About existence-as-consistency
Mikhail Katz
katzmik at math.biu.ac.il
Thu Jul 8 01:13:17 EDT 2021
Dear William Tait,
In connection with the issue of consistency being sufficient for
mathematical existence, you mentioned Cantor's well-known comment
regarding mathematics and its essential "freedom". There are two
reasons why attempts to paint Cantor as a visionary anticipating
Hilbert's insights into Formalism and "consistency as existence" need
to be placed in context.
1. In addition to publishing an alleged proof that infinitesimals are
inconsistent, Cantor described infinitesimals not merely as 'paper
numbers' but as
(a) an abomination;
(b) the cholera bacillus of mathematics (this was at the time of the
cholera plague in Europe).
So apparently some forms of mathematics were "freer" than others, in
Cantor's view. To give an example, somebody who describes, say,
Scholze's perfectoids in those terms (the terms used by Cantor) today
would have difficulty maintaining a reputation as a defender of
mathematical "freedom".
2. As is well known, Cantor's set-theoretic ideas encountered
resistance among his contemporary mathematicians. What is perhaps
less known is the fact that Cantor's attempt to defend his theory was,
paradoxically, far less convincing than Leibniz's as an antecedent of
Hilbert's Formalism. Instead of disassociating mathematics from
metaphysics and working to relax the requirement that mathematical
entities must have physical counterparts, as Leibniz did, Cantor on
the contrary developed a peculiar form of metaphysics to justify his
theory, and adhered as much as possible to the straitjacket of a
correspondence to physical reality. There is a perceptive analysis of
Cantor's rather old-fashioned metaphysics in Hill's article, which I
summarized in my MathSciNet review freely accessible here:
https://u.cs.biu.ac.il/~katzmik/hill17MSN.pdf
In sum, if one is looking for historical antecedents of Hilbert's idea
that 'mathematical consistency is enough', then Leibniz seems a better
bet than Cantor.
Cheers, Misha Katz
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