Hi, Clem, nice to meet you. Great post!
clem said:
Proofs by contradiction are a standard part of math and physics.
I know of no proof of any uniqueness theorem that is not by contradiction.
I don't know how you could start.
Of course you are correct. Proofs by contradiction are used and relied upon every day. When a short proof by contradiction is possible, it often represents a very efficient and natural way to shore up a theorem. Humans often
think in proofs by contradiction. ("Hmm. It must be Saturday today because on Sunday the bank I see open would be closed.")
However, mathematicians prefer direct proofs for a couple of reasons. First, when one arrives at a contradiction, one has little insight into which of the premises "broke" the consistency. It's entirely possible that the initial axioms were themselves inconsistent before adding in the converse of the theorem to be proved. A proof by contradiction doesn't easily reveal that. For this reason, proofs by contradiction are less trustworthy as the number of axioms relied upon grows, and they're less suitable to prove nontrivial theorems. ("Oh, it's really Monday? I had assumed it was the weekend.")
Second, which is what I was really driving at, is that direct proofs often require greater exploration of the mathematical structures under study. While this makes it harder, there can be real payoff in comprehension. If I recall correctly, the first proof I ever saw of the Intermediate Value Theorem was a proof by contradiction. Years later, I saw a direct proof that relied upon the Heine-Borel theorem, which rests on the deep properties of the real numbers. Now if my goal was to
use the Intermediate Value Theorem, then the quicker proof suffices. But if my goal was to understand
why it's true, and to find other sets besides the real numbers where it still might be true, then the direct proof was harder but worth it.
None of which, of course, takes anything away from your insight that proofs by contradiction are a standard part of math and physics. But, if I could put on my computer programming hat for a moment, they are the "goto" statements of mathematics: Economical at the assembly code level, but considered harmful in higher order languages.
I'm trying to think of a direct uniqueness proof, and what comes to mind is that the inverse of an element of a group is unique. If x is an element of a group, then let y and z be its inverse, meaning that xy = xz = 1. You could do a proof by contradiction, assume that y \neq z, and then show that leads to a contradiction. Or you could just use the group axioms to prove that y = z directly.
xy = xz
yxy = yxz
1y = 1z
y = z
On the other hand, I can't think of a direct proof of the irrationality of \sqrt 2 off the top of my head, but the reductio ad absurdam proof is elegant and accessible.
