Cantor's Controversies: Resolving Divisive Theories in Mathematics

[greswd]

Cantor was severely criticised by many brilliant mathematicians of his time, many of whom we still regard as brilliant today.

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

 

[HallsofIvy]

Primarily because Cantor did not express his ideas very well or very clearly. What we have now are his ideas expressed by other people in clearer and more logical terms.

 

[Hornbein]

Cantor was severely criticised by many brilliant mathematicians of his time, many of whom we still regard as brilliant today.

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

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.

 

[micromass]

At the time even proof by contradiction was controversial.

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!

 

[Hornbein]

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!

The intuitionist movement was in full swing, which denies proof by contradiction.

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.

 

[micromass]

The intuitionist movement was in full swing, which denies proof by contradiction.

Intuitionism does not deny proof by contradiction, it denies the law of excluded middle. There is a subtle difference.
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.

 

[Hornbein]

Intuitionism does not deny proof by contradiction, it denies the law of excluded middle. There is a subtle difference.

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.

 

[micromass]

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

 

[Hornbein]

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

Beats me. Not that I care.

 

[greswd]

Primarily because Cantor did not express his ideas very well or very clearly. What we have now are his ideas expressed by other people in clearer and more logical terms.

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

 

[greswd]

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

Sorry, this thread is not about proving but about why brilliant men couldn't use reason to reach a conclusion.

 

[TeethWhitener]

I know this is an old thread, but I couldn't resist:

It seems to me that proof by contradiction relies on the law of excluded middle.

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## is defined as ##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).

 

[Erland]

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

The ordinary proof is perfectly constructivistic. But I suppose that was your point.

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## is defined as ##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).

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.

 

[greswd]

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...

I'm sorry but could you please stay on topic?

 

[TeethWhitener]

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.

Sorry, in the first part I was showing that you can get proof by contradiction in classical logic 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.

I'm sorry but could you please stay on topic?

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

Sorry, this thread is not about proving but about why brilliant men couldn't use reason to reach a conclusion.

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.

 

[greswd]

Sorry, in the first part I was showing that you can get proof by contradiction in classical logic 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.

So wait, Cantor introduced new axioms?

 

[micromass]

So wait, Cantor introduced new axioms?

No, Cantor didn't think axiomatically.

 

[TeethWhitener]

So wait, Cantor introduced new axioms?

No, Cantor didn't think axiomatically.

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.

 

[zinq]

Infinity is a mind-boggling concept, especially the first time you run into it. Some mathematicians of Cantor's time had a lot of trouble thinking about an infinity the way that Cantor did (and that modern mathematicians do) because they were familiar only with infinity as a "process", like for example in the definition of a sequence of points approaching a limit.

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.

 

[lavinia]

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.

This is from the Wikipedia article on Kronecker

"He criticized Cantor's work on set theory, and was quoted by Weber (1893) as having said, "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk" ("God made the integers, all else is the work of man.")."

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]"

 

[lavinia]

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.

 

[lavinia]

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."

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.

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.

 

[greswd]

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.

ultimately, what was the resolution to Cantor's controversy?

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

 

[zinq]

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.

 

[greswd]

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 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.

What about opposers like Wittgenstein, Weyl and Poincare?

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

 

[Erland]

What about opposers like Wittgenstein, Weyl and Poincare?

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

There is only one right answer, but all don't agree what it is!

 

[greswd]

There is only one right answer, but all don't agree what it is!

But how come?

 

[Erland]

But how come?

I was actaully kidding a little but not entirely... But you deserve a more serious reply...

If you adopt the platonist viewpoint, there is only one answer to every question

https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Platonism

but we might not be able to find it or prove it. Also, not all mathematicians are platonists.

For example, take the Continuum hypothesis. This can neither be proved or disproved from the axioms of set theory (ZFC or NBG), provided that the theory is consistent. Still, a platonist would argue that it must be either true or false in an absolute sense, as a description of a true mathematical universe, just that we cannot know which it is, at least not just using the axioms of ZFC or NBG.
A non-platonist would not agree, and say that the issue of its truth or falsity is not meaningful in an absolute sense.

