[FOM] Certificates are fully practical

Arnold Neumaier 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.

Arnold Neumaier

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