The Philosophy of Mathematics

The philosophy of mathematics is the branch of philosophy whose task is to reflect on, and account for the nature of mathematics. this is a special case of the task of epistemology which is to account for human knowledge in general. The philosophy of mathematics addresses such questions as : What is the basis for mathematical knowledge? What is the nature of mathematical truth? What characterises the truths of mathematics? What is the justification for their assertion? Why are the truth of mathematics necessary truths?.
A widely adopted approach to epistemology, is to assume that knowledge in any field is represented by a set of propositions, together with a set of procedures for verifying them, or providing a warrant for their assertion. On this basis, mathematical knowledge consists of a set of propositions together with their proofs. Since mathematical proofs are based on reason alone, without recourse to empirical data, mathematical knowledge is understood to be the most certain of all knowledge. Traditionally the philosophy of mathematics has seen its task as providing a foundationfor the certainty of mathematical knowledge. That is, providing a system into which mathematical knowledge can be cst to systematically establish its truth. This depends on an assumption, which is widely adopted, implicity if not explicity.

Assumption
The role of the philosophy of mathematics is to provide a systematic and absolutely secure foundation for mathematical knowledge, that is for mathematical truth. This assumption is the basis of foundationism, the doctrine that the function of philosophy of mathematics is ti provide certain foundations for the mathematical knowledge. Foundationism is bound up with the absolutist view of mathematical knowledge, for it regards the task of justifying this view to be central to the philosophy of mathematics.

Source :
Ernest, Paul. 1991. The Philosophy of Mathematics Education. London, UK : RoutledgeFalmer, Taylor & Francis Group.