Who is right? Can we know that?

 

[greswd]

a platonist would argue that it must be either true or false in an absolute sense, as a description of a true mathematical universe, just that we cannot know which it is, at least not just using the axioms of ZFC or NBG.
A non-platonist would not agree, and say that the issue of its truth or falsity is not meaningful in an absolute sense.

do you know what happened to accomplished mathematicians but prominent opposers to Cantor like Wittgenstein, Weyl and Poincare?

 

[Erland]

do you know what happened to accomplished mathematicians but prominent opposers to Cantor like Wittgenstein, Weyl and Poincare?

I wasn't aware that anything in particular happened to them.

 

[greswd]

I wasn't aware that anything in particular happened to them.

they opposed Cantor vehemently, I wonder if they ever changed their minds about it

 

[Erland]

they opposed Cantor vehemently, I wonder if they ever changed their minds about it

No, sorry, I don't know anything more about that than what can be found on the net.

 

[Erland]

I found this about Wittgenstein. It seems that he changed his opinions more often than I change my shirt...

http://plato.stanford.edu/entries/wittgenstein-mathematics/

• 1. Wittgenstein on Mathematics in the Tractatus

• 2. The Middle Wittgenstein's Finitistic Constructivism
  • 2.1 Wittgenstein's Intermediate Constructive Formalism
  • 2.2 Wittgenstein's Intermediate Finitism
  • 2.3 Wittgenstein's Intermediate Finitism and Algorithmic Decidability
  • 2.4 Wittgenstein's Intermediate Account of Mathematical Induction and Algorithmic Decidability
  • 2.5 Wittgenstein's Intermediate Account of Irrational Numbers
  • 2.6 Wittgenstein's Intermediate Critique of Set Theory

• 3. The Later Wittgenstein on Mathematics: Some Preliminaries
  • 3.1 Mathematics as a Human Invention
  • 3.2 Wittgenstein's Later Finitistic Constructivism
  • 3.3 The Later Wittgenstein on Decidability and Algorithmic Decidability
  • 3.4 Wittgenstein's Later Critique of Set Theory: Non-Enumerability vs. Non-Denumerability
  • 3.5 Extra-Mathematical Application as a Necessary Condition of Mathematical Meaningfulness
  • 3.6 Wittgenstein on Gödel and Undecidable Mathematical Propositions

• 4. The Impact of Philosophy of Mathematics on Mathematics

 

[Stephen Tashi]

One speculation about why mathematicians of Cantor's era would tend to be anti-Cantor is that they had only recently gotten calculus on a reliable footing by eliminating the notion of infinity as an isolated concept. For example, if we look at rigorous definitions of the form "The limit of [something] as [something else] approaches infinity", such definitions do not contain any definition of "infinity" as an isolated concept. Nor do they contain any definition of "approaches". The phrases being defined suggest that they have some common language interpretation where individual nouns and verbs have an independent meaning, but the rigorous definitions do not define each word involved. The rigorous definitions replace any independent notion of "infinity" by using quantifiers such as "for each" and "there exists".

 

[Nagase]

they opposed Cantor vehemently, I wonder if they ever changed their minds about it

As far as I know, Wittgenstein never changed his mind regarding his finitism (if I'm not mistaken, he only accepted as legitimate a very weak form of arithmetic known as Primitive Recursive Arithmetic); similarly, I don't think Poincaré changed his mind about his "intuitionism", which, incidentally, should be sharply distinguished from Brouwer's. As for Weyl, although he came to embrace Brouwer's views for some time (much to Hilbert's chagrin), he did change his mind about it. The thing is, at the beginning of the movement, it seemed that one could get very far with the simple intuitionistic tools at one's disposal. It quickly became clear, however, that restricting analysis to what can only be accomplished by intuitionistic methods was a very restrictive move (just an example: the intermediate value theorem is not a theorem in an intuitionistic setting). In particular, it seems that Weyl's work on theoretical physics convinced him of the need to go beyond intuitionistically acceptable methods, so he ended up (more or less) accepting Hilbert's point of view.