Oct 13, 2012

The Relevance of Kant’s Theory of Knowledge to Contemporary Foundation of Mathematics




By Marsigit

The relevance of Kant’s theory of knowledge to the contemporary foundation of mathematics can be traced from the notions of contemporary writers. Jørgensen, K.F.(2006) admits that a philosophy of mathematics must square with contemporary mathematics as it is carried out by actual mathematicians.


This leads him to define a very general notion of constructability of mathematics on the basis of a generalized understanding of Kant's theory of schema. Jørgensen, K.F. further states that Kant’s theory of schematism should be taken seriously in order to understand his Critique. It was science which Kant wanted to provide a foundation for. He says that one should take schematism to be a very central feature of Kant's theory of knowledge.
Meanwhile, Hanna, R. insists that Kant offers an account of human rationality which is essentially oriented towards judgment. According to her, Kant also offers an account of the nature of judgment, the nature of logic, and the nature of the various irreducibly different kinds of judgments, that are essentially oriented towards the anthropocentric empirical referential meaningfulness and truth of the proposition. Further, Hanna, R. indicates that the rest of Kant's theory of judgment is then thoroughly cognitive and non-reductive. In Kant , propositions are systematically built up out of directly referential terms (intuitions) and attributive or descriptive terms (concepts), by means of unifying acts of our innate spontaneous cognitive faculties. This unification is based on pure logical constraints and under a higher-order unity imposed by our faculty for rational self-consciousness. Furthermore all of this is consistently combined by Kant with non-conceptualism about intuition, which entails that judgmental rationality has a pre-rational or proto-rational cognitive grounding in more basic non-conceptual cognitive capacities that we share with various non-human animals. In these ways, Hanna, R. concludes that Kant’s theory of knowledge is the inherent philosophical interest, contemporary relevance, and defensibility remain essentially intact no matter what one may ultimately think about his controversial metaphysics of transcendental idealism.
Meanwhile, Hers R. insists that at the bottom tortoise of Kant’s synthetic a priori lies intuition. In the sense of contemporary foundation of mathematics, Hers R. notifies that in providing truth and certainty in mathematics Hilbert implicitly referred Kant. He. pointed out that, like Hilbert, Brouwer was sure that mathematics had to be established on a sound and firm foundation in which mathematics must start from the intuitively given. The name intuitionism displays its descent from Kant’s intuitionist theory of mathematical knowledge. Brouwer follows Kant in saying that mathematics is founded on intuitive truths. As it was learned that Kant though geometry is based on space intuition, and arithmetic on time intuition, that made both geometry and arithmetic “synthetic a priori”. About geometry, Frege agrees with Kant that it is synthetic intuition. Furthermore, Hers R. indicates that all contemporary standard philosophical viewpoints rely on some notions of intuition; and consideration of intuition as actually experienced leads to a notion that is difficult and complex but not inexplicable. Therefore, Hers R suggests that a realistic analysis of mathematical intuition should be a central goal of the philosophy of mathematics.
In the sense of very contemporary practical and technical mathematics works Polya G in Hers R. states:
Finished mathematics presented in a finished form appears as purely demonstrative, consisting of proofs only. Yet mathematics in the making resembles any other human knowledge in the making. You have to guess a mathematical theorem before you prove it; you have to have the idea of the proof before you carry through the details. You have to combine observations and follow analogies; you have to try and try again.

