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mathematics is a body of absolute and certain knowledge. In contrast
71
, conceptual change philosophies assert that mathematics is corrigible, fallible and a changing social product.
Lakatos
72
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 do
73
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, Lakatos
74
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; todays mathematical Platonisms descend in a clear line from the doctrine of Ideas in Plato .
Platos philosophy of mathematics
75
came from the Pythagoreans, so mathematical Platonism ought to be Pythago-Platonism. Meanwhile, Wilder R.L. contends that
Platonism
76
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 objects
77
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.
71
Ernest, P, 2004. “Social Constructivism As A Philosophy Of Mathematics:Radical Constructivism Rehabilitated?
Retrieved 2004 http:www.google.comernest
72
Lakatos in Ernest, P. “Social Constructivism As A Philosophy Of Mathematics:Radical Constructivism Rehabilitated?
Retrieved 2004 http:www.google.comernest
73
Ibid.
74
Ibid.
75
Hersh, R., 1997, “What is Mathematics, Really?”, London: Jonathan Cape, p.9
76
Wilder,R.L., 1952, “Introduction to the Foundation of Mathematics”, New York, p.202
77
Hersh, R., 1997, “What is Mathematics, Really?”, London: Jonathan Cape, pp.9
24
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, Platonists
78
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 dont have the problem with truths because objects in the ideal part of mathematics have
properties. Instead the Platonists
79
have an epistemological problem viz. one can have no knowledge of objects in the ideal part of mathematics; they cant 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 Platos division of numbers into substantial and quantitative,
shows that numbers are structured and conceived in opposition to geometrical entities. In particular
80
, numbers are constituted as a synthetic unity of indivisible, discrete units, whereas geometrical objects are continuous and do not consist of indivisible parts. For
Platonists
81
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 Platonisms differences with constructivism;
and, there is also a connection between Platonism and set theory. Various degrees of Platonism
82
can be described according to what totalities they admit and whether they treat these totalities as themselves mathematical objects.
The most elementary kind of Platonism
83
is that which accepts the totality of natural numbers i.e. that which applies the
78
Nikulin, D., 2004, “Platonic Mathematics: Matter, Imagination and Geometry-Ontology, Natural Philosophy and Mathematics in Plotinus, Proclus and Descartes
”, Retrieved 2004 http:www. amazon.comexec obidosAZIN075461574wordtradecom
79
Folkerts, M., 2004, “Mathematics in the 17th and 18th centuries”, Encyclopaedia Britannica, Retrieved 2004 http:www.google.search
80
Nikulin, D., 2004, “Platonic Mathematics: Matter, Imagination and Geometry-Ontology, Natural Philosophy and Mathematics in Plotinus, Proclus and Descartes
”, Retrieved 2004 http:www.amazon. com exec obidosAZIN075461574wordtradecom
81
Wilder, R.L., 1952, “Introduction to the Foundation of Mathematics”, New York, p.202
82
Ibid.p.2002
83
Ibid. p.2002
25
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 logics 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. Wilder R.L
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asserts that Platonists tend to perceive that mathematicians can not invent mathematics, because everything is already there; he can only discover. Our
mathematical knowledge
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is objective and unchanging because its knowledge of objects external to us, independent of us, which are indeed changeless. For Plato
86
the Ideals, including numbers, are visible or tangible in Heaven, which we had to leave in order to be
born. Yet most mathematicians and philosophers of mathematics continue to believe in an independent, immaterial abstract world-a remnant of Platos Heaven, attenuated, purified,
bleached, with all entities but the mathematical expelled. Platonists explain mathematics by a separate universe of abstract objects, independent of the material universe. But how do
the abstract and material universes interact? How do flesh-and-blood mathematicians acquire the knowledge of number?
b. Logicism
