Dear Parents and fellow Ugandans, this is sweet !! First of all, Mathematician N. J. Wildberger is a University of Toronto Alumnus. Which is wonderful; our beloved UoT !! For *Primary and Secondary schools* teachers he is stressing a branch of Mathematics called '*Foundations of Mathematics*' ( Foundations of Knowledge/Theory of knowledge). At Makerere they never used to teach '*Foundations of Mathematics*'. Their three-year degrees are crappy and lousy. So for our School Maths teachers back in Uganda this is a fantastic resource .Our Maths Teachers also should check out videos for his *Linear Algebra*; and others for his '*Rational Trigonometry*'======================== Is N. J. Wildberger a joke or a genius when he claims that mathematics, in its current form, is a hoax? Wildberger is an Associate Professor of Mathematics at UNSW ( Prof N J Wildberger Personal Pages <http://web.maths.unsw.edu.au/%7Enorman/index.html> ) author of the book DIVINE PROPORTIONS : Rational Trigonometry to Universal Geometry ( Rational trigonometry <http://en.wikipedia.org/wiki/Rational_trigonometry> ).
He claims that mathematics in its current form has serious foundational issues and proposes new foundations which he explains in detail in his video series: Is he a joke or a genius? As a blog ( Dirty Rotten Infinite Sets and the Foundations of Math <http://scientopia.org/blogs/goodmath/2007/10/15/dirty-rotten-infinite-sets-and-the-foundations-of-math/#more-529> ) puts it: *" This isn't the typical wankish crackpottery, but rather a deep and interesting bit of crackpottery. "* Wildberger questions the very basics on which contemporary mathematics rests. He claims that the notion of infinite sets, functions, real numbers, even angles in trigonometry are problematic concepts which can't be taken for granted. He follows a constructionist approach based on integers and rational numbers. Instead of membership in sets and functions he speaks of types and expressions. He is not prepared to accept infinite sets or 'functions' as perfect completed entities - he would rather treat a mathematical object only something that can be constructed. In this view things like the Banach–Tarski paradox <http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox> are not interesting or isolated mathematical facts without relevance to the foundations of mathematics, rather they are completely counter intuitive results, even worse: they show that the foundations on which they are based are plain wrong. He also looks at known things from a fresh perspective. For example he creates the type 'PolyInt' ( integer 'poly number' ) based on integers as an abstract representation of polynomials. Without reference to real numbers, sets, functions, even without any visual representation of 'tangents to curves' he is able to introduce the idea of calculus and Taylor expansion purely in a finitistic algebraic way, where coefficients in the Taylor expansion ( called 'subderivatives' ) emerge as a scaled version of the Pascal array ( and the expansion therefore contains no 'factorials' ). Comment1 <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax#> 5 Answers Ask to Answer <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax#> [image: David Joyce] <http://www.quora.com/David-Joyce-11>David Joyce <http://www.quora.com/David-Joyce-11>, Professor of Mathematics at Cl... (more) <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax#> 50 upvotes by Adam Merberg <http://www.quora.com/Adam-Merberg> (Ph.D. student in mathematics at UC Berkeley), Anurag Bishnoi <http://www.quora.com/Anurag-Bishnoi> (Ph.D. student in Mathematics at Ghent University.), Alon Amit <http://www.quora.com/Alon-Amit> (PhD in Mathematics; Mathcircler.), Joachim Pense <http://www.quora.com/Joachim-Pense> (Got a degree in Mathematics.), (more) <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax#> I'm looking at Norman's videos now. The entire series is listed at MathFoundations - YouTube <https://www.youtube.com/playlist?list=PL5A714C94D40392AB&feature=view_all> The first 12 videos are the standard introduction to the theory of natural numbers using the Dedekind/Peano axioms. Then comes rational numbers. The 14th video with Ford circles is worth while if you haven't seen them before. In the 15th he comments on math education in primary schools. It's the 16th, "why infinite sets don't exist" where it should get interesting, but it doesn't. He offers no actual criticism of infinite sets, but does mention that there were mathematicians that rejected them when they were originally proposed. In the 17th he describes very large numbers and concludes with the statement: "There's no right that we have to say that we have to say we understand all the natural numbers. We don't. And there's no need to pretend that we do." I don't find that a serious concern. Whenever I study a new topic I don't understand much of anything about it, but I don't reject it due to my ignorance. You can't use ignorance to justify the existence or nonexistence of something. Several nice videos on Euclid's geometry follow. In video 22, he mentions Hilbert's formalism as a foundation of geometry to replace the incomplete foundation that Euclid used. He claims that "for high school teachers [Hilbert's foundation] was