The writer of this dissertation perceives that we may examine above notions in the frame work of Kant’s theory of knowledge to prove that it is relevant to the current practice of mathematics. We found some related key words to Kant’s notions e.g. “presented”, “appears”, “human knowledge”, ”observation” and “analogies”. We may use Kant’s notions to examine contemporary practice of mathematics e.g. by reflecting metaphorical power of the “myth” of the foundation of contemporary mathematics. Hers R. listed the following myth: 1) there is only one mathematics- indivisible now and forever, 2) the mathematics we know is the only mathematics there can be, 3) mathematics has a rigorous method which yields absolutely certain conclusion, 4) mathematical truth is the same for everyone.
Meanwhile, Mrozek, J. (2004) in “The Problems of Understanding Mathematics” attempts to explain contemporary the structure of the process of understanding mathematical objects such as notions, definitions, theorems, or mathematical theories. Mrozek, J. distinguishes three basic planes on which the process of understanding mathematics takes place: first, understanding the meaning of notions and terms existing in mathematical considerations i.e. mathematician must have the knowledge of what the given symbols mean and what the corresponding notions denote; second, understanding concerns the structure of the object of understanding wherein it is the sense of the sequences of the applied notions and terms that is important; and third, understanding the 'role' of the object of understanding - consists in fixing the sense of the object of understanding in the context of a greater entity - i.e., it is an investigation of the background of the problem. Mrozek, J. sums up that understanding mathematics, to be sufficiently comprehensive, should take into account at least three other connected considerations - historical, methodological and philosophical - as ignoring them results in a superficial and incomplete understanding of mathematics.
Furthermore, Mrozek, J. recommends that contemporary practice in mathematics could investigate properly, un-dogmatically and non-arbitrarily the classical problems of philosophy of mathematics as it was elaborated in Kant’s theory of knowledge. According to him, it implies that teaching mathematics should not consist only in inculcating abstract formulas and conducting formalized considerations; we can not learn mathematics without its thorough understanding. Mrozek, J. sums up that in the process of teaching mathematics, we should take into account both the history and philosophy (with methodology) of mathematics i.e. theory of knowledge and epistemological foundation of mathematics, since neglecting them makes the understanding of mathematics superficial and incomplete.

References:

Jørgensen, K.F., 2006, “Philosophy of Mathematics” Retrieved 2006
2Hanna, R., 2004, “Kant's Theory of Judgment”, Stanford Encyclopedia of Philosophy, Retrieved 2004,
3 Ibid.
4 Ibid.
5 Ibid.
6 Hersh, R., 1997, “What is Mathematics, Really?”, London: Jonathan Cape, p.162
7 Ibid.p.162
8 Ibid. p.162
9 In Hersh, R., 1997, “What is Mathematics, Really?”, London: Jonathan Cape, p.162
10 Ibid. p. 216
11Ibid. p.216
12Mrozek, J.,2004, “The Problems of Understanding Mathematics” University of Gdańsk, Gdańsk, Poland. Retieved 2004
13 Ibid.
14Ibid.
15Ibid.
16Ibid.

11 comments:

  1. Kartika Pramudita
    17701251021
    PEP S2 B
    Hal yang saya pahami adalah tentang relevansi teori Kant dengan dasar matematika kontemporer. Banyak matematikawan yang setuju dengan teori Kant bahwa ilmu pengetahuan diperoleh dengan cara sintetik a priori. Selain itu ada pula matematikawan yang sepakat bahwa matematika memiliki landasan dan landasan matematika adalah pengetahuan intuitif. Matematika dibangun berlandaskan intuisi. Hal yang dapat dipetik adalah untuk belajar matematika maka harus belajar tentang filosofi, sejarah dan seluk beluk matematika sehingga dapat mencari keterkaitan dan relevansinya dan dapat berpikir matematika secara mendalam.

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  2. I Nyoman Indhi Wiradika
    17701251023
    PEP B

    Artikel ini menjelaskan tentang relevansi teori Kant terhadap pembelajaran dan pondasi matematika. Kant yang identic dengan gagasan sintetik a priori nya dianggap sesuai dalam penerapan pembelajaran matematika. Bahwa sesungguhnya dalam bermatematika tidak hanya berkutat pada rumus atau formula yang abstrak dan formal, tetapi pembelajaran matematika haruslah dipelajari dan ditemukan oleh pengalaman siswa.

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  3. Nama: Dian Andarwati
    NIM: 17709251063
    Kelas: Pendidikan Matematika (S2) Kelas C

    Assalamu’alaikum.Kant mengemukakan bahwa ilmu matematika merupakan contoh yang paling cemerlang tentang bagaimana akal murni berhasil bisa memperoleh kesuksesannya dengan bantuan pengalaman. Kant bisa mengatakan tanpa berlebihan bahwa banyak logika mengikuti jalur tunggal sejak awal, dan bahwa sejak Aristoteles itu tidak harus menelusuri kembali satu langkah. Kant mengatakan bahwa logika silogisme adalah untuk semua tampilan lengkap dan sempurna

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  4. Yusrina Wardani
    17709251057
    PPs PMAT C 2017
    Kant berpendapat bahwa matematika merupakan cara logis yang salah atau benarnya dapat ditentukan tanpa mempelajari dunia empiris dan matematika murni merupakan cabang dari logika, konsep matematika dapat di reduksikan menjadi konsep logika

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  5. Yusrina Wardani
    17709251057
    PPs PMAT C 2017
    Matematika dikenal dengan ilmu deduktif. Ini berarti proses pengerjaan matematika harus bersifat deduktif. Matematika tidak menerima generalisasi berdasarkan pengamatan, tetapi harus berdasarkan pembuktian deduktif. Meskipun demikian untuk membantu pemikiran pada tahap-tahap permulaan seringkali kita memerlukan bantuan contoh-contoh khusus atau ilustrasi geometris.

