In article <firstname.lastname@example.org>, email@example.com wrote: > You say, I'm forced to act like I'm outside of integers at the start, > but what if there were an integer solution to FLT? > > Then wouldn't your objection fall away?
One proof of Fermat's result that primes congruent to 1 modulo 4 can be written as the sum of the squares of two integers uses complex numbers (in particular, Gaussian integers). You are proving a result about integers, there are integer solutions for the result, yet you go outside to complex numbers (and you have to specify that you are going out to complex numbers so that you can use their properties).
There is also an interesting proof of Gauss law of quadratic reciprocity that uses complex numbers and trigonometric functions like sin and cos. But, Gauss law is a statement about integers. You still need to prove the properties of complex numbers before you use them and also prove the properties of the trig functions sin and cos before you use them (e.g., that they are periodic with period 2*pi). You cannot just assume such functions exists. For example, there are doubly periodic functions in a complex variable, but not a triply periodic function.
> We casually deal with the first case and happily use sqrt(2) all over > the place without caring about the higher abstraction,
You and some others may not care, but technically the use of sqrt(2) needs to be justified.
> and act as if > the operator with its object is the actual number, > which is like using 1+1 for 2,
It is not like it at all.
Depending upon how you define natural numbers (say Peano axioms), 1+1 is already guaranteed to exist since it represents the successor of 1 and every number has a successor number and 1 is a number. Since 1+1 exists, we can give it a name like 2 and define properties like "even" that apply to it or properties like "odd" that don't apply to it.
Furthermore, the "2" can be eliminated by replacing it with "1+1" since it is only a name for something that already exits, and nothing would be lost in stating theorems or giving proofs. You cannot do that with your sqrt(2): what does it refer to that you could replace it with?
> except that it is more convenient than the alternative.