Harmonic Analysis.
Euler's equation is perfectly understood and proven, what do you mean?
@@HDitzzDH Yes, Thanks. It was 5 years ago. I've taken Math A since then :-) I still feel that Dr. Wildberger fundamental right about the problems with real numbers :-) Happy New Year.
@@HDitzzDH But Eulers equation still has two "nasty" transcendental numbers in it :-)
Do you agree that the notion of real number is directly connected to the notion of infinite divisibility of length?
ie: if we do away with real number, then we have to do away with infinite divisibility of length...
No I do not agree, since first of all `length' is a problematic notion, and not necessarily involved in discussion of the nature of the number line, and secondly because the continuum as the rational number line has inherent in it the possibility of division of segments or intervals into arbitrarily large numbers. If we want to divide [0,1] into 10^10 pieces, we can do that. I am not sure however if that is what you mean by `infinite divisibility'.
It is a great exercise to ask, before asking a question: do I know precisely what all the words that I am using here mean?
njwildberger "No I do no agree, since first of all `length' is a problematic notion, and not necessarily involved in discussion of the nature of the number line, and secondly because the continuum as the rational number line has inherent in it the possibility of division of segments or intervals into arbitrarily large numbers. "
I was referring geometric length in Euclidean geometry, not the number line rational or otherwise. I see the purpose of real numbers as giving a quantifier to euclidean length...
Do you think the notion of length needs to removed from Euclidean geometry?
otakurocklee Somewhat surprisingly, length is NOT a part of Euclidean geometry, in the following sense: Euclid in his book The Elements did not introduce a notion of length of line segments. This is a modern attribution to Euclid. Why did Euclid not introduce such a concept? My guess is because he well understood the difficulties with having a prior number system in which lengths can have values.
Complex analysis definitely goes in the direction of algebraic calculus and topology always goes in the combinatorial direction. Professors talk and youtubers work. I've started thinking about definitions of a manifold.
The notion of a manifold is an enormous challenge. It's a great thing to think about.
So what's the length of the diagonal of a square of size 1, in your theory ?
Totally agree almost on everything, but perhaps I'm even more radical.
I think there shouldn't be rational or natural numbers, just numbers. Those numbers can have certain conditions such as the limitations which natural numbers have in comparison to rational numbers. Of course you can still use the name natural number, but the mathematics is all around just the definition of a number.
Then the logical steps/results which evolve from this definition would be mathematics.
Finally a scholar who dares.
Why are we always trying to make all shapes into squares.
Though they may be rectangles, triangles, curves etc. We're always asking what is the root, as if all things should revert to square.
One is slave to the standard one sets. And the givens you define today, if not controlled, will invariably be your frustration tomorrow.
There is a fundamental reason why a rectangle has unequal sides & that reason gives a unique meaning to what it is to be a rectangle.
Same with a circle, its a unique object, a type, complete with its own geometry & math.
Seems to me that there is a math-by-type logic embedded in the construct of each type of object.
And unifying by ignorantly setting single standards, generalizes over the separate & unique.
We got no clue what the common basic building block(s) of physical objects are (if there is only one) yet we claim its nature
Type-by-type, with respect given to uniqueness may be the way to go. And if approximation is our fate, let it be with at least sensible.
Its funny I refer to exact concrete answers as sticking the landing. Im with you on that Professor.
To the person who asked how to find the length of a diagonal of a unit square, my first question is, "How thick are the sides?" Remember they cannot be infinitesimal in size, since we're now rejecting such fantasies.
When you have a definite thickness to the sides, length becomes either approximate (no real numbers needed) or reliant on what exactly you're measuring. It's no longer obvious that the length of a diagonal is incommensurable with that of the legs.
Furthermore, it would have to be the case that the picture in physics of fields existing on a manifold is not the way to go. Some people work on theories of quantum gravity where our spacetime is hoped to emerge as an approximation of some discrete structure. In this picture, `squares' would only be an approximation with no fundamental place in physics.
obviously God-Volume who trascends finite dimensional space produces finite Finite-surfaces as God-Volume rotates and intersects itself.
Disclaimer! I am NOT a mathematician! Root 2 is often shown as the most easily provable "real" number. There are many videos showing proofs by contradiction all over TH-cam channels. We want to assume that the diagonal of a square of side-length 1 has a length, and it's easy to show that it ought to be some value, namely "root 2", and the units of measurement don't matter.... But aren't we "measuring" sidelengths according to the typical Cartesian co-ordinate system, aligned to a right-angled pair of axes, generally referred to as x and y, or horizontal and vertical? Well, almost any measure of length along a diagonal in this reference frame will lead to some root, some irrational or "real" number...
Ok. Accepting this conclusion, let's look at the intersection of the hypotenuse with either of the legs: two lines meeting at a 45° angle. Now, imagine measuring the length of the two sides which intersect. And we take more and more measurements of a finer and finer scale, to try to refine the measurement of the hypotenuse. The side-length reads 1 unit but to finer abd finer precision. Fine. But the measurement of the hypotenuse will never intersect perfectly with the end of the side-length!! Even with an "infinite" number of repetitions, a diagonal "unit" can never equal a vertical "unit" in a Cartesian set-up, unless you rotate one to match the other. And even if you did, you will never come to a fine enough measurement to make one expressable in terms of the other.
