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foundation of mathematics. The most important of these was the discovery of inconsistencies, or paradoxes, in the foundations of mathematics; this represents the
starting point of the modern philosophy of mathematics.
1. Ontology of Mathematics
Ross, D.S. 2003 states that there are some ontological questions in the Philosophy of Mathematics:
What is the origin of mathematical objects? In what way do mathematical objects exist? Have they always been present as Platonic abstractions, or do they require a
mind to bring them into existence? Can mathematical objects exist in the absence of matter or things to count?.
Since the beginning of Western philosophy
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, there are important philosophical problems: Do numbers and other mathematical entities exist independently on human
cognition? If they exist dependently on human cognition then how do we explain the extraordinary applicability of mathematics to science and practical affairs? If they exist
independently on human cognition then what kind of things are they and how can we know about them? And what is the relationship between mathematics and logic? The first
question is a metaphysical question with close affinities to questions about the existence of other entities such as universals, properties and values. According to many philosophers, if
such entities exist then they do beyond the space and time, and they lack of causal powers. They are often termed abstract as opposed to concrete entities. If we accept the existence
of abstract mathematical objects then an adequate epistemology of mathematics must explain how we can know them; of course, proofs seem to be the main source of
justification for mathematical propositions but proofs depend on axioms and so the question of how we can know the truth of the axioms remains.
It is advocated especially by Stuart Mill J. in Hempel C.G. 2001 that mathematics itself is an empirical science which differs from the other branches such as astronomy,
physics, chemistry, etc., mainly in two respects: its subject matter is more general than that of any other field of scientific research, and its propositions have been tested and confirmed
to a greater extent than those of even the most firmly established sections of astronomy or
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Ibid.
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physics. According to Stuart Mill J., the degree to which the laws of mathematics have been born out by the past experiences of mankind is so unjustifiable that we have come to think
of mathematical theorems as qualitatively different from the well confirmed hypotheses or theories of other branches of science in which we consider them as certain, while other
theories are thought of as at best as very probable or very highly confirmed and of course this view is
open to serious objections. While Hempel C.G. himself acknowledges that, from empirical hypothesis, it is
possible to derive predictions to the effect that under certain specified conditions, certain specified observable phenomena will occur; the actual occurrence of these phenomena
constitutes confirming evidence. It
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was concluded that an empirical hypothesis is theoretically un-confirmable that is possible to indicate what kind of evidence would
disconfirm the hypothesis; if this is actually an empirical generalization of past experiences, then it must be possible to state what kind of evidence would oblige us to concede the
hypothesis not generally true after all. The mathematical propositions are true simply by virtue of definitions or of similar stipulations which determine the meaning of the key terms
involved. Soehakso, RMJT, guides that mathematics validation naturally requires no empirical evidence; they can be shown to be true by a mere analysis of the meaning
attached to the terms which occur in. The exactness and rigor of mathematics
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means that the understanding of mathematics follows the logical development of important peculiar
mathematical methods, and of course is acquainted with major results especially in “foundations”.
The validity of mathematics, as it was stated by Hempel C.G., rests neither on its
alleged self-evidential character nor on any empirical basis, but it also derives from the stipulations which determine the meaning of the mathematical concepts, and that the
propositions of mathematics are therefore essentially true by definition. The rigorous
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development of a mathematical theory proceeds not simply from a set of definitions but rather from a set of non-definitional propositions which are not proved within the theory.
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In Hempel, C.G., 2001, “On the Nature of Mathematical Truth”, Retrieved 2004 http:www. ltn.lv~podniekgt.htm
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Soehakso, RMJT, 1989, “Some Thought on Philosophy and Mathematics”, Yogyakarta: Regional Conference South East Asian Mathematical Society, p.3
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In Hempel, C.G., 2001, “On the Nature of Mathematical Truth”, Retrieved 2004 http:www. ltn.lv~podniekgt.htm
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Hempel states that there are the postulates or axioms of the theory and formulated in terms of certain basic or primitive concepts for which no definitions are provided within the theory.
The postulates themselves represent implicit definitions of the primitive terms while the postulates do limit, in a specific sense, the meanings that can possibly be ascribed to the
primitives, any self-consistent postulate system admits.
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Once the
primitive terms and the postulates
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have been laid down the entire theory is completely determined. Hence, every term of the mathematical theory is definable in
terms of the primitives, and every proposition of the theory is logically deducible from the postulates. Hempel adds that to be entirely precise, it is necessary to specify the principles
of logic used in the proof of the propositions; these principles can be stated quite explicitly and fall into primitive sentences or postulates of logic. Accordingly, any fact that we can
derive from the axioms needs not be an axiom; anything that we cannot derive from the axioms and for which we also cannot derive the negation might reasonably added as an
axiom. Hempel concludes that by combining the analyses of the aspects of the Peano system, the thesis of logicism was accepted that Mathematics is a branch of logic due to all
the concepts of mathematics i.e. arithmetic, algebra, and analysis can be defined in terms of four concepts of pure logic and all the theorems of mathematics can be deduced from
those definitions by means of the principles of logic.
2. Epistemology of Mathematics
