Creating new concepts in mathematics: freedom and limitations. The case of Category Theory

Main Article Content

Zbigniew Semadeni
https://orcid.org/0000-0002-8655-3364

Abstract

In the paper we discuss the problem of limitations of freedom in mathematics and search for criteria which would differentiate the new concepts stemming from the historical ones from the new concepts that have opened unexpected ways of thinking and reasoning.


We also investigate the emergence of category theory (CT) and its origins. In particular we explore the origins of the term functor and present the strong evidence that Eilenberg and Carnap could have learned the term from Kotarbiński and Tarski.

Article Details

How to Cite
Semadeni, Z. (2020). Creating new concepts in mathematics: freedom and limitations. The case of Category Theory. Philosophical Problems in Science (Zagadnienia Filozoficzne W Nauce), (69), 33–65. Retrieved from https://www.zfn.edu.pl/index.php/zfn/article/view/512
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References

Atiyah, M., 2002. Mathematics in the 20th century. Bulletin of the London Mathematical Society [Online], 34(1), pp.1–15. Available at: https://doi.org/10.1112/S0024609301008566 [visited on 29 October 2020].

Baszmakowa, I.G., 1975. Grecja starożytna. Kraje hellenistyczne i imperium rzymskie. In: A.P. Juszkiewicz, ed. Historia matematyki od czasów najdawniejszych do początku XIX stulecia. Vol. 1: Od czasów najdawniejszych do początku czasów nowożytnych (S. Dobrzycki, Trans.). Warszawa: Państwowe Wydawnictwo Naukowe, pp.64–168.

Beth, E.W. and Piaget, J., 1966. Mathematical Epistemology and Psychology. Dordrecht: D. Reidel Publishing Company.

Cantor, G., 1883. Uber unendliche, lineare Punktmannigfaltigkeiten. Mathematische Annalen [Online], 21(4), pp.545–591. Available at: https://doi.org/10.1007/BF01446819 [visited on 29 October 2020].

Carnap, R., 1929. Abriss der logistik: mit besonderer Berucksichtigung der relationstheorie und ihrer Anwendungen, Schriften zur Wirtschaftswissenschaftlichen Forschung Bd. 2. Wien: Verlag von Julius Springer.

Carnap, R., 1934. Logische Syntax der Sprache, Schriften zur Wissenschaftlichen Weltauffassung Bd. 8. Wien: Verlag von Julius Springer.

Dedekind, R., 1872. Stetigkeit und irrationale Zahlen [Online]. Braunschweig: Friedrich Vieweg & Sohn. Available at: <http://www.rcin.org.pl/publication/13029>.

Dedekind, R., 1888. Was sind und was sollen die Zahlen? [Online]. Braunschweig: Friedrich Vieweg & Sohn. Available at: <http://www.digibib.tu-bs.de/?docid=00024927> [visited on 30 October 2020].

Dydak, J., 2012. Ideas and Influence of Karol Borsuk. Wiadomości Matematyczne [Online], 48(2), p.81. Available at: https://doi.org/10.14708/wm.v48i2.305 [visited on 30 October 2020].

Eilenberg, S. and Mac Lane, S., 1945. General theory of natural equivalences. Transactions of the American Mathematical Society [Online], 58, pp.231–294. Available at: https://doi.org/10.1090/S0002-9947-1945-0013131-6 [visited on 30 October 2020].

Eilenberg, S. and Steenrod, N.E., 1952. Foundations of Algebraic Topology, Princeton Mathematical Series 15. Princeton: Princeton University Press.

Feferman, A.B. and Feferman, S., 2004. Alfred Tarski: Life and Logic. Cambridge [etc.]: Cambridge University Press.

Ferreiros, J., 1999. Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics, Science Networks. Historical Studies vol. 23. Basel: Birkhauser.

Freudenthal, H., 1984. The Implicit Philosophy of Mathematics: History and Education. In: Z. Ciesielski and C. Olech, eds. Proceedings of the International Congress of Mathematicians, August 16-24, 1983, Warszawa. Warszawa; Amsterdam [etc.]: PWN-Polish Scientific Publishers; North-Holland, pp.1695–1709.

Freyd, P., 1964. Abelian Categories: An Introduction to the Theory of Functors. New York: Harper & Row.

Gelman, R. and Gallistel, C.R., 1978. The Child’s Understanding of Number. Cambridge Mass; London: Harvard University Press.

Goldblatt, R., 1984. Topoi – The Categorial Analysis of Logic. North-Holland.

Hausdorff, F., 1914. Grundzuge der Mengenlehre [Online]. Leipzig: Von Veit & Comp. Available at: <http://gallica.bnf.fr/ark:/12148/bpt6k9736980b> [visited on 30 October 2020].

Heller, M., 2015. Bóg i geometria: gdy przestrzeń była Bogiem. Krakow: Copernicus Center Press.

Hewitt, E. and Ross, K.A., 1963. Abstract Harmonic Analysis. Vol. 1: Structure of Topological Groups, Integration Theory, Group Representations, Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit Besonderer Berucksichtigung der Anwendungsgebiete Bd. 115. Berlin: Springer.

Jackowski, S., 2015. Samuel Eilenberg – wielki matematyk z Warszawy. Wiadomości Matematyczne [Online], 50(1), pp.21–43. Available at: https://doi.org/10.14708/wm.v50i1.651 [visited on 1 November 2020].

Kan, D.M., 1958. Adjoint functors. Transactions of the American Mathematical Society [Online], 87(2), pp.294–294. Available at: https://doi.org/10.1090/S0002-9947-1958-0131451-0 [visited on 1 November 2020].

Kelley, J.L., 1955. General Topology, University Series in Higher Mathematics. Princeton, N.J.: D. Van Nostrand Co.

