Mathematical Rigour

Abstract

Above all other mental exercises, mathematics has the advantage that with each statement you can decide whether it is right or wrong. Whether a mathematical paper is good or bad, important or unimportant, original or trifling, is not as easy to decide but there can be no doubt about it being right or wrong. Already in physics it is more difficult to tell this, unless it is a kind of physics that is practically mathematics. A physical theory is expected to be confirmed in reality but even if it is not confirmed it need not be wrong but can have been applied less appropriately. In all sciences except mathematics the criterion of usefulness is often more relevant than a criterion of truth that actually nobody knows for sure how to handle. The reason is that on no other science can be imposed so strong a deductive structure as on mathematics. In mathematics it is not only certain whether the result is correct but even — or actually solely — whether it has been founded correctly. This is what is called mathematical rigour. It is the measuring-rod of mathematics, and if we teach mathematics we use it also.

Would it be believable that the logicians always proceeded from the general to the particular, as the rules of formal logic seemed to prescribe? By this way they would never have been able to extend the frontiers of science; scientific conquests are only made by generalization.

When mathematical science becomes rigorous it assumes an artificial character which cannot be overlooked. It forgets about its historic origin: it shows how problems can be solved but not how and why they are posed.

This shows that logic does not suffice; the science of proofs is not the whole science, intuition is assigned a complementary part, I would say, as counterpart or antidote of logic.

H. Poincaré, C.R. 2ème Congrès Int. des math. Paris, 1900, 127

H. Poincaré, C.R. 2ème Congrès Int. des math. Paris 1900, 123-4