The philosophy of mathematics attempts to explain both the nature of mathematical facts and entities, and the way in which we have our knowledge of both. Modern philosophy of mathematics began with the foundational studies of Cantor, Dedekind, and K. T. W. Weierstrass in the late 19th century. It received its fundamental impulse from the work of Frege and Russell on the relations between numbers, sets, and logic. The logicist programme in the philosophy of mathematics, culminating in Russell and Whitehead's great Principia Mathematica of 1910–13, attempted to reduce mathematics to logic, in the sense of proving that all of mathematics could be represented in a system whose axioms were simply axioms of pure logic. It is generally thought that the programme failed over the need to admit sets as entities subject to their own particular axioms, whose logical status was not at all obvious. At the beginning of the 20th century the place of set theory in mathematics was well established, but the contradictions of naive set theory, and the sheer scale of the full transfinite hierarchy, divided mathematicians into several camps, reflecting different philosophies of mathematics. Full-scale realism or Platonism takes mathematical entities to be real, independent objects of study about which discoveries (and mistakes) can be made. Constructivism takes it that we ourselves construct what we talk about. Formalism assimilates mathematics to the purely syntactic process of following proof procedures, without any question being raised about the interpretation of the theorems proved. The Achilles heel of formalism has always been the application of mathematics in our best descriptions of the world, so realism and constructivism have emerged as the main contrasting ideologies. The division has mathematical consequences. For the realist there is no problem in admitting completed infinite collections, whether or not we have an effective method of specifying their members. For the constructivist this will be inadmissible.
Quite apart from this issue modern debates on the epistemology of mathematics have been infected by the general flight from the category of a priori knowledge. Suggestions taking its place include the general conventionalism associated with Wittgenstein, the view that abstract objects such as sets are not so frightening after all, and indeed are in fact perceptible, and the view that the general success of science, of which mathematics is an indispensable part, affords an argument for the truth of mathematics.