Relationship between mathematics and physics

Short description: Relationship between fields of study

A cycloidal pendulum is isochronous, a fact discovered and proved by Christiaan Huygens under certain mathematical assumptions.[1] Mathematics was developed by the Ancient Civilizations for intellectual challenge and pleasure. Surprisingly, many of their discoveries later played prominent roles in physical theories, as in the case of the conic sections in celestial mechanics.

The relationship between mathematics and physics has been a subject of study of philosophers, mathematicians and physicists since antiquity, and more recently also by historians and educators.[2] Generally considered a relationship of great intimacy,[3] mathematics has been described as "an essential tool for physics"[4] and physics has been described as "a rich source of inspiration and insight in mathematics".[5]

Some of the oldest and most discussed themes are about the main differences between the two subjects, their mutual influence, the role of mathematical rigor in physics, and the problem of explaining the effectiveness of mathematics in physics.

In his work Physics, one of the topics treated by Aristotle is about how the study carried out by mathematicians differs from that carried out by physicists.[6] Considerations about mathematics being the language of nature can be found in the ideas of the Pythagoreans: the convictions that "Numbers rule the world" and "All is number",[7][8] and two millennia later were also expressed by Galileo Galilei: "The book of nature is written in the language of mathematics".[9][10]

## Contents

* 1 Historical interplay
* 2 Physics is not mathematics
* 2.1 Mathematics is physics
* 3 Role of rigor in physics
* 4 Philosophical problems
* 5 Education
* 6 See also
* 7 References
* 8 Further reading
* 9 External links

## Historical interplay

Before giving a mathematical proof for the formula for the volume of a sphere, Archimedes used physical reasoning to discover the solution (imagining the balancing of bodies on a scale).[11] Aristotle classified physics and mathematics as theoretical sciences, in contrast to practical sciences (like ethics or politics) and to productive sciences (like medicine or botany).[12]

From the seventeenth century, many of the most important advances in mathematics appeared motivated by the study of physics, and this continued in the following centuries (although in the nineteenth century mathematics started to become increasingly independent from physics).[13][14] The creation and development of calculus were strongly linked to the needs of physics:[15] There was a need for a new mathematical language to deal with the new dynamics that had arisen from the work of scholars such as Galileo Galilei and Isaac Newton.[16] The concept of derivative was needed, Newton did not have the modern concept of limits, and instead employed infinitesimals, which lacked a rigorous foundation at that time.[17] During this period there was little distinction between physics and mathematics;[18] as an example, Newton regarded geometry as a branch of mechanics.[19]

Non-Euclidean geometry, as formulated by Carl Friedrich Gauss, János Bolyai, Nikolai Lobachevsky, and Bernhard Riemann, freed physics from the limitation of a single Euclidean geometry.[20] A version of non-Euclidean geometry, called Riemannian geometry, enabled Albert Einstein to develop general relativity by providing the key mathematical framework on which he fit his physical ideas of gravity.[21]

In the 19th century Auguste Comte in his hierarchy of the sciences, placed physics and astronomy as less general and more complex than mathematics, as both depend on it.[22] In 1900, David Hilbert in his 23 problems for the advancement of mathematical science, considered the axiomatization of physics as his sixth problem. The problem remains open.[23]

In 1930, Paul Dirac invented the Dirac delta function for quantum mechanics which produced a single value when used in an integral. The mathematical rigor of this function was in doubt until the mathematician Laurent Schwartz developed on the theory of distributions.[24] The mathematical formulation of quantum mechanics was further formalized by John von Neumann's 1932 book, Mathematical Foundations of Quantum Mechanics. According to Freeman Dyson, this book was largely ignored for a while because mathematics and physics communities were distant at the time.[25]

Connections between the two fields sometimes only require identifying similar concepts by different names, as shown in the 1975 Wu–Yang dictionary,[26] that related concepts of gauge theory with differential geometry.[27]: 332