Now, mathematically we assume that a geometric "line" has no width, and is a "one-dimensional" object. If so, how could a diagonal line ever hope to intersect with a straight line? One would be completely and utterly unable to conceive or interact with the other
Unless a line has some width, even speaking of "Intersection" is meaningless, it seems to me. I can accept the (for want of a better term) "Absolute length" of the Euclidean point of view of a line segment which has no dependance on any underlying grid. But as soon as you combine the Cartesian Plane with 1-dimensional, abstract, lines or line segments, how can they possibly intersect, or for that matter, even interact at all?
The applied math POV to choose some level of precision makes sense to me, but to speak of 2 1d objects "touching"... How?
Jeremy Nasmith I think sqrt(2) is easily proved to be not rational. But that doesn’t prove it’s “real” unless this is the definition of “real” in which case that is an axiom, not a proof.
As far as I can tell this video is questioning the coherence of the definition of real numbers.
'Completed' infinite sets e.g. N, Q, R, C are heuristics, one does not need to actually take a position on the truthiness or ontological status of 'actual infinity' versus potential infinity. Also the use of 'completed' infinite sets is valuable especially for pedagogical reasons.
In a discussion about things that are imaginary, which are numbers, can you honestly put limits on the imagination, e.g. claiming that infinite sets are past the set of what-is-possible-to-think-of-with-the-imagination? It seems to me that you are saying that a completed infinity is a 'lie of the imagination'. But i could be misinterpreting your position.
If I was pressed to take a position, without thinking too deeply into it, the concept of infinity does not seem incoherent. It seems to be another order or level of imagination. unless you are claiming that we must always tie mathematics to generalizations about real objects (sets of trees, sets of sheep, etc) , and since no one has experienced infinity, from whence comes this generalization of large sets come from. But this is like an extreme empiricist position which seems unnecessary. Since when were mathematicians bothered by reality, they are philosophical by nature and daring in their ability to imagine.
From an empirical point of view it seems that infinity is taking the idea of 'and so on' and bracketing it. In most instances of the use of infinity, it can be substituted for 'arbitrary large' or 'arbitrarily small' or arbitrarily close expressions. To approximate irrational numbers we use functions that wait for the call of how much precision you want. What is π? What is √2, they are functions that have a certain type of arbitrary precision when requested ( so far i have avoided talk about infinity).
"The natural numbers are infinite" - that just means that given any number you think is the biggest because you grew tired of counting, i can find a number bigger than it.
"The prime numbers are infinite" - that just means that given any prime number, no matter how big, I can give you a procedure to find a bigger prime or has a prime factor that is bigger than your prime.
But how do you account for ordinals 0,1,2, ... w, w+1, ... , then i guess it would correspond to multiple levels of bracketing , like a nested function (makes me think of C programming, the variables are deallocated when the function is returned).
There does seem to be a need for completed infinities in proofs by contradiction , e.g. cantor's real number uncontability theorem.
There is a lot to think about here. Maybe infinity is a lie after all, an imagination lie. but its a useful lie because we can pack a lot of information into this lie. As a society we realized that god does not exist (maybe speaking for myself here), and now infinity does not exist. What's left? only the 'and so on'
I hope I didn't strawman your position. its a little hard to pinpoint your views because you haven't said straightforwardly that 'infinity is incoherent' or 'infinity is a lie' or 'infinity is a nonsense word' (because it doesnt refer to any coherent concept).
Maybe the best analogy for infinity is as a do-while loop procedure, infinity( ), that never terminates. I don't have to actually call this function,i.e., i never have to run infinity ( ) , but i know it's there.
I don't understand this, but I think that this has connections wit you subject of the last weeks:
Ideas and Explorations : Brouwer's Road to Intuitionism
authors Kuiper, Johannes John Carel
source Utrecht University Repository (2004)
full text [Full text]
document type Dissertation
discipline Wijsbegeerte
abstract This dissertation is about the initial period of Brouwer's role in the foundational debate in mathematics, which took place during the first decades of the twentieth century. His intuitionistic and constructivistic attitude was a reaction to logicism (Russell, Couturat) and to Hilbert's formalism. Brouwer's own dissertation (1907) is a first introduction to his intuitionism, which was the third movement in the foundational debate. This intuitionism reached maturity from 1918 onwards, but one of my aims is to show that there are demonstrable traces of this new development of mathematics as early as 1907 and even before, viz. in his personal notes, which are composed of his numerous ideas in the field of mathematics and philosophy. To mention some important ones:
1. Mathematics is entirely independent of language. Mathematics is created by the individual mind (Brouwer certainly is a solipsist) and the role of language is limited to that of communication a mathematical content to others and is also useful for one's own memory.
