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.

13 comments:

  1. Kartika Pramudita
    17701251021
    PEP S2 B

    Bukti dalam matematika didasarkan pada definisi, aksioma, dan demonstrasi. Konsep dalam matematika berasal dari definisi. Definisi tidak perlu untuk dibuktikan. Dengan menggunakan definisi dapat muncul teorema-teorema yang harus dibuktikan kebenarannya. Matematika harus dikontruksi berdasarkan intuisi sehingga menghasilkan matematika yang bersifat sintetik a priori. Dalam membangun landasan matematika, Kant melibatkan rasio dan empiris.

    ReplyDelete
  2. I Nyoman Indhi Wiradika
    17701251023
    PEP B

    Dalam artikel ini kant menjelaskan bahwa geometri dan filsafat merupakan dua hal yang berbeda, meskipun mereka sama-sama digolongkan dalam sains alami. Matematika memiliki aksioma-aksioma untuk mengungkapkan kondisi dari intuisi apriori, yang bergantung kepada skema konsepsi murni tentang intuisi eksternal dapat terjadi misalnya; Antara dua titik hanya memungkinkan untuk sebuah garis lurus.

    ReplyDelete
  3. Arung Mega Ratna
    17709251049
    PPs PMC 2017


    Metode matematika adalah bukan bagian dari filsafat. Meskipun geometri (matematika) dan filsafat sama-sama merupakan bagian dari ilmu alam namun keduanya berbeda dalam hal perosedur. Salah satu contohnya yaitu bukti matematika bersandar pada definisi, aksioma, dan demonstrasi namun tidak satu pun dari bentuk-bentuk ini dapat digunakan atau ditiru dalam filsafat.

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  4. Latifah Fitriasari
    17709251055
    PPs PM C

    Proses sosial sedangkan siswa bersifat hidup yang membutuhkan pendidikan untuk membangun hidupnya sehingga siswa perlu adanya proses mengkonstruksi pengetahuannya sendiri merupakan pendidikan matematika. Dari proses sosial siswa dapat mengkonstruksi pengetahuannya dan untuk mengkonstruksi pengetahuannya siswa memerlukan intuisi. Maka pendidikan matematika harus membantu perkembangan konstruksi pengetahuan melalui keterkaitan aktif dan interaksi siswa serta dapat mengebangkan intuisi matematika siswa.

    ReplyDelete
    Replies
    1. Matematika sebagai ilmu adalah mungkin jika kita mampu menemukan intuisi murni sebagai landasannya. Kant menyatakan bahwa pengetahuan manusia muncul dari dua sumber utama dalam benak yakni fakultas penerimaan kesan-kesan inderawi (sensibility) dan fakultas pemahaman (understanding) yang membuat keputusan-keputusan tentang kesan-kesan inderawi yang diperoleh melalui fakultas pertama. Menurut Kant, matematika sebagai ilmu adalah mungkin jika konsep matematika dikontruksi berdasarkan intuisi keruangan dan waktu. Kontruksi konsep matematika berdasar intuisi ruang dan waktu akan menghasilkan matematika sebagai ilmu yang bersifat “sintetik a priori”.

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  5. Rahma Dewi Indrayanti
    17709251038
    PPS Pendidikan Matematika Kelas B

    Menurut Kant, 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.

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  6. Yusrina Wardani
    17709251057
    PPs PMAT C 2017
    Matematika adalah bahasa yang melambangkan serangkaian makna dari pernyataan yang ingin kita sampaikan. Lambang–lambang matematika bersifat artifisial yang baru mempunyai arti setelah sebuah maknadiberikan padanya. Tanpa itu maka matematika hanya merupakan kumpulan rumus-rumus yang mati.

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  7. Pendidikan matematika merupakan proses sosial sedangkan siswa bersifat hidup yang membutuhkan pendidikan untuk membangun hidupnya sehingga siswa perlu adanya proses mengkonstruksi pengetahuannya sendiri. Melalui proses sosial siswa dapat mengkonstruksi pengetahuannya dan untuk mengkonstruksi pengetahuannya siswa memerlukan intuisi. Oleh karenanya pendidikan matematika harus membantu perkembangan konstruksi pengetahuan melalui keterkaitan aktif dan interaksi siswa serta dapat mengembangkan intuisi matematika siswa.

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  8. Junianto
    PM C
    17709251065

    Kant membacakan bahwa metode matematis tidak dijaga dalam lingkup filsafat dengan sedikit keuntungan bahwa geometri dan filsafat adalah dua hal yang sangat berbeda, walaupun mereka berjalan beriringan di bidang sains alam, dan oleh karena itu, prosedur yang dapat dilakukan seseorang tidak pernah ditiru oleh yang lain. Secara objek memang sangatlah berbeda antara geometri dan filsafat karena memang terlalu jauh jaraknya. Geometri merupakan ilmu matematika tentang konsep ruang. Sebenarnya geometri merupakan turunan konsep ruang dalam filsafat namun karena filsafat mengkaji secara umum maka keterkaita antar keduanya cukup jauh.

    ReplyDelete
  9. Tri Wulaningrum
    17701251032
    PEP S2 B

    Metode dalam matematika yang dibahas melalui kacamata filsafat. Dua bagian yang berbeda, akan tetapi saling bertalian satu dengan yang lainnya. Pada artikel di atas dikupas tentang gagasan pikir Kant mengenai metode matematika berikut juga pembuktian dalam matematika. Kant mengklaim bahwa semua pengetahuan matematis kita berhubungan dengan kemungkinan intuisi, karena hanya dengan intuisi saja seorang pembelajar mampu mendatangkan sebuah objek dalam pikrian mereka. Pembahasan intuisi pada metode matematika menurut Kant sekali lagi membawa kita juga pada statement bahwa matematika merupakan pengetahuan yang diperoleh lewat sintetik apriori.

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

    Assalamu’alaikum. Menurut Kant, bukti matematika bersandar pada definisi, aksioma, dan demonstrasi. Kant mengklaim bahwa semua pengetahuan matematis kita berhubungan dengan kemungkinan intuisi. Kant menjelaskan bahwa tidak ada prinsip sintetis yang didasarkan pada konsepsi dapat segera dipastikan, karena diperlukan perantara untuk menghubungkan dua konsepsi kejadian dan penyebabnya.

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  11. Auliaul Fitrah Samsuddin
    17709251013
    PPs P.Mat A 2017
    Terima kasih atas postingannya, Prof. Metode matematika mengandung demonstrasi karena matematika tidak menyimpulkan kognisinya dari konsepsi, melainkan dari konstruksi konsepsi, yakni dari intuisi, yang bisa saja bersifat apriori dengan hubungannya dengan konsepsi.

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  12. Dewi Thufaila
    17709251054
    Pendidikan Matematika Pascasarjana C 2017

    Assalamualaikum.wr.wb
    Menurut Imanuael Kant, matematika sebagai ilmu adalah jika konsep matematika dikontruksi berdasarkan intuisi ke ruang dan waktu. Matematika tidak dikembangkan hanya dengan konsep aposteriori Sebab jika demikian matematika akan bersifat empiris. Tetapi data Empiris yang diperoleh dari pengalaman diperlukan untuk menggali konsep-konsep matematika yang bersifat apriori.
    Wassalamualaikum.wr.wb

    ReplyDelete