On 9/19/2013 4:34 PM, Dan Christensen wrote: > On Thursday, September 19, 2013 5:04:44 PM UTC-4, Peter Percival wrote: >> >>>> The first principle is the definition of x^y, but when I prove 0^0 = 1 >> >>>> >> >>>> from the definition in two lines you object. What you seem to object to >> >>>> >> >>>> is that it is just two lines. >> >>>> >> >>> >> >>> So, why don't we just include Goldbach's Conjecture in the axioms of number theory and move on? >> >> >> >> Because we don't yet know that Goldbach's conjecture is true. But a >> >> more significant point so far as the discussion of 0^0 is concerned is >> >> this: Goldbach's conjectute isn't a definition. >> > > 0^0=1 can be seen as just another conjecture. >
Whatever "number theory" is, its current representation arises from the "foundational crisis" introduced by non-Euclidean geometry. The "arithmetization of mathematics" had been a response to the uncertainty introduced by this development.
The attempt to give mathematics a more secure logical foundation had been ongoing through this period, possibly beginning with Bolzano. In one direction, it led to Cantor and Frege. In another it led to Whitehead and Russell through De Morgan.
When Hilbert had been confronted with the problem of indemonstrable infinities with respect to his formalistic axiomatics, he directed his attention to the arithmetization of proofs (defeated by Goedel and Gentzen).
The prominent role of arithmetic in modern foundations is a response to historical developments.
The moral of this little story is that the continuity of polynomials in real analysis had been one part of the "insecure" mathematics which one would expect to be "grounded" by foundational theories.
One can, perhaps, formulate useful systems using different conventions. But, you are arguing for the (metaphysically) "correct" interpretation of a term relative to an established theory.
Under such conditions, you may not suddenly invoke the arbitrariness surmised from the phrase "just another".