FOM: December 1 - December 22, 1998

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Re: FOM: certainty

F. Xavier Noria wrote:
> Dear FOMers,
>  | > Randy Pollack wrote:
>  | > >
>  | > > Vladimir Sazonov said he is "a permanent opponent of those who assert
>  | > > existence of absolute mathematical truth."  I don't know what
>  | > > "absolute truth" means.
>  | >
>  | > I too! Does anybody know?
>  | >
>  |  I don't either. But "2+2=4" seems to be absolutely true.
>  |
>  | Andrzej Trybulec
> I cannot understand what "2" or "+" or "4" or, even, "=" would mean, I'm
> afraid. I am sorry that I cannot figure out what the "set of the natural
> numbers" could be and what "truth" concerning that concept would signify.
> Nevertheless, we would agree if your claim was that "+(ss0, ss0) = ssss0"
> is PA-demonstrable or the like.

Yes, this is a very good syntactic analysis of this big problem. What 
about semantics? Take, e.g. two drops of water (or vodka or what you 
like) + again two drops. The result will be 2 + 2 = 1 (one big drop). 

Martin Davis wrote:

> I fail to understand why the formulas of PA, the set of axioms, and the
> notion of a proof in PA are considered to be easier to understand than the
> set of natural numbers and its members.

Experimentally (I mean strings of symbols to be physical objects like 
sequences of pebbles), numbers and syntactic objects have the same 
nature. Then the real fact 2 + 2 = 4 (understood in terms of pebbles) 
and provability of "+(ss0, ss0) = ssss0" are, however of different 
complexities, very similar. Of course the question on absolute truth 
is worth to consider with respect to more complicated situations. 
Thus, we should distinguish syntax and semantics. Then syntax is much 
more simple (still like real pebbles). However, semantics, i.e. the 
set of "all" (what does it mean "all"?) natural numbers, etc., is 
something *imaginary* and therefore rather vague. As to me personally, 
I am not sure that I can completely control my imaginations and 
fantasies *if it is not by a formal system*. Moreover, I have various 
versions of my imagination of natural numbers (not necessarily with 
the drops of vodka) for which not all axioms of PA hold. I do not know 
how to distinguish among all of my imaginations the unique ("standard"?, 
the "best"?) one. Thus, (for me) the general notion of truth w.r.t. 
any my imagined world of natural numbers is inevitably uncertain (even 
despite the fact that FLT was eventually decided, I hope in PA). 

In other words, I have no sufficient evidence in favour of non-vague 
notion of natural numbers. 

This is about me. If anybody have some radically different, not such 
a vague understanding of this situation, I would be happy to learn. 

But I think that it is even quite unnecessary to have "standard model" 
or "absolute truth" if we have very reliable formalizations (like PA) 
of our (more or less) vague fantasies. The philosophy is very simple. 
We have ANY kind of reality (pebbles, or our fantasies, or what we like) 
and develop formal methods which help us to approach to this reality 
much better than without these formal methods. Formal methods *regulate, 
organize, govern and strengthen* our thinking (and imagination)

> Best to all,
> Martin

> Have a good Christmas time!
> -- Xavier

and happy New Year!

Vladimir Sazonov

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