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

Arnon Avron aa at tau.ac.il
Fri Sep 5 03:33:11 EDT 2003

I shall be abroad for two conferences in the next two weeks, and this
will put an end to this sudden blitz of postings from my direction. I want 
before this to "clean my table". So here is an important (to me) remark
I would like to make concerning Podnieks' postings and point of view:

In his posting from August 30 Podnieks wrote:

"I would propose 
voting of all FOM members for or against "formalism" (against or 
for "mysticism") according to the above definition by Prof.Davis. Let us 
establish two parliament groups and elect speakers? I would vote 
for "formalism". Of course, Godel, Post and Penrose would vote "against". But, 
I suspect, Einstein, von Neumann and Kolmogorov would vote "for". "

I protest that Podnieks  describes the debate whether there is
true arithmetical sentence unprovable in PA as a clear-cut debate between 
Formalism and Platonism. I myself will vote for Formalism rather than 
Platonism with respect to Set Theory, and I always have had strong formalist
tendencies (because I was, and still am,  expecting 100%
certainty in  at least some parts of mathematics). 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.  I changed my views accordingly. 

 Set Theory, especially Cantor's style, and even Classical analysis 
as it is curently concieved and taught, are completely different stories
and I am still a formalist with respect to both (not to the same
degree with respect to both).  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.

Arnon Avron

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