8 ms·

Wildberger is a recurring subject on /r/badmathematics including this recent story. He's almost a crank and has some very very strange views on mathematics. He

Wildberger is a recurring subject on /r/badmathematics including this recent story. He's almost a crank and has some very very strange views on mathematics. He strongly objects to the use of infinity and proofs and mathematics that involve it, which includes real numbers. If I understand correctly, he goes even farther and claims even really big but finite numbers don't "exist". And through this he claims to have "resolved" the Goldbach conjecture and other strange things.

coldtea 9y ago
The kind of crank with a PhD from Yale that has taught at Toronto uni, Stanford etc? Or is it impossible to have contrarian views to basic tenets of a human-made axiomatic system such as math without being called a blasphemer? >If I understand correctly, he goes even farther and claims even really big but finite numbers don't "exist". And through this he claims to have "resolved" the Goldbach conjecture and other strange things. Does he just "claim" or does he give proof? Because if he has the proper system, and he proves his theorems based on his axioms, and does that correctly, that's not much different than e.g. non euclidean geometries.
- Houshalter 9y ago
  Smart people can be cranks. In fact they are far more likely to be so, perhaps because they overestimate their own ability. See e.g. http://lesswrong.com/lw/j8/the_crackpot_offer/ http://lesswrong.com/lw/j8/the_crackpot_offer/ In any case I didn't call him a crank, but many actual mathematicians do. >Does he just "claim" or does he give proof? Because if he has the proper system, and he proves his theorems based on his axioms... I might be misremembering this, but I think his argument was just that "computers have tested all possible counterexamples up to some big number, therefore it's true. All this talk of different axiom systems is giving ultrafinitism vastly more credit than it's due. It's not a different but coherent mathematical system. It's a "philosophy" based entirely on refusing to accept a result because it feels counterintuitive. And then coming to vastly more counterintuitive and absurd results as a consequence of trying to resolve it. I mean they really believe that there exists a biggest number and there are no numbers bigger than that. Even regular finitists are kind of absurd, denying the existence of pi or the square root of 2.
  - pbhjpbhj 9y ago
    Under some interpretations of physics there is a finite number of objects in the universe, hence a largest possible cardinality of things that is finite. Even sets like the number of possible states of the Universe appear to me to be finite, though so large as to be intractable to my imagination. A conceptual circle and a real circle differ markedly because of the lumpiness of matter, quantisation, uncertainty. Every real circle is an approximation (yay Plato!). Pi doesn't appear to me to be real, as in writ in matter/energy -- it's a ratio of measurements of an imaginary article, circles don't exist in our experiential 4-space. Irrational numbers like root-of-2 just fall out of basic maths though, there's not going to be a real line with that measure just as there's no line with a predefined unit measure (ie unless you define that line at that point in time as your unit, but a real line isn't even straight). I'd be really interested in refutation of this -- it's one of those positions I've held since Uni, at least, but never explored as I've never seen it challenged/supported nor named.
    - revelation 9y ago
      What do you mean "refutation"? This is making you look as much of a crank as they are. There is no imaginary requirement to have a direct counterpart in the physical world, in "matter/energy". Infinity, axiom of choice, .. are all tools. We use them because their presence gives us other helpful results. None of it has anything to do with "lumpiness of matter", "matter/energy", "experiental 4-space", "real line".
      - pbhjpbhj 9y ago
        Perhaps I am a "crank". Refutation as in "you overlooked this standard point that undermines your position", like maybe "polarised light describes a perfect circle", or "the gravitational field of a black-hole (when shaved) is a sphere". AFAIK there's nothing truly infinite, nor perfectly geometric, that we can experience in our physical spacetime -- I'd like to hear from those who can show me that's wrong. The world itself isn't axiomatic, if certain theories/systems that are unfalsified show parts of it are infinite/etc. -- despite that just being a model -- that's an interesting result.