Knorr, W.R., 1975. The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry, Synthese Historical Library vol. 15. Dordrecht: D. Reidel.

Kotarbiński, T., 1929. Elementy teorji poznania, logiki formalnej i metodologji nauk. Lwow: Wydawn. Zakładu Narodowego im. Ossolińskich.

Kromer, R., 2007. Tool and Object: A History and Philosophy of Category Theory, Science Networks. Historical Studies vol. 32. Basel [etc.]: Birkhauser.

Król, Z., 2019. Category theory and philosophy. In: M. Kuś and B. Skowron, eds. Category Theory in Physics, Mathematics, and Philosophy [Online], Springer Proceedings in Physics 235. Springer International Publishing, pp.21–32. Available at: https://doi.org/10.1007/978-3-030-30896-4 [visited on 1 November 2020].

Kuratowski, K., 1933. Topologie. 1, Espaces Metrisables, Espaces Complets, Monografje Matematyczne t. 3. Warszawa; Lwow: s.n.

Lakatos, I., 1976. Proofs and Refutations: The Logic of Mathematical Discovery. Ed. by J. Worrall and E. Zahar. Cambridge [etc.]: Cambridge University Press.

Lawvere, F.W., 1963. Functorial semantics of algebraic theories. Proceedings of the National Academy of Sciences of U.S.A. [Online], 50, pp.869–872. Available at: <https://www.pnas.org/content/pnas/50/5/869.full.pdf> [visited on 29 October 2020].

Lawvere, F.W., 1964. An elementary theory of the category of sets. Proceedings of the National Academy of Sciences of the U.S.A. [Online], 52(6), pp.1506–1511. Available at: https://doi.org/10.1073/pnas.52.6.1506 [visited on 1 November 2020].

Lawvere, F.W., ed., 1972. Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics 274. Berlin; Heidelberg; New York: Springer.

Łukasiewicz, J., 1910. Uber den Satz des Widerspruchs bei Aristoteles. Bulletin International de l’Academie des Sciences de Cracovie, 1-2, pp.15–38.

Mac Lane, S., 1950. Duality for groups. Bulletin of the American Mathematical Society [Online], 56(6), pp.485–516. Available at: <https://projecteuclid.org/euclid.bams/1183515045> [visited on 29 October 2020].

Mac Lane, S., 1971. Categories for the Working Mathematician, Graduate Texts in Mathematics 5. New York: Springer-Verlag.

Mac Lane, S., 1986. Mathematics: Form and Function. New York [etc.]: Springer-Verlag.

Mac Lane, S., 1988. Concepts and Categories in Perspective. In: P. Duren, R. Askey and U. Merzbach, eds. A Century of Mathematics in America, Part I [Online], History of Mathematics 1. Providence, R.I.: American Mathematical Society, pp.323–365. Available at: <https://www.ams.org/publicoutreach/math-history/hmath1-maclane25.pdf> [visited on 29 October 2020].

Mac Lane, S., 2002. Samuel Eilenberg and categories. Journal of Pure and Applied Algebra [Online], 168(2-3), pp.127–131. Available at: https://doi.org/10.1016/S0022-4049(01)00092-5 [visited on 29 October 2020].

Mitchell, B., 1965. Theory of Categories, Pure and Applied Mathematics. A Series of Monographs and Textbooks 17. New York; London: Academic Press.

Piaget, J. and Garcia, R.V., 1989. Psychogenesis and the history of science (H. Feider, Trans.). New York: Columbia University Press.

Samuel, P., 1948. On universal mappings and free topological groups. Bulletin of the American Mathematical Society [Online], 54(6), pp.591–598. Available at: <https://projecteuclid.org/euclid.bams/1183512049> [visited on 29 October 2020].

Semadeni, Z., 2015. Transgresje poznawcze jako istotna cecha rozwoju matematyki. In: R. Murawski, ed. Filozofia matematyki i informatyki. Krakow: Copernicus Center Press, pp.65–90.

Semadeni, Z., 2018. Platonizujący konceptualizm w matematyce. In: R. Murawski and J. Woleński, eds. Problemy filozofii matematyki i informatyki. Poznań: Wydawnictwo Naukowe UAM, pp.77–95.

Shannon, C.E., 1936. A Symbolic Analysis of Relay and Switching Circuits [Online]. Thesis. Massachusetts Institute of Technology. Available at: <https://dspace.mit.edu/handle/1721.1/11173> [visited on 2 November 2020].

Skowron, B., manuscript. Was Saunders Mac Lane a platonist? [unpublished].

Tall, D., 2013. How Humans Learn to Think Mathematically: Exploring the Three Worlds of Mathematics, Learning in Doing: social, cognitive, and computational perspectives. Cambridge: Cambridge University Press.

Tarski, A., 1933. Pojęcie prawdy w językach nauk dedukcyjnych, Prace Towarzystwa Naukowego Warszawskiego. Wydział III: Nauk Matematyczno-Fizycznych 34. Warszawa: nakładem Towarzystwa Naukowego Warszawskiego, z zasiłku Ministerstwa Wyznań Religijnych i Oświecenia Publicznego.

Thurston, W.P., 1990. Mathematical Education. Notices of the Amer- ican Mathematical Society, 37(7), pp.844–850.

Waerden, B.L.v.d., 1930. Moderne Algebra. T. 1, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit Besonderer Berucksichtigung der Anwendungsgebiete Bd. 33. Berlin: Verlag von Julius Springer.

Youschkevitch, A.P., 1976. The concept of function up to the middle of the 19th century. Archive for History of Exact Sciences [Online], 16(1), pp.37–85. Available at: https://doi.org/10.1007/BF00348305 [visited on 2 November 2020].