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

[Lucas Wiman]

 >The idea is that we limit the universal quantifiers in FLT. Obviously
 >it has only the force of being true for integers below a certain
 >level. But this certain level is so high that one can somewhat
 >reasonably take the view that we don't care about integers any
 >higher. The essential content survives, since it seems no easier to
 >establish FLT for all integers below a very very very large finite
 >level, than to establish FLT for all integers.

Perhaps it is, perhaps it's not.  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?

This brings up an interesting philosophical issue (at least to any 
mathematicians on the list who are tired of mass and count terms).  I've 
recently been doing some work on graph Ramsey theory.  I've found that 
all results which I've been able to prove, I've first proved using on a 
certain tree with seven vertices (for people who don't know what Ramsey 
theory is, seven is a really big number).  This graph seems to be 
totally "universal" in this respect, but why?  My ability to express a 
construction for this graph is restricted so that anything I can say 
about this graph is so general and universal that it works for all 
graphs.  Why should this be so?  The existence of such numbers is 
certainly not guaranteed for all problems, though I think it is for the 
tractable ones, *and hence for all possible problems of mathematics 
understood as a psycho-social enterprise*.

People who work in any branch of mathematics can tell you that much of 
the interesting work and deep conjectures are guided by the behavior of 
a few easy to understand examples.  When a nice property of these 
examples generalizes to all objects (or an easily definable class of 
objects), often a beautiful theorem is generated.  Yet our ability to 
understand the beauty of a certain kind of theorem, or perhaps even to 
generalize results (and hence to do mathematics) may be caused by 
limitations placed on our ability to express mathematical notions.

Say there was some kind of super-human being, with the ability to 
compute and remember large things, so they could deal with an enormous 
number of possible counterexamples, but in all other ways like humans.  
Could such a being do mathematics as we understand it?  I think that the 
answer is probably no.  Such a being would find itself unable to "see" 
regularities which we see quite naturally, since these regularities 
would be lost in a sea of possible regularities.  If this being could 
practice mathematics at all, it would probably find our mathematics 
either hopelessly pedestrian (dealing with such pathetic special cases), 
or perhaps astoundingly general (seeing such broad generalities).  On 
the other hand, looking at its mathematics, we might find its 
mathematics either hopelessly convoluted (dealing with inelegant, 
incomprehensible statements--proving FLT by examining 10^10^10 cases or 
something) or astoundingly general (seeing such "high level" 
regularities).  Probably a little bit of each would be true.

**Begin Hot Air**
Perhaps Friedman's "finite foundations" project should take something 
like this into consideration, for example by thinking about the 
iterations of some "easily understood" notions, and cutting them off at 
some point.  We would thus come up with the class of "easily definable" 
natural numbers (or finite sets).  I think in such a system:

(1) Notions of mathematical beauty or elegance might become *formal* 
properties.
(2) Each proof of a statement would be "surveyable"
(3) Since the proofs of statements would have "surveyability" 
requirements, pseudo-Goedel statements would creep up.  The system might 
be "surveyably incomplete", and consistency might be "surveyably 
unprovable."
**End Hot Air**

- Lucas Wiman