Mar 8, 2011

Elegi Menggapai "Kant on Mathematical Judgment"




By Marsigit
Yogyakarta State University

In his Critic of Pure Reason Kant mentions that a judgment is the mediate cognition of an object; consequently it is the representation of a representation of it. In every judgment there is a conception which applies to his last being immediately connected with an object. All judgments 1 are functions of unity in our representations. A higher representation is used for our cognition of the object, and thereby many possible


cognitions are collected into one. Hanna R. learns that in term of the quantity of judgments Kant captures the basic ways in which the comprehensions of the constituent concepts of a simple monadic categorical proposition are logically combined and separated.
For Kant 2, the form “All Fs are Gs” is universal judgments, the form “Some Fs are Gs” is particular judgments. Tthe form “This F is G” or “The F is G” is singular judgments. A simple monadic categorical judgment 3 can be either existentially posited or else existentially cancelled. Further, the form “it is the case that Fs are Gs” (or more simply: “Fs are Gs”) is affirmative judgment. The form “no Fs are Gs” is negative judgments, and the form “Fs are non-Gs” is infinite judgments. Kant's pure general logic 4 includes no logic of relations or multiple quantification, because mathematical relations generally are represented spatiotemporally in pure or formal intuition, and not represented logically in the understanding. True mathematical propositions, for Kant 5, are not truths of logic viz. all analytic truths or concept-based truths, but are synthetic truths or intuition-based truths. Therefore, according to Kant 6, by the very nature of mathematical truth, there can be no such thing as an authentically “mathematical logic.”
For Kant 7, in term of the relation of judgments, 1-place subject-predicate propositions can be either atomic or molecular; therefore, the categorical judgments repeat the simple atomic 1-place subject-predicate form “Fs are Gs”. The molecular hypothetical judgments 8 are of the form “If Fs are Gs, then Hs are Is” (or: “If P then Q”); and molecular disjunctive judgments are of the form “Either Fs are Gs, or Hs are Is” (or: “Either P or Q”). The modality of a judgment 9 are the basic ways in which truth can be assigned to simple 1-place subject-predicate propositions across logically possible worlds--whether to some worlds (possibility), to this world alone (actuality), or to all worlds (necessity). Further, the problematic judgments 10 are of the form “Possibly, Fs are Gs” (or: “Possibly P”); the ascertoric judgments are of the form “Actually, Fs are Gs” (or: “Actually P”); and apodictic judgments are of the form “Necessarily, Fs are Gs” (or: “Necessarily P”).
Mathematical judgments 11 are all synthetical; and the conclusions of mathematics, as is demanded by all apodictic certainty, are all proceed according to the law of contradiction. A synthetical proposition can indeed be comprehended according to the law of contradiction, but only by presupposing another synthetical proposition 12from which it follows, but never in itself. In the case of addition 7 + 5 = 12, it 13 might at first be thought that the proposition 7 + 5 = 12 is a mere analytical judgment, following from the concept of the sum of seven and five, according to the law of contradiction. However, if we closely examine the operation, it appears that the concept of the sum of 7+5 contains merely their union in a single number, without its being at all thought what the particular number is that unites them.
Therefore, Kant 14 concludes that the concept of twelve is by no means thought by merely thinking of the combination of seven and five; and analyzes this possible sum as we may, we shall not discover twelve in the concept. Kant suggests that first of all, we must observe that all proper mathematical judgments are a priori, and not empirical. According to Kant 15, mathematical judgments carry with them necessity, which cannot be obtained from experience, therefore, it implies that it contains pure a priori and not empirical cognitions. Kant, says that we must go beyond these concepts, by calling to our aid some concrete image [Anschauung], i.e., either our five fingers, or five points and we must add successively the units of the five, given in some concrete image [Anschauung], to the concept of seven; hence our concept is really amplified by the proposition 7 + 5 = I 2, and we add to the first a second, not thought in it”17. 18 Ultimately, Kant concludes that arithmetical judgments are therefore synthetical. According to Kant, 16 we analyze our concepts without calling visual images (Anscliauung) to our aid. We can never find the sum by such mere dissection. Further, Kant argues that all principles of geometry are no less analytical.
Kant 19 illustrates that the proposition “a straight line is the shortest path between two points”, is a synthetical proposition because the concept of straight contains nothing of quantity, but only a quality. Kant then claims that the attribute of shortness is therefore altogether additional, and cannot be obtained by any analysis of the concept; and its visualization [Anschauung] must come to aid us; and therefore, it alone makes the synthesis possible. Kant 20 confronts the previous geometers assumption which claimed that other mathematical principles are indeed actually analytical and depend on the law of contradiction. However, he strived to show that in the case of identical propositions, as a method of concatenation, and not as principles, e. g., “a=a”, “the whole is equal to itself”, or “a + b > a”, and “the whole is greater than its part”. Kant 21 then claims that although they are recognized as valid from mere concepts, they are only admitted in mathematics, because they can be represented in some visual form [Anschauung].

