[FOM] Platonism and Formalism

[Karlis Podnieks]

----- Original Message ----- 
From: "Torkel Franzen" <torkel at sm.luth.se>
To: <fom at cs.nyu.edu>
Cc: <torkel at sm.luth.se>
Sent: Saturday, September 06, 2003 8:35 AM
Subject: Re: [FOM] Platonism and Formalism (a reply to Podnieks)


> ...
>We must examine how people actually work, how they argue, how
> they apply mathematics, what kind of considerations seem to guide
> their thinking in practice.
> ...
> A consistently non-Platonistic point of view with regard
> to mathematics is much like a consistently skeptical view of human
> knowledge: it has a certain appeal to the intellect, but it has no
> apparent relation to how people, including professed non-Platonists or
> skeptics, actually go about their business.
>
> ---
> Torkel Franzen

I tried to explain this "practical" aspect of the mathematical Platonism in
my old paper "Platonism, Intuition, and the Nature of Mathematics"
(http://www.ltn.lv/~podnieks/gt1.html):

FOR HUMANS, Platonist thinking is the best way of working with stable
self-contained systems (the "true" subject of mathematics - at least, for
me). Thus, a correct philosophical position of a mathematician should be:
a) Platonism - on working days - when I'm doing mathematics (otherwise, my
"doing" will be inefficient),
b) Advanced Formalism - on weekends - when I'm thinking "about" mathematics
(otherwise, I will end up in mysticism).
(Of course, the initial version of this aphorism is due to Reuben Hersh).

For me, such a position is neither schizophrenia, nor perversity - it is
determined by the very nature of mathematics. And this is why I would ask
FOMers having bigger mathematical intuitions than my own to try a high level
("Corfield style") explanation of the last remaining Platonist illusion:

PROBLEM
Which properties of structures and methods used in mathematics and
metamathematics are leading to the illusion that the natural number system
is
a stable and unique mathematical structure that exists independently of any
axioms and cannot be defined by using axioms?

Best wishes,
Karlis.Podnieks at mii.lu.lv
www.ltn.lv/~podnieks
Institute of Mathematics and Computer Science
University of Latvia