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. Mathematics70 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 contrast71, conceptual change
philosophies assert that mathematics is corrigible, fallible and a changing social product.
Lakatos72 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 do73 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, Lakatos74 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.
a. Platonism
Hersh R. issues that Platonism is the most pervasive philosophy of mathematics;
today's mathematical Platonisms descend in a clear line from the doctrine of Ideas in Plato .
Plato's philosophy of mathematics75 came from the Pythagoreans, so mathematical
"Platonism" ought to be "Pythago-Platonism." Meanwhile, Wilder R.L. contends that
Platonism76 is the methodological position which goes with philosophical realism regarding
the objects mathematics deals with. However, Hersh R. argues that the standard version of
Platonism perceives mathematical entities exist outside space and time, outside thought
and matter, in an abstract realm independent of any consciousness, individual or social.
Mathematical objects77 are treated not only as if their existence is independent of cognitive
operations, which is perhaps evident, but also as if the facts concerning them did not
involve a relation to the mind or depend in any way on the possibilities of verification,
concrete or "in principle." On the other hand, Nikulin D. (2004) represents that Platonists tend to perceive that
mathematical objects are considered intermediate entities between physical things and
neotic, merely thinkable, entities. Accordingly, Platonists78 discursive reason carries out its
activity in a number of consecutively performed steps, because, unlike the intellect, it is not
capable of representing an object of thought in its entirety and unique complexity and thus
has to comprehend the object part by part in a certain order. Other writer, Folkerts M.
specifies that Platonists tend to believed that abstract reality is a reality; thus, they don't
have the problem with truths because objects in the ideal part of mathematics have
properties. Instead the Platonists79 have an epistemological problem viz. one can have no
knowledge of objects in the ideal part of mathematics; they can't impinge on our senses in
any causal way.
According to Nikulin D., Platonists distinguish carefully between arithmetic and
geometry within mathematics itself; a reconstruction of Plotinus' theory of number,
which embraces the late Plato's division of numbers into substantial and quantitative,
shows that numbers are structured and conceived in opposition to geometrical entities.
In particular80, numbers are constituted as a synthetic unity of indivisible, discrete units,
whereas geometrical objects are continuous and do not consist of indivisible parts. For
Platonists81 certain totalities of mathematical objects are well defined, in the sense that
propositions defined by quantification over them have definite truth-values. Wilder
R.L.(1952) concludes that there is a direct connection between Platonism and the law of
excluded middle, which gives rise to some of Platonism's differences with constructivism;
and, there is also a connection between Platonism and set theory. Various degrees of
Platonism82 can be described according to what totalities they admit and whether they treat
these totalities as themselves mathematical objects. The most elementary kind of
Platonism83 is that which accepts the totality of natural numbers i.e. that which applies the law of excluded middle to propositions involving quantification over all natural numbers.
Wilder R.L. sums up the following:
Platonism says mathematical objects are real and independent of our knowledge;
space-filling curves, uncountable infinite sets, infinite-dimensional manifolds-all the
members of the mathematical zoo-are definite objects, with definite properties,
known or unknown. These objects exist outside physical space and time; they were
never created and never change. By logic's law of the excluded middle, a
meaningful question about any of them has an answer, whether we know it or not.
According to Platonism, mathematician is an empirical scientist, like a botanist