2. The ur-intuition of the 'move of time', that is, the experience that two events are not coinciding, is the most fundamental basis of all mathematics. A separate space intuition (Kant) is not needed.
3. A strict constructivism. Only that what is constructed by the individual mind counts as a mathematical object.
4. Logic only describes the structure of the language of mathemqatics. Hence logic comes after mathematics, instead of being its basis.
5. An axiomatic foundation is rejected by Brouwer. Axioms only serve the purpose of describing concisely the properties of a mathematical construction.
These five items have far-reaching consequences for Brouwer's mathematical building. To mention the most relevant ones:
- The only possible cardinalities for sets are: finite, denumerably infinite, denumerably infinite unfinished and the continuum.
- The continuum is not composed of points (Aristotle already said so), but is given to us in its entirety in the ur-intuition. It can be turned into an everywhere dense measurable continuum by constructing a rational scale on it.
- Cantor's second number class does not exist as a finished totality for Brouwer, since there is no conceivable closure for the elements of this class.
- The continuum problem is a trivial one: Every well-defined subset of the continuum is finite, denumerably infinite, or has the cardinality of the continuum.
Finally, in my dissertation the sixth chapter is devoted to Brouwer's view on the application of mathematics to the human evironment and on his outlook on man and on human society in general (chapter 2 of Brouwer's dissertation). His opinion about humanity turns out to be a pessimistic one: All man's effort, when applying mathematics to the surrounding world, is aimed at a domination over his environment and over his fellow men.
keywords constructive second number class, intuitionism, ur-intuition of mathematics, possible cardinalities, logic and mathematics, solipsism, continuum, objectivity, apriori, actual infinite
Why run it to 1004444...
If length is truly quantized then it makes no sense to talk about irrationals anyway - to me the question becomes whether we want our pure mathematics to capture the real mathematical universe, or talk about going further at the expense of seemingly useless accuracy.
I think this overhaul has a lot to do with reality vs a "super reality" accuracy with our math... the irony being the latter is so error ridden.
There really has to be a systematic study on identities. There is so much 'heavenly assistance' in proving your UHG laws. The identities need to be numbered and named systematically somehow. Perhaps we can start with (a+b)^2 = a^2 + b^2 + 2ab and name it (1) and then the binomial theorem 1.1, as its generalization.
Yes we do need a much more systematic and thorough understanding of identities. It's a subject that has only really come into view in the last few decades, with computers helping us find identities much more efficiently.
you forgot, 'SUPPOSE' we have a real number alpha...
Yes that's another common term that invokes wishful dreaming, thanks.
As a physicist I don't care about it. As long I am able to integrate and to make derivatives I am happy with real numbers.
Wouldn't it be great if there was a mathematical manipulative that helped the young guide themselves towards the foundations: (your)multi-set theory, chromogeometry, UHG/rattrig.
Maybe puzzles can be very useful here. Think about all the video games that have been made and the mathematics that have been used to make them...
But maybe we need the finite fields at the heart and not the rational numbers.
You know that the finite fields are key, we all wish you would stand on them.
Can you solve a rubiks cube? Have you used a rubiks cube to study the theories you have developed?
I think you might see that this will help you guide us, we need a lot of help. Maybe if you showed us how a rubiks cube can model the continuum very nicely using chromogeometry, rat trig, UHG, multi-set theory...
You have really brought us out of the darkness... but keep in mind what motivated you to rat trig.
What motivates me towards the foundations of mathematics is a HONEST conversation with students.
Wait, you're okay with octonions, but not complex numbers? Octonions are built on complex numbers, aren't they?
@Jo Reven, Complex numbers are fine, as long as you define them properly, meaning as combinations a+bi where a and b are rational numbers. Similarly the true octonions are built from rational numbers. The fictional realm of "real numbers" ought to play no role in careful algebra, where we aspire to making computations on the page, completely precisely and correctly.
@@WildEggmathematicscourses Thank you for clarifying this with a quick reply!
I love when someone like him reveals a naked truth
Yes, yes. "Dihedrons and their conjugates" -> looking at Bott Periodity from a Information Theoretic (Computer Science like perspective)
more work needed on Math Itself
edgeoforder.org/mathitself.html
This is starting to be a habit.
Georg kantor was an important mathematician. He showed the idea og infitity is working precisely ;-)
I feel sorry for Cantor.
mental illness
Dear Norman. Since you are going to talk about axioms of real numbers, could you please have following commentary at the very end of your next MF120 video:
" Now we know that with the least upper bound property every positive real number has a square root, that is again a real number. But, of course, that is not the case with _my_ _precious_ _rational_ _numbers_ [now you should smile like Gollum]. It is delusional, wishful thinking to even ask what is the square root of a number. There is no such calculation! It makes no sense! It makes no sense!! ... "
Could the number of problematic fields of mathematics built upon real numbers be.... infinite? :)
Hi Dr. Wildberger,I've watched all of your math history videos, and I've gotten through many of your math foundations videos. I have learned a lot from you, and I respect people who are brave enough to challenge the status quo. It is therefore with great hesitation that I must disagree with your position on real numbers. I posted a video reply here: th-cam.com/video/qiZP5sakM9M/w-d-xo.html.