## Physics is not mathematics

See also: Deductive reasoning and Inductive reasoning

Despite the close relationship between math and physics, they are not synonyms. In mathematics objects can be defined exactly and logically related, but the object need have no relationship to experimental measurements. In physics, definitions are abstractions or idealizations, approximations adequate when compared to the natural world. In 1960, Georg Rasch noted that no models are ever true, not even Newton's laws, emphasizing that models should not be evaluated based on truth but on their applicability for a given purpose.[28] For example, Newton built a physical model around definitions like his second law of motion 𝐅=m𝐚 based on observations, leading to the development of calculus and highly accurate planetary mechanics, but later this definition was superseded by improved models of mechanics.[29] Mathematics deals with entities whose properties can be known with certainty.[30] According to David Hume, only statements that deal solely with ideas themselves—such as those encountered in mathematics—can be demonstrated to be true with certainty, while any conclusions pertaining to experiences of the real world can only be achieved via "probable reasoning".[31] This leads to a situation that was put by Albert Einstein as "No number of experiments can prove me right; a single experiment can prove me wrong."[32] The ultimate goal in research in pure mathematics are rigorous proofs, while in physics heuristic arguments may sometimes suffice in leading-edge research.[33] In short, the methods and goals of physicists and mathematicians are different.[34] Nonetheless, according to Roland Omnès, the axioms of mathematics are not mere conventions, but have physical origins.[35]

### Mathematics is physics

A well-known dictum of the Russian and Soviet mathematician Vladimir Arnold is "Mathematics is the part of physics where experiments are cheap".[36][37] While the phrase generated controversy and even parodies, Arnold defended it.[38] Mathematicians Arthur Jaffe and Frank Quinn have noted trends in mathematics towards more focus on intuition even at the cost of rigor and suggest this trend is due to interactions between math and physics.[39] In the framework of Willard van Orman Quine's epistemological holism, our beliefs, even in mathematics, are subjected to the "tribunal of experience", just like in physics.[40][41]

## Role of rigor in physics

See also: Physics:Theoretical physics and Mathematical physics

Rigor is indispensable in pure mathematics.[42] 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.[33][43][44][45]

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.[46]: ix [47]: 2

Some mathematicians, such as Arthur Jaffe and Frank Quinn, argue that non-rigorous mathematical work can sometimes bring benefits too.[39]

## 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).[48]
* 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.[49]
* What is the geometry of physical space?[50]
* What is the origin of the axioms of mathematics?[51]
* How does the already existing mathematics influence in the creation and development of physical theories?[52]
* Is arithmetic analytic or synthetic? (from Immanuel Kant, see Analytic–synthetic distinction)[53]
* 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)[54]
* Do Gödel's incompleteness theorems imply that physical theories will always be incomplete? (from Stephen Hawking)[55][56]
* Is mathematics invented or discovered? (millennia-old question, raised among others by Mario Livio)[57]

## Education

In recent times the two disciplines have most often been taught separately, despite all the interrelations between physics and mathematics.[58] 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.[59][60] 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.[61]

## See also

* Relationship between chemistry and physics
* Non-Euclidean geometry
* Kepler's laws of planetary motion
* Saving the phenomena
* The Unreasonable Effectiveness of Mathematics in the Natural Sciences
* Mathematical universe hypothesis
* Zeno's paradoxes
* Mathematical model
* Empiricism
* Mathematics of general relativity
* Bourbaki
* Experimental mathematics
* History of Maxwell's equations
* Philosophy of mathematics § Platonism
* History of astronomy
* Why Johnny Can't Add
* Mathematical formulation of quantum mechanics
* Scientific modelling
* All models are wrong

