domingo, 25 de febrero de 2007

Proofs and Refutations (I)

Imre Lakatos
The British Journal for the Philosophy of Science, 1963

A conjecture (as an outcome after much trial and error) which have passed many different tests (trying to falsify it), suggests that it could be proved. In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. Lakatos proposes to use the word proof for a thought-experimental which leads to a decomposition of the original conjecture into sub-conjectures (or lemmas), instead of using it in the sense of a `guarantee of certain truth’.

Lakatos outlines that informal mathematics grows through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations. Therefore, the proposed concept of proof deploys the conjecture on a wider front so that the criticism has more targets and more opportunities for counterexamples. Lakatos makes a distinction within the counterexamples: (1) The local counterexamples are examples which refute sub-conjectures, and (2) the global counterexamples are examples that refute the main conjecture. While a local counterexample is a criticism of the proof, but not of the conjecture, a global counterexample is a criticism of the conjecture, but not necessarily of the proof.

When a local counterexample emerges, we don’t have to scrap the proof, is better to improve it and replace the false sub-conjecture by a slightly modified one that will stand up to the criticism. Because of the improvement we might obtain implausible conjectures -matured in criticism- that might hit on the truth.

On the other hand, the refutations by counterexamples depend on the meaning of the terms in question. If counterexamples are to be an objective criticism, we have to agree on the meaning of the terms. One can eliminate any counterexample by ad hoc redefinitions; for that reason the last ones are frequently proposed and argued when counterexamples emerge.

Finally, because of the proof concept here presented, we are not perturbed at finding a counterexample to a proved conjecture; we must try to set out to prove a false conjecture.

‘Prove all things; holds fast which is good’