[FOM] Formal nature of mathematics

[Vladimir Sazonov]

Quoting Dmytro Taranovsky <dmytro at MIT.EDU> Sat, 11 Feb 2006:

> Vladimir Sazonov wrote,
>> For example, I ASSERT, not as a mere speculation, that there is a
>> simple, but rather unusual FORMAL system of axioms and proof
>> rules in which a (semi)set of natural numbers 0,1,2,... < 1000
>> is FORMALLY definable which is PROVABLY closed under successor
>> and is also upper bounded by the number 1000. Quite intuitive
>> informal examples of such semisets from our real world are well
>> known (e.g. presented by P. Vopenka).
>
> I think Sazonov meant 2^1000 instead of 1000,

In the above citation I really mean 1000 (or even 100). Have you read 
my last posting
http://www.cs.nyu.edu/pipermail/fom/2006-February/009746.html
with all the details, except formal derivations?

In fact, I should note that in discussions on feasible numbers there is 
a serious danger to run into useless speculations like "how many devils 
can fit at the end of a needle". We cannot avoid consideration of 
FORMAL details like those from my last posting. The point is that 
mathematical intuition does not exist alone, in a pure or absolute 
(platonic?) form without supporting formalism(s). This is the meaning 
assumed when I say "Formal nature of mathematics" and, I believe, this 
gives the proper meaning of the formalist view on mathematics.

Thus, again, is anything unclear in my previous posting cited above? 
Please note that it is mathematically quite precise, except probably 
some my faults in the exposition. Does it resolve your doubts on 1000?


Vladimir Sazonov

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