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Cantor's proofs (he gave multiple) are un-controversial. There's absolutely zero question about them from any reputable mathematician. One COULD take issue wit

by xamuel 8y ago

Cantor's proofs (he gave multiple) are un-controversial. There's absolutely zero question about them from any reputable mathematician. One COULD take issue with the wording: what Cantor demonstrated is that there is no injection from the reals to the naturals (i.e., no way to assign a natural number to every real with no repeats). Anyone who disputes this is a quack. The layman controversy comes from our choice to describe this situation in English as, "There are more reals than naturals". That's merely a shorthand for the more precise statement. People get bent out of shape because they mistake the shorthand itself as some deep philosophical claim, rather than looking at what it actually means.

btilly 8y ago
You overstate your case. The proof that there is a bijection between the reals and the power set of the natural numbers depends on Cantor's bijection theorem. As you can verify from https://en.wikipedia.org/wiki/Constructivism_(mathematics) https://en.wikipedia.org/wiki/Constructivism_(mathematics) or many other sources, that proof and theorem has been rejected by many reputable mathematicians over time. Most notably including Brouwer.
- xamuel 8y ago
  Constructivism is subtler than that (the wikipedia intro is misleading). When an intuitionist says that we can't use the principle of excluded middle, they mean it more like in a functional programming sense: if we have two recipes for a cake, one of which requires a proof of X, and one of which requires a disproof of X, we cannot combine those recipes with a proof of "X or not X" and bake a cake. Intuitionists noticed that (in a sense that can be formalized), if you do mathematics while "pretending" that the law of excluded middle is doubtful, then all your proofs become constructive. There is a misconception among laymen, who see these mathematicians who are so pretending for a pragmatic purpose, and mistakenly think these mathematicians are so pretending out of philosophical principles. That's never or almost never the case. I can't speak for Brouwer's "religious" beliefs but what I can say is: if he attempted to teach students "It isn't always true that (P or not P)", without appropriate disclaimers that by saying that he's actually saying something very subtle and precise--then his math department would be obligated to stop him from misleading those students.
  - btilly 8y ago
    I have no idea what your background is, but my understanding is exactly opposite of yours. You are shoehorning people who think something very different from what you think into the framework of how you think people should think about it. The first "pure existence proof" by contradiction was due to David Hilbert in 1888. The now-named Hilbert Basis Theorem resolved a famous problem introduced by Paul Gordan. Paul Gordan's famous response was, "Das ist nicht Mathematik. Das ist Theologie." (That is not Mathematics. That is Theology.) In the debates that followed in subsequent decades, the question was whether or not we could sensibly talk about the existence of something we have no way to find or verify. Brouwer wasn't just making a subtle point in the 1920s that is palatable to modern mathematicians. He was rejecting the symbol game that was Hilbert's Formalism as meaningless nonsense. Brouwer's school lost. So it is true that any Constructivist today does have to present a weakened form in public that fits your description. But historically they meant exactly what they said. And what they said is that proving that there is an unfindable contradiction in an infinite set is not an acceptable proof of existence. And what they believed was that allowing such unsound reasoning methods would lead to contradictions. (This belief has since been disproven, but in the 1920s nobody can be faulted for having been genuinely concerned about it.) To gain a better understanding of the times, I would recommend two books. The first is Hilbert! by Constance Reid. It is a biography of David Hilbert, and does a surprisingly good job of describing the fundamental epistemological issues that lead him to Formalism. The other is The Mathematical Experience by Davis and Hersch.
    - xamuel 8y ago
      I'm not a math historian, so I'll take your word for it about the math history. I should have qualified that my statements apply to contemporary mathematics. Paradox: Cantor's theorem is uncontroversial in mathematics. But the controversialness of Cantor's theorem is uncontroversial in history of mathematics. :) I, for one, am glad Brouwer's school was defeated: I wouldn't want to choose a mathematical denomination like people choose their church denomination. EDIT: Thinking about it deeper, it does make me wonder if we haven't all already chosen a mathematics denomination, and just not realized it. It's fun to imagine an alternate reality where Brouwer won and Hilbertists are forced to couch their theorems with elaborate contortions about "when I say X exists I really mean that a Hilbert-style proof that X exists exists"...

