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What was so divisive about his theories? How did we resolve it? How could the best minds not do so back then?

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What was so divisive about his theories? How did we resolve it? How could the best minds not do so back then?

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HallsofIvy

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His main enemy was Kronecker. K thought that infinity was a bogus concept, so naturally he didn't like what Cantor was up to. At the time even proof by contradiction was controversial.

What was so divisive about his theories? How did we resolve it? How could the best minds not do so back then?

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Was it? Even Euclid uses proof by contradiction many times in his Elements, and nobody had an objection to that! It is true that nonconstructive methods were controversial, but that is not the same as proof by contradiction being controversial!At the time even proof by contradiction was controversial.

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The intuitionist movement was in full swing, which denies proof by contradiction.Was it? Even Euclid uses proof by contradiction many times in his Elements, and nobody had an objection to that! It is true that nonconstructive methods were controversial, but that is not the same as proof by contradiction being controversial!

But I have to admit that what I had in mind was existence proofs, which are often proofs by contradiction. Hilbert was criticized for them.

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Intuitionism does not deny proof by contradiction, it denies the law of excluded middle. There is a subtle difference.The intuitionist movement was in full swing, which denies proof by contradiction.

And intuitionism as we know it today was started by Brouwer as a reaction to Cantor. So it cannot be true that the intuitionist movement was in full swing. There is pre-intuitionism too, which included Borel, Poincare, Lebesgue and perhaps Kronecker. You might be referring to this. But they had no problem with the law of excluded middle.

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It seems to me that proof by contradiction relies on the law of excluded middle. NOT(NOT(A)) => A. The distinction is too subtle for me.Intuitionism does not deny proof by contradiction, it denies the law of excluded middle. There is a subtle difference.

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How would you prove that ##\sqrt{2}## is not rational in constructivism?

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How would you prove that ##\sqrt{2}## is not rational in constructivism?

Beats me. Not that I care.

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Did prominent critics like Poincare, Weyl and Wittgenstein eventually turn around?

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Sorry, this thread is not about proving but about why brilliant men couldn't use reason to reach a conclusion.How would you prove that ##\sqrt{2}## is not rational in constructivism?

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TeethWhitener

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This doesn't have to be true. If we define proof by contradiction as ##(\neg p \rightarrow \bot ) \models p##, then we can get there without the law of the excluded middle through the conjunction expression of a conditional. Very roughly: ##(\neg p \rightarrow \bot) \leftrightarrow (\neg(\neg p \wedge \neg\bot))##. Modus ponens gives us the right side of the biconditional, which simplifies to ##\neg(\neg p \wedge \top)##. A simple truth table shows that this expression is only valid for ##p## true.It seems to me that proof by contradiction relies on the law of excluded middle.

EDIT: In fact, Wikipedia is telling me that ##\neg p## is

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Erland

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The ordinary proof is perfectly constructivistic. But I suppose that was your point.How would you prove that ##\sqrt{2}## is not rational in constructivism?

The last is correct. But can't follow you before that. It is not obvious that the biconditional is valid in intutionistic logic, and truth tables cannot be used there.This doesn't have to be true. If we define proof by contradiction as ##(\neg p \rightarrow \bot ) \models p##, then we can get there without the law of the excluded middle through the conjunction expression of a conditional. Very roughly: ##(\neg p \rightarrow \bot) \leftrightarrow (\neg(\neg p \wedge \neg\bot))##. Modus ponens gives us the right side of the biconditional, which simplifies to ##\neg(\neg p \wedge \top)##. A simple truth table shows that this expression is only valid for ##p## true.

EDIT: In fact, Wikipedia is telling me that ##\neg p## isdefinedas ##p \rightarrow \bot## in intuitionistic logic, so it seems that proof by contradiction is baked in at the outset (or at least my model of proof by contradiction).

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I'm sorry but could you please stay on topic?I know this is an old thread, but I couldn't resist:

This doesn't have to be true. If we define proof by contradiction as ##(\neg p \rightarrow \bot ) \models p##, then we can....

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TeethWhitener

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Sorry, in the first part I was showing that you can get proof by contradiction inThe last is correct. But can't follow you before that. It is not obvious that the biconditional is valid in intutionistic logic, and truth tables cannot be used there.

Alright, if you insist , but I warn you, this is going to get into philosophy very quickly.I'm sorry but could you please stay on topic?

