Among sciences, mathematics has a unique relation to philosophy. Many
philosophers have taken mathematics to be the paradigm of knowledge, and the reasoning
employed in some mathematical proofs are often regarded as rational thought, however
mathematics is also a rich source of philosophical problems which have been at the centre
of epistemology and metaphysics. Since antiquity1, philosophers have envied their ideas as
the model of mathematical perfection because of the clarity of its concepts and the certainty
of its conclusions.
Many efforts should be done if we strive to elaborate philosophy and foundation of
mathematics. Philosophy of mathematics covers the discussion of ontology of
mathematics, epistemology of mathematics, mathematical truth, and mathematical
objectivity. While the foundation of mathematics engages with the discussion of ontological
foundation, epistemological foundation which covers the schools of philosophy such as
platonism, logicism, intuitionism, formalism, and structuralism.
Philosophy of mathematics, as it was elaborated by Ross D.S. (2003), is a
philosophical study of the concepts and methods of mathematics. According to him,
philosophy of mathematics is concerned with the nature of numbers, geometric objects, and
other mathematical concepts; it is concerned with their cognitive origins and with their
application to reality. Further, it addresses the validation of methods of mathematical
inference. In particular, it deals with the logical problems associated with mathematical
infinitude. Meanwhile, Hersh R. (1997) thinks that Philosophy of mathematics should
articulate with epistemology and philosophy of science; but virtually all writers on
philosophy of mathematics treat it as an encapsulated entity, isolated, timeless, a-historical,
inhuman, connected to nothing else in the intellectual or material realms.
Philip Kitcher2 indicates that the philosophy of mathematics is generally supposed to
begin with Frege due to he transformed the issues constituting philosophy of mathematics.
Before Frege, the philosophy of mathematics was only "prehistory." To understand Frege,
we must see him as a Kantian. To understand Kant we must see his response to Newton,
Leibniz, and Hume. Those three philosophers go back to Descartes and through him they back to Plato. Platos is a Pythagorean. The thread from Pythagorean to Hilbert and Godel
is unbroken. A connected story from Pythagoras to the present is where the foundation
came from. Although we can connect the thread of the foundation of mathematics from the
earlier to the present, we found that some philosophers have various interpretation on the
nature of mathematics and its epistemological foundation.
While Hilary Putnam in Hersh, R. (1997), a contemporary philosopher of
mathematics, argues that the subject matter of mathematics is the physical world and not
its actualities, but its potentialities. According to him, to exist in mathematics means to exist
potentially in the physical world. This interpretation is attractive, because in facts
mathematics is meaningful, however, it is unacceptable, because it tries to explain the clear
by the obscure. On the other hand, Shapiro in Linebo, O states that there are two different
orientations of relation between mathematical practice and philosophical theorizing; first, we
need a philosophical account of what mathematics is about, only then can we determine
what qualifies as correct mathematical reasoning; the other orientation of mathematics is an
autonomous science so it doesn’t need to borrow its authority from other disciplines.
On the second view3, philosophers have no right to legislate mathematical practice
but must always accept mathematicians’ own judgment. Shapiro insists that philosophy
must also interpret and make sense of mathematical practice, and that this may give rise to
criticism of oral practice; however, he concedes that this criticism would have to be internal
to mathematical practice and take ‘as data that most of contemporary mathematics is
correct’. Shapiro confesses whether mathematicians should really be regarded as
endorsing philosophical theorizing will depend on what is meant by ‘accurately represents
the semantic form of mathematical language’. If the notions of semantic form and truth
employed in philosophical theorizing are understood in a deflationary way, it is hard to see
how philosophical theorizing can go beyond mathematicians’ claim that the realist principles
are literally true. On the other hand, Stefanik, 1994, argues whether the philosophy of
mathematics most fruitfully pursued as a philosophical investigation into the nature of
numbers as abstract entities existing in a platonic realm inaccessible by means of our
standard perceptual capacities, or the study of the practices and activities of
mathematicians with special emphasis on the nature of the fundamental objects that are the
concern of actual mathematical research.