[FOM] Davis on Platonism

[Arnon Avron]

On Wed, May 21, 2008 at 11:31:17AM -0700, Martin Davis wrote:
> It is alleged that the Russell paradox of the set 
> of all sets not belonging to themselves is a 
> problem for Platonists. The most famous Platonist 
> of them all, Gödel, pointed out that the sets 
> that occur in mathematics are those built up from 
> the empty set by iterating the power set 
> operation indefinitely (so-called iterative 
> notion of set - going back to Zermelo's 1930 
> paper).  Since none of these sets are members of 
> themselves, what the paradox tells us is that there is no universal set.

This is not so simple.

First of all, before Russell paradox was discovered, the full
comprehension principle was part of of the Platonists' mathematics.
It was used freely by Cantor and others - until it caused troubles
(in the book of Bar-Hillel, Fraenkel and Levy on the foundations
of mathematics it is the main part of what they call "the ideal
calculus"). So at least part of the original Platonists' intuitions
was destroyed by Russell paradox. Moreover: some version of the
comprehension principle is still a must for Platonists. So
they have to provide a natural, intuitive substitute,
and persuade themselves that this substitute exactly
match their "intuitions". To my best judgement, they failed,
and their intuitions and principles are still incoherent
(note that I am not saying "inconsistent"). Thus on the one hand
they know that there is no universal set. On the other hand
they believe (most of them, at least) in the existence
of inaccessible cardinals. Why? As far as I understand, because
of the basic intuition that one can form a set of all the sets
that can be constructed by any collection of acceptable constructions.
This is extremely close in spirit to the intuition behind
the full comprehension principle.

Second, the basic, historic goal of FOM was not descriptive,
but normative. Had the fact that mathematicians used something 
successfully been sufficient justification for using that 
something then there would have been no need to justify the 
use of complex (or even negative) numbers, or the use of infinitesimals
(which are in fact still very much in use in practice). So the fact
that the sets  that occur in mathematics are those built up from 
the empty set by iterating the power set  operation indefinitely
is not a justification for using the power set  operation.
The original justification, as far as I understand, was given
by the comprehension principle. Again, this justification *was* 
(not "allegedly") destroyed by Russel paradox, and I have encountered
no alternative convincing justification.

Arnon Avron