In all of the videos I've watched I have been hoping that you would address the elephant in the room with regards to integers. You haven't. I wonder if it is because this elephant represents the fatal flaw in your argument to do away with irrational numbers entirely, or if it is because you haven't considered this difficulty seriously.The problem with your position is that the ontology of existence is continuous. The proof of this lies in the fact that discreet entities (those entities that can be successfully described using integers) arise from continuous entities. However the reverse is not true, you can never have a continuous entity arise from the discreet.
For example, the intersection of two continuous lines is a discreet entity called a point, which can be described with an integer. However, no matter how many points you have you can never create a continuous line, the best you can do is have a row of points placed side by side. Even with an "infinite" number of points, you will never have a continuous object, because this would constitute a contradiction.This is the deep reason why there exists geometrical relationships that are physically impossible to describe with integers. The relationship of a circle (a continuous entity) to the line that bisects it (another continuous entity) is an example. Another example is the relationship to the line that makes up the side of a square (lines are continuous entities) to the line that runs diagonal through a square (also made up of lines, which are continuous). In these cases, it is logically impossible to describe these relationships with integers because integers can only fully describe that which is not continuous.
Because the discreet is in reality a sub-category of the continuous, and integers can only be used to describe the discreet, you are left with a system that is strictly limited to this sub-category. When faced with a relationship that falls outside of the limitations of your system you simply claim that anything outside your limited system does not exist. For example, when graphing the square root function, you draw circles anywhere an integer cannot describe the continuous, claiming that the point does not exist. The reality is that the point lies in a place on the continuum that cannot be described in terms of a unit. Not only does this magnitude exist, it MUST exist, or the curve wouldn't be continuos in the first place.This is a defining characteristic of the continuous.
A CONTINUOUS ENTITY IS THAT WHICH HAS MAGNITUDES THAT CANNOT BE DESCRIBED WITH INTEGERS. IF EVERY MAGNITUDE COULD BE DESCRIBED WITH INTEGERS, THEN THERE IS NO REASON TO ASSUME THAT IT IS CONTINUOUS.
This is also the deep meaning behind the Greek's proof regarding the Pythagorean theorem. The Greeks proved simply that integers can only describe part of continuous objects, and therefore any system based on integers (including yours) will be likewise limited. You have tried to get around this either by simply claiming that anything outside your limited integer-based system does not exist (and it certainly does) or you have employed techniques that mask the problem (such as "quadrants"). A quadrant only demonstrates that you can square a magnitude that is not describable with integers to find one that is. It still does nothing to express the true magnitude of the original incommensurable side itself.
So how do we express the incommensurable with honesty? Certainly not by trying to deny its existence or mask it through some operation such as squaring it. Conventional mathematics has developed the use of non-integer based symbols to express that which cannot be expressed with integers or any system based on integers (such as yours).What you are attempting to do is worse than what you are trying to "fix". You are attempting to sell a contradiction.
I write this with deep respect. I have learned a lot from you, and it gives me no pleasure to disagree.
Respectfully,
Mark Botirius
+Mark Botirius Hi Mark, Thanks for the thoughtful and extensive comment. I suppose my main reply is that: existence is usually not the prime issue. The prime issue is usually always definition.
The difficulty with the continuous is that it is highly challenging to even define what we mean by it. There is also something of that difficulty in defining the discrete, almost always in terms of natural numbers or something close to them. In my view, this is the essential reason why the discrete trumps the continuous when we inquire into the foundations of mathematics_ it is just a lot simpler defining arithmetic with natural numbers than it is defining arithmetic with rational numbers. And that is orders of magnitude easier than defining arithmetic with "real numbers".
So I would ask you to consider your comment in the following light: which of the terms that you are using has a prior accepted definition? Without such prior definitions, I am afraid the argument drifts towards the philosophical.
+Streets, blocks, experimental music
Hello,
You asked, " If you have an exact theory of geometry (with all fundamental definitions), can you refer to that?"
The basis for my comments are ancient and go all the way back to the Greeks. This controversy is not new. It has existed for millennia. Dr. Wildberger has simply brought it back into view, which is good, because with the invention of analytical geometry and the Cartesian plane, we have forgotten some very fundamental truths established by Aristotle long ago. Like Dr. Wildberger, the Pythagoreans insisted that everything was number. Later, Aristotle corrected the Pythagorean view by correctly asserting that everything was not number, there was a distinction between number and magnitude. For Aristotle, number referred to the discreet only, whereas magnitude referred to the continuous.
According to Aristotle, continuous objects are not made up of points. By definition they can't be, because that would be a contradiction. Integers (i.e. rational numbers), being a discreet entity (by definition) can only refer to some point on a continuous object. However, because the object is continuous, there will always exist some magnitude that lies outside of any two points, The magnitude certainly exists, however it is simply not possible to express it in terms of a unit (i.e. a number)
In other words, the discreet is contained WITHIN the continuous, NOT the other way around.