by xamuel 8y ago

btilly 8y ago

You overstate your case. The proof that there is a bijection between the reals and the power set of the natural numbers depends on Cantor's bijection theorem. As you can verify from https://en.wikipedia.org/wiki/Constructivism_(mathematics) https://en.wikipedia.org/wiki/Constructivism_(mathematics) or many other sources, that proof and theorem has been rejected by many reputable mathematicians over time. Most notably including Brouwer.

xamuel 8y ago
Constructivism is subtler than that (the wikipedia intro is misleading). When an intuitionist says that we can't use the principle of excluded middle, they mean it more like in a functional programming sense: if we have two recipes for a cake, one of which requires a proof of X, and one of which requires a disproof of X, we cannot combine those recipes with a proof of "X or not X" and bake a cake. Intuitionists noticed that (in a sense that can be formalized), if you do mathematics while "pretending" that the law of excluded middle is doubtful, then all your proofs become constructive. There is a misconception among laymen, who see these mathematicians who are so pretending for a pragmatic purpose, and mistakenly think these mathematicians are so pretending out of philosophical principles. That's never or almost never the case. I can't speak for Brouwer's "religious" beliefs but what I can say is: if he attempted to teach students "It isn't always true that (P or not P)", without appropriate disclaimers that by saying that he's actually saying something very subtle and precise--then his math department would be obligated to stop him from misleading those students.
- btilly 8y ago
  I have no idea what your background is, but my understanding is exactly opposite of yours. You are shoehorning people who think something very different from what you think into the framework of how you think people should think about it. The first "pure existence proof" by contradiction was due to David Hilbert in 1888. The now-named Hilbert Basis Theorem resolved a famous problem introduced by Paul Gordan. Paul Gordan's famous response was, "Das ist nicht Mathematik. Das ist Theologie." (That is not Mathematics. That is Theology.) In the debates that followed in subsequent decades, the question was whether or not we could sensibly talk about the existence of something we have no way to find or verify. Brouwer wasn't just making a subtle point in the 1920s that is palatable to modern mathematicians. He was rejecting the symbol game that was Hilbert's Formalism as meaningless nonsense. Brouwer's school lost. So it is true that any Constructivist today does have to present a weakened form in public that fits your description. But historically they meant exactly what they said. And what they said is that proving that there is an unfindable contradiction in an infinite set is not an acceptable proof of existence. And what they believed was that allowing such unsound reasoning methods would lead to contradictions. (This belief has since been disproven, but in the 1920s nobody can be faulted for having been genuinely concerned about it.) To gain a better understanding of the times, I would recommend two books. The first is Hilbert! by Constance Reid. It is a biography of David Hilbert, and does a surprisingly good job of describing the fundamental epistemological issues that lead him to Formalism. The other is The Mathematical Experience by Davis and Hersch.
  - xamuel 8y ago
    I'm not a math historian, so I'll take your word for it about the math history. I should have qualified that my statements apply to contemporary mathematics. Paradox: Cantor's theorem is uncontroversial in mathematics. But the controversialness of Cantor's theorem is uncontroversial in history of mathematics. :) I, for one, am glad Brouwer's school was defeated: I wouldn't want to choose a mathematical denomination like people choose their church denomination. EDIT: Thinking about it deeper, it does make me wonder if we haven't all already chosen a mathematics denomination, and just not realized it. It's fun to imagine an alternate reality where Brouwer won and Hilbertists are forced to couch their theorems with elaborate contortions about "when I say X exists I really mean that a Hilbert-style proof that X exists exists"...