## 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.

### Similar content being viewed by others

## Notes

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.

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

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.

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).

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.

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

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*.

## References

Azzouni, J. (2005). Is there still a sense in which mathematics can have foundations? In G. Sica (Ed.),

*Essays on the foundations of mathematics and logic*. Italy: Polimetrica.Bailly, Francis, & Longo, Giuseppe. (2011).

*Mathematics and the natural sciences: The physical singularity of life*., Advances in computer science and engineering London: Imperial College Press.Bangu, S. (2008). Reifying mathematics? Prediction and symmetry classification.

*Studies in History and Philosophy of Modern Physics*,*39*, 239–258.Bangu, S. (2012). Symmetry. In R. Batterman (Ed.),

*The Oxford handbook of philosophy of physics, chapter 8*(pp. 287–317). New York: Oxford University Press.Benacerraf, P. (1965) What numbers could not be.

*The Philosophical Review, lxxiv*:47–73.**Reprinted in [10]**.Benacerraf, P. (1973). Mathematical truth.

*The Journal of Philosophy, 70*:661–680.**Reprinted in [10]**.Benacerraf, P., & Putnam, H. (Eds.). (1983).

*Philosophy of mathematics: Selected readings*(2nd ed.). Cambridge: Cambridge, MA.Brading, K., & Castellani, E. (Eds.). (2003).

*Symmetries in physics: Philosophical reflections*. New York: Cambridge University Press.Brading, K., & Castellani, E. (2007). Symmetries and invariances in classical physics. In J. Butterfield & J. Earman (Eds.),

*Handbook of the philosophy of science: Philosophy of physics*(pp. 1331–1364). Amsterdam: Elsevier.Brading, K., & Castellani, E. (2013) Symmetry and symmetry breaking. In E. N. Zalta (Ed.),

*The stanford encyclopedia of philosophy*. Spring 2013 edn. http://plato.stanford.edu/archives/spr2013/entries/symmetry-breaking/Earman, John. (2004). Laws, symmetry, and symmetry breaking: Invariance, conservation principles, and objectivity.

*Philosophy of Science*,*71*, 1227–1241.Epp, S. S. (2011). Variables in mathematics education. In P. Blackburn, H. van Ditmarsch, M. Manzano, & F. Soler-Toscano (Eds.),

*Tools for teaching logic*(pp. 54–61). New York: Springer.Feynman, R. (1967).

*The character of physical law*. Cambridge, MA: MIT.Fillion, N. (2012).

*The reasonable effectiveness of mathematics in the natural sciences*. Ph.D. thesis, The University of Western Ontario.Frenkel, E. (2013).

*Love and math: The heart of hidden reality*. New York: Basic Books.Galilei, G. (1953).

*Dialogue concerning the two chief world systems*. Berkeley, Los Angeles: University of California Press.Grosholz, E. R. (2000). The partial unification of domains, hybrids, and the growth of mathematical knowledge. In E. Grosholz & H. Breger (Eds.),

*The growth of mathematical knowledge*(pp. 81–91). Dordrecht: Kluwer Academic Publishers.Heeffer, A., & van Dyck, M. (Eds.). (2010).

*Philosophical aspects of symbolic reasoning in early modern mathematics*. London: College Publications.Kitcher, P. (1976). Explanation, conjunction, and unification.

*Journal of Philosophy*,*73*, 207–212.Krömer, R. (2007).

*Tool and object: A history and philosophy of category theory*. Basel: Birkhäuser.Lakatos, I. (1976).

*Proofs and refutations*. New York: Cambridge University Press.Lawvere, W. (1964). An elementary theory of the category of sets.

*Proceedings of the National Academy of Sciences of the United States of America*,*52*, 1506–1511.Lederman, L., & Hill, C. T. (2004).

*Symmetry and the beautiful universe*. Amherst, NY: Prometheus Books.Longo, G. (2015). Synthetic philosophy of mathematics and natural sciences, conceptual analyses from a grothendieckian perspective: Reflections on synthetic philosophy of contemporary mathematics.

*Speculations: Journal of Speculative Realism*, (VI), 207–267.Marquis, J.-P. (1995). Category theory and the foundations of mathematics: Philosophical excavations.

*Synthese*,*103*(3), 421–447.Marquis, J.-P. (2009).

*From a geometrical point of view: A study of the history and philosophy of category theory. Logic, epistemology and the unity of science*. London: Springer.Mazur, J. (2014).

*Enlightening symbols: A short history of mathematical notation and its hidden powers*. Princeton, NJ: Princeton University Press.Serfati, Michel. (2005).

*La Révolution symbolique. La constitution de l’écriture symbolique mathématique*. Paris: Éditions Pétra.Steiner, Mark. (1995). The applicabilities of mathematics.

*Philosophia Mathematica*,*3*(3), 129–156.Steiner, Mark. (1998).

*The applicability of mathematics as a philosophical problem*. Cambridge, MA: Harvard.Steiner, M. (2005). Mathematics—Application and applicability. In S. Shapiro (Ed.),

*The Oxford handbook of philosophy of mathematics and logic, chapter 20*(pp. 625–650). New York: Oxford Univ.Stenger, Victor. (2006).

*The comprehensible cosmos: Where do the laws of physics come from?*. Amherst, NY: Prometheus Books.Van Fraassen, Bas C. (1989).

*Laws and symmetry*. New York: Oxford University Press.Weinberg, S. (1992).

*Dreams of a final theory*. New York: Vintage.Wigner, E. P. (1960). The unreasonable effectiveness of mathematics in the natural sciences.

*Communications in Pure and Applied Mathematics*, 13(1), February 1960. Reprinted in*Symmetries and Reflections*. Indiana University Press, Bloomington, 1967.Yanofsky, N. S. (2010). Galois theory of algorithms. arXiv:hep-lat/1011.0014.

Yanofsky, N. S., & Zelcer, M. (2013). Mathematics via symmetry. arXiv:hep-lat/1306.4235v1.

Zee, A. (1990). The effectiveness of mathematics in fundamental physics. In R. E. Mickens (Ed.),

*Mathematics and Science*(pp. 307–323). Singapore: World Scientific.

## 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).

## Author information

### Authors and Affiliations

### Corresponding author

## Rights and permissions

## About this article

### Cite this article

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

Published:

Issue Date:

DOI: https://doi.org/10.1007/s10699-016-9486-7