[FOM] 164: Foundations with (almost) no axioms

[Vladik Kreinovich]

Lucas Wiman wrote:

If the number is sufficiently large, 
then yes, this is true of FLT, but I would guess that this number is 
context sensitive.  For example, in finite Ramsey theory, "large" on the 
input of the Ramsey functions might be numbers less than 15, while large 
on the output end (in terms of what is provable) might be in the 
hundreds of thousands of digits.  It is known that odd perfect numbers 
must be bigger than 10^300, and there are a slew of other conditions on 
what it must be.  Is 10^300 a big number?  Goldbach's conjecture is 
known to hold for all even numbers up through 2*10^16.  Is that a big 
number?  I don't think that mathematicians find the proof that an odd 
perfect number must be really, really big to be as satisfying as a 
proof.  Say that a mathematician proved that any odd perfect number must 
be bigger than 10^10^10^10^10...10 (exponent iterated 10^10^10  times).  
While I think that mathematicians interested in the problem would find 
that an interesting result, they would distinguish it very carefully 
from an actual proof that no such number exists.  The reason for this 
distinction is an interesting problem in philosophy (or perhaps 
psychology).  Comments?

Comment: 

I think the distinction is more pragmatic than psychological.

In mathematics, when a result is proven, we can use it in other proofs, and 
this is often one of the main uses of proofs. In this reduction, we often 
reduce a property of n to some other property of a larger number say 2^n or 
even 2^{2^n}. If this second property is proven for all integers, then all such 
reductions are safe and lead to the proof of the original statement. If the 
second property is only proven for feasible integers, then we must first check 
that the reduction preserves feasibility, so this second proof is much less 
useful in this very pragmatic sense.