DOI: https://doi.org/10.17072/2078-7898/2021-4-528-540

Pythagoreanism in modern philosophy of mathematics

Elena V. Kosilova
Candidate of Philosophy, Associate Professor
of the Department of Ontology and GnoseologyLomonosov Moscow State University,
27/4, Lomonosovsky av., Moscow, 119991, Russia;
e-mail: implicatio@yandex.ru
ORCID: https://orcid.org/0000-0002-2261-7680

There are two opposing approaches in the ontology of mathematics — realistic (mathematics does not depend on the human being) and constructivist (mathematical objects are created by mathematicians) ones. P. Benacerraf objected to the realistic approach by stating that mathematics is non-causal in nature and, accordingly, it is impossible to reach it from the physical world. The objection to constructivism is based solely on the idea of the effectiveness of mathematics in applied sciences. In the 20th century, the constructivist approach was prevailing. By the end of the century, this prevalence coincided with the general tendency: a skeptical attitude towards scientific knowledge in the philosophy of science. However, at the same time, science began to play a huge role in life. This inevitably led to the emergence of realistic ontologies. The philosophy of mathematics by Q. Meillassoux, M. Tegmark, R. Penrose represents a modern kind of realism. The paper deals with Meillassoux’s theory and arrives at a conclusion about its closeness to Pythagoreanism. Mathematical knowledge gives an opportunity to get out of the circle of correlationism, provided that mathematics is understood purely formally, not intuitively. The paper shows that logic and physics cannot be contingent if mathematics is reliable. The coincidence of mathematical structures with physical ones was previously called the «pre-established harmony between mathematics and physics». Now this is explained by the fact that the universe is organized according to mathematical laws. Thus, a new Pythagoreanism appears. It differs from Platonism in that in Platonism, mathematics is an autonomous world, while in Pythagoreanism it is built into the physical world and determines its laws. The article shows the need to apply the Aristotelian ontology of matter and form. Mathematics is a form, while physical embodiment requires matter. The flow of time in the physical world is associated with matter, as well as the presence of causality in it. The author comes to a conclusion that acceptance of the idea that the universe is arranged according to mathematical laws leads to the idea of the identity of being and thought.

Keywords: realism, Pythagoreanism, Aristotelianism, identity of being and thought, Q. Meillassoux, M. Tegmark, R. Penrose.

Acknowledgements

The research was supported by the Interdisciplinary Scientific and Educational School of Moscow University «Brain, Cognitive Systems, Artificial Intelligence».

References

Aristotle (1934). Metafizika [Metaphysics] Moscow, Leningrad: SOTSEKGIZ Publ., 348 p.

Benacerraf, P. (1983). Mathematical truth. Philosophy of Mathematics: Selected readings. Cambridge: Cambridge University Press, pp. 403–420. DOI: https://doi.org/10.1017/cbo9781139171519.022

Bloor, D. (2012). [Can there be an alternative mathematics?]. Sociologiya vlasti [Sociology of Power]. No. 6–7, pp. 150–177.

Bueno, O. Nominalism in the philosophy of mathematics. The Stanford Encyclopedia of Philosophy, ed. by E.N. Zalta. (Fall 2020 Edition). Available at: https://plato.stanford.edu/archives/fall2020/entries/nominalism-mathematics/ (accessed: 15.02.2021).

Burov, A. and Burov, L. (2016). Genesis of a Pythagorean universe. Trick or truth? The mysterious connection between physics and mathematics. Cham: Springer Publ., pp. 157–170. DOI: https://doi.org/10.1007/978-3-319-27495-9_14

Dirac, P.A.M. (1940). The relation between mathematics and physics. Proceedings of the Royal Society of Edinburgh. Vol. 59, pt. 2, pp. 122–129. DOI: https://doi.org/10.1017/s0370164600012207

Flores Peña, G.R. (2018). [The sign without meaning]. Revista Espiral. Available at: http://revistaespiraltijuana.org/2018/06/09/el-signo-sin-significado-gerardo-r-flores/ (accessed 12.08.2019).

