FOM: certainty

[Vladimir Sazonov]

Arnon Avron wrote:
> 
> On Wed, 23 Dec 1998 0:52:36 +0100 Xavier Noria wrote:
> 
> >
> >    We agree what is right about naturals and what not. All our N's have an
> >    associative addition, and prime and composite numbers and questions to
> >    know the answer, but I think that _truth_ have nothing to do with it.
> >
> >    Our N's satisfy that 2 + 2 is equal to 4, but, to my mind, this is our
> >    _agreement_ about our abstraction. Saying "2 + 2 = 4 is true" sounds
> >    quite different to me.
> >
> 
> Fine. You have convinced me. From now on I shall never say that
> " `2 + 2 = 4' is true in N". I shall say instead  that " `2 + 2 = 4' is right
> about  N". 

What is the difference?? On the other hand, "2 + 2 = 4 is (absolutely) 
true" sounds (in the context of our discussion) indeed completely 
different and, I believe, meaningles. This does not mean that, say, 
it makes sense to discuss before students in the class not devoted 
to FOM on where and why "2 + 2 = 4" is true. 

> In fact, I shall be even more careful and say that it is
> right about *my* N!
> 
>   I am glad that you agree with me about what is right about my N.
> Unfortunately, I dont feel that I have the right to say what is right
> about *your* N's. How can I know? The only thing I can say for sure is that
> since you (and Vladimir Sazonov) have several N's, you have at least one
> which is different from my own poor, lonely N (this only follows, I admit,
> from what is right about *my* logic. You may of course have several other
> logics!). So how can we agree about your N's with which I will never
> possibly have any acquaintance?

How much agreement do you want to achieve? 
Xavier Noria actually replied you. Let me say a bit more explicit: 
any "owner" of some N will tell you some axioms and proof rules on 
his N. That is essentially all what one mathematician can inform 
to another (plus may be some epsilon of his non-formalized intuition). 


Vladimir Sazonov