Nov 1, 2012

Kant on Mathematical Method




By Marsigit
Yogyakarta State University

Kant’s notions of mathematical method can be found in “The Critic Of Pure Reason: Transcendental Doctrine Of Method; Chapter I. The Discipline Of Pure Reason, Section I. The Discipline Of Pure Reason In The Sphere Of Dogmatism”. Kant recites that mathematical method is unattended in the sphere of philosophy by the least advantage that geometry and philosophy are two quite different things, although they go hand in hand in the field of natural science, and, consequently, that the procedure of the one can never be imitated by the other.


According to Kant 1, the evidence of mathematics rests upon definitions, axioms, and demonstrations; however, none of these forms can be employed or imitated in philosophy in the sense in which they are understood by mathematicians. Kant 2 claims that all our mathematical knowledge relates to possible intuitions, for it is these alone that present objects to the mind.

An a priori or non-empirical conception contains either a pure intuition that is it can be constructed; or it contains nothing but the synthesis of possible intuitions, which are not given a priori. Kant 3 sums up that in this latter case, it may help us to form synthetical a priori judgements, but only in the discursive method, by conceptions, not in the intuitive, by means of the construction of conceptions.

On the other hand, Kant 4 explicates that no synthetical principle which is based upon conceptions, can ever be immediately certain, because we require a mediating term to connect the two conceptions of event and cause that is the condition of time-determination in an experience, and we cannot cognize any such principle immediately and from conceptions alone.

Discursive principles are, accordingly, very different from intuitive principles or axioms. In his critic, Kant 5 holds that empirical conception can not be defined, it can only be explained. In a conception of a certain number of marks or signs, which denote a certain class of sensuous objects, we can never be sure that we do not cogitate under the word which.

The science of mathematics alone possesses definitions. According to Kant 6, philosophical definitions are merely expositions of given conceptions and are produced by analysis; while, mathematical definitions are constructions of conceptions originally formed by the mind itself and are produced by a synthesis.
Further, in a mathematical definition 7 the conception is formed; we cannot have a conception prior to the definition. Definition gives us the conception. It must form the commencement of every chain of mathematical reasoning.

In mathematics 8, definition can not be erroneous; it contains only what has been cogitated. However, in term of its form, a mathematical definition may sometimes error due to a want of precision. Kant marks that definition: “Circle is a curved line, every point in which is equally distant from another point called the centre” is faulty, from the fact that the determination indicated by the word curved is superfluous.

For there ought to be a particular theorem, which may be easily proved from the definition, to the effect that every line, which has all its points at equal distances from another point, must be a curved line (see Figure 22.)- that is, that not even the smallest part of it can be straight. 9

Kant (1781) in “The Critic Of Pure Reason: 1. AXIOMS OF INTUITION, The principle of these is: All Intuitions are Extensive Quantities”, illustrates that mathematics have its axioms to express the conditions of sensuous intuition a priori, under which alone the schema of a pure conception of external intuition can exist e.g. "between two points only one straight line is possible", "two straight lines cannot enclose a space," etc.

These 10 are the axioms which properly relate only to quantities as such; but, as regards the quantity of a thing, we have various propositions synthetical and immediately certain (indemonstrabilia) that they are not the axioms. Kant 11 highlights that the propositions: "If equals be added to equals, the wholes are equal"; "If equals be taken from equals, the remainders are equal"; are analytical, because we are immediately conscious of the identity of the production of the one quantity with the production of the other; whereas axioms must be a priori synthetical propositions.

On the other hand 12, the self-evident propositions as to the relation of numbers, are certainly synthetical but not universal, like those of geometry, and for this reason cannot be called axioms, but numerical formulae. Kant 13 proves that 7 + 5 = 12 is not an analytical proposition; for either in the representation of seven, nor of five, nor of the composition of the two numbers; “Do I cogitate the number twelve?” he said.

