Axiomatism and computational positivism. Two mathematical cultures in pursuit of exact sciences

Abstract

It is argued here that the mathematical approach to the exact sciences has historically appeared to contain two largely distinct cultures (which nevertheless overlap to some extent). One of these takes the deduction of 'certain' conclusions from clearly stated axioms or models as the primary objective; the other considers number the primary concept, and emphasises computation and algebra, conforming to unambiguous rules. A philosophy that may be called computational positivism, whose goal is to make computation agree with observation, appears to have been characteristic of Indian (and apparently Babylonian) astronomy. The interactions between these two cultures have played a key role in the history of science, and seem set to continue to do so in the future as well.