From: Brian Holtz [brian@holtz.org]
Sent: Saturday, April 20, 2002 8:29 AM
To: alt.atheism.moderated
Subject: Re: finite number of sentences

"Paul Holbach" <paulholbachSPAMBAN@freenet.de> wrote:

> what does it mean for the number 4 to be
> there, ie to exist?

A number like four is a concept which exists explicitly as
that which is in common to all four-ish acts of contemplation
and implicitly in the axioms and rules of the mathematical
system(s) in which it is believed to be producible.

> we keep on regarding those numbers as existent. But in
> virtue of what is a formalist entitled to do so?

In virtue of their theoretical producibility. The issue of
course then becomes whether one subscribes to the Law of
the Excluded Middle, the Law of Double Negation, the
Axiom of Infinity, the Axiom of Choice, etc.

> the received philosophical view on mathematics
> is not formalism or semioticism, which are variants of good old
> nominalism (sometimes with a more or less conceptualistic flavour),
> but Platonism, which philosophy doesnīt have any trouble in dealing
> with existential claims (maybe thatīs the reason why it is so
> appealing to most mathematicians)

I'm about as impressed by the platonism of mathematicians as
I am by the mysticism of physicists.

> The mathematical Platonist holds that infinite sets have
> always been there mind-independently [..]
> Against this, the mathematical constructivist holds that
> only those sets exist which have actually
> been constructed or are at least theoretically constructible.

You omit the (in my judgment preferred) option of Logicism:
the thesis that mathematics can be derived from or reduced to
pure logic.
--
brian@holtz.org
http://humanknowledge.net