Oct 10, 2012

Elegi Menggapai 'The Foundations of Mathematics'




By Marsigit
Yogyakarta State University

The search for foundations of mathematics is in line with the search for philosophical foundation in general. The aspects of the foundation of mathematics can be traced through the tread of philosophical history and mathematics history as well. Hersh, R. elaborates that the foundations of mathematics 1 have ancient roots; the philosophers behind Frege are Hilbert, Brouwer, Immanuel Kant. The philosopher behind Kant is


Gottfried Leibniz; the philosophers behind Leibniz are Baruch Spin¬oza and Rene Descartes. The philosophers behind all of them are Thomas Aquinas, Augustine of Hippo, Plato, and the great grandfather of foundationism-Pythagoras.
The term "foundations of mathematics" 2 is sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory and model theory. The search for foundations of mathematics is however also the central question of the philosophy of mathematics: on what ultimate basis can mathematical statements be called "true"? Hersh R. describes that , the name "foundationism" was invented by a prolific name-giver, Imre Lakatos. Gottlob Frege, Bertrand Russell, Luitjens Brouwer and David Hilbert, are all hooked on the same delusion that mathematics must have a firm foundation; however, they differ on what the foundation should be. The currently foundation of mathematics 3 is, characteristically as more formalistic approach, based on axiomatic set theory and formal logic; and therefore, all mathematical theorems today can be formulated as theorems of set theory.
Hersh R. (1997) writes that the truth of a mathematical statement, in the current view, is then nothing but the claim that the statement can be derived from the axioms of set theory using the rules of formal logic. However, this formalistic 4 approach does not explain several issues such as: why we should use the axioms we do and not some others, why we should employ the logical rules we do and not some other, why "true" mathematical statements appear to be true in the physical world. The above mentioned notion of formalistic truth could also turn out to be rather pointless; it is perfectly possible that all statements, even contradictions, can be derived from the axioms of set theory. Moreover, as a consequence of Gödel's second incompleteness theorem, we can never be sure that this is not the case.

References:
1 Hersh, R., 1997, “What is Mathematics, Really?”, London: Jonathan Cape, p. 91
2 …….., “Foundations of mathematics”, Retrieved 2004 < http://www.wikipedia.org/>
3 Hersh, R., 1997, “What is Mathematics, Really?”, London: Jonathan Cape, p.91
4 Ibid. p. 256

2 comments:

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

    Pembahasan mengenai fondasi matematika dapat ditelusuri melalui jejak rekam sejarah filsafat dan sejarah matematika itu sendiri. Pencarian akan fondasi matematika menjadi pertanyaan utama dari filsafat matematika yaitu atas dasar atau fondasi apa pernyataan matematika dikatakan “benar”. Hal ini lah yang masih menjadi perdebatan para matematikawan dan para filsuf terhadap apa yang menjadi fondasi tunggal dalam matematika. Namun, saat ini fondasi matematika lebih bersifat pendekatan formalistik meski konsekuensinya juga terjadi inkonsisten dan kontradiktif dalam matematika dimana hal tersebut seharusnya dihindari.

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  2. Wahyu Lestari
    16709251074
    PPs Pendidikan Matematika 2016 Kelas D

    dari artikel di jelaskan Hersh R. (1997) menulis bahwa kebenaran pernyataan matematis, dalam pandangan saat ini, tidak lain adalah pernyataan bahwa pernyataan tersebut dapat diturunkan dari aksioma teori yang ditetapkan menggunakan aturan logika formal. Namun, pendekatan formalistik ini tidak menjelaskan beberapa masalah seperti: mengapa kita harus menggunakan aksioma yang kita lakukan dan bukan yang lain, mengapa kita harus menggunakan peraturan logis yang kita lakukan dan bukan yang lain, mengapa pernyataan matematika "sebenarnya" tampak seperti Benar di dunia fisik. Gagasan tentang kebenaran formalistik yang disebutkan di atas juga bisa berubah menjadi tidak ada gunanya; Sangat mungkin bahwa semua pernyataan, bahkan kontradiksi, dapat diturunkan dari aksioma teori himpunan. Selain itu, sebagai konsekuensi dari teorema ketidaklengkapan Gödel yang kedua, kita tidak dapat memastikan bahwa ini tidak terjadi.

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