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  6. Tri Wulaningrum
    17701251032
    PEP S2 B

    Gagasan Kant mengenai pondasi dan pembelajaran matematika membawa saya kembali pada artikel sebelumnya yang menyebutkan bahwa pengetahuan menurut Kant berdasarkan sintetik a priori. Pada akhir artikel di atas, saya menjumpai pernyataan bahwa dalam proses pengajaran matematika, guru harus memperhitungkan beberapa aspek yang melekat pada suatu kegiatan belajar mengajar, baik sejarah maupun filsafat (dengan metodologi) matematika, yaitu teori pengetahuan dan landasan epistemologis matematika. Landasan pengetahuan tidak boleh kita tinggalkan, karena jika kita mengabaikannya, maka pemahaman matematika yang diperoleh bersifat dangkal dan tidak lengkap.

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  7. Nama: Hendrawansyah
    NIM: 17701251030
    S2 PEP 2017 Kelas B

    Assalamualaikum wr wb

    Daya sensitif dan kepekaannya terhadap matematika memberikan pengaruh yang besar bagi dunia ilmu pengetahuan . Khant merasa bahwa ada yang yang kurang lengkap dalam matematika. Maka untuk mencapai sesuatu yang komplesk dengan memberikan gagasan melalui berpikir kritisnya dengan menjembatani perdebatan antara kaum idealis dan realis.Khant memuluskan jalan bagi para ilmuwan, mungkin saya istilahkan dengan revolusi pikiran.

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  8. Latifah Fitriasari
    PM C

    Manusia dalam mencari ilmu pengetahuan menggunakan berbagai cara, ada yang menggunakan cara-cara yang terstruktur dengan rapi dan secara sadar. Munculnya teori pengetahuan dari Immanuel Kant, sebagai landasan epistemologis dari pengetahuan , dipengaruhi paling tidak oleh pengaruh dua aliran epistemologi yang masing-masing berakar pada pondasi empiris dan pondasi rasionalis. Pembelajaran matematika kontemporer itu adalah pemecahan masalah menjadi sentralnya pembelajaran matematika dan banyak melibatkan alat-alat teknologi canggih seperti kalkulator dan komputer.

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  9. Uswatun Hasanah
    17701251022
    S2 PEP B

    Melalui bacaan di atas saya memperoleh informasi terkait pemahaman konsep matematiak dijadikan sebagai fokus utamanya. Ada berbagai cara yang dapat ditempuh oleh individu saat mempelajari struktur teorinya. Salah satunya adalah dengan memperdalam intuisinya. Intuisi yang membuat seseorang dapat mengenali sebuah konsep dari mulai yang bersifat ideal dan kompleks hingga praktis dan dangkal. Seseorang bisa saja membuat formulasi konsep matematika dalam dirinya sendiri berdasarkan pada pengalaman yang telah dialaminya.

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  10. Isoka Amanah Kurnia
    17709251051
    PPs Pendidikan Matematika 2017 Kelas C

    Mrozek menyatakan bahwa praktek kontemporer dalam matematika seperti yang diuraikan dalam teori pengetahuan Kant, menunjukkan bahwa mengajar matematika tidak hanya dalam menanamkan rumus-rumus abstrak kita tidak bisa belajar matematika tanpa pemahaman menyeluruh. Selanjutnya menurut Mrozek dalam proses mengajar matematika, kita harus memperhitungkan sejarah dan filsafat (dengan metodologi) matematika yaitu teori pengetahuan dan landasan epistemologis matematika.

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  11. Irham Baskoro
    17709251004
    S2|Pendidikan Matematika A 2017|UNY

    Uraian di atas menggambarkan mengenai relevansi pemikiran Immanuel Kant dalam matematika kontemporer. Seperti yang dinyatakan oleh Mrozek, J. bahwa ia menganjurkan bahwa praktik kontemporer dalam matematika dapat menyelidiki secara benar, tidak dogmatis dan non-sewenang-wenang masalah klasik filsafat matematika. Hal ini sesuai dengan yang diuraikan dalam teori pengetahuan Imanuel Kant.

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