Oct 10, 2012

Elegi Menggapai "Kant on the Basis Validity of Mathematical Knowledge"

By Marsigit
Yogyakarta State University

According to Wilder R.L., Kant's philosophy of mathematics can be interpreted in a constructivist manner and constructivist ideas that presented in the nineteenth century-notably by Leopold Kronecker, who was an important for a runner of intuition¬ism-in opposition to the tendency in mathematics toward set-theoretic ideas, long before the paradoxes of set theory were discovered. In his philosophy of mathematics , Kant

supposed that arithmetic and geometry comprise synthetic a priori judgments and that natural science depends on them for its power to explain and predict events. As synthetic a priori judgments , the truths of mathematics are both informative and necessary; and since mathematics derives from our own sensible intuition, we can be absolutely sure that it must apply to everything we perceive, but for the same reason we can have no assurance that it has anything to do with the way things are apart from our perception of them.
Kant believes that synthetic a priori propositions include both geometric propositions arising from innate spatial geometric intuitions and arithmetic propositions arising from innate intuitions about time and number. The belief in innate intuitions about space was discredited by the discovery of non-Euclidean geometry, which showed that alternative geometries were consistent with physical reality. Kant perceives that mathematics is about the empirical world, but it is special in one important way. Necessary properties of the world are found through mathematical proofs. To prove something is wrong, one must show only that the world could be different. While , sciences are basically generalizations from experience, but this can provide only contingent and possible properties of the world. Science simply predicts that the future will mirror the past.
In his Critic of Pure Reason Kant defines mathematics as an operation of reason by means of the construction of conceptions to determine a priori an intuition in space (its figure), to divide time into periods, or merely to cognize the quantity of an intuition in space and time, and to determine it by number. Mathematical rules , current in the field of common experience, and which common sense stamps everywhere with its approval, are regarded by them as mathematical axiomatic. According to Kant , the march of mathematics is pursued from the validity from what source the conceptions of space and time to be examined into the origin of the pure conceptions of the understanding. The essential and distinguishing feature of pure mathematical cognition among all other a priori cognitions is, that it cannot at all proceed from concepts, but only by means of the construction of concepts.
Kant conveys that mathematical judgment must proceed beyond the concept to that which its corresponding visualization contains. Mathematical judgments neither can, nor ought to, arise analytically, by dissecting the concept, but are all synthetical. From the observation on the nature of mathematics, Kant insists that some pure intuition must form mathematical basis, in which all its concepts can be exhibited or constructed, in concreto and yet a priori. Kant concludes that synthetical propositions a priori are possible in pure mathematics, if we can locate this pure intuition and its possibility. The intuitions which pure mathematics lays at the foundation of all its cognitions and judgments which appear at once apodictic and necessary are Space and Time. For mathematics must first have all its concepts in intuition, and pure mathematics in pure intuition, it must construct them. Mathematics proceeds, not analytically by dissection of concepts, but synthetically; however, if pure intuition be wanting, it is impossible for synthetical judgments a priori in mathematics.
The basis of mathematics actually are pure intuitions, which make its synthetical and apodictically valid propositions possible. Pure Mathematics, and especially pure geometry, can only have objective reality on condition that they refer to objects of sense. The propositions of geometry are not the results of a mere creation of our poetic imagination, and that therefore they cannot be referred with assurance to actual objects; but rather that they are necessarily valid of space, and consequently of all that may be found in space, because space is nothing else than the form of all external appearances, and it is this form alone where objects of sense can be given. The space of the geometer is exactly the form of sensuous intuition which we find a priori in us, and contains the ground of the possibility of all external appearances. In this way geometry be made secure, for objective reality of its propositions, from the intrigues of a shallow metaphysics of the un-traced sources of their concepts.
Kant argues that mathematics is a pure product of reason, and moreover is thoroughly synthetical. Next, the question arises: Does not this faculty, which produces mathematics, as it neither is nor can be based upon experience, presuppose some ground of cognition a priori, which lies deeply hidden, but which might reveal itself by these its effects, if their first beginnings were but diligently ferreted out? However, Kant found that all mathematical cognition has this peculiarity: it must first exhibit its concept in a visual intuition and indeed a priori, therefore in an intuition which is not empirical, but pure. Without this mathematics cannot take a single step; hence its judgments are always visual, viz., intuitive; whereas philosophy must be satisfied with discursive judgments from mere concepts, and though it may illustrate its doctrines through a visual figure, can never derive them from it.

Wilder, R. L. , 1952, “Introduction to the Foundation of Mathematics”, New York, p.205
2 Ibid.205
3 Wegner, P., 2004, “Modeling, Formalization, and Intuition.” Department of Computer Science. Retrieved 2004
4 Posy, C. ,1992, “Philosophy of Mathematics”, Retreived 2004
5 Ibid.
6 Kant, I., 1781, “The Critic Of Pure Reason: SECTION III. Of Opinion, Knowledge, and Belief; CHAPTER III. The Arehitectonic of Pure Reason” Translated By J. M. D. Meiklejohn, Retrieved 2003
7 Ibid.
8 Kant, I, 1783, Prolegomena To Any Future Methaphysics, Preamble, p. 19
9 Ibid. p. 21
10 Kant, I, 1783, “Prolegomena to Any Future Metaphysic: First Part Sect. 7”, Trans. Paul Carus. Retrieved 2003
12Kant, I, 1783, “Prolegomena to Any Future Metaphysic: First Part Sect.10”, Trans. Paul Carus. Retrieved 2003
15Kant, I, 1783, “Prolegomena to Any Future Metaphysic: First Part Sect.12 Trans. Paul Carus. Retrieved 2003
16Kant, I, 1783, “Prolegomena to Any Future Metaphysic: REMARK 1 Trans. Paul Carus. Retrieved 2003
19Wikipedia The Free Encyclopedia. Retrieved 2004
21Kant, I, 1783, “Prolegomena to Any Future Metaphysic: First Part Of The Transcendental Problem: How Is Pure Mathematics Possible? Sect. 6. p. 32
22Immanuel Kant, Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible? Sect. 7.p. 32


