[FOM] Certificates are fully practical
Arnold.Neumaier at univie.ac.at
Sat Sep 28 03:58:37 EDT 2013
On 09/27/2013 01:07 AM, Alan Weir wrote:
> In general, I remain to be convinced that there is a sense of 'in principle possible' which is at all helpful in philosophy of mathematics, one in which, for example, all finite proofs are 'in principle' graspable but no countably infinite one is. And my point is not, of course, that infinitary proofs are unproblematic, from a naturalistic, anti-platonist standpoint. Rather it is that all but a tiny fragment of finitary proofs are problematic from that perspective; the suggestion is that it is unclear why a satisfactory resolution of that problem (if there is one) will apply only to finitary idealisations and not infinitary ones. Hope that's clearer.
From a mathematician's point of view, what counts is not whether or not
there is a proof of a statement - there are countless unproved
conjectures in mathematics.
Instead, what counts is that whenever someone hands you a putative
certificate for a proof (which therefore necessarily is finite and quite
small compared to the examples you had mentioned), a mgroup of
athematicians (possibly aided by computers) can verify or disprove the
associatited claim with an algorithm that can be executed by them in a
reasonable time, small enough that they don't lose interest.
No mathematician cares (unless they work in mathematical logic or set
theory) what happens with proofs that are significantly longer than
that. Thus a foundation of mathematics that guarantees this - for the
collection of concepts thaey are using and generating with their
definitions - is fully adequate.
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