 Although the proposition 14 is synthetical, it is nevertheless only a singular proposition. In so far as regard is here had merely to the synthesis of the homogeneous, it cannot take place except in one manner, although our use of these numbers is afterwards general. Kant then exemplifies the construction of triangle using three lines as the following:

The statement: "A triangle can be constructed with three lines, any two of which taken together are greater than the third" is merely the pure function of the productive imagination, which may draw the lines longer or shorter and construct the angles at its pleasure; therefore, such propositions cannot be called as axioms, but numerical formulae
15

Kant in “The Critic Of Pure Reason: II. Of Pure Reason as the Seat of Transcendental Illusory Appearance, A. OF REASON IN GENERAL”, enumerates that mathematical axioms 16 are general a priori cognitions, and are therefore rightly denominated principles, relatively to the cases which can be subsumed under them. While in “The Critic Of Pure Reason: SECTION III. Of Opinion, Knowledge, and Belief; CHAPTER III.

The Arehitectonic of Pure Reason”, Kant propounds that 17mathematics may possess axioms, because it can always connect the predicates of an object a priori, and without any mediating term, by means of the construction of conceptions in intuition. On the other hand, in “The Critic Of Pure Reason: CHAPTER IV. The History of Pure Reason; SECTION IV. The Discipline of Pure Reason in Relation to Proofs” , Kant designates that in mathematics, all our conclusions may be drawn immediately from pure intuition. Therefore, mathematical proof must demonstrate the possibility of arriving, synthetically and a priori, at a certain knowledge of things, which was not contained in our conceptions of these things.

All 18 the attempts which have been made to prove the principle of sufficient reason, have, according to the universal admission of philosophers, been quite unsuccessful. Before the appearance of transcendental criticism, it was considered better to appeal boldly to the common sense of mankind, rather than attempt to discover new dogmatical proofs. Mathematical proof 19 requires the presentation of instances of certain concepts.

These instances would not function ex¬actly as particulars, for one would not be entitled to assert anything concerning them which did not follow from the general concept. Kant 20 says that mathematical method contains demonstrations because mathematics does not deduce its cognition from conceptions, but from the construction of conceptions, that is, from intuition, which can be given a priori in accordance with conceptions. Ultimately, Kant 21 contends that in algebraic method, the correct answer is deduced by reduction that is a kind of construction; only an apodeictic proof, based upon intuition, can be termed a demonstration.

References:

Kant, I., 1781, “The Critic Of Pure Reason: Transcendental Doctrine Of Method; Chapter I. The Discipline Of Pure Reason, Section I. The Discipline Of Pure Reason In The Sphere Of Dogmatism”, Translated By J. M. D. Meiklejohn, Retrieved 2003 ).
2 Ibid.
3 Ibid.
4 Ibid.
5 Kant, I., 1781, “The Critic Of Pure Reason: Transcendental Doctrine Of Method, Chapter I, Section I .”, Translated By J. M. D. Meiklejohn, Retrieved 2003
6 Ibid.
7 Ibid.
8 Ibid.
9 Ibid.
10 Kant, I., 1781, “The Critic Of Pure Reason: 1. AXIOMS OF INTUITION, The principle of these is: All Intuitions are Extensive Quantities”, Translated By J. M. D. Meiklejohn, Retrieved 2003).
11Ibid.
12Ibid.
13Ibid.
14Ibid.
15Ibid.
16Kant, I., 1781, “The Critic Of Pure Reason: II. Of Pure Reason as the Seat of Transcendental Illusory Appearance, A. OF REASON IN GENERAL”, Translated By J. M. D. Meiklejohn, Retrieved 2003).
17Kant, 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)
18Kant, I., 1781, “The Critic Of Pure Reason: CHAPTER IV. The History of Pure Reason; SECTION IV. The Discipline of Pure Reason in Relation to Proofs” Translated By J. M. D. Meiklejohn, Retrieved 2003)
19Kant in Wilder, R. L. , 1952, “Introduction to the Foundation of Mathematics”, New York
20Kant, I., 1781, “The Critic Of Pure Reason: Transcendental Doctrine Of Method, Chapter I, Section I .”, Translated By J. M. D. Meiklejohn, Retrieved 2003).
21 Ibid.