Real numbers are simply the method employed by modern mathematics to account for what lies outside the realm of integers. Of course, real numbers do not make sense in terms of rational numbers. How could they? Real numbers are trying to express the continuous to human beings that can really know only the discreet. But then again, they aren't trying to make sense in terms of rational numbers, hence the reason why we place them in a category by themselves.
In other words, in this very ancient debate, it appears to me that Dr. Wildberger has taken the side of the Pythagoreans, and I have chosen the side of Aristotle.
Which side are you on?
Mark Botirius
+Streets, blocks, experimental music
Hello,
Thanks for the reply. I want you to know that this is a very important discussion. Unfortunately, I have a report to write for my immunology class and I have to study my organic chemistry at the moment (I am a microbiology student), so I will not have the time to give as thorough of a reply as I would like, and I may need to put our conversation on hold for a week or two while I complete my assignments. If it is ok with you, hopefully we can continue the discussion a few weeks from now when my load is not as heavy.
I did not want to leave your questions unanswered, however, so I hope this very brief reply will be sufficient until a later date when I have more time.
Regarding Aristotle, I don't know if you will be able to find the information you seek on the internet, so I wanted to share with you where I get my information from. For example, on page 342 of Thomas Little Heath's book "A History of Greek Mathematics" we learn that Aristotle held that the continuous "could not be made up of indivisible parts; the continuous is that which the boundary or limit between the two consecutive parts, where they touch, is one and the same, and which, is kept together, which is not possible if the extremities are two and not one."
Heath not only informs us of Aristotle's mathematics, but as the name of the book implies, he walks through the mathematics of the Greek's from what is considered to be the first mathematician/philosopher, Thales. I've owned a copy of this book for quite some time (recommended by Dr. Wildberger in his math history videos as well), maybe you can find one at your local library.
In addition, I used to own pdf's of Aristotle's works on my Ipad, but a while back, they mysteriously disappeared, and I haven't replaced them. I intend to buy paper copies when I am not so financially distressed in the future. But, of course, most of my information came directly from the works of Aristotle himself.
Lastly, I want to point out that I agree with your points regarding the theoretical world we imagine in our thought experiments and the actual real world around us (that they should agree). My interest, and the reason that I care about this topic so deeply, is that my concern is that we understand the actual world, and that our mathematics reflect the actual world. However, this is a very deep and complex issue. If we are able to continue our discussion at a later date, I will state my case as to why I hold that, existence qua existence is continuous. This ultimately will bring the discussion into the realm of physics, and unfortunately I am no physicist (I am no mathematician either), but I will do my best.
I look forward to continuing our discussion at a later date. Until then.
Mark Botirius
+Mark Botirius
Norman addressed all this numerous consideration in this series.
That's why MF is so lengthy, too lengthy for comfortable comprehension to my taste.
You didn't watch key videos or didn't listen/followed his arguments and references.
>My interest, and the reason that I care about this topic so deeply, is that my concern is that we understand the actual world, and that our mathematics reflect the actual world., this is a very deep and complex issue.
It has been formalized by Turing, modern mathematicians just ignore him and stick with mysteries of platonic objects.
+pieinth3sky
Hello,
>You didn't watch key videos or didn't listen/followed his arguments and references
On the contrary, I made it through 45 of his Math Foundations videos (I've participated by leaving comments on several of them). I simply didn't have the time to watch all of them. There are hundreds! However, I've seen several videos make the same claims as the one that you bought to my attention. Claims for example, that there exists magnitudes where a circle and it's diameter do not actually intersect because we are unable to express the relationship in terms of rational numbers. The video you brought to my attention, much later in the series than I had the time to wade through, did not make any new claims, and therefore my counter arguments were not affected. Thanks for bringing it to my attention however.
> just ignore him and stick with mysteries of platonic objects.
Thanks for the advice. Have a great day!
Mark Botirius
I fail to see any reason to believe that a noticable number of mathematicians will stop using real numbers any time soon. Furthermore, I am still under the impression that the "problems" with real numbers touched upon here are philosophical and should be discussed in the context of set theory.
I am hoping that the realization that the theory of real numbers is logically flawed, as I have at length described in the last dozen or so videos, will provide a strong inducement for mathematicians to start to re-examine their assumptions.
but math is kind of (strict) philosophy
philosophical stone Or is philosophy a kind of (loose) mathematics?
njwildberger I will meditate on this =}
njwildberger Something can be logically flawed and still be useful in practice. If real numbers offer a utility and ease of use that can't be found in the alternatives mathematicians will continue to use them, flawed or not.