- mannykannot 9y ago
  Post-euclidean geometers did not claim that Euclid was wrong, but if the article is to be believed, this guy is saying everyone else is wrong, apparently with arguments that are sketchy at best. He is not being called a blasphemer, he is being dismissed with reasons. There is a difference.
  - coldtea 9y ago
    >Post-euclidean geometers did not claim that Euclid was wrong, but if the article is to be believed, this guy is saying everyone else is wrong, apparently with arguments that are sketchy at best. When everybody else believed that euclidean geometry was the only one possible, someone that insisted otherwise would also be labelled a crank (and he would be right in calling others wrong).
    - mannykannot 9y ago
      You are making an invalid comparison here. In your possibly hypothetical example (were the founders of non-euclidean geometry dismissed as cranks by their peers?) your lone post-euclidean would be justified in calling the others wrong for dismissing his mathematics, and also justified in denying their claim that Euclidean geometry was the only geometry, but he would be wrong if he were to say that Euclidean geometry itself was wrong. Wildberger is doing the equivalent of the latter, apparently. I am sure you understand that being labeled a crank is not evidence of actually having greater insight. Instead of arguing from a dubious analogy of uncertain applicability, you would be much more convincing if you first took the specific objections to Wildberger's position from the article, and refuted them.
szemet 9y ago
He strongly objects to the use of infinity and proofs and mathematics that involve it, which includes real numbers He is not alone, this is a known branch of mathematics called finitism: https://en.wikipedia.org/wiki/Finitism https://en.wikipedia.org/wiki/Finitism
- agumonkey 9y ago
  Thanks for this link !
- danieltillett 9y ago
  The problem is not so much infinities in mathematics (after all mathematics is just a game with any rules you care to create), but if mathematics with infinities should be used when modelling the real world. I am certainly happy to be corrected, but I don’t think we have found any event in nature that requires infinities to explain it.
  - rlpb 9y ago
    Geometry? Pi is an infinite series, as are the trigonometric functions. How can you explain the circumference of a circle without invoking pi?
    - srean 9y ago
      As long as \epsilon != 0 one can make rational_of(\Pi, \epsilon) an arbitrarily close approximation of \Pi, without the need for infinity. In fact that is precisely what we use in practice, by that I mean computation. Ultimately it comes down to a trade off. Which axiomatic system does one to work in ? Introducing infinity does have its fair share of warts (by warts I mean consequences that flies in the face of intuition).
      - rlpb 9y ago
        How do you approximate pi in this way without making use of an infinite series? Or are you saying that an approximation that doesn't use an infinity is fine even if an infinity was used to arrive at it?
bergoid 9y ago
In mathematics education, I always disliked the way inifinity is shoehorned into the different sets of numbers and treated as "just another number", so that you can write shorthands like: 1/∞=0, instead of the more accurate ∀ε>0, ∃c>0: x>c ⇒ 1/x<ε Infinity is not a number. It's a limit. Treating it as a number only confuses real understanding of it.
- jacobolus 9y ago
  It makes sense to treat ∞ ≡ 1:0 as a “number” in the context of projective geometry, the Riemann sphere, etc. The quarrel Wildberger has is not with the proportion 1:0, but with the concept of a limit.
- tnzn 9y ago
  Every single prof I have had have forbidden my class to write 1/infinity=0...
- leephillips 9y ago
  In 1960 Abraham Robinson invented the hyperreal number system, which contains infinitely small and large numbers. They behave as you would expect. The branch of analysis (including calculus) based on the hyperreals is called "nonstandard analysis". So you can develop calculus without the limit concept using this number system.

