zlacker

>>jeremy+(OP)
Mathematician Edward Nelson has a very interesting take on "big" numbers. He claims that the exponential function is not necessarily total, and a number like 2^1000000000000 might not actually exist. The reason he singles out exponentiation is that the reasoning that exponential numbers are "real" numbers, is circular (impredicative). According to Nelson, speaking about such numbers might lead to condradictions, just like speaking about "the set of all sets that don't contain themselves" leads to a contradiction.

He also relates these issues to Christian philosophy, which I find very interesting. In particular, he claims that the a priori belief in the objects defined by Peano Arithmetic, is equivalent to worshipping numbers, as the Pythagoreans did.

I think this is the best starting point if you're interested in reading about his ideas: https://web.math.princeton.edu/~nelson/papers/warn.pdf

>>swatow+im
I'll read the paper, it sounds interersting. But isn't all Math "circular" -- Goedels Second Incompleteness Theorem yeah?

http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_the...

>>hellba+Nm
Even if an axiom system could prove its own consistency, that wouldn't be any less circular - we could believe T because T proves that T is consistent - but if T were inconsistent then it might still prove that T was consistent.

Most of us take some axioms and believe them, ultimately because they correspond to physical experience - if you take a pile of n pebbles and add another pebble to the pile, you get a pile of n+1 pebbles, and n+1 is always a new number that doesn't =0 (that is, our pile looks different from a pile of 0 pebbles). Maybe there's some (very large) special n where this wouldn't happen - where you'd add 1 and get a pile of 0 pebbles, or where you can have n pebbles but you can't have n sheep. But so far we haven't found that. So far the universe remains simple, and the axioms of arithmetic are simpler than conceivable alternatives.