Yogyakarta State University
The epistemological foundation of mathematics elicits the status and foundation of mathematical knowledge by examining the basis of mathematical knowledge and the certainty of mathematical judgments. Nikulin D. (2004) enumerates that ancient philosophers perceived that mathematics and its methods could be used to
describe the natural world. Mathematics 1 can give knowledge about things that cannot be otherwise and therefore has nothing to do with the ever fluent physical things, about which there can only be a possibly right opinion. While Ernest P. explains that Absolutist philosophies of mathematics, including Logicism, Formalism, Intuitionism and Platonism perceive that mathematics is a body of absolute and certain knowledge. In contrast 2, conceptual change philosophies assert that mathematics is corrigible, fallible and a changing social product.
Lakatos 3 specifies that despite all the foundational work and development of mathematical logic, the quest for certainty in mathematics leads inevitably to an infinite regress. Contemporary, any mathematical system depends on a set of assumptions and there is no way of escaping them. All we can do 4 is to minimize them and to get a reduced set of axioms and rules of proof. This reduced set cannot be dispensed with; this only can be replaced by assumptions of at least the same strength. Further, Lakatos 5 designates that we cannot establish the certainty of mathematics without assumptions, which therefore is conditional, not absolute certainty. Any attempt to establish the certainty of mathematical knowledge via deductive logic and axiomatic systems fails, except in trivial cases, including Intuitionism, Logicism and Formalism.
Nikulin, D., 2004, “Platonic Mathematics: Matter, Imagination and Geometry-Ontology, Natural Philosophy and Mathematics in Plotinus, Proclus and Descartes”, Retrieved 2004
2Ernest, P, 2004. “Social Constructivism As A Philosophy Of Mathematics:Radical Constructivism Rehabilitated? Retrieved 2004
3 Lakatos in Ernest, P. “Social Constructivism As A Philosophy Of Mathematics:Radical Constructivism Rehabilitated? Retrieved 2004