6 comments:

  1. Nama : Irna K.S.Blegur
    Nim : 16709251064
    kelas : PM D 2016(PPS)


    “Kritik der reinen Vernunft (Critique of Pure Reason)” melalui karya ini, Kant terbangun dari ‘tidur dogmatik’ –nya dan mulai membangun aliran filsafat yang disebut sebagai Kritisisme Kantian. Kritisisme merupakan filsafat yang diawali dengan menyelidiki kemampuan dan batas – batas nalar. Bagi Kant, kritisisme merupakan jawaban terhadap dogmatisme. Dogmatisme menganggap pengetahuan objektif sebagai hal yang terjadi dengan sendirinya. Sebagai aliran filsafat, dogmatisme percaya sepenuhnya pada kemampuan nalar dan mendasarkan pandangannya pada kaidah – kaidah a priori tanpa bertanya apakah nalar memahami hakikatnya sendiri, yakni jangkauan dan batas – batas kemampuannya.

    Pembahasan Kant dalam Critique of Pure Reason utamanya berkenaan dengan konsep analisis transendental. Bagian karya ini membahas konsep analisis transendental yang merupakan bagian terpenting dari Critique of Pure Reason. Meski demikian, tentu hal ini bukan merupakan satu – satunya konsep yang harus dibahas.
    Terdapat beberapa kesulitan dalam usaha mentranslasikan karya klasik Kant yang berlatar belakang kebudayaan Jerman abad ke–18. Di samping itu, beberapa gaya penulisan Kant juga terkesan sangat kaku dan sulit untuk dimengerti. Kant cenderung menyampaikan gagasannya secara eksak dan hati – hati. Hal – hal inilah yang umumnya menjadi kesulitan bagi pembaca pemula dalam memahami Critique of Pure Reason.
    Dalam Teori Pengetahuannya, Immanuel Kant berusaha meletakkan dasar epistemologis bagi matematika untuk menjamin bahwa matematika memang benar dapat dipandang sebagai ilmu. Kant menyatakan bahwa metode yang benar untuk memperoleh kebenaran matematika adalah memperlakukan matematika sebagai pengetahuan a priori. Menurut Kant, secara spesifik, validitas obyektif dari pengetahuan matematika diperoleh melalui bentuk a priori dari sensibilitas kita yang memungkinkan diperolehnya pengalaman inderawi. Namun, perkembangan matematika pada dua abad terakhir telah memberikan tantangan yang cukup signifikan terhadap pandangan Immanuel Kant ini.

    References:
    Epistemologi Matematika oleh Marsigit
    Herho, Sandy. H.S. (2016). Critique Of Pure Reason: Sebuah Pengantar. Bandung : Perkumpulan Studi Ilmu Kemasyarakatan ITB

    ReplyDelete
  2. Anwar Rifa’i
    PMAT C 2016 PPS
    16709251061

    Kant berpendapat bahwa metode matematika tanpa pengawasan di bidang filsafat adalah dua hal yang berbeda meskipun mereka berjalan beriringan di bidang ilmu pengetahuan alam. Kant 1 berpendapat bahwa matematika bersandar pada definisi, aksioma, dan demonstrasi. Namun tidak satupun dari bentuk-bentuk ini dapat digunakan atau ditiru dalam filsafat dalam arti dimana mereka dipahami oleh ahli matematika. Kant 2 mengklaim bahwa semua pengetahuan matematika berhubungan dengan intuisi. Kant 3 menyatakan bahwa harus dibentuk suatu penilaian apriori sintesis tetapi hanya dalam metode diskursif. Kant 4 secara eksplisit menyatakn bahwa ada prinsi kimis yang didasarkan pada konsepsi. Kant 5 menyatakan bahwa konsepsi empiris tidak dapat didefinisikan, hanya dapat dijelaskan. Kant 6 mendefinisikan bahwa eksposisi yang diberikan konsepsi awal yang dibentuk oleh pikiran sendiri.

