[FOM] Semantical realism without ontological realism in mathematics

[Bill Taylor]

Panu Raatikainen writes:

-> He says that it is possible to be e.g. a neo-Fregean platonist about 
-> mathematical objects but nevertheless deny that they have any properties 
-> other than those we are capable of recognizing, and that it is also 
-> possible to be a Dedekindian who maintain that mathematical objects are 
-> free creations of human mind but may nevertheless have, once created, 
-> properties independently of our capacity to recognize them.    

I find both these positions almost incoherent.

-> platonist about 
-> mathematical objects but nevertheless deny that they have any properties
-> other than those we are capable of recognizing,

I presume this means "capable of recognizing fairly quickly".  If so, what
about the Platonic object  2347...341  which presumably either is or isn't
a prime right now, but we won't find out for sure for several weeks?
This object seems to contravene the excerpt above.

OC if he means...
  "capable of recognizing EVER, by means we may not even know about yet",
                  ...then this is so vague as to be virtually useless.

OTOH...

->        maintain that mathematical objects are 
-> free creations of human mind but may nevertheless have, once created, 
-> properties independently of our capacity to recognize them.  

This seems to say that once created, they are COMPELLED to have certain
properties, which seems to say that IF created WILL HAVE certain properties,
which seems to say that they have them "even before" they have been created,
which is Platonism (realism) anyway.

So these views are close to being incoherent, IMHO.

But having said that, I must also admit that all this debate (including mine)
is "mere" philosophical arguments, so has very little, (no?), significance.  
This "mereness" is given away by the observation that, whichever side we come 
down on, it will have NO EFFECT on our mathematics, thus is all a non-issue.

But please tell me where I'm wrong.

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             Bill Taylor        W.Taylor at math.canterbury.ac.nz
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    For every philosopher, there exists an equal and opposite philosopher.
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