[FOM] Platonism and Formalism

[Harvey Friedman]

Reply to Gaifman.

On 9/25/03 6:48 PM, "Haim Gaifman" <hg17 at columbia.edu> wrote:
                
> ... The points I am going to make
> might have been touched upon by other messages
> I have not seen, but I shall make it anyway.
>  
> ...But I  cannot see  why you think that this
> should make Platonists uncomfortable.
> 
> On the contrary. 
> 
> Platonists claim that there are objective
> mathematical facts, but they do NOT claim
> that such facts (or such interesting facts) are knowable
> to us. 
> 
> The result you predict shows clearly
> that if we believe that every statement is either
> true or false in V(9), then we should also accept that
> certain facts are inaccessible to our knowledge.
> 
> This situation strengthens the view that mathematical truths
> as *facts*, whose status does not derive from
> our ability to prove them, or verify them in this
> way or another. ... Your message provides a more
> concrete and dramatic way of supporting this point.
> 
> It is true that the independence results in set theory have
> given trouble to Platonists. But this is due to the
> way they have been proven, rather than the results themselves.

The kind of uncomfortableness that Platonists could or should have, that I
have in mind, is of a different nature. Or at least, non Platonists might
use it against the Platonists, feeling that it OUGHT to make the Platonists
uncomfortable. 

Specifically, I am referring to the following.

On the one hand, it should comfort Platonists a lot that there will be
essential and valued uses of large cardinal principles, in the realm of
V(9), that they accept, or are very inclined to accept, like various
standard large cardinal axioms in the ZFC context.

HOWEVER, the results should also show that even sharper and cleaner and more
valued consequences of a similar general nature are made, in the realm of
V(9), from various large cardinal principles that are firmly rejected by
Platonists ­ on the grounds that they outright contradict the axiom of
choice. 

It has been known for some time that from the point of view of arithmetic
consequences, the strongest principles that have been studied are
inconsistent with the axiom of choice, and so the Platonists cannot argue
that their arithmetic consequences are true by appealing to the truth of
these principles (since they contradict the axiom of choice).

I suspect that this difference in strength will be clearly manifest in
various specific valued consequences that they have for V(9), where the
axioms inconsistent with the axiom of choice are substantially more
effective for deriving sharper and more satisfying results for V(9) than
axioms that are believed to be consistent with the axiom of choice.

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 can¹t 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.

Harvey Friedman

 





  
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