[FOM] Platonism and Formalism (a reply to Podnieks)

[Todd Wilson]

On Fri, 5 Sep 2003, Arnon Avron "cleaned his table" by mentioning the
old debate between formalists and Platonists, a debate that is very
unlikely to be settled here, as Martin Davis suggests.  However, Avron
admits that, although he wants to be a formalist, he had a change of
heart in this debate when it came to the natural numbers:

> I became a "Platonist" with repect to the natural numbers when I
> realized two things: that even a formalist should accept some
> mathematical statements as true or false in an absolute sense,
> namely: statement of the form that a certain formal sentence is
> provable in a certain formal system (and so also sentences stating
> that a given formal system is consistent or not). Otherwise
> formalism makes no sense. Second (I believe that this was one of
> Hilbert's insights): that statements of this sort and elementary
> arithmetical statements (concerning the natural numbers) are on
> exactly the same level (something that Martin Davis has pointed out
> again in his reply to Kanovei). So I found that one cannot be both
> intelectually honest and at the same time a fanatic formalist
> concerning N.

Without taking any definite stand here on this debate, I would just
like to point out that Avron can still be a "fanatic formalist
concerning N", keeping his intellectual honesty in the face of the
worries he has expressed, and thereby save himself from selling out to
the Platonists.  For, there is no reason why a formalist need be a
Platonist concerning his formal systems.  Formal systems are simply a
means of organizing and exploring the consequences of our thinking
about mathematical topics, making it easier to keep from getting lost
or confused in complicated reasoning, thereby magnifying our own
mental capabilities in a way similar to the way a telescope or
microscope magnifies our visual capabilities.  Formal systems are also
sufficiently mathematical themselves to be the object of further
mathematical thinking, which can in turn be guided by other formal
systems, multiplying the magnification effect.  To use formal systems
in these ways, it is not necessary to commit to the belief that every
mathematical statement concerning these systems, for example their
consistency, is decided or even meaningful -- or even that all of the
infinite number of formulas and proofs that the system engenders
exists as a complete totality -- except in the context of other formal
systems.

I read Kanovei and others as saying the same thing, but perhaps
another voice added to the chorus will help Avron from losing hope for
formalism concerning even N.

> So I would like to ask Podnieks not to call "platonist" everyone who
> explicitly expresses opinions different from his opinions, and not
> to present Einstein, von Neumann and Kolmogorov as sharing his views
> (something I strongly doubt!) because they have supported *some*
> formalist ideas.

I read Podnieks's remarks on having a vote among representatives of
different philosophical positions as sarcastic and a defense of
Kanovei's being pigeonholed by Davis's one-sentence summary of where
he was "stuck".  But perhaps it is I who has misread?

-- 
Todd Wilson                               A smile is not an individual
Computer Science Department               product; it is a co-product.
California State University, Fresno                 -- Thich Nhat Hanh