[FOM] Platonism and Formalism

[Haim Gaifman]

On 9/26/03  Harvey Friedman wrote:
....

> To summarize: the most effective way to get the cleanest, sharpest, 
> strongest information about V(9) is expected to be through the use of 
> large cardinal principles that are blatantly rejected by Platonists as 
> false. The plausible large cardinal principles, from the point of view 
> of the Platonists, give similar information, but weaker and less 
> satisfying, about V(9).
>
> A related point: if the Platonists want to justify the large cardinal 
> principles consistent with choice on the basis of their consequences 
> for V(9), then they must say why such an argument can't be used to 
> also justify the large cardinal principles inconsistent with choice, 
> as the consequences for V(9) are even cleaner, more satisfying, and 
> stronger.
>
> I am not a Platonist. I am also not an anti Platonist. So if the 
> Platonists tell me that they are or are not uncomfortable, then I will 
> accept that.
>
>  
>  
>
First, to clarify the terminology:
For the purpose of this discussion, I use 'Platonist' for someone who 
believes
 that mathematical statements in some language,
which allegedly has some intended interpretation (some structure)
have objective truth values,  True or False,
irrespective of our knowledge.  If one wants to use
'realist' instead of 'Platonist', that is fine with me.
I do not think that going into finer distinctions will be
helpful here.

Obviously, one's position will depend on the theory
in question. My original message was about Platonism
concerning arithmetic (the standard model), or very large
finite structures.

Harvey, your message is about Platonism in set theory
("Cantor's universe"), which is a much stronger position.
 The results you have been reporting for some time
have to do with implications from various versions of set theory
to arithmetic. But for this you do not need to
assume these set theories, but only that these
theories have omega-models; sometimes mere consistency will
suffice.

Say, T is some version of set theory, and A is some arithmetical
statement implied by it. Then A is implied by the assumption that
T has an omega model. If  A is Pi_1 then it is implied
by the assumption that T is consistent. This is the case
of the statement concerning V(9).

A Platonist about set theory can believe that AC
(the axiom of choice) is true, but that its negation
has transitive---and a fortiori omega---models. In fact,
Cohen's original results give him good grounds
for the second belief.

A set theorist may have also grounds
for believing that Reinhardt's axiom is consistent.
But for this you need to have   well developed
intuitions that come from high expertise in this
area. 

In the case of Peano's arithmetic, however,
the obvious ground for believing in its consistency is
one's belief in the standard model.

Finally, my remarks are intended only to
clarify one's options. I have not so far
expressed either a Platonist or an anti-Platonist
position.

Haim Gaifman
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