## 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

1. ULFA LU'LUILMAKNUN
16709251022
S2 Pendidikan Matematika 2016 Kelas B

Assalamualaikum Wr.Wb.

Kant menyarankan kita harus mengamati bahwa semua keputusan matematika yang tepat adalah bersifat a priori dan tidak empiris. Penilaian matematika membawa mereka pada kebutuhan yang tidak dapat diperoleh dari pengalaman. Kant mengatakan bahwa kita harus memahami konsep-konsep sebelum melakukan penilaian. Dengan bantuan gambar atau visual akan memudahkan untuk memahami konsep.

Wassalamualaikum Wr.Wb.

2. ROFI AMIYANI
S2 P.MAT A 2016
16709251004

Menurut Kant, semua penilaian analitik adalah apriori. Penilaian sintetis mungkin posteriori atau apriori. Sebuah penilaian apriori mungkin analitik atau sintetis. Sebuah penilaian posteriori selalu sintetis. Menurut Kant, kebenaran sintetik apriori termasuk kebenaran matematika dan kebenaran ilmu alam. Semua penilaian matematika adalah sintetis, dan semua keputusan matematis yang tepat adalah a priori.

3. Nilza Humaira Salsabila
16709251026
Pendidikan Matematika kelas B PPs 2016

Assalamu’alaikum Wr. Wb.
Berdasarkan elegi di atas, logika umum Kant yang murni mencakup bukan logika dari relasi dan beberapa kuantifikasi, karena relasi matematika umumnya direpresentasikan spatiotemporally dalam intuisi formal maupun intuisi murni, dan tidak merepresentasikan secara logis dalam pemahaman. Proposisi matematika yang benar menurut Kant adalah bukan berdasarkan kebenaran dari logika, kebenaran analitis atau kebenaran konsep, tetpi kebenaran sintetik atau kebenaran intuisi.
Wassalamu’alaikum Wr. Wb.

16701251001
PPs PEP B 2016

Kant memberi kontribusi karena memberi jalan tengah bahwa putusan matematika bersifat sintetik apriori, yaitu putusan yang pertama-tama diperoleh secara a priori dari pengalaman, tetapi konsep yang diperoleh tidaklah bersifat empiris melainkan bersifat murni. Pengetahuan geometri yang bersifat sintetik a priori menjadi mungkin jika dan hanya jika konsep keruangan dipahami secara transcendental dan menghasilkan intuisi a priori.

5. Wan Denny Pramana Putra
16709251010
PPs Pendidikan Matematika A

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.

6. Budi Yanto
16709251024
P. Mat S2 Kelas B 2016
Kant membedakan aspek judgments menjadi tiga yaitu pertama putusan analitis a priori, dimana predikat tidak menambah sesuatu yang baru pada subyek, karena termasuk di dalamnya (misalnya, setiap benda menempati ruang). Kedua putusan sintesis aposteriori yaitu berdasarkan pengalaman indrawi manusia misalnya pernyataan kursi itu bagus. Ketiga putusan sintesis a priori juga, misalnya putusan yang berbunyi segala kejadian mempunyai sebab.

7. Fitri Ayu Ningtiyas
16709251037
S2 P.Mat B UNY 2016

Menurut Kant, konsep matematika merupakan ilmu yang bersifat sintetik a priori. Metode sintetik dengan metode analitik dan konsep a priori dengan a posteriori. Selanjutnya, Kant mengatakan bahwa matematika tidak dikembangkan hanya dengan konsep a posteriori sebab jika demikian matematika akan bersifat empiris. Pada akhirnya Kant mengatakan bahwa intuisi menjadi inti dan kunci bagi pemahaman dan konstruksi matematika.

8. Rhomiy Handican
16709251031
PPs Pendidikan Matematika B 2016

Menurut Kant, penilaian adalah sesuatu hal yang kompleks, meliputi a) penilaian terhadap objek baik secara langsung (intuisi) atau secara tidak langsung (konsep), b) konsep yang didasarkan salah satu dari mereka objek atau konsep konstituen lainnya, c) konsep logis murni dan masuk ke kesimpulan logis menurut hukum murni, d) melibatkan suatu aturan dan penerapan aturan untuk objek oleh intuisi, e) mengungkapkan proposisi benar atau salah, f) memediasi pembentukan keyakinan, dan g) bersatu dan sadar diri.

9. Erlinda Rahma Dewi
16709251006
S2 PPs Pendidikan Matematika A 2016

Menurut Kant, judgement (pernyataan keputusan) adalah kognisi perantara suatu objek. Dalam judgement ada konsep yang memegang banyak (representasi), dan bahwa di antara banyak ini juga memahami representasi yang diberikan, yang kemudian segera dirujuk ke objek . Semua judgement adalah fungsi dari persatuan di antara representasi kita, karena bukan sebuah representasi langsung yang lebih tinggi, yang memahami dirinya dan representasi lain di bawahnya, digunakan untuk kognisi objek, dan banyak kemungkinan kognisi dipadukan.

10. Aprisal
16709251019
PPs S2 Pendidikan Matematika Kelas A 2016

Assalamu alaikum

Kant berpendapat bahwa analitik sekaligus dapat menyelesaikan permasalahan dan menyatukan karakteristik-karakteristik yang berbeda, di mana kebenaran adalah analitik hanya dalam kasus yang dapat diubah menjadi suatu kebenaran logis oleh adanya subtitusi atau pergantian. Sementara kebenaran logis adalah kebenaran yang dibuktikan dari logika saja. Di sisi lain Kalderon mengklaim bahwa sebuah kebenaran logis adalah kontradiksi diri.

Waalaikum salam wr.wb

marsigitina@yahoo.com, marsigitina@gmail.com, marsigit@uny.ac.id