The problem here is that you're assuming there's a "right" conclusion to be drawn. You're committing the same error these brilliant men did. According to logic, there's only what can be derived from the axioms. The axioms themselves are, of course, assumed to be true. The mathematicians who rejected Cantor's results were forced to look for sets of axioms that didn't allow Cantor's claims to succeed logically. But the set of axioms you choose is true by default, and the extent to which you can reach a conclusion about which axioms are best is based more on the utility of the axioms you choose than logical truth. As a really simple example, you could choose a set of axioms that gave you an inconsistent system, and within your model the axioms would be true, but an inconsistent system is practically useless. So you could say that choosing a certain set of axioms is incorrect, or wrong, but you're basing that conclusion on extra-logical considerations.Sorry, this thread is not about proving but about why brilliant men couldn't use reason to reach a conclusion.

EDIT: (and here comes the philosophy...) A lot of this hinges on equivocation between "true" as "valid within a logical framework" and some metaphysical notion of truth. These distinctions were not nearly as clear in Cantor's day as they are now, and we generally just say "valid" now to make it clear that we are referring to a truth value within a particular framework.

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So wait, Cantor introduced new axioms?Sorry, in the first part I was showing that you can get proof by contradiction inclassicallogic without relying on excluded middle. I don't know much about intuitionistic logic. As an aside, Wiki also says you can get ##\neg(\neg p \wedge \neg\bot)## from ##\neg p \rightarrow \bot## in intuitionistic logic. Again, someone more knowledgeable can speak to this.

Alright, if you insist , but I warn you, this is going to get into philosophy very quickly.

The problem here is that you're assuming there's a "right" conclusion to be drawn. You're committing the same error these brilliant men did. According to logic, there's only what can be derived from the axioms. The axioms themselves are, of course, assumed to be true. The mathematicians who rejected Cantor's results were forced to look for sets of axioms that didn't allow Cantor's claims to succeed logically. But the set of axioms you choose is true by default, and the extent to which you can reach a conclusion about which axioms are best is based more on the utility of the axioms you choose than logical truth. As a really simple example, you could choose a set of axioms that gave you an inconsistent system, and within your model the axioms would be true, but an inconsistent system is practically useless. So you could say that choosing a certain set of axioms is incorrect, or wrong, but you're basing that conclusion on extra-logical considerations.

EDIT: (and here comes the philosophy...) A lot of this hinges on equivocation between "true" as "valid within a logical framework" and some metaphysical notion of truth. These distinctions were not nearly as clear in Cantor's day as they are now, and we generally just say "valid" now to make it clear that we are referring to a truth value within a particular framework.

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No, Cantor didn't think axiomatically.So wait, Cantor introduced new axioms?

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TeethWhitener

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So wait, Cantor introduced new axioms?

Cantor himself might not have thought axiomatically (I don't know; I'm not a historian), but his work and the controversy surrounding it is probably the major reason why people in the late 19th/early 20th century hammered on axiomatic thinking so much.No, Cantor didn't think axiomatically.

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Also, because an axiomatic set theory had not yet been developed, *some* of Cantor's discoveries rested on a weak foundation. Wikipedia explains that in particular, his discovery of the cardinal numbers — the series of alephs — was flawed for this reason. I would say flawed, but still utterly brilliant in its departure from what was then conventional in mathematics.

Yet the mathematician David Hilbert — who around 1900 was widely considered the best mathematician in the world — strongly supported Cantor's work, as did most mathematicians of the 20th century. After all, it is entirely reasonable to ask whether there exists a one-to-one correspondence between two sets, such as the integers and the real numbers.

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lavinia

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This is from the Wikipedia article on KroneckerHis main enemy was Kronecker. K thought that infinity was a bogus concept, so naturally he didn't like what Cantor was up to. At the time even proof by contradiction was controversial.

"He criticized Cantor's work on set theory, and was quoted by Weber (1893) as having said, "

This is from the Wikipedia article on Cantor

"The objections to Cantor's work were occasionally fierce: Henri Poincaré referred to his ideas as a "grave disease" infecting the discipline of mathematics,[8] and Leopold Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth."[9] Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum."