Gironi, F. (2011). Meillassoux’s speculative philosophy of science: Contingency and mathematics. Pli: The Warwick Journal of Philosophy. Vol. 22, pp. 25–60.

Grygiel, W. (2018). On the adequacy of qualifying Roger Penrose as a complex Pythagorean. Philosophical Problems in Science. No. 65, pp. 61–84.

Hahn, H. (1972). [Crisis of intuition]. Matematiki o matematike [Mathematicians on mathematics]. Moscow: Znanie Publ., pp. 25–42.

Hilbert, D. (1998). [Cognition of nature and logic]. Znanie-sila [Knowledge is power]. No. 1, pp. 55–62.

Hohol, M. (2009). Roger Penrose — pythagorean of complex numbers? Semina Scientiarum. Vol. 8, pp. 79–90.

Hohol, M. (2019). Foundations of geometric cognition. London, New York: Routledge Publ., 204 p. DOI: https://doi.org/10.4324/9780429056291

Horsten, L. Philosophy of mathematics. The Stanford Encyclopedia of Philosophy, ed. by E.N. Zalta. (Spring 2019 Edition). Available at: https://plato.stanford.edu/archives/spr2019/entries/philosophy-mathematics/ (accessed: 15.02.2021).

Husserl, E. (1996). Nachalo geometrii. Vvedenie Zhaka Derrida [Origin of geometry. Introduction by J. Derrida]. Moscow: AdMarginem Publ., 269 p.

Husserl, E. (2001). Sobraniye sochineniy. T. 3(1): Logicheskie issledovaniya (T. 2, ch. 1) [Collection of works. Vol. 3(1): Logical research (Vol. 2, pt. 1)]. Moscow: Gnozis Publ., 576 p. 

Król, Z. (2006). Matematyczny i hermeneutyka [Mathematical and hermeneutics]. Warsaw: IFIs PAN Publ., 245 p.

Lektorskiy, V.A. (2005). [Kant, radical constructivism and constructive realism in epistemology]. Voprosy filosofii. No. 8, pp. 11–21.

Meillassoux, Q. (2012). Iteration, reiteration, repetition: a speculative analysis of the meaningless sign. Available at: https://cdn.shopify.com/s/files/1/0069/6232/files/Meillassoux_Workshop_Berlin.pdf (accessed 17.07.2021).

Meillassoux, Q. (2015). Posle konechnosti: esse o neobkhodimosti kontingentnosti [After finitude: An essay on the necessity of contingency]. Moscow: Kabinetnyy Uchenyy Publ., 196 p.

Penrose, R. (2007). Put’ k real’nosti ili Zakony, upravlyayushchiye Vselennoy. Polnyy putevoditel’ [The Road to Reality: A Complete Guide to the Laws of the Univers]. Moscow: ICR Publ., 912 p. 

Tall, D. (2002). Advanced mathematical thinking. New York: Kluwer Academic Publ., 310 p.

Tegmark, M. (2014). Nasha matematicheskaia vselennaia [Our mathematical universe]. Moscow: Korpus, AST Publ., 592 p.

Tegmark, M. (2008). The mathematical universe. Foundations of Physics. Vol. 38, iss. 2, pp. 101–150. DOI: https://doi.org/10.1007/s10701-007-9186-9

Vizgin, V.P. (2004). Dogmat very fizika-teoretika [Article of faith of the theoretical physicist]. Available at: http://realigion.me/article/23643.html (accessed 17.07.2021).

Wigner, E. (1968). [The unreasonable effectiveness of mathematics in the natural sciences]. Uspekhi fizicheskikh nauk[Advances in Physical Sciences]. Vol. 94, iss. 3, pp. 535–546.

Received: 05.03.2021. Revised: 28.08.2021. Accepted: 24.08.2021

For citation:

Kosilova E.V. [Pythagoreanism in modern philosophy of mathematics]. Vestnik Permskogo universiteta. Filosofia. Psihologia. Sociologia [Perm University Herald. Philosophy. Psychology. Sociology], 2021, issue 4, pp. 528–540 (in Russian). DOI: https://doi.org/10.17072/2078-7898/2021-4-528-540