FOM: Re: foundationalism (fwd)

Martin Davis davism at
Fri Sep 25 15:22:23 EDT 1998

On Thu, 24 Sep 1998, Reuben Hersh wrote:

> 	Here in foundations the story was different.  There were three
> schools, logicism, formalism, and intuitionism.  All sought to
> repair the foundations of mathematics, after the damage they had suffered
> froom the Antinomies.  None of them succeeded in their mission.  In the 
> course of an unsuccessful philosophic quest, they all created some
> original and important mathematics.

Bad history. Logicism was begun by Frege; his work was completed and his
magnum opus at the printers when he learned of the Russell paradox. The
effort to avoid methods though to be illegitimate because of
non-constructivity or use of impredicative definitions goes back at least
to Kronecker. The Weierstrass-Cantor-Dedekind foundation for analysis 
involved modes of thought troubling to many mathemticians including
Poincare, Borel, and Weyl and leading to rich philosophical
discussions. There is no reason to think that Brouwer cared especially
about the antinomies: he drew the line well below the level at which they
became issues. Hilbert certainly was concerned with the antinomies. But he
was at least at much bothered by the attack on the core of modern analysis
by Brouwer and Weyl.
> 	I was bothered not so much by the impasse in which
> foundationalist philosophy of mathematics found itself.
>  I was much more disturbed by the obvious (to me) fact that
> all three were incredible.  The pictures of mathematics they
> offered did not at all resemble the mathematics I knew as student,
> teacher, and researcher.

> 	Platonism (including its special case, logicism) invoked
> a mathematical world eternal, unchanging, and independent of human
> actions.  How this ghost world related to the material world wasn't
> even recognized as a problem!  (Later I learned about Benacerraf
> throwing the same problem at his fellow philosophers of math.)

Logicism as a movement was mainly concerned to provide a seamless
development of mathematics beginning with purely logical notions and
eventually leading up to the totality of mathematical discourse. As such
it can be followed without any necessary ontological commitments. If (as
Dedekind seems to have) one thinks of "set" as a logical notion, the
logicist program has been a great success, tacitly followed by
mathematicians from Halmos to Bourbaki.
> 	Formalism, removing or denying the content, the meaning, from
> mathematics, was equally indigestible, for in my experience the
> meaning of mathematical sentences, words, and ideas was exactly what
> made them interesting, what made it worth the trouble to analyze them. 
>  Trying to explain math by making it meaningless was to me just as
> absurd as trying to understand literature by making it meaningless.

This shows the problem of trying to understand these things from
undergraduate textbooks rather than by going back to the real sources.
This is a weak parody of what Hilbert was trying to accomplish. By proving
the consistency of a formal system in which ordinary mathematics was
embedded, by methods that such as Brouwer and Weyl would find
unobjectionable, Hilbert hoped to refute their criticisms. For this
purpose it was crucial to represent things in a purely formal symbolic

For Hilbert this was an instance of the tried-and-true method of ideal
elemenmts in which the expansion of the mathematical universe is
accomplished by a consistency proof. Since with arithmetic, he was down to
mathematical bedrock, a formalist reconstruction of mathematics was
needed. But Hilbert never "denied" the content of mathematics.


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