completely impossible." I doubt that high school teachers ever knew or used Hilbert's foundations. His statement "If David Hilbert could not come up with a good system for the foundations of geometry..." is also unjustified. Hilbert's system is a good system for the foundations of geometry. He mentions that geometry is not taught in many high schools and argues that it needs to be taught again. I agree. Next come videos 24–28 on a basis for geometry. He uses the plane coordinatized by rational numbers. A point in it has two rational numbers as its two coordinates. That's fine for lines. You might think it would cause a problem with the distance between two points [image: (a,b)] and [image: (c,d)] because that distance is [image: \sqrt{(c-a)^2+(d-b)^2}] which is not usually a rational number. It's not a real problem, however, since you can use the square of the distance which is rational. At worst, it makes things hard to express using the square of the distance instead of the distance itself. There are problems with this approach. For example, there are no equilateral triangles in this geometry. Lines that we normally consider to intersect circles don't (since they don't intersect at points with rational coordinate). Likewise, circles don't always intersect. So, for example, Euclid's Proposition 1 in Book I is false in rational-coordinate geometry. As to the question, is this a hoax? No, but it's not geometry as we know it. Restricting mathematics to only rational numbers is crippling. No equilateral triangles, lines and circles that don't intersect, and that's only the beginning. Still, for applications in number theory, rational-coordinate geometry is appropriate. There are 78 videos in the series, and I've only looked at portions of some of them. There's a lot of well-presented mathematics. The unusual way Wildberger looks at mathematics has some merit to it. Updated 23 Apr <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax/answer/David-Joyce-11> . Downvote <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax#> Comments1+ <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax#> Related Questions - - Genius and Geniuses: Which is more impressive, Terence Tao's PhD in Mathematics at the age of 20 or Stephen Wolfram's PhD in Theoretical Physics at 20? <http://www.quora.com/Genius-and-Geniuses/Which-is-more-impressive-Terence-Taos-PhD-in-Mathematics-at-the-age-of-20-or-Stephen-Wolframs-PhD-in-Theoretical-Physics-at-20> - Mathematical Puzzles: How many 5-letter words can be formed using 3 letters if repetition of numbers is allowed? How can you generalize this solution for any n > m ... (continue) <http://www.quora.com/Mathematical-Puzzles/How-many-5-letter-words-can-be-formed-using-3-letters-if-repetition-of-numbers-is-allowed-How-can-you-generalize-this-solution-for-any-n-m-such-that-using-m-letters-words-of-length-n-can-be-formed> - Mathematics: If three points are chosen at random on a circle, and a triangle is formed by taking the three chosen points as its vertices, what is the prob... (continue) <http://www.quora.com/Mathematics/If-three-points-are-chosen-at-random-on-a-circle-and-a-triangle-is-formed-by-taking-the-three-chosen-points-as-its-vertices-what-is-the-probability-that-the-triangle-thus-formed-contains-the-center-of-the-circle-within-it> [image: Hans Hyttel] <http://www.quora.com/Hans-Hyttel>Hans Hyttel <http://www.quora.com/Hans-Hyttel>, Associate Professor 29 upvotes by David Joyce <http://www.quora.com/David-Joyce-11> (Professor of Mathematics at Clark University), Anurag Bishnoi <http://www.quora.com/Anurag-Bishnoi> (Ph.D. student in Mathematics at Ghent University.), Rajeev Ramachandran <http://www.quora.com/Rajeev-Ramachandran>, (more) <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax#> N.J. Wildberger is neither a joke nor a genius. It appears to me that he is re-discovering some ideas from constructive mathematics and is relating them to the teaching of mathematics. Not only is there a link to Brouwer's intuitionism, but Wildberger's worries about infinite sets also appear to be related to the finitism of Leopold Kronecker. In other words: The concerns are legitimate but they have already been addressed a long time ago (and, in my opinion, much more convincingly), about 100 years ago when mathematics was undergoing its so-called *foundational crisis. *See the exposition by Evan Warner from Stanford for more about this. <http://math.stanford.edu/%7Eebwarner/SplashTalk.pdf> What is new in Wildberger's work is – as far as I can tell – his concern about how a constructive approach to maths should influence how we teach the subject. Written 21 Apr <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax/answer/Hans-Hyttel> . Downvote <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax#> Comment1 <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax#> [image: Emily Czinege] <http://www.quora.com/Emily-Czinege>Emily Czinege <http://www.quora.com/Emily-Czinege> 14 upvotes by David Joyce <http://www.quora.com/David-Joyce-11> (Professor of Mathematics at Clark University), Jordan Aaron Mandel <http://www.quora.com/Jordan-Aaron-Mandel>, Abhinav Jangda <http://www.quora.com/Abhinav-Jangda>, (more) <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax#> *Tl;dr *As far as I can see, his frameworks are kind-of rigorous, but his objections to standard calculus are moot and his own definitions are too restricted, avoiding real numbers and infinite sets. In essence, he builds a new maths on intuition instead of set theory, making for some superficial rigor, but removing the power of mathematics. He basically reinvents set theory in a more intuitive but less general way. ------------------------------ He starts of by giving a definition of numbers and arithmetic, which I think can be a good thing; schools often start out by taking some ideas for granted, and some students have trouble just accepting that basis. His definition is not particularly rigorous, but very intuitive, combining the standard "real-world" three-apples-and-seven-apples-make-ten-apples intuition with some basic ideas from a more rigorous framework, without actually invoking that complex monster. However, his videos stray more and more from actual mathematics. The pattern is to take a canonical mathematical idea, tell us what he thinks is wrong with it, and define a more restricted analogue that he finds more natural, using only rational numbers. It's actually commendable how neat his framework is for rational numbers. As an example, video 106 is about limits. After giving the standard epsilon-delta definition, while complaining that it is too hard to understand that many higher education programs even shy away from it (those must be some really crappy math programs then), he explains why it is incorrect: - *The term 'sequence' is not well-defined, so we can't define limits* This is false, a sequence is a function [image: \mathbb{N}\rightarrow X], for some set X. He claims in other videos that the function's definition is that it is a 'rule or procedure' from one set to another, the problem being that a 'rule or procedure' is not well-defined. The real definition of functions is of course based on set theory. But he avoids saying what's wrong with set theory religiously, instead saying what's wrong with the intuition that people have from the definitions that follow from it. - *The definition uses real numbers epsilon, A and sequence values* According to him, sequences and limits are often defined in a real context, and real numbers are defined as limits of sequences of rational numbers, making a circle. But it is perfectly correct to define rational sequences and limits first (even for epsilon, as the definitions with rational or real epsilon are equivalent), and define real numbers from there. Here again, he removes a set theory definition on the grounds of the intuition behind it being unclear. - *How to verify the definition?* If it was hard to verify, this would still not be a flaw in the definition. But it's not even that hard to verify; giving a relation between epsilon and N is a good proof. We do not need to give some kind of "infinite table", I have no idea why he would think that. He invokes the idea of infinity in weird places where it is not relevant, and shies away from it when it is relevant. - *Do we need to prove this association or just state it?* Prove, obviously. ?? And the proof is not even related to the definition. - *Do we need absolute values? Especially problematic in higher dimensions.* False. Limits are extended to 'higher dimensions' easily by using the vector norm. They are also natural in more general metric spaces, and even in topological (Hausdorff) spaces. This is part of the beauty of the epsilon-delta definition, not a flaw. He himself defines limits on "polynumber sequence-ons" (which are his version of rational polynomials and sequences), so that the limit of p(n) is A iff there exist m ("start") and k ("scale") so that [image: -k/n \le p(n)-A \le k/n]. The advantage is that you only need to find m and k, not an actual relation. It doesn't work for sequences that converge more slowly than [image: 1/n], but there are no such "polynumber sequence-ons", so it works nicely. Of course, you still need to fill an "infinite table" to give a formal proof for all n, so it doesn't even take away the original alleged problem. Written 21 Apr <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax/answer/Emily-Czinege> . Downvote <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax#> Comments1+ <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax#> [image: Anonymous]Anonymous Neither, he's simply opinionated. Anyone with "foundational" gripes will feel like they have a thorn in their shoe. For others "hoax" is too strong a word. Written 21 Apr <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax/answers/4846902> . Downvote <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax#> Comment <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax#> [image: Anonymous]Anonymous 1 upvote by MelroseVick MelroseVick <http://www.quora.com/MelroseVick-MelroseVick>. Neither. He's what you get if you took everything John Gabriel on Quora says, take out the wrong bits, take out the polarizing insults,add in a college degree, a Ph.D. and tenure at a university. Written 22 Apr <http://www.quora.com/Is-N-J-Wildberger-a-joke-or-a-genius-when-he-claims-that-mathematics-in-its-current-form-is-a-hoax/answers/4853875> . 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