coldtea 9y ago

The kind of crank with a PhD from Yale that has taught at Toronto uni, Stanford etc? Or is it impossible to have contrarian views to basic tenets of a human-made axiomatic system such as math without being called a blasphemer? >If I understand correctly, he goes even farther and claims even really big but finite numbers don't "exist". And through this he claims to have "resolved" the Goldbach conjecture and other strange things. Does he just "claim" or does he give proof? Because if he has the proper system, and he proves his theorems based on his axioms, and does that correctly, that's not much different than e.g. non euclidean geometries.

Houshalter 9y ago
Smart people can be cranks. In fact they are far more likely to be so, perhaps because they overestimate their own ability. See e.g. http://lesswrong.com/lw/j8/the_crackpot_offer/ http://lesswrong.com/lw/j8/the_crackpot_offer/ In any case I didn't call him a crank, but many actual mathematicians do. >Does he just "claim" or does he give proof? Because if he has the proper system, and he proves his theorems based on his axioms... I might be misremembering this, but I think his argument was just that "computers have tested all possible counterexamples up to some big number, therefore it's true. All this talk of different axiom systems is giving ultrafinitism vastly more credit than it's due. It's not a different but coherent mathematical system. It's a "philosophy" based entirely on refusing to accept a result because it feels counterintuitive. And then coming to vastly more counterintuitive and absurd results as a consequence of trying to resolve it. I mean they really believe that there exists a biggest number and there are no numbers bigger than that. Even regular finitists are kind of absurd, denying the existence of pi or the square root of 2.
- pbhjpbhj 9y ago
  Under some interpretations of physics there is a finite number of objects in the universe, hence a largest possible cardinality of things that is finite. Even sets like the number of possible states of the Universe appear to me to be finite, though so large as to be intractable to my imagination. A conceptual circle and a real circle differ markedly because of the lumpiness of matter, quantisation, uncertainty. Every real circle is an approximation (yay Plato!). Pi doesn't appear to me to be real, as in writ in matter/energy -- it's a ratio of measurements of an imaginary article, circles don't exist in our experiential 4-space. Irrational numbers like root-of-2 just fall out of basic maths though, there's not going to be a real line with that measure just as there's no line with a predefined unit measure (ie unless you define that line at that point in time as your unit, but a real line isn't even straight). I'd be really interested in refutation of this -- it's one of those positions I've held since Uni, at least, but never explored as I've never seen it challenged/supported nor named.
  - revelation 9y ago
    What do you mean "refutation"? This is making you look as much of a crank as they are. There is no imaginary requirement to have a direct counterpart in the physical world, in "matter/energy". Infinity, axiom of choice, .. are all tools. We use them because their presence gives us other helpful results. None of it has anything to do with "lumpiness of matter", "matter/energy", "experiental 4-space", "real line".
    - pbhjpbhj 9y ago
      Perhaps I am a "crank". Refutation as in "you overlooked this standard point that undermines your position", like maybe "polarised light describes a perfect circle", or "the gravitational field of a black-hole (when shaved) is a sphere". AFAIK there's nothing truly infinite, nor perfectly geometric, that we can experience in our physical spacetime -- I'd like to hear from those who can show me that's wrong. The world itself isn't axiomatic, if certain theories/systems that are unfalsified show parts of it are infinite/etc. -- despite that just being a model -- that's an interesting result.
mannykannot 9y ago
Post-euclidean geometers did not claim that Euclid was wrong, but if the article is to be believed, this guy is saying everyone else is wrong, apparently with arguments that are sketchy at best. He is not being called a blasphemer, he is being dismissed with reasons. There is a difference.
- coldtea 9y ago
  >Post-euclidean geometers did not claim that Euclid was wrong, but if the article is to be believed, this guy is saying everyone else is wrong, apparently with arguments that are sketchy at best. When everybody else believed that euclidean geometry was the only one possible, someone that insisted otherwise would also be labelled a crank (and he would be right in calling others wrong).
  - mannykannot 9y ago
    You are making an invalid comparison here. In your possibly hypothetical example (were the founders of non-euclidean geometry dismissed as cranks by their peers?) your lone post-euclidean would be justified in calling the others wrong for dismissing his mathematics, and also justified in denying their claim that Euclidean geometry was the only geometry, but he would be wrong if he were to say that Euclidean geometry itself was wrong. Wildberger is doing the equivalent of the latter, apparently. I am sure you understand that being labeled a crank is not evidence of actually having greater insight. Instead of arguing from a dubious analogy of uncertain applicability, you would be much more convincing if you first took the specific objections to Wildberger's position from the article, and refuted them.

szemet 9y ago

He strongly objects to the use of infinity and proofs and mathematics that involve it, which includes real numbers He is not alone, this is a known branch of mathematics called finitism: https://en.wikipedia.org/wiki/Finitism https://en.wikipedia.org/wiki/Finitism

agumonkey 9y ago
Thanks for this link !
danieltillett 9y ago
The problem is not so much infinities in mathematics (after all mathematics is just a game with any rules you care to create), but if mathematics with infinities should be used when modelling the real world. I am certainly happy to be corrected, but I don’t think we have found any event in nature that requires infinities to explain it.
- rlpb 9y ago
  Geometry? Pi is an infinite series, as are the trigonometric functions. How can you explain the circumference of a circle without invoking pi?
  - srean 9y ago
    As long as \epsilon != 0 one can make rational_of(\Pi, \epsilon) an arbitrarily close approximation of \Pi, without the need for infinity. In fact that is precisely what we use in practice, by that I mean computation. Ultimately it comes down to a trade off. Which axiomatic system does one to work in ? Introducing infinity does have its fair share of warts (by warts I mean consequences that flies in the face of intuition).
    - rlpb 9y ago
      How do you approximate pi in this way without making use of an infinite series? Or are you saying that an approximation that doesn't use an infinity is fine even if an infinity was used to arrive at it?

bergoid 9y ago

In mathematics education, I always disliked the way inifinity is shoehorned into the different sets of numbers and treated as "just another number", so that you can write shorthands like: 1/∞=0, instead of the more accurate ∀ε>0, ∃c>0: x>c ⇒ 1/x<ε Infinity is not a number. It's a limit. Treating it as a number only confuses real understanding of it.

jacobolus 9y ago
It makes sense to treat ∞ ≡ 1:0 as a “number” in the context of projective geometry, the Riemann sphere, etc. The quarrel Wildberger has is not with the proportion 1:0, but with the concept of a limit.
tnzn 9y ago
Every single prof I have had have forbidden my class to write 1/infinity=0...
leephillips 9y ago
In 1960 Abraham Robinson invented the hyperreal number system, which contains infinitely small and large numbers. They behave as you would expect. The branch of analysis (including calculus) based on the hyperreals is called "nonstandard analysis". So you can develop calculus without the limit concept using this number system.