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  3. Windi Agustiar Basuki
    16709251055
    S2 Pend. Mat Kelas C – 2016

    Immanuel Kant berpendapat bahwa matematika merupakan suatu penalaran yang bersifat mengkonstruksi konsep-konsep secara sintetik a priori dalam konsep ruang dan waktu secara umum yang pada akhirnya dianggap mendasari matematika. Kemudian, dalam pembuktian matematika itu sendiri Kant memberikan tanggapan bahwa hal itu bersandar pada definisi, aksioma, dan demonstrasi. Dalam definisi matematika konsepsi terbentuk, sehingga konsep didapatkan setelah adanya definisi. Matematika memiliki aksioma untuk mengekspresikan kondisi intuisi sensual apriori, di mana saja skema konsepsi murni dari intuisi eksternal berada. Sedangkan matematika berisi demonstrasi karena matematika membangun konsepsi yaitu dari intuisi, yang berarti tidak menyimpulkan kognisi dari konsepsi.

    ReplyDelete
  4. Saepul Watan
    16709251057
    S2 P.Mat Kelas C 2016

    Bismilahir rahmaanir rahiim..
    Assalamualaikum wr..wb...

    Artikel ini memaparkan gagasan Kant mengenai metode Matematika. Metode matematika merupakan proses berpikir menggunakan prinsip-prisip matematika dalam menyelesaikan setiap permasalahan dan bagaimana kemampuan pemecahan masalah matematika yang kita miliki. Permasalahan yang ada menimbulkan keingintahuan kita untuk memecahkannya sehingga menimbulkan pertanyaan yang akan dicari bagaimana solusi dalam pemecahan masalah tersebut. Solusi-solusi tersebut berawal dari bagaimana kita membuat identifikasi, hipotesis, menganalisis, dan pada ahkirnya menarik kesimpulan dari setiap permasalahan yang ada.

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  5. Sumandri
    16709251072
    S2 Pendidikan Matematika D 2016

    Immanuel Kant berusaha meletakkan dasar bagi matematika untuk menjamin bahwa matematika memang benar dapat dipandang sebagai ilmu. Kant menyatakan bahwa metode yang benar untuk memperoleh kebenaran matematika adalah memperlakukan matematika sebagai pengetahuan a priori. Menurut Kant, secara spesifik, validitas obyektif dari pengetahuan matematika diperoleh melalui bentuk a priori dari sensibilitas kita yang memungkinkan diperolehnya pengalaman inderawi.

    ReplyDelete
  6. Wahyu Lestari
    16709251024
    PPs P.Matematika Kelas D

    dari artikel di atas, The Arehitectonic of Pure Reason ", Kant mengemukakan bahwa 17 matematika dapat memiliki aksioma, karena ia selalu dapat menghubungkan predikat objek secara apriori, dan tanpa istilah perantara, melalui konstruksi konsepsi dalam intuisi. Di sisi lain, dalam "Kritik Alasan Murni: BAB IV. Sejarah Alasan Murni; BAGIAN IV. Disiplin Murni Alasan Berkaitan dengan Bukti ", Kant menunjuk bahwa dalam matematika, semua kesimpulan kita dapat ditarik langsung dari intuisi murni. Oleh karena itu, bukti matematis harus menunjukkan kemungkinan untuk tiba, secara sintetis dan apriori, dengan pengetahuan tertentu tentang hal-hal, yang tidak terkandung dalam konsepsi kita tentang hal-hal ini.

    Semua 18 upaya yang telah dilakukan untuk membuktikan asas alasan yang cukup, menurut pengakuan para filsuf universal, sangat tidak berhasil. Sebelum munculnya kritik transendental, dianggap lebih baik mengajukan banding dengan berani kepada akal sehat umat manusia, daripada mencoba menemukan bukti dogmatika baru. Bukti matematis 19 mensyaratkan penyajian contoh konsep tertentu.

    Contoh-contoh ini tidak akan berfungsi secara khusus sebagai hal yang spesifik, karena seseorang tidak berhak untuk menyatakan sesuatu mengenai hal-hal yang tidak mengikuti konsep umum. Kant 20 mengatakan bahwa metode matematika mengandung demonstrasi karena matematika tidak menyimpulkan kognisi dari konsepsi, namun dari konstruksi konsepsi, yaitu dari intuisi, yang dapat diberikan secara apriori sesuai dengan konsepsi. Pada akhirnya, Kant 21 berpendapat bahwa dalam metode aljabar, jawaban yang benar disimpulkan oleh pengurangan yang merupakan semacam konstruksi; Hanya bukti apodeictic, berdasarkan intuisi, bisa disebut demonstrasi.

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