A transcendental analysis of mathematics: The constructive nature of mathematics
- DOI
- 10.5922/0207-6918-2016-1-2
- Pages
- 16-33
Abstract
Kant’s transcendental philosophy (transcendentalism) focuses on both the human method of cognition in general [CPR, B 25] and certain types of cognition aimed at justifying their objective significance. This article aims to explicate Kant’s understanding (resp. justification) of the abstract nature of mathematical knowledge (cognition) as the “construction of concepts in intuition” (see: “to construct a concept means to exhibit a priori the intuition corresponding to it”; [CPR, A 713/В 741], which is “thoroughly grounded on definitions, axioms, and demonstrations” [CPR, A 726/В 754]. Unlike specific ‘physical’ objects, mathematical objects are of abstract nature and they are introduced using Hume’s principle of abstraction. Based on the doctrine of schematism, Kant develops an original theory of abstraction: Kant’s schemes serve as a means to construct mathematical objects, as an “action of pure thought" [CPR, B 81]. A ‘constructive’ understanding of mathematical acts going back to Euclid’s genetic method is an important innovation introduced by Kant. This understanding is at the heart of modern mathematical formalism, intuitionism, and constructivism. Within Kant’s constructivism, mathematics can be described as a two-tier system, which suggests a “shift” from the level of concepts of the understanding to the level of sensual intuition, where mathematical acts are performed, followed by a subsequent return to the initial level. On this basis, the author develops a theory of transcendental constructivism (pragmatism). In particular, Kant's ‘intuitionism’ of mathematics can be understood as structural properties of mathematical language or its ‘logical space’ (Wittgenstein; cf. mathematical structuralism). In his theory, Kant distinguishes between two types of constructing — ostensive (geometric) and symbolic (algebraic). The paper analyses these types and shows that modern mathematical structures are a combination and intertwining of both. The author also identifies a third type — logical constructing [in proving theorems], which inherits the features of both Kant's types.
Reference
1. Aristotle, 2011, in: Shiffman, M. De Anima: On the Soul, (Newburyport, MA: Focus Publishing.
2. Frege, G. 1884, Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl. Breslau. (English: The Foundations of Arithmetic: the logical-mathematical Investigation of the Concept of Number).
3. Galileiy, G, 1964,Izbrannye trudy v 2 t. M.: Nauka, 1964. v.1.
4. Galileiy, G., 1987, Probirnyh del master. M. Nauka.
5. Gil'bert. D., 1998, Aksiomaticheskoe myshlenie //Ego j`e. Izbrannye trudy, T. 1, M. : Faktorial.
6. Gil'bert, D., 1948 O ponjatii chisla // Ego j`e. Osnovanija geometrii M.—L., OGIZ.
7. Gudsteiyn,R. L., 1961, Matematicheskaja logika. M. : Izd-vo IL.
8. Hintikka, J. 1978, Surface Information and Depth Information. In: Logico-Epistemologic Research.
9. Kant I., 1994, Kritika chistogo razuma. In: Kant I. Sob. soch. v 8-mi tt. V. 3. M.: CHoro.
10. Katrechko, S. L. 2003, K voprosu ob apriornosti matematicheskogo znanija [On the question of a priori mathematical knowledge]. In: Matematika i opyt [Mathematics and Experience]. M. : MGU, s. 545—574.
11. Katrechko, S. L. 2007, Modelirovanie rassuj`deniiy v matematike: transcendental'nyiy podhod [Modeling reasoning in mathematics: transcendental approach]. In: Modeli rassuj` deniiy — 1 : Logika i argumentacija. Kaliningrad : Izd. RGU im. I. Kanta, 2007. s. 63—90.
12. Katrechko, S. L. 2008, Transcendental'naja filosofija matematiki [Transcendental philosophy of mathematics]. In: Vestnik Moskovskogo universiteta. Serija 7 «Filosofija [Philosophy] », № 2, 2008. M. : Izd—vo MGU, s. 88—106.
13. Katrechko S. L., 2011, Abstraktnaja priroda logiko-matematicheskogo znanija i prirashenie informacii. In: Sed'mye Smirnovskie chtenija. M. : Sovremennye tetradi, s. 176—178.
14. Katrechko, S. L. 2013, Platonovskiiy chetyrehchastnyiy otrezok (Linija): Platon i Kant o prirode (specifike) matematicheskogo znanija [Plato’s Divided Line: Plato and Kant about the nature (specific) of the mathematics]. In: Vestnik RHGA, T. 14, vyp. 3, 2013. s. 172—177.
15. Katrechko, S. L. 2014а, Transcendental'nyy analiz matematicheskoy deiatel'nosti: abstraktnye (matematicheskie) ob'ekty, konstrukcii i dokazatel'stva [Transcendental analysis of mathematics: abstract (mathematical) objects, constructions and proofs]. In: Dokazatel'stvo: ochevidnost', dostovernost' i ubeditel'nost' v matematike [Proof: evidence, credibility and convincing sequences in mathematics. Moscow Study in the Philosophy of Mathematics], Moscow, s. 86—120.
16. Katrechko, S. L. 2014в, Matematika kak «rabota» s abstraktnymi ob"ektami: ontologo— transcendental'nyiy status matematicheskih abstrakciiy [Mathematics as a "job" with abstract objects: ontological-transcendental status of mathematical abstractions]. In: Matematika i real'nost' [Mathematics and reality]. Trudy Moskovskogo seminara po filosofii matematiki. M., Izd-vo MGU, s. 421—452.
17. Katrechko S. L., 2015, Transcendental'nyiy analiz matematiki: abstraktnaja priroda matematicheskogo znanija. In: Kantovskiiy sbornik [The Kantovsky sbornik], 2015, № 2 (52). s. 16—31 (http://journals. kantiana. ru/kant_collection/2017/5885/).
18. Klini S., 1957, Vvedenie v metamatematiku, M. : IL.
19. Koiyre A., 1985, Ocherki po istorii filosofskoiy mysli, M.
20. Maslov S. JU., 1986, Teorija deduktivnyh sistem i ee primenenija. M. : Radio i svjaz'.
21. Lakatos, I. 1976, Proofs and Refutations. Cambridge: Cambridge University Press.
22. Novoselov M. M., 2000, «Abstrakcija», «abstraktnyiy ob"ekt». In: Novoiy filosofskoiy enciklopedii [The new philosophical encyclopedia]: http://iph. ras. ru/elib/0019.html
23. Novoselov M. M. Logika abstrakciiy (metodol. analiz). M. : IFRAN.
24. Parsons, C. 2008, Mathematical Thought and Its Objects, Cambridge Univ. Press.
25. Rosen, G. 2001, Аbstract Objects; URL: http://plato. stanford. edu/entries/abstractobjects/
26. SHul'pekov V. A., 2014, Instrumental'naja struktura matematicheskih postroeniiy. In: Dokazatel'stvo: ochevidnost', dostovernost' i ubeditel'nost' v matematike [Proof: evidence, credibility and convincing sequences in mathematics. Moscow Study in the Philosophy of Mathematics], Moscow, s. 331—335.
27. Smirnov, V. A. 2001, Geneticheskiiy metod postroenija nauchnoiy teorii //Logikofilosofskie trudy V. A. Smirnova. M. Editoral URSS, s. 417—438. \
28. Vinberg E., 2002, Kurs algebry, M., Faktorial Press.