Oct 10, 2012

Elegi Menggapai "Platonism as the Epistemological Foundation of Mathematics"




By Marsigit
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?

References:
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
8 Ibid.p.2002
9 Ibid. p.2002
10 Ibid.p.202
11Ibid.p.202
12Hersh, R., 1997, “What is Mathematics, Really?”, London: Jonathan Cape, pp.12

9 comments:

  1. Heni Lilia Dewi
    16709251054
    Pascasarjana Pendidikan Matematika 2016/ Kelas C

    What is the real mathematics? Plato’s idea said that abstract objects have nothing to do with mathematics. Platonists conclude that mathematical knowledge is a priori, which means that mathematical knowledge is not based on the truth of the senses. According to Plato, the ultimate reality is what we think with our reason.

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  2. Cendekia Ad Dien
    16709251044
    PPs Pendidikan Matematika Kelas C 2016

    Objek matematika menurut Platonisme adalah nyata dan bebas dari pengetahuan kita karena itu pengetahuan matematika kita bersifat objektif dan tidak berubah. Platonis cenderung menganggap bahwa matematika itu sebenarnya sudah ada dan belum diketahui serta bukan benar-benar baru yang belum ada sehingga hanya perlu dilakukan penemuan (discovery).

    ReplyDelete
  3. Wahyu Lestari
    16709251074
    PPs Pendidikan Matematika 2016 Kelas D

    Platonisme adalah filsafat matematika yang paling meluas; Platonisme matematika turun dalam garis yang jelas dari doktrin Gagasan di Plato.Platonis menjelaskan matematika oleh alam semesta abstrak yang terpisah, independen dari alam material.

    ReplyDelete
  4. PUTRI RAHAYU S
    S2 PENDIDIKAN MATEMATIKA_D 2016
    16709251070

    Menurut Plato, Filsafat matematika berasal dari Pythagorean, sehingga matematika "Platonisme" seharusnya "Pythago-Platonisme." Sementara itu, Wilder RL berpendapat bahwa Platonisme 2 adalah posisi metodologis yang berjalan dengan realisme filosofis mengenai penawaran obyek matematika dengan. Namun, Hersh R. berpendapat bahwa versi standar Platonisme merasakan entitas matematika ada di luar ruang dan waktu, pikiran dan materi di luar, di independen ranah abstrak setiap kesadaran, individu atau sosial.

    ReplyDelete
  5. Sylviyani Hardiarti
    16709251069
    S2 Pendidikan Matematika Kelas D 2016

    Platonisme matematika menjadi topik perdebatan yang paling hangat dalam filsafat matematika selama beberapa dekade terakhir. Platonisme matematika dapat didefinisikan sebagai gabungan dari tiga tesis/pernyataan berikut:
    Keberadaan : Adanya benda-benda matematis.
    Keabstrakan : Objek matematika yang abstrak.
    Independen : Objek matematika adalah independen dari tingkat kecerdasan dan bahasa, pola pikir, dan praktik.
    Platonisme harus dibedakan dari pandangan Plato. Beberapa pihak dalam perdebatan kontemporer tentang Platonisme membuat klaim penafsiran yang kuat tentang pandangan Plato. Meskipun pandangan yang kita sebut 'Platonisme' terinspirasi oleh teori terkenal Plato tentang bentuk-bentuk abstrak dan kekal, Platonisme sekarang didefinisikan dan diperdebatkan secara independen dari inspirasi asli sejarah.

    ReplyDelete
  6. Sehar Trihatun
    16709251043
    S2 Pend. Mat Kelas C – 2016

    Dalam pandangan platonisme, objek-objek matematika itu nyata, dan tidak terikat dalam ruang dan waktu. Platonis beranggapan bahwa matematika itu berada diluar jangkauan pemikiran kita, sehingga matematika itu tidak diciptakan dan tidak pernah berubah. Objek-objek matematika itu telah ada, hanya saja keberadaannya kadang dapat kita ketahui dan tidak kita ketahui. Seorang matematikawan tidak mungkin dapat menciptakan matematika, mereka hanya bisa menemukannya, karena matematika itu sendiri bukanlah sesuatu yang ada karena manusia tetapi memang matematika sudah ada hanya bagaimana manusia bisa menemukannya atau tidak.

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  7. Lihar Raudina Izzati
    16709251046
    P. Mat C 2016 PPs UNY

    Platonism adalah paham yang diciptakan oleh Plato. Plato sangat besar pengaruhnya dalam perkembangan filsafat matematika. Menurut Plato, matematika adalah bentuk pengetahuan yang paling sempurna. Menurut Plato, indra tidak memberi pengetahuan yang benar. Objek matematika dalam pandangan Plato adalah realitas yang bebas dari pikiran manusia.

    ReplyDelete
  8. Primaningtyas Nur Arifah
    16709251042
    Pend. Matematika S2 kelas C 2016
    Assalamu’alaikum. Platonisme adalah posisi metodologis yang sejalan dengan realisme filosofis mengenai objek yang dihadapi matematika. Platonisme memandang entitas matematis ada di luar ruang dan waktu, di luar pemikiran dan materi, dalam alam abstrak yang independen dari setiap kesadaran, individu atau sosial. Platonis membedakan secara cermat antara aritmatika dan geometri dalam matematika itu sendiri; Sebuah rekonstruksi teori bilangan Plotinus, yang mencakup pembagian bilangan Plato akhir menjadi substansial dan kuantitatif, menunjukkan bahwa jumlah tersebut disusun dan dipahami sebagai oposisi terhadap entitas geometris.

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  9. Helva Elentriana
    16709251068
    PPS Pend Matematika Kelas D 2016

    Aliran ini berasal dari Plato dan pengikutnya seperti Frege, Russell, Cantor, Bernays, Hardy, dan Godel. Ernest (1995) mengatakan bahwa aliran platonisme memandang bahwa objek dan struktur matematika mempunyai keberadaan yang riil yang tidak bergantung kepada manusia, dan bahwa mengerjakan matematika adalah suatu proses penemuan tentang hubungan keberadaan sebelumnya. Kegiatan matematika adalah proses menemukan hubungan-hubungan yang telah ada di alam semesta.

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