To show and to prove

Carlos Torres

Jaime O. Falcon Vege

pp. 249-264

in: Robert S. Cohen (ed), Mexican studies in the history and philosophy of science, Berlin, Springer, 1995

Abstract

It is often said that mathematics are a deductive science. Inasmuch as this claim puts forward its demonstrative character, it has two failures: first, it does not account for the creative or heuristic side of mathematics (for example, how are theorems discovered?); second, it omits the fact that a great part of mathematical knowledge is based upon sense evidence, not on proof. With this we mean that there are — in the last instance, for in practice the process is quite mixed — two kinds of "truth" in mathematics: those based on sense-perception and those inferred from other propositions by logical deduction. Those, we say, are shown , the latter are proven.