Cantor's Controversies: Resolving Divisive Theories in Mathematics

[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.

 

[bahamagreen]

It was not so long ago (18th century) that some mathematicians frowned on negative numbers... Some thought they did not exist and should not be used (e.g., Francis Maseres, William Friend) ; others thought they could be used if they appeared during a calculation if they subsequently disappeared before the calculation was completed.

 

[chiro]

Mathematicians look for consistency amongst abstraction (or variation).

In order to do this though the abstraction in its largest form has to be considered and this has happened many times.

One good example is differential geometry.

It was thought that flat geometry was the only geometry. Even those who working on it (like Gauss) kept it secret and Gauss had political clout in the community.

Cantor is no different to what many other leaps of faith in mathematics are - he basically considered something beyond what was considered to be only (or likely) consistent and eventually mathematicians came around to see that it could be (or was).

Same with complex numbers.

Same with higher dimensional algebras.

It's the same story again and again - basically what is assumed to be consistent in its entirety is found out to be wrong and a new form of abstraction is discovered.

Cantor was a victim of this mentality just like many other mathematicians who promoted abstraction beyond what was considered possible and consistent have experienced and it probably (unfortunately) won't be the last.

 

[greswd]

Mathematicians look for consistency amongst abstraction (or variation).

In order to do this though the abstraction in its largest form has to be considered and this has happened many times.

One good example is differential geometry.

It was thought that flat geometry was the only geometry. Even those who working on it (like Gauss) kept it secret and Gauss had political clout in the community.

Cantor is no different to what many other leaps of faith in mathematics are - he basically considered something beyond what was considered to be only (or likely) consistent and eventually mathematicians came around to see that it could be (or was).

Same with complex numbers.

Same with higher dimensional algebras.

It's the same story again and again - basically what is assumed to be consistent in its entirety is found out to be wrong and a new form of abstraction is discovered.

Cantor was a victim of this mentality just like many other mathematicians who promoted abstraction beyond what was considered possible and consistent have experienced and it probably (unfortunately) won't be the last.

do you know what happened to all the mathematicians who opposed Cantor? when did his ideas get mainstream acceptance?

 

[Nagase]

do you know what happened to all the mathematicians who opposed Cantor? when did his ideas get mainstream acceptance?

About those opposed to Cantorian set theory, I gave you a (partial?) answer in post #35; do you have any further questions about that?

As for when Cantor's ideas became mainstream, I'd say probably in the 30's, when it became clear the limits of logic set theory in general (due to Gödel's theorems). One nice indication of this shift is Tarski's 1935 postscript to his 1933 paper on truth. In the 1933 paper, Tarski still worked inside the framework of (roughly Russellian) type theory, with the restrictions imposed by that (noticed that the 1933 paper was actually written during 1929-1931, so before Gödel published his results). In the postscript, however, he indicates that he now prefers to work inside a framework which accepts the transfinite hierarchy and which is thus much closer to Cantorian set-theory. This is indicative that in the early to mid 30's, Cantorian set theory, formalized as a first-order theory, became much more mainstream.

What occasioned this shift? There are a number of factors here. Certainly, one important factor was the recognition of first-order logic as an ideal framework for such theories. Today, it's common to present ZFC in a first-order theory, so that we often forget the fact that, when Zermelo proposed his axioms in 1908, his preferred framework was second-order logic----indeed, until Weyl's and Hilbert's work in the 1910's, almost nobody saw a clear distinction between first and second order logic (take a look, e.g., at Frege's Begriffsschrift from 1879, where there's no distinction between first and second order instantiation). Even Skolem's defense of first-order logic in the early 1920's was met with skepticism. So it seems that it was only after Gödel's groundbreaking work that first-order gained in prominence, which also paved the way for a more systematic investigation of Cantorian set theory.

Other important factors were the results of investigations, by Tarski, von Neumann, Zermelo, and others, into the consequences of adopting a Cantorian framework. For instance, we now take for granted the von Neumann ordinals and transfinite recursion, but before von Neumann formulated these in the 1920's, there wasn't any rigorous theory of the ordinals and their properties. Similarly, the discovery by Tarski and others of several equivalents to the Axiom of Choice certainly made it easier for others to swallow it, specially when it was discovered that many who opposed it actually employed it implicitly in their work. Relatedly, classical descriptive set theory was just being born, with the impressive results by Lebsegue, Lusin, Suslin, and others on analytic and Borel sets. Together, these developments established set theory as an important tool in the investigation of classical mathematical areas, such as analysis and topology.

Finally, I'd also add that Gödel's work on the independence of the Continuum Hypothesis and the Axiom of Choice from ZFC (and other theories such as NBG) were also crucial in establishing set theory as an important area of investigation in its own right. They showed that set theory was not only a rich tool to be used, but also a field with its own questions and research program (e.g. inner model theory). So, by 1939, when Gödel announced his results, I'd say that Cantorian set theory was firmly established as a solid and mature research program. Considering how young it was by then, that's certainly an impressive feat!

Incidentally, if you're interested in pursuing these issues in depth, I'd strongly recommend that you read José Ferreirós's Labyrinths of Thought, a very detailed and comprehensive account of the development of set theory.

 

[greswd]

Thanks for the comprehensive and informative response.

About those opposed to Cantorian set theory, I gave you a (partial?) answer in post #35; do you have any further questions about that?

I wanted to ask chiro what he knows about the issue.

It is sad that a very brilliant mind like Poincare never came around to Cantor's ideas.

 

[chiro]

I'm not sure on specifics but usually what happens is when something reaches "political critical mass" (meaning enough people accept something to make it popular enough to be considered and then eventually verified) it gains the "main-stream" acceptance.

This happens all the time and you see it with all kinds of people from Galileo to Kepler and even people like Grassmann and Cantor (who both had ideas who were opposed for various reasons).

This is what politics does - people with political clout decide things (even if they shouldn't) because they have political clout.

Even mathematics, science, and engineering have politics and this often stops progress of many sorts from happening at the pace it should ideally happen at.

The thread has said that Cantor had supporters (like Hilbert) but in mathematics people have to reach political critical mass to take it seriously and then decide whether they want to individually verify the ideas and this is what is needed for adoption in mathematics - basically enough "votes" from people independently going through it and verifying what is being proven.