[FOM] Certainty in mathematical proofs

Andrew Boucher Helene.Boucher at wanadoo.fr
Sat Oct 20 01:21:12 EDT 2007

On  19 Oct 2007, at 12:45 PM, Arnon Avron wrote:

> But even without exact line
> there are obvious cases  of the two sorts. Thus the fact
> that there are infinitely many primes was absolutely
> certain at the time of Euclid, it is still
> absolutely certain today, and so it will  remain forever.

Could you explain what you mean by "absolutely certain"?  It does not  
seem to me that "there are infinitely many primes" is absolutely  

> (Of course,
> there will always be strange people who deny it.
> The only thing one can do about it is to ignore them),

Surely you realize that many people think it is appropriate to ignore  
predicativism as well.  Is there any special, non-circular status in  
the determination that certain people (or certain views) are to be  
ignored other than that it's *you* making this determination?   (This  
is a genuine question; I'm just trying to understand your position.)

> The only way to change
> the status of a proposition to "absolutely certain" is by
> *proving* it on the basis of absolutely certain axioms,
> using absolutely certain methods of proofs.

How do you determine what are "absolutely certain axioms"?  The inner  
blinding light?

> (1) If pushed with determination, than there will
> be no escape from the claim that certainty is
> at the end a personal matter, (2) and that one
> can never be certain about what is certain.
> (3) If we leave the issue at that, then there is no
> point in discussing FOM.

I myself have always taken "certain" to have psychological content,  
and so indeed what is certain to you may not be certain to me.  So,  
under my interpretation of 'certain,' (1) is true.  But (2) does not  
follow at all, since obviously everyone is certain about what he is  
certain.  And (3) also doesn't follow as well, as there are still  
many things to discuss about FOM, since at the least the question,  
"What is the status of arithmetic?" can be speaking of other  
classifications besides certain/not-certain (e.g. analytic/synthetic).

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