## References

1. ↑ Jed Z. Buchwald; Robert Fox (10 October 2013). The Oxford Handbook of the History of Physics. OUP Oxford. p. 128. ISBN 978-0-19-151019-9. https://books.google.com/books?id=1SxoAgAAQBAJ&pg=PA128.
2. ↑ Uhden, Olaf; Karam, Ricardo; Pietrocola, Maurício; Pospiech, Gesche (20 October 2011). "Modelling Mathematical Reasoning in Physics Education". Science & Education 21 (4): 485–506. doi:10.1007/s11191-011-9396-6. Bibcode: 2012Sc&Ed..21..485U.
3. ↑ Francis Bailly; Giuseppe Longo (2011). Mathematics and the Natural Sciences: The Physical Singularity of Life. World Scientific. p. 149. ISBN 978-1-84816-693-6. https://books.google.com/books?id=7-dGHyIyI-AC&pg=PA149.
4. ↑ Sanjay Moreshwar Wagh; Dilip Abasaheb Deshpande (27 September 2012). Essentials of Physics. PHI Learning Pvt. Ltd.. p. 3. ISBN 978-81-203-4642-0. https://books.google.com/books?id=-DmfVjBUPksC&pg=PA3.
5. ↑ Atiyah, Michael (1990). "On the Work of Edward Witten". International Congress of Mathematicians. Japan. pp. 31–35. http://www.mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0031.0036.ocr.pdf.
6. ↑ Lear, Jonathan (1990). Aristotle: the desire to understand (Repr. ed.). Cambridge [u.a.]: Cambridge Univ. Press. p. 232. ISBN 978-0-521-34762-4. https://archive.org/details/aristotledesiret0000lear/page/232.
7. ↑ Gerard Assayag; Hans G. Feichtinger; José-Francisco Rodrigues (10 July 2002). Mathematics and Music: A Diderot Mathematical Forum. Springer. p. 216. ISBN 978-3-540-43727-7. https://books.google.com/books?id=bjsD8ClsFKEC&pg=PA216.
8. ↑ Al-Rasasi, Ibrahim (21 June 2004). "All is number". King Fahd University of Petroleum and Minerals. http://faculty.kfupm.edu.sa/math/irasasi/Allisnumber.pdf.
9. ↑ Aharon Kantorovich (1 July 1993). Scientific Discovery: Logic and Tinkering. SUNY Press. p. 59. ISBN 978-0-7914-1478-1. https://books.google.com/books?id=vMFc43w0FfEC&pg=PA59.
10. ↑ Kyle Forinash, William Rumsey, Chris Lang, Galileo's Mathematical Language of Nature .
11. ↑ Arthur Mazer (26 September 2011). The Ellipse: A Historical and Mathematical Journey. John Wiley & Sons. p. 5. ISBN 978-1-118-21143-4. Bibcode: 2010ehmj.book.....M. https://books.google.com/books?id=twWkDe1Y9YQC&pg=SA5-PA28.
12. ↑ Shields, Christopher (2023), Zalta, Edward N.; Nodelman, Uri, eds., Aristotle (Winter 2023 ed.), Metaphysics Research Lab, Stanford University, https://plato.stanford.edu/entries/aristotle/, retrieved 2024-11-11
13. ↑ E. J. Post, A History of Physics as an Exercise in Philosophy, p. 76.
14. ↑ Arkady Plotnitsky, Niels Bohr and Complementarity: An Introduction, p. 177.
15. ↑ Roger G. Newton (1997). The Truth of Science: Physical Theories and Reality. Harvard University Press. pp. 125–126. ISBN 978-0-674-91092-8. https://archive.org/details/truthofscienceph00newt.
16. ↑ Eoin P. O'Neill (editor), What Did You Do Today, Professor?: Fifteen Illuminating Responses from Trinity College Dublin, p. 62.
17. ↑ Rédei, M. "On the Tension Between Physics and Mathematics". J Gen Philos Sci 51, pp. 411–425 (2020). https://doi.org/10.1007/s10838-019-09496-0
18. ↑ Timothy Gowers; June Barrow-Green; Imre Leader (18 July 2010). The Princeton Companion to Mathematics. Princeton University Press. p. 7. ISBN 978-1-4008-3039-8. https://books.google.com/books?id=ZOfUsvemJDMC&pg=PA7.
19. ↑ David E. Rowe (2008). "Euclidean Geometry and Physical Space". The Mathematical Intelligencer 28 (2): 51–59. doi:10.1007/BF02987157.
20. ↑ Read, Charlotte (1968). "Alfred Korzybski: His contributions and their historical development". The Polish Review (University of Illinois Press) 13 (2): 5–13. ISSN 00322970.
21. ↑ ""Riemann, Georg Friedrich Bernhard" Complete Dictionary of Scientific Biography". https://www.encyclopedia.com/people/science-and-technology/mathematics-biographies/bernhard-riemann#2830903674.
