3 ms·

I'm not sure "try not to use [proof by contradiction] unless you really need it" is the optimal conclusion. In "Meta Math"[1], Gregory Chaitin explores the ques

I'm not sure "try not to use [proof by contradiction] unless you really need it" is the optimal conclusion. In "Meta Math"[1], Gregory Chaitin explores the question: "What makes a mathematical concept desirable?" Simplicity was the solution. The best theorems seem obvious in retrospect, because they are reduced into relationships between existing concepts. I would recommend trying all angles of attack when searching for a proof. If it starts to take too long or you hit a dead end, try a different angle. Even if you discover a proof, try your other opening moves, look for a more elegant solution. I believe Wolfram Alpha converges on this solution by performing some sort of weighted, breadth-first search on the permutations of an equation. I highly recommend "Meta Math"[1] if you are interested in theories about entropy and algorithms that generate proofs. [1] ("Meta Math", Gregory Chaitin, http://arxiv.org/pdf/math/0404335.pdf http://arxiv.org/pdf/math/0404335.pdf)

eru 11y ago
> I'm not sure "try not to use [proof by contradiction] unless you really need it" is the optimal conclusion. Oh, that's only a guideline for presenting your proves. Not for coming up with them in the first place. Mathematicians do exactly what you are suggesting already. The first prove one finds is usually quite ugly, and then comes the simplification. Writers call it editing, programmers refactoring.
- EGreg 11y ago
  Many proofs by contradiction are simply results of wrapping a constructive proof: Suppose premises/axioms (B, C, ...) which don't depend on the value of A. Then, B=true AND C=true ... implies A=true by some logical argument, such as A Or Not (B And C And ...) = True In this case, the proof by contradiction is Assume A=False But B=True, C=True, ... (Since they dont depend on A) Apply above constructive argument So A is true, A CONTRADICTION Proofs by contradiction can often be of this type so the first and last lines can be removed. The question is, what other types can there be?

eru 11y ago

> I'm not sure "try not to use [proof by contradiction] unless you really need it" is the optimal conclusion. Oh, that's only a guideline for presenting your proves. Not for coming up with them in the first place. Mathematicians do exactly what you are suggesting already. The first prove one finds is usually quite ugly, and then comes the simplification. Writers call it editing, programmers refactoring.

EGreg 11y ago
Many proofs by contradiction are simply results of wrapping a constructive proof: Suppose premises/axioms (B, C, ...) which don't depend on the value of A. Then, B=true AND C=true ... implies A=true by some logical argument, such as A Or Not (B And C And ...) = True In this case, the proof by contradiction is Assume A=False But B=True, C=True, ... (Since they dont depend on A) Apply above constructive argument So A is true, A CONTRADICTION Proofs by contradiction can often be of this type so the first and last lines can be removed. The question is, what other types can there be?