Here is how Dedekind proves strictly mathematically that there are infinite systems
66. Theorem. There exist infinite systems.
Proof My own realm of thoughts, i. e., the totality S of all things, which
can be objects of my thought, is infinite. For if s signifies an element of S, then
is the thought s0, that s can be object of my thought, itself an element of S. If
we regard this as transform _(s) of the element s then has the transformation
_ of S, thus determined, the property that the transform S0 is part of S; and
S0 is certainly proper part of S, because there are elements in S (e. g., my own
ego) which are different from such thought s0 and therefore are not contained in
S0. Finally it is clear that if a, b are different elements of S, their transforms a0,
b0 are also different, that therefore the transformation _ is a distinct (similar)
transformation (26). Hence S is infinite, which was to be proved.
The point is in the first sentence. He actually said that infinite sets exist because he is able to think of infinite number of things. He even mentioned his ego in this very scientific proof. As opposed to him, Cantor stated that actually infinite sets exist because God is actually infinite. I never thought of God as a set but there you are.
Excellent, thank you for sharing with us this waffily and completely inadequate argument of Dedekind.
njwildberger
If you don't like Dedekind's cuts, go and read (Super boring): Edmund Landau - Foundations of Analysis.
If you don't like Cauchy sequences, go and read (Super interesting): Terence Tao - Analysis 1.
Juho x If you have read them, please report to us: do they address the concrete issues that I have raised in my recent videos? But I will be talking about those books in some future video discussing more absent theories of `real numbers'.
njwildberger Our axioms origin how our mind understand nature. Existence surpasses human computability (e.g. two lines _really_ meet, whether you like it or not.)
Juho x "Our axioms origin how our mind understand nature"
Who said that? No need for a link, just the name would do, please.
Construction by description only is a main tenet of postmodern left philosophy, i.e,, woke and cancel culture. Is there a causal connection between the idea of real numbers being real and postmodern leftist political philosophy? Is real number theory an overreach of Kantian/Aristotelian Reason? And the procedural processes of logic?
@Jim Lucas Those are great questions!
@@njwildberger Prof. Wildberger, I never dreamed I would engage in discourse with you! The next step should be a 3way talk with philosophy prof. Stephen R.C. Hicks. Thank you!
Constructive Mathematics has made beautiful progress recently & Intuitionistic Type Theory has had some elegant software, go downoad COQ and Isabelle people!
zeno's paradox solved to see we want
I completely understand that the rationals are more beautiful than the "real" numbers. Yet, it is unavoidable that we deal with irrationals like the square root of 2 or pi. Before, you said the best way of going about these numbers were to bound them, like Archimedes did. Though doesn't this take away from the exactness we are looking for?
In applied mathematics, we will always need approximate values of sqrt(2) or pi. That is different from postulating that these are exact numbers belonging to some distant `real number field' whose definitions elude us but that we insist form the basis of most mathematical thought.
As we progress through this series, your dependence on `real numbers' like `sqrt(2)' and `pi' will naturally diminish.
But surely, we will always have circle, and we will always have diagonals of a unit square. How could they go away?
Swiftclaw123 Having a circle, and having a length of that circle, are two quite different things.
True, though if we want to study circles without their circumferences or areas, then we simply aren't doing them justice. Similarly with squares and their diagonals.
Swiftclaw123 As we will see, the quadrance of a line segment (the sum of the squares of the vector coordinates) is a more than adequate replacement for its length. It is more accurate, more algebraic, and one can compute with it in a finite precise way, actually over a general field. It also extends to more general bilinear forms. So we should not be too quick in assuming that the transcendental notions that are currently used are necessarily the best ones: often they demonstrably are not!
Genius ! A personal crusade... I approve.
When keepin it real goes wrong...
No, no, no Norman.. "Boo" to matrix algebras. Please look in to geometric (multivector) algebra. If you value elegance and naturalness in mathematics matrices are abominable. All the matrix algebras can be reexpressed far more "natively" with the Spin groups embedded in geometric algebra. See David Hestenes' work, and the papers from the Cambridge GA Research Group.
While I think GCH algebra (a much better term than "geometric algebra", please consider it) has lots of potential, I see it as a more advanced tool. To establish it simply and logically in an elementary setting--without making key assumptions, is not so easy!
You keep saying that the real number system/theory is logically flawed. But in what way, can this be shown by a proof, logical or mathematical? Is it possible to show a contradiction in the system, or is "logically flawed" just an opinion that it might be shown logically inconsistent in the future, but we are not really sure which is right at the moment?
I certainly see both advantages and disadvantages with real numbers as compared to rationals. One disadvantage is the not too exact correspondence to the non-dimensional points in the line, where rationals are exact and irrationals are, well, not so exact--in that there are many different points corresponding to single irrational real numbers due to the completeness property (pairing up all the points on the line with irrationals where ther are no corresponding rationals).
Maybe the completeness property is what is bothering you most, the lack of precision in the reals compared to the rationals? If that is the case, I don't see a solution in other number systems, because they work in similar fashion to the reals, but maybe one could construct a better, "more complete" number system that could represent all points on a line with separate numbers. Until then, maybe a hyperrational system might give good enoughapproximations to do math with, extending the rationals with infinitesimals and infinites. That way you would get perfect precision AND infinity in math!
Have you been watching any of the other videos in this series?
