Relating to ontological foundation of mathematics, Litlang (2002) views that in
mathematical realism, sometimes called Platonism, the existence of a world of
mathematical objects independent of humans is postulated; not our axioms, but the very
real world of mathematical objects forms the foundation. The obvious question, then, is:
how do we access this world? Some modern theories in the philosophy of mathematics
deny the existence of foundations in the original sense. Some theories tend to focus on
mathematical practices and aim to describe and analyze the actual working of
mathematicians as a social group. Others try to create a cognitive science of mathematics,
focusing on human cognition as the origin of the reliability of mathematics when applied to
the 'real world'. These theories51 would propose to find the foundations of mathematics only
in human thought, not in any 'objective' outside construct, although it remains controversial. Litlang indicates that although mathematics might seem the clearest and most
certain kind of knowledge we possess, there are problems just as serious as those in any
other branch of philosophy. It is not easy to elaborate the nature of mathematics and in
what sense do mathematics propositions have meaning?. Plato52 believes, in Forms or
Ideas, that there are eternal capable of precise definition and independent of perception.
Plato includes, among such entities, numbers and the objects of geometry such as lines,
points or circles which were apprehended not with the senses but with reason. According
to Plato53, the mathematical objects deal with specific instances of ideal Forms. Since the
true propositions of mathematics54 are true of the unchangeable relations between
unchangeable objects, they are inevitably true, which means that mathematics discovers
pre-existing truths out there rather than creates something from our mental predispositions;
hence, mathematics dealt with truth and ultimate reality.
Litlang (2002) indicates that Aristotle disagreed with Plato. According to Aristotle,
Forms were not entities remote from appearance but something that entered into objects of
the world. That we abstract mathematical object does not mean that these abstractions
represent something remote and eternal. However, mathematics is simply reasoning about
idealizations. Aristotle55 looks closely at the structure of mathematics, distinguishing logic,
principles used to demonstrate theorems, definitions and hypotheses. Litlang implies that
while Leibniz brought together logic and mathematics, Aristotle uses propositions of the
subject- predicate form. Leibniz argues that the subject contains the predicate; therefore the
truths of mathematical propositions are not based on eternal or idealized entities but based
on their denial is logically impossible.
According to Leibniz56, the truth of mathematics is not only of this world, or the world
of eternal Forms, but also of all possible worlds. Unlike Plato, Leibniz sees the importance
of notation i.e. a symbolism of calculation, and became very important in the twentieth
century mathematics viz. a method of forming and arranging characters and signs to
represent the relationships between mathematical thoughts. On the other hand, Kant57
perceives that mathematical entities were a-priori synthetic propositions on which it provides the necessary conditions for objective experience. According to Kant58,
mathematics is the description of space and time; mathematical concept requires only selfconsistency,
but the construction of such concepts involves space having a certain
structure.
On the other hand, Frege, Russell and their followers59 develop Leibniz's idea that
mathematics is something logically undeniable. Frege60 uses general laws of logic plus
definitions, formulating a symbolic notation for the reasoning required. Inevitably, through
the long chains of reasoning, these symbols became less intuitively obvious, the transition
being mediated by definitions. Russell61 sees the definitions as notational conveniences,
mere steps in the argument. While Frege sees them as implying something worthy of
careful thought, often presenting key mathematical concepts from new angles. For
Russell62, the definitions had no objective existence; while for Frege, it is ambiguous due to
he states that the definitions are logical objects which claim an existence equal to other
mathematical entities.
Eves H. and Newsom C.V. write that the logistic thesis is that mathematics is a
branch of logic. All mathematical concepts are to be formulated in terms of logical concepts,
and all theorems of mathematics are to be developed as theorems of logic. The distinction
between mathematics and logic63 becomes merely one of practίcal convenience; the
actual reduction of mathematical concepts to logical concepts is engaged in by
Dedekind (1888) and Frege (1884-1903), and the statement of mathematical theorems
by means of a logical symbolism as undertaken by Peano (1889-1908). The logistic thesis
arises naturally from the effort to push down the foundations of mathematics to as deep a
level as possible.64 Further, Eves H. and Newsom C.V. (1964) state:
The foundations of mathematics were established in the real number system, and
were pushed back from the real number system to the natural number system, and
thence into set theory. Since the theory of classes is an essential part of logic, the
idea of reducing mathematics to logίc certainly suggests itself.