References:
Kant, I., 1781, “The Critic Of Pure Reason: Transcendental Analytic, Book I, Section 1, Ss 4.”, Translated By J. M. D. Meiklejohn, Retrieved 2003
2Hanna, R., 2004, “Kant's Theory of Judgment”, Stanford Encyclopedia of Philosophy, Retreived 2004,
3 Ibid.
4 Ibid.
5 Ibid.
6 Ibid.
7 Ibid.
8 Ibid.
9 Ibid.
10 Ibid.
11Kant, I, 1783, “Prolegomena to Any Future Metaphysic, p. 15
12Ibid. p. 16
13Ibid. p. 18
14Ibid. p.18
15Ibid. p. 19
16Ibid.p.20
17Ibid. p.21
18Ibid. p.21
19Ibid p.22
20Ibid. p.22
21Ibid. p.23

7 comments:

  1. Elli Susilawati
    16709251073
    Pmat D pps16

    Immanuel Kant dalam filsafat matematika adalah aliran logistik. Kant berpendapat bahwa matematika merupakan cara logis (logistik) yang salah atau benarnya dapat ditentukan tanpa mempelajari dunia empiris.Matematika murni merupakan cabang dari logika, konsep matematika dapat di reduksikan menjadi konsep logika.

    ReplyDelete
  2. Wahyu Berti Rahmantiwi
    PPs Pendidikan Matematika Kelas C 2016
    16709251045

    Sebuah penilaian terdapat kognisi perantara dari suatu objek. Penilaian matematis bergantung pada tujuannya ketika digunakan, tujuan yang akan dicapai berasal dari pengalaman dan pengetahuan masing-masing individu. Pengalaman yang dimiliki individu mengandung kognisi murni apriori dan tidak empiris. Banyak konsep yang harus digunakan ketika akan mengubah dari bentuk lambang ke dalam dunia nyata. Dengan demikian, kita harus banyak-banyak menggali pengetahuan yang ada dengan membaca.

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  3. Gamarina Isti Ratnasari
    17709251036
    Pendidikan Matematika Kelas B(S2)

    Menurut Kant sebuah penilaian adalah kognisi perantara dari suatu objek dan semua penilaian adalah fungsi kesatuan dalam representasi. Proposisi matematika sejati, bukanlah kebenaran logika tapi semua kebenaran analitik atau kebenaran berbasis konsep merupakan kebenaran sintetis atau kebenaran berbasis intuisi. Selanjutnya menutut Kant semua penilaian matematis yang tepat bersifat apriori, dan bukan bersifat empiris karena penilaian matematis membawa serta kebutuhan mereka, yang tidak dapat diperoleh dari pengalaman, oleh karena itu, ini menyiratkan bahwa ia mengandung kognisi murni apriori dan tidak empiris. Selain itu semua prinsip geometri lebih ke analitis seperti proposisi "garis lurus adalah jalur terpendek antara dua titik", adalah proposisi sintetis karena konsep lurus tidak mengandung kuantitas apapun, namun hanya kualitasnya.

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  4. Dimas Candra Saputra, S.Pd.
    17709251005
    PPs PM A 2017

    Assalamualaikum Prof,
    Menurut Kant, penilaian menghubungkan antara pemahaman dan alasan tentang suatu objek. Penilaian pertama-tama diperoleh secara a priori dari pengalaman, tetapi konsep yang diperoleh tidaklah bersifat empiris melainkan bersifat murni. Pada setiap penilaian terdapat konsepsi yang diaplikasikan pada hubungan secara cepat dengan objek. Semua penilaian merupakan fungsi dari keseluruhan dari representasi. Representasi yang lebih tinggi digunakan untuk kognisi kita terhadap objek, sehingga banyak kemungkinan kognisi untuk dikumpulkan menjadi satu. Menurutnya, semua penilaian matematika adalah sintesis dan semua kesimpulan matematika diperoleh berdasarkan hukum kontradiksi.

    ReplyDelete
    Replies
    1. Sebagai contoh penjumlahan 7 + 5 = 12, mungkin pertama kali dipikirkan bahwa proposisi tersebut tidak lebih dari penilaian analitis, dengan mengikuti konsep penjumlahan 7 dan 5, berdasarkan hukum kontradiksi. Namun, jika kita mempertimbangkan operasi tersebut secara lebih dekat, tampak bahwa konsep penjumlahan 7 + 5 hanya mengandung kesatuan angka tunggal.

      Delete
  5. Elsa Susanti
    17709251024
    S2 Pendidikan Matematika 2017 Kelas B

    Penilaian matematika menurut Kant adalah fungsi kesatuan dalam representasinya. Representasi yang lebih tinggi digunakan untuk mengkognisikan objek. Penilaian matematika dapat diklasifikasikan dalam berbagai cara. Menutut Kant hal pertama yang harus kita pahami bahwa semua penilaian matematis yang tepat bersifat apriori, dan tidak bersifat empiris. Penilaian matematis tidak dapat diperoleh dari pengalaman. Sebuah penilaian apriori haruslah logis dan dengan universalitas yang ketat, sedangkan penilaian posteriori tidak ditandai oleh keharusan logis atau universalitas mutlak.

    ReplyDelete
  6. Widuri Asmaranti
    17709251035
    S2 Pend Matematika B 2017

    Pada elegi ini Kant menyebutkan bahwa sebuah penilaian adalah kognisi perantara dari suatu objek, akibatnya adalah representasi dari objek itu. Representasi yang lebih tinggi digunakan untuk kognisi objek kita, dan dengan demikian banyak kemungkinan kognisi dikumpulkan menjadi satu. Hanna R. mengetahui bahwa dalam hal jumlah penilaian, Kant menangkap cara dasar di mana pemahaman konsep konstituen tentang proposisi kategoris monadik sederhana digabungkan dan dipisahkan secara logis.

    ReplyDelete