  1. Ibrohim Aji Kusuma
    S2 PMA 2018

    Kant memandang bahwa matematika adalah tentang ilmu empiris. Sifat yang diperlukan dari dunia ditemukan melalui bukti matematika. Untuk membuktikan ada sesuatu yang salah, kita harus menunjukkan bahwa dunia hanya bisa berbeda. Sementara, ilmu pada dasarnya generalisasi dari pengalaman, tetapi hal ini dapat memberikan kemungkinan sifat dunia. Sains hanya memprediksi bahwa masa depan akan mencerminkan masa lalu.

  2. Yoga Prasetya
    S2 Pendidikan Matematika UNY 2018 A
    Filsafat matematika merupakan pemikiran yang reflektif oleh seseorang. Kant menyampaikan bahwa penilaian matematis harus melampaui konsep yang mengandung visualisasi terkait. Penilaian matematis tidak dapat, atau seharusnya muncul secara analitis, dengan membedah konsep, tetapi semuanya sintetik. Dari pengamatan pada sifat matematika, Kant juga menegaskan bahwa beberapa intuisi murni harus membentuk dasar matematis, di mana semua konsepnya dapat dikonstruksi dalam konkresi dan belum a priori. Kant juga berpendapat bahwa matematika adalah produk akal murni, dan terlebih lagi adalah benar-benar sintetik.

  3. Fany Isti Bigo
    PPs UNY PM A 2018

    Elegi ini menjelaskan pandangan Kant mengenai validitas dasar pengetahuan matematika. Menurut Kant matematika merupakan suatu penalaran yang bersifat mengkonstruksi konsep-konsep secara sintetik apriori dalam konsep ruang dan waktu. Dalam tulisan ini dijelaskan pula bahwa sebagai penilaian sintetik apriori, kebenaran matematika yang informatif dan diperlukan; dan karena matematika berasal dari intuisi yang masuk akal, kita dapat benar-benar yakin bahwa itu harus berlaku untuk segala sesuatu yang kita rasakan, tetapi untuk alasan yang sama kita dapat memiliki jaminan bahwa hal itu ada hubungannya dengan hal-hal yang terpisah dari persepsi kita tentang matematika.

  4. Fabri Hidayatullah
    S2 Pendidikan Matematika B 2018

    Kebenaran metamatika menurut Kant. Kant menyatakan bahwa penilaian matematika harus dilakukan melebihi konsep yang berhubungan dengan visualisasi. Penilaian matematika tidak dapat dihasilkan secara analitis dengan membedah konsep, tetapi dengan sintesis. Kant menyatakan bahwa kebenaran barisan matematika ialah dari sumber konsepsi ruang dan waktu yang dipertimbangkan ke dalam konsepsi murni asli dari pemahama. Perbedaan yang menojol dari sifat-sifat dari pengetahuan matematika murni dibandingkan dengan pengetahuan kognisi yang lain ialah bahwa tidak semua matematika murni dapat di hasilkan dari konsep, tetapi hanya dari makna pembentukan konsep.

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  6. Amalia Nur Rachman
    S2 Pendidikan Matematika B UNY 2018

    Dasar matematika adalah intuisi murni, dimana membuat proposisi sintetis dan kemungkinan valid. Menurut Kant, semua kognisi matematika merupakan intuisi murni dan tidak empiris yang harus menunjukkan konsep dalam intuisi visual dan apriori. Penilaian visual dalam penilaian filsafat itu intuitif. Sedangkan penilaian diskursif sebuah konsep dan selalu menggambarkan doktrin melalui sosok visual.

  7. Septia Ayu Pratiwi
    S2 Pendidikan Matematika 2018

    Kebenaran matemematika bersifat informative dan berasal dari intuisi kita. Sehingga matematika berhubungan dengan pengalaman-pengalaman yang telah kita alami. Kant berpendapat bahwa matematika murni dari akal pikiran kita dan jika lebih dari itu maka matematika adalah sintesis. Untuk membuktikan matematika ini dibutuhkan konsep-konsep visual dan dan kebenaran a priori. Oleh karena itu, matematika itu tidak empiris melainkan ilmu murni yang melibatkan intuisi di dalamnya.

  8. Janu Arlinwibowo
    PEP 2018

    Pemikiran Kant memberikan dampak besar terhadaf dunia. Keberaniannya mengambil tengahan antara kaum realis dan idealis membuat dirinya mampu memikirkan kombinasi akal dan empiris. Pemikiran Kant nampaknya juga memberikan pengaruh besar terhadap perkembangan dunia pendidikan matematika. Pemikiran Kant mengarah ke pola pikir intuitif yang sejalan dengan terkonstruksinya ilmu atau konsep matematika dalam pikiran.