Not all of them. Maybe the answer is in a video i haven't seen.
MisterrLi As several people have pointed out in comments of prior videos, it is unreasonable to ask someone who is critical of a subtely flawed theory to come up with a contradiction. There are other, much more common, ways in which logical weaknesses manifests themselves: in a lack of explicit examples and computations, in arguments which border on the philosophical, in confusions about terminology. If I point out weaknesses in your theory of real numbers, you have an obligation to defend your position and answer the charges, don't you think?
In modern mathematics, contradictions are finessed away as semi-amusing paradoxes. For example, in a recent video, I have shown that with a Cauchy sequence point of view, all real numbers are in fact the same. Is this a contradiction?
njwildberger "Is this a contradiction?" If you really did show that the Cauchy sequences all lead to the same number (or something in that direction), that would certainly qualify as a contradiction, debunking the real number system as we know it. Why not publish that? If, on the other hand, it is all about which axioms or definitions to pick in the first place, it is less convincing. Is ZFC ok?
Not really sure what you mean by "logical weakness" if the system can't be shown to be false in some way. Well, I do agree that the real number theory could be explained much clearer. This is something I enjoy doing, visualize math concept using images and animations to make them easier to grasp. With the addition of interactivity and textual info of course.
MisterrLi Actually I did publish that result! It was published on this TH-cam channel, on Nov 23, 2014, in the video
MF114: Real numbers as Cauchy sequences don't work! with link th-cam.com/video/3cI7sFr707s/w-d-xo.html.
Things are correct. Concrete. Complete. Not a take over.
This is choking
is this guy a mathematician or a sociologist? sometimes i get mad and yell at cantor until i put on some schoenberg and reconsider the radical subject. anyway, he comes off a little schizoid in the videos with the constant video cuts, and some of his comments were insightful whereas others were strange. perhaps some of this will occur in progression but there are 'famous potholes' that have gotten in the way of some of the more radical shifts he suggests. the psychology of mathematics is perhaps going to become more adept at its methods, and theological mathematics is a much stronger social force than many pseudorationals expect (prosperity gospel is a booming business and number theory portal for the masses). philosophy will probably expand into a nightmare. high science will be safe from major inconsistencies. the economics will likely mitigate anti-traditional approaches up to the point of futility. "alternate" opinions will not have access to mathematics, but may bubble up 'math-isms' which have huge importance in fields not conventionally labelled mathematics.
Mr. Wildberger, you "hate" the concept of real numbers and "love" the rational numbers. Thinking about that, it seems to me that what you hate is infinity, because when the numerators amd denumerators of real numbers become larger and larger you naturally end up having real numbers??
+Juha Immonen Let's say that I am crusading against imprecision. I have no problem with infinity, in the same way I have no problem with ghosts. If someone is able to define either of these terms, then we can have a discussion about whether or not such things exist. But first we must clarify what it is we are talking about.
njwildberger Thank you for the reply! One question: If we have the right triangle for ex. with sides 3, 4 and 5, do you find it meaningfull to speak about the lengths of all sides? If we then have a more general right triangle with two integer or rational side lengths, would you say that we might have a third side which has no length at all ? :-)
Tiny nitpick: you pronounce David Hestenes’s last name a bit differently than he pronounces it himself. You can hear him say it right at the beginning of this lecture, th-cam.com/video/ItGlUbFBFfc/w-d-xo.html
First of all I want to say I'm a long time fan of your videos and to be quite honest I've really enjoyed and learnt alot from them, even and also from the lectures in which you present this problem of real numbers. But I want to say something that could refresh your way of thinking and teaching.
I think you are too much focused on this fixation of yours that real numbers must be eradicated. And by being too narrowly focused on this one problem makes your vision of other problems more blurry. I'm definitely not saying you are blind to other problems in mathematics but I think priorities must be made clear before presenting anything at all. I think TH-cam is a great platform to learn any subject, like mathematics, and I would so much like to see more videos from you that really teach mathematics.
Videos such as this one I think should be a side project. You can't just state a number 2 to students before teaching that 1 + 1 makes 2. In a simple sense. Like someone said earlier, this seems more like philosophy almost. You have a great teaching style and would be really happy to REALLY learn mathematics. Teach us the way, master!
Sir ,, I really admire and feel satisfied to see some one who is an anomaly in this closed , no room for discussion kind of world of Mathematics.
Let me ask you something . Have you been interested in Indian Spiritual and Mystical Traditions ever ?
+Tejas Natu Actually I am an admirer of Indian spiritual traditions. But I don't think that has much to do with my approach to mathematics.
heavy!
A radical remedy is needed if we want to brush irrational numbers under the carpet: The notion of _distance_ is not defined in a plane, since it's pure 1-dimensional concept. More 2-dimensional is to talk about _area_. This works since square of a rational number is again rational, and taking square roots is not even defined! Now we can actually construct everything in *direct* and *finite* fashion, so no need of fictitious infinities or _argumentum_ _ad_ _absurdum_ or wishful thinking in our proofs!