and further down in the Wikipedia article

"Mathematicians from three major schools of thought (constructivism and its two offshoots, intuitionism and finitism) opposed Cantor's theories in this matter. For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea that nonconstructive proofs such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that constructive proofs are required. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at the decision via a different route than constructivism. Firstly, Cantor's argument rests on logic to prove the existence of transfinite numbers as an actual mathematical entity, whereas intuitionists hold that mathematical entities cannot be reduced to logical propositions, originating instead in the intuitions of the mind.[8] Secondly, the notion of infinity as an expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infinite set.[59] Mathematicians such as Brouwer and especially Poincaré adopted an intuitionist stance against Cantor's work. Citing the paradoxes of set theory as an example of its fundamentally flawed nature, Poincaré held that "most of the ideas of Cantorian set theory should be banished from mathematics once and for all."[8] Finally, Wittgenstein's attacks were finitist: he believed that Cantor's diagonal argument conflated the intension of a set of cardinal or real numbers with its extension, thus conflating the concept of rules for generating a set with an actual set.[10]"

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lavinia

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Also from the Wikipedia article on Cantor(see post #20)

"Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God.[6] In particular, Neo-Thomist thinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity".[60] Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake:"

From the Wikipedia article on "actual infinity:

"**Actual infinity** is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the abstractionof actual infinity involves the acceptance of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given objects. This is contrasted with **potential infinity**, in which a non-terminating process (such as "add 1 to the previous number") produces an unending "infinite" sequence of results, but each individual result is finite and is achieved in a finite number of steps."

At core in all of the criticisms of Cantor's theory seems to be the denial that an infinity could actually exist - except perhaps God .

It would seem that these ideas must deny the existence of limits since a limit is a completed infinity - by definition. E.G. the number 1 is the completion of all Cauchy sequences that converge to 1.

"Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God.[6] In particular, Neo-Thomist thinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity".[60] Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake:"

From the Wikipedia article on "actual infinity:

"

At core in all of the criticisms of Cantor's theory seems to be the denial that an infinity could actually exist - except perhaps God .

It would seem that these ideas must deny the existence of limits since a limit is a completed infinity - by definition. E.G. the number 1 is the completion of all Cauchy sequences that converge to 1.

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lavinia

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This seems to show that Cantor's theory was misunderstood since there is no largest infinite set and even if there were a supreme infinite, it may not be unique.Also from the Wikipedia article on Cantor(see post #20)

"Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God.[6] In particular, Neo-Thomist thinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity."

For instance, one might try to resurrect St. Anselm's proof of the existence of God using set theory with the following argument.

The set containing all sets of virtues is partially ordered by inclusion. Given any chain of sets of virtues, the union of the sets in the chain is also a set of virtues so every chain has a maximal element. The Hausdorff Maximal Principle now asserts that there exists a set of virtues for which there is none greater. This is Anselm's God.

Sadly, this maximal element may not be unique and also the collection of all sets of virtues may be too big to be a set.

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ultimately, what was the resolution to Cantor's controversy?Cantor himself might not have thought axiomatically (I don't know; I'm not a historian), but his work and the controversy surrounding it is probably the major reason why people in the late 19th/early 20th century hammered on axiomatic thinking so much.

Did these opposing great men of mathematics eventually stop their opposition?

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Most mathematicians and people who use mathematics in their everyday work have accepted the existence of infinities. Yes, Cantor's set theory did give rise to axiomatic set theory, particularly in light of paradoxes like Russell's paradox. (Which is: Let X be the set of all sets not containing themselves. Then X cannot contain itself, for then it would have to not contain itself. But X also cannot **not** contain itself, for then it would have to contain itself.)

A number of axiom systems for set theory were proposed, and the one that is almost universally accepted these days is ZFC, which stands for Zermelo-Frankel plus Axiom of Choice: https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory.

Some, but not many, mathematicians are Intuitionists or Constructivists today and continue to resist the idea of infinity.

A number of axiom systems for set theory were proposed, and the one that is almost universally accepted these days is ZFC, which stands for Zermelo-Frankel plus Axiom of Choice: https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory.

Some, but not many, mathematicians are Intuitionists or Constructivists today and continue to resist the idea of infinity.

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What about opposers like Wittgenstein, Weyl and Poincare?Most mathematicians and people who use mathematics in their everyday work have accepted the existence of infinities. Yes, Cantor's set theory did give rise to axiomatic set theory, particularly in light of paradoxes like Russell's paradox. (Which is: Let X be the set of all sets not containing themselves. Then X cannot contain itself, for then it would have to not contain itself. But X also cannotnotcontain itself, for then it would have to contain itself.)

A number of axioms for set theory were proposed, and the one that is most widely accepted these days is ZFC, which stands for Zermelo-Frankel plus Axiom of Choice: https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory.

Some, but not many, mathematicians are Intuitionists or Constructivists today and continue to resist the idea of infinity.

That's so amazing. I didn't know that mathematics still had such a conundrum. I thought there was only one right answer. LOL

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