Yogyakarta State University
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 theories 1 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?.
Plato 2 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 Plato 3, the mathematical objects deal with specific instances of ideal Forms.
Since the true propositions of mathematics 4 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.
Aristotle 5 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 Leibniz 6, 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, Kant 7 perceives that mathematical entities were a-priori synthetic propositions on which it provides the necessary conditions for objective experience.
According to Kant 8, mathematics is the description of space and time; mathematical concept requires only self-consistency, but the construction of such concepts involves space having a certain structure.
On the other hand, Frege, Russell and their followers 9 develop Leibniz's idea that mathematics is something logically undeniable.
Frege 10 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.
Russell 11 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 Russell 12, 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 logic 13 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. 14
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.
The logistic thesis is thus an attempted synthesization suggested why an important trend in the history of the application of the mathematical method.
Meanwhile, Litlangs determines that in geometry, logic is developed in two ways.
The 15 first is to use one-to-one correspondences between geometrical entities and numbers.
Lines, points, circle, etc. are matched with numbers or sets of numbers, and geometric relationships are matched with relationships between numbers.
The second is to avoid numbers altogether and define geometric entities partially but directly by their relationships to other geometric entities.
Litlangs comments that such definitions are logically disconnected from perceptual statements so that the dichotomy between pure and applied mathematics continues.
It is somewhat paralleling Plato's distinction between pure Forms and their earthly copies.
Accordingly, alternative self-consistent geometries can be developed, therefore, and one cannot say beforehand whether actuality is or is not Euclidean; moreover, the shortcomings of the logistic procedures remain, in geometry and in number theory.
Furthermore, Litlangs (2002) claims that there are mathematicians perceiving mathematics as the intuition of non-perceptual objects and constructions.
According to them, mathematics is introspectively self-evident and begins with an activity of the mind which moves on from one thing to another but keeps a memory of the first as the empty form of a common substratum of all such moves.
Next, he states:
Subsequently, such constructions have to be communicated so that they can be repeated clearly, succinctly and honestly. Intuitionist mathematics employs a special notation, and makes more restricted use of the law of the excluded middle viz. that something cannot be p' and not-p' at the same time. A postulate, for example, that the irrational number pi has an infinite number of unbroken sequences of a hundred zeros in its full expression would be conjectured as un-decidable rather than true or false. 16
The law of excluded middle, tertium non datur in Latin, states that for any proposition P, it is true that “P or not P”. For example, if P is “Joko is a man” then the inclusive disjunction “Joko is a man, or Joko is not a man” is true.
P not P P or not P
True False True
False True True
Litlangs (2002), further, adds that different writers perceive mathematics as simply what mathematicians do; for them, mathematics arises out of its practice, and must ultimately be a free creation of the human mind, not an exercise in logic or a discovery of preexisting fundamentals.
Mathematics 17 does tell us, as Kant points out, something about the physical world, but it is a physical world sensed and understood by human beings.
On the other hand 18, relativists remind that nature presents herself as an organic whole, with space, matter and time.
Humans have in the past analyzed nature, selected certain properties as the most important, forgotten that they were abstracted aspects of a whole, and regarded them thereafter as distinct entities; hence, for them, men have carried out mathematical reasoning independent of sense experience.
2Litlangs, 2002-2004, “Math Theory” Retrieved 2004
7 Eves, H and Newsom, C.V., 1964, “An Introduction to the Foundation & Fundamental Concepts of Mathematics”, New York: Holt, Rinehart and Winston, p.70
10 Litlangs, 2002-2004, “Math Theory” Retrieved 2004
13Eves, H and Newsom, C.V., 1964, “An Introduction to the Foundation & Fundamental Concepts of Mathematics”, New York: Holt, Rinehart and Winston, p. 286
15Litlangs, 2002-2004, “Math Theory” Retrieved 2004
18Eves, H and Newsom, C.V., 1964, “An Introduction to the Foundation & Fundamental Concepts of Mathematics”, New York: Holt, Rinehart and Winston, p.289