The rise of the foundational research in mathematics

Abstract

It is no pure and simple truism to say that the philosophy of mathematics has always existed in Western culture, at least since philosophy itself came into being. In fact, not only were most of the early Greek philos ophers mathematicians as well, but it is also well known that speculation within the Greek systems of thought was very often based on problems of a mathematical nature. Soon mathematics, besides being a source of philosophical problems and of their solutions, became also a direct object of philosophical enquiry, to such an extent that the early forms of philosophy of science were in fact forms of the philosophy of mathematics. This sort of privileged link between mathematics and philosophy has never been interrupted in Western thought, and even today there are philosophies that are in various ways influenced by mathematical thought (from the philosophy of Russell to that of Husserl, from Wittgenstein to Carnap and logical empiricism), while a large amount of research is being directed to the study of the 'philosophy of mathematics' proper. Underneath this undeniable continuity, however, considerable differ ences emerge between the ancient and modern conceptions of the philos ophy of mathematics, and some problems, once considered of primary importance, seem now to have almost completely lost their value. To mention only a few of them, let us recall certain classical questions of the old philosophy of mathematics: What are numbers? What kind of relationship exists between mathematics and the structure of the world? How can the property of universality and necessity of mathematical truths be explained? In what sense is mathematical infinity to be con ceived? Is mathematics man's invention or man's discovery? By attributing such questions to the past, we do not mean to confine them to ages far distant from our own. Even Dedekind, for example, entitled one of his most celebrated essays: "What are numbers and what must they be?" ; and Frege devoted the whole of his profound logico-math ematical research to the effort of pointing out the essence of the natural number to mention only these two great authors. However, the theories