“When are two proofs essentially the same?”

Tim Gowers asks When are two proofs essentially the same?

For example, it is often possible to convert a standard inductive proof into a proof by contradiction that starts with the assumption that is a minimal counterexample.

He gives examples proving, that every natural number can be factored into primes and that is irrational.

In the first case, you go from to or you assume a smallest number for which such a factorization does not exist and decide whether it is prime or composite. (In either case the induction hypothesis respectively the proof that the set of such numbers is non-empty are ommited.)

In the second case we assume for a rational representation using either coprime or minimal. Turning ending with a contradiction in either case.