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The Role of Symmetry in Mathematics

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Abstract

Over the past few decades the notion of symmetry has played a major role in physics and in the philosophy of physics. Philosophers have used symmetry to discuss the ontology and seeming objectivity of the laws of physics. We introduce several notions of symmetry in mathematics and explain how they can also be used in resolving different problems in the philosophy of mathematics. We use symmetry to discuss the objectivity of mathematics, the role of mathematical objects, the unreasonable effectiveness of mathematics and the relationship of mathematics to physics.

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Notes

  1. For philosophical introductions to symmetry see the Introduction in Brading and Castellani (2003), Brading and Castellani (2007), Brading and Castellani (2013) and Bangu (2012). For a popular introduction to the physical issues see Lederman and Hill (2004). For an interesting philosophical view of symmetry in biology see Bailly and Longo (2011). One should also read Longo (2015) for a fascinating perspective about symmetry in modern mathematics.

  2. See Bangu (2008) for a related discussion of the discovery of the \(\Omega ^{-}\) particle.

  3. Whether or not there are laws of nature at all or whether they should be eliminated in favor of symmetries is a matter of considerable controversy among philosophers of science. See van Fraassen (Van Fraassen 1989; Earman 2004) for stronger and weaker versions of eliminationist views on this issue. Our account is agnostic about this.

  4. The function of variables (and types) have a long and interesting history. We are not saying that the reason for their creation was to foster the notion of symmetry of semantics. However, variables as they are now, are helpful for dealing with symmetry of semantics. For more on the history of variables and symbols in mathematics see Mazur (2014), Heeffer and van Dyck (2010) and Serfati (2005).

  5. Frege’s influence on this definition should be evident. A finite number for Frege consists of the equivalence class of the finite sets where two sets are equivalent if there is an isomorphism from one set to another. When we talk of the equivalence class 5 we are ignorant of the set of the equivalence class under discussion; we may be talking about 5 apples or 5 cars.

  6. Steiner (Steiner 2005) treats those fields as applications of mathematics.

  7. Exactness comes from the fact that there are no counterexamples. This, in turn, leads to symbolization in mathematics. If we can replace one entity by another, we might as well call the entity x.

  8. Various authors (e.g. Steiner 1998; Fillion 2012) now distinguish between various problems of the applicability of mathematics. We confine our remarks to what we take to be Wigner’s (1967) original question of why mathematics can be used at all with respect to the physical world.

  9. Philip Kitcher (e.g. Kitcher 1976) touts the importance of these types of cases for mathematics and uses them in the service of demonstrating the existence of mathematical explanation. Emily Grosholz has studied domain unifications in mathematics extensively. See e.g. Grosholz (2000).

  10. For more about the relationship of category theory and invariance of syntax of semantics see the appendix of Yanofsky and Zelcer (2013). See also Krömer (2007) and Marquis (2009) for more on the history and philosophy of category theory and it relationship to symmetry.

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Acknowledgments

We wish to thank our friend and mentor Distinguished Professor Rohit Parikh for helpful conversations and for much warm encouragement. Thanks also to Jody Azzouni, Sorin Bangu, Nicolas Fillion, André Lebel, Jim Lambek, Guisseppe Longo, Jean-Pierre Marquis, Jolly Mathen, Alan Stearns, Andrei Rodin, Mark Steiner, Robert Seely, K. Brad Wray, Gavriel Yarmish, and four anonymous referees, who were all extremely helpful commenting on earlier drafts. N. Y. would also like to thank Jim Cox and Dayton Clark for many stimulating conversations on these topics. He acknowledges support for this project from a PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York. This work was also supported by a generous “Physics of Information” grant from The Foundational Questions Institute (FQXi).

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Authors and Affiliations

  1. Department of Computer and Information Science, Brooklyn College, CUNY, Brooklyn, NY, USA

    Noson S. Yanofsky

  2. Department of Philosophy, SUNY Oswego, Oswego, NY, USA

    Mark Zelcer

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Correspondence to Mark Zelcer.

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Yanofsky, N.S., Zelcer, M. The Role of Symmetry in Mathematics. Found Sci 22, 495–515 (2017). https://doi.org/10.1007/s10699-016-9486-7

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