(This was all parody)
I was starting to think you were beginning to make sense!
njwildberger So was I {:?) . I am sorry Juho, think of infinity as being stuck in an obsessional compulsive loop, or, think of it as mathematical delusion. Ok, they are not infinite because we die before we get to the largest number.
njwildberger
Why should I be converted? With analysis we (all) can do difficult-convoluted-geometrical-calculations and get the perfect answer within ε, with ε as small as we wish. It's not an ugly approximation -- it's pure mathematics! But if you don't know what the notion _limit_ means, only God can help you.
Altought in _continious_ math everything goes _smooth_ (pun int.) I'm willing to belive that physical world is discrete: energy in quantized (_quanta_), charge is quantized (_elementary_ _charge_), matter is quantized (_quark_), so it's likely that time and space are quantized too. Maybe that fundamental scale (Planck's lenght/time) is the realm where "discrete calculus" is needed.
But changing the notion of metric with every dimension for just to salve the rational numbers... well, that's just bullshit...
Juho x Pure mathematics based on a lie? Just because it is simple? What happened to the quest for the truth? But who cares for the truth if belief in God makes ones everyday life easier (and also I will live on after death, tralalala...)
In one of analysis books the author gives the usual definition of an ordered pair (a,b)=: {{a},{a,b}}. He calls this definition "startling" (of course it is totally senseless) but then he praises it because it is easy to use it to prove basic properties of ordered pairs. What kind of mathematical thinking is that?!?!?! Just because a lie makes life easier does not make it excusable.
Aleksandar Ignjatovic said "the usual definition of an ordered pair (a,b)=: {{a},{a,b}} ... (of course it is totally senseless)"
You don't accept that finite sets exist?
If you do accept that finite sets exists, why would {{3},{3,7}} be totally senseless?
This guy is crazy
Dude has a book called "DIVINE PROPORTIONS"
❤
You won’t readily find a willingness for debate on all these ideas. What you are proposing requires nothing short of a radical reformulation of modern mathematics. Instead of constantly banging your head against the wall and talking and complaining about it till you’re blue in the face, start working on your own comprehensive treaties on these subjects and show people by example what could be done in all those areas that you had pointed out. Then it would be up to the mathematics community to decide which approach it preferes to follow. However, if your aim is to recruit converts to do the work for you, you’d be seriously swimming against the tide.
@NothingMaster, Swimming against the tide is no problem, often the view is better. If you check out the Algebraic Calculus One course, or the Universal Hyperbolic Geometry or Wild Trig series, you will see that I am doing just what you propose--showing by example that there is a simpler and better mathematics. But at the same time I can also pleasantly point out that the current form of pure mathematics is logically very weak, and explain exactly how. As for whether or not pure mathematicians currently can appreciate these concerns, that is much less important than providing a road for future mathematicians to explore. Forward!
Thank you for the MF* videos.
Your "new math" reminds me something.
Galois re-defines the essence of symmetry thanks to group action over sets* (and then over fields). So groups becames the bricks of Mathematics. Now also of physics, from General relativity to Supersymmetry. Algebra instead of Geometry. The central role of the rational and finite fields in your approach reminds me this (old) "new language".
Boole vision about algebraic logic. The Boolean Algebra is one of my favorite (old) "new language". One of the rivers that flows from it is the concept of partially ordered set, and then the "Universal Algebra" (thanks to Whitehead and then G. Birkhoff). Algebra instead of Logic. The central role in your "new math" of "algebraic structures that can be defined by identities" (this is the definition of varieties in Universal Algebra) reminds me this (old) "new language".
One of the most strange and beautifull theory that cames out in this context is the "Pointless Topology", in which we focus on lattice instead on neighborhood. Poset instead of Topology. This reminds me your work on Rational Trigonometry: quadrance and spread instead of length and angle. In rational trigonometry we can regain length and angles but also in pointless topology we can define "points". This is a difficult message to universalize.
Also the central role of Algebraic Geometry in your Rational Trigonometry (as your articles points out) is in the direction of "Algebra instead something". Working on the (e.g. qradrance and spread), in an affine or projective context, is one standard strategy of translation between other languages to Algebra.
Other powerfull "translation tools" are Linear Algebra and Computability**. You cite them.
Stanley (article of 1989, book: Topics in Algebraic Combinatorics, 2013) works with Linear Algebra to prove that the Boolean lattice is Sperner. Algebra instead Combinatorics.
I think that your "new math" lives in the present, not in the future. The previous examples deal all with the question: can we overcame Analysis? All these theories wants to subtract something to Analysis.
All the previous theories (and Analysis) lives together, and we can learn from them.
*Is not exactly a group action over sets as we now define it in 2014.
** The "hash table revolution" in Computer Science give us functions as "first-class" object. Is Object-Oriented programming "the way" to overcame floating point? I don't think so. Big integer algebra (also thanks to DFT) is mature?
pointless
Pointless topology is an elegant topic indeed.
you are my god
Real mathematicians do it without reals.