Yogyakarta State University
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 mathematics 1 came from the Pythagoreans, so mathematical "Platonism" ought to be "Pythago-Platonism."
Meanwhile, Wilder R.L. contends that Platonism 2 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 3 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, Platonists 4 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 Platonists 5 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 particular 6, 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 7 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 Platonism 8 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 9 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 knowl¬edge; 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.
Wilder R.L 10 asserts that Platonists tend to perceive that mathematicians can not invent mathematics, because everything is already there; he can only discover.
Our mathematical knowledge 11 is objective and unchanging because it's knowledge of objects external to us, independent of us, which are indeed changeless.
For Plato 12 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 Plato's 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?
Hersh, R., 1997, “What is Mathematics, Really?”, London: Jonathan Cape, p.9
2Wilder,R.L., 1952, “Introduction to the Foundation of Mathematics”, New York, p.202
3 Hersh, R., 1997, “What is Mathematics, Really?”, London: Jonathan Cape, pp.9
4 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/ obidos/AZIN/075461574/wordtradecom>
5Folkerts, M., 2004, “Mathematics in the 17th and 18th centuries”, Encyclopaedia Britannica, Retrieved 2004
6Nikulin, D., 2004, “Platonic Mathematics: Matter, Imagination and Geometry-Ontology, Natural Philosophy and Mathematics in Plotinus, Proclus and Descartes”, Retrieved 2004
7Wilder, R.L., 1952, “Introduction to the Foundation of Mathematics”, New York, p.202
9 Ibid. p.2002
12Hersh, R., 1997, “What is Mathematics, Really?”, London: Jonathan Cape, pp.12