[FOM] Platonism and Formalism

Karlis Podnieks Karlis.Podnieks at mii.lu.lv
Thu Sep 18 02:27:38 EDT 2003

----- 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"

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:

Which properties of structures and methods used in mathematics and
metamathematics are leading to the illusion that the natural number system
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
Institute of Mathematics and Computer Science
University of Latvia

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