Abstract

Even though mathematics is deductive reasoning all along, it does not depend on logic. On the contrary, formal logic is mathematical. Despite what is often claimed, mathematics is not the science of quantity, but the science of necessary conclusions. It reasons from pure hypotheses, explores their possibilities in imagination and draws apodictic conclusions. Obtaining new theoretical results—as opposed to mere “corollaries” of previous results—asks for more than mere deduction of analytical consequences by analysis of the general concepts involved. It requires representing these general concepts into singular diagrams and operating transformations on them in order to reveal necessary properties or relations which had not yet been perceived. By making these properties or relations the objects of new judgments, mathematics practices “abstraction” in a sense that does not reduce to the psychic operation of reserving one's attention to certain features of a percept at the expense of others.