Theos-Talk Archives (January 1997 Message tl00160)

theos-l@vnet.net wrote:

> ----------------------------------------------------------------------
> 
> Date: Wed, 8 Jan 97 10:31:26 -0800
> From: Tim Maroney <maroney@apple.com>
> To: <theos-l@vnet.net>
> Subject: Re: mathematical philosophy
> Message-ID: <199701081832.KAA61544@scv3.apple.com>
> 
> >You made a definative statement about the nature of mathematical laws.
> >This is a claim that has been extensively discussed in the field, and is
> >widely agreed to have been fully resolved earlier in the century.
> 
> I'm sorry, but I have to differ. The subject is by no means considered
> resolved. To the extent that there is a dominant philosopher in the area,
> it would have to be the rather cranky but well-regarded Karl Popper, who
> insisted in no equivocal terms that mathematical laws pre-exist their
> discovery (while at the same time indulging in some self-contradictory
> screeds attacking a straw man version of Platonic idealism). If your
> mathematician friends are telling you that this issue was settled by
> Godel's Theorem, they are not accurately respresenting the state of
> mathematical philosophy.
> 
> I should mention, les I be accused of bias, that I consider mathematical
> laws to be psychological phenomena and not physical realities. I just
> can't go along with a statement that the world has come to share my
> position. It hasn't.
> 
> Tim Maroney
> 
> ------------------------------

I agree with Tim Maroney that the question is unresolved in the sense 
that competing philosophical theories are quite viable.

The psychological aspect of mathematical discovery/invention is also a 
relavant subject about which whole books have been written (e.g., The 
Psychology of Invention in the Mathematical Field, by Jacques Hadamard), 
and which ties in with the attitude of the practicing mathematician.  
Nevertheless, Pure Mathematics exists as a subject aside from any 
considerations of psychology.

Certain intuitive mathematicians approach the subject from the point of 
view that they are capturing a glimpse of an already existant reality, 
and thus, they are making a "discovery".  If there is indeed a Platonic 
realm of Ideals, and a way of getting there, then pure mathematics is a 
voyage of discovery.  But another point of view will see the entire 
enterprise as the free creation of the human mind, a kind of art, a kind 
of poetry.  So just because we invent some mathematics doesn't mean 
there is anything "out there" which already existed, and which we have 
now found.  

These are questions which must be attacked by the professional 
philosophers, and in the mean time, mathematicians will keep on doing 
mathematics.