---
title: "Relationship between mathematics and physics"
chunk: 2/2
source: "https://en.wikipedia.org/wiki/Relationship_between_mathematics_and_physics"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T03:36:15.509993+00:00"
instance: "kb-cron"
---

Rigor is indispensable in pure mathematics. But many definitions and arguments found in the physics literature involve concepts and ideas that are not up to the standards of rigor in mathematics.
For example,
Freeman Dyson characterized quantum field theory as having two "faces". The outward face looked at nature and there the predictions of quantum field theory are exceptionally successful. The inward face looked at mathematical foundations and found inconsistency and mystery. The success of the physical theory comes despite its lack of rigorous mathematical backing.
Some mathematicians, such as Arthur Jaffe and Frank Quinn, argue that non-rigorous mathematical work can sometimes bring benefits too.

== Philosophical problems ==
Some of the problems considered in the philosophy of mathematics are the following:

Explain the effectiveness of mathematics in the study of the physical world: "At this point an enigma presents itself which in all ages has agitated inquiring minds. How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?" —Albert Einstein, in Geometry and Experience (1921).
Clearly delineate mathematics and physics: For some results or discoveries, it is difficult to say to which area they belong: to the mathematics or to physics.
What is the geometry of physical space?
What is the origin of the axioms of mathematics?
How does the already existing mathematics influence in the creation and development of physical theories?
Is arithmetic analytic or synthetic? (from Immanuel Kant, see Analytic–synthetic distinction)
What is essentially different between doing a physical experiment to see the result and making a mathematical calculation to see the result? (from the Turing–Wittgenstein debate)
Do Gödel's incompleteness theorems imply that physical theories will always be incomplete? (from Stephen Hawking)
Is mathematics invented or discovered? (millennia-old question, raised among others by Mario Livio)

== Education ==
In recent times the two disciplines have most often been taught separately, despite all the interrelations between physics and mathematics. This led some professional mathematicians who were also interested in mathematics education, such as Felix Klein, Richard Courant, Vladimir Arnold and Morris Kline, to strongly advocate teaching mathematics in a way more closely related to the physical sciences. The initial courses of mathematics for college students of physics are often taught by mathematicians, despite the differences in "ways of thinking" of physicists and mathematicians about those traditional courses and how they are used in the physics courses classes thereafter.

== See also ==

== References ==

== Further reading ==
Darling, David (1993-07-14). Equations of Eternity. Hyperion. ISBN 1-56282-875-4.
Arnold, V. I. (1999). "Mathematics and physics: mother and daughter or sisters?". Physics-Uspekhi. 42 (12): 1205–1217. Bibcode:1999PhyU...42.1205A. doi:10.1070/pu1999v042n12abeh000673. S2CID 250835608.
Arnold, V. I. (1998). "On teaching mathematics". Russian Mathematical Surveys. 53 (1). Translated by A. V. Goryunov: 229–236. Bibcode:1998RuMaS..53..229A. doi:10.1070/RM1998v053n01ABEH000005. S2CID 250833432. Archived from the original on 28 April 2017. Retrieved 29 May 2014.
Atiyah, M.; Dijkgraaf, R.; Hitchin, N. (1 February 2010). "Geometry and physics". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 368 (1914): 913–926. Bibcode:2010RSPTA.368..913A. doi:10.1098/rsta.2009.0227. PMC 3263806. PMID 20123740.
Boniolo, Giovanni; Budinich, Paolo; Trobok, Majda, eds. (2005). The Role of Mathematics in Physical Sciences: Interdisciplinary and Philosophical Aspects. Dordrecht: Springer. ISBN 978-1-4020-3106-9.
Colyvan, Mark (2001). "The Miracle of Applied Mathematics" (PDF). Synthese. 127 (3): 265–277. doi:10.1023/A:1010309227321. S2CID 40819230. Retrieved 30 May 2014.
Dirac, Paul (1938–1939). "The Relation between Mathematics and Physics". Proceedings of the Royal Society of Edinburgh. 59 Part II: 122–129. Retrieved 30 March 2014.
Feynman, Richard P. (1992). "The Relation of Mathematics to Physics". The Character of Physical Law (Reprint ed.). London: Penguin Books. pp. 35–58. ISBN 978-0-14-017505-9.
Hardy, G. H. (2005). A Mathematician's Apology (PDF) (First electronic ed.). University of Alberta Mathematical Sciences Society. Archived from the original (PDF) on 9 October 2021. Retrieved 30 May 2014.
Hitchin, Nigel (2007). "Interaction between mathematics and physics". ARBOR Ciencia, Pensamiento y Cultura. 725. Retrieved 31 May 2014.
Harvey, Alex (2012). "The Reasonable Effectiveness of Mathematics in the Physical Sciences". General Relativity and Gravitation. 43 (2011): 3057–3064. arXiv:1212.5854. Bibcode:2011GReGr..43.3657H. doi:10.1007/s10714-011-1248-9. S2CID 121985996.
Neumann, John von (1947). "The Mathematician". Works of the Mind. 1 (1): 180–196. (part 1) (part 2).
Poincaré, Henri (1907). The Value of Science (PDF). Translated by George Bruce Halsted. New York: The Science Press.
Schlager, Neil; Lauer, Josh, eds. (2000). "The Intimate Relation between Mathematics and Physics". Science and Its Times: Understanding the Social Significance of Scientific Discovery. Vol. 7: 1950 to Present. Gale Group. pp. 226–229. ISBN 978-0-7876-3939-6.
Vafa, Cumrun (2000). "On the Future of Mathematics/Physics Interaction". Mathematics: Frontiers and Perspectives. USA: AMS. pp. 321–328. ISBN 978-0-8218-2070-4.
Witten, Edward (1986). Physics and Geometry (PDF). Proceedings of the International Conference of Mathematicians. Berkeley, California. pp. 267–303. Archived from the original (PDF) on 2013-12-28. Retrieved 2014-05-27.
Eugene Wigner (1960). "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". Communications on Pure and Applied Mathematics. 13 (1): 1–14. Bibcode:1960CPAM...13....1W. doi:10.1002/cpa.3160130102. S2CID 6112252. Archived from the original on 2011-02-28. Retrieved 2014-05-27.

== External links ==
Gregory W. Moore – Physical Mathematics and the Future (July 4, 2014)
IOP Institute of Physics – Mathematical Physics: What is it and why do we need it? (September 2014)
Feynman explaining the differences between mathematics and physics in a video available on YouTube