22. ↑ Bourdeau, Michel (2023), Zalta, Edward N.; Nodelman, Uri, eds., Auguste Comte (Spring 2023 ed.), Metaphysics Research Lab, Stanford University, https://plato.stanford.edu/entries/comte/, retrieved 2024-11-08
23. ↑ Gorban, A. N. (2018-04-28). "Hilbert's sixth problem: the endless road to rigour" (in en). Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376 (2118). doi:10.1098/rsta.2017.0238. ISSN 1364-503X. PMID 29555808.
24. ↑ Lamb, Evelyn (2018-04-24). "The coevolution of physics and math" (in en). symmetry magazine. https://www.symmetrymagazine.org/article/the-coevolution-of-physics-and-math.
25. ↑ Dyson, Freeman (2013-02-01). "A Walk through Johnny von Neumann’s Garden" (in en). Notices of the American Mathematical Society 60 (02): 154. doi:10.1090/noti942. ISSN 0002-9920. https://www.ams.org/jourcgi/jour-getitem?pii=noti942.
26. ↑ Wu, Tai Tsun; Yang, Chen Ning (1975-12-15). "Concept of nonintegrable phase factors and global formulation of gauge fields" (in en). Physical Review D 12 (12): 3845–3857. doi:10.1103/PhysRevD.12.3845. ISSN 0556-2821. https://link.aps.org/doi/10.1103/PhysRevD.12.3845.
27. ↑ Zeidler, Eberhard (2008-09-03) (in en). Quantum Field Theory II: Quantum Electrodynamics: A Bridge between Mathematicians and Physicists. Springer Science & Business Media. ISBN 978-3-540-85377-0. https://books.google.com/books?id=Pk9yyC239scC&q=dictionary.
28. ↑ Rasch, Georg (1960). Probabilistic Models for Some Intelligence and Attainment Tests. Danish Institute for Educational Research. pp. 37. ISBN 978-0-598-55451-2. https://books.google.com/books?id=aB9qLgEACAAJ.
29. ↑ Feynman, Richard P. (2011). "Characteristics of Force". The Feynman lectures on physics. Volume 1: Mainly mechanics, radiation, and heat (The new millennium edition, paperback first published ed.). New York: Basic Books. ISBN 978-0-465-02493-3. https://www.feynmanlectures.caltech.edu/I_12.html.
30. ↑ Ernest, Paul (2003). The philosophy of mathematics education. Studies in mathematics education (Reprint ed.). New York: Routledge. ISBN 978-1-85000-667-1.
31. ↑ Russell, Paul, ed (2016). The Oxford Handbook of Hume. Oxford University Press. pp. 34, 94. ISBN 978-0-19-049392-9.
32. ↑ Fundamentals of Physics \- Volume 2 - Page 627, by David Halliday, Robert Resnick, Jearl Walker (1993)
33. ↑ 33.0 33.1 MICHAEL ATIYAH ET AL. "RESPONSES TO THEORETICAL MATHEMATICS: TOWARD A CULTURAL SYNTHESIS OF MATHEMATICS AND THEORETICAL PHYSICS, BY A. JAFFE AND F. QUINN. https://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00503-8/S0273-0979-1994-00503-8.pdf"
34. ↑ Redish, Edward F.; Kuo, Eric (2015-07-01). "Language of Physics, Language of Math: Disciplinary Culture and Dynamic Epistemology". Science & Education 24 (5): 561–590. doi:10.1007/s11191-015-9749-7. ISSN 1573-1901.
35. ↑ Roland Omnès (2005) Converging Realities: Toward a Common Philosophy of Physics and Mathematics p. 38 and p. 215
36. ↑ Gusein-Zade, Sabir M.; Varchenko, Alexander N (December 2010), "Obituary: Vladimir Arnold (12 June 1937 – 3 June 2010)", Newsletter of the European Mathematical Society 78: 28–29, http://www.ems-ph.org/journals/newsletter/pdf/2010-12-78.pdf
37. ↑ Euler M. (2004). "The role of experiments in the teaching and learning of physics" [JB]. Proceedings of the International School of Physics (Research on Physics Education), pp. 175–221. https://doi.org/10.3254/978-1-61499-012-3-175
38. ↑ Arnold, V. I. (1999). "Mathematics and physics: mother and daughter or sisters?". Physics-Uspekhi 42 (12): 1205–1217. doi:10.1070/pu1999v042n12abeh000673. Bibcode: 1999PhyU...42.1205A. .
39. ↑ 39.0 39.1 Jaffe, Arthur; Quinn, Frank (1993). ""Theoretical mathematics": toward a cultural synthesis of mathematics and theoretical physics" (in en). Bulletin of the American Mathematical Society 29 (1): 1–13. doi:10.1090/S0273-0979-1993-00413-0. ISSN 0273-0979. https://www.ams.org/bull/1993-29-01/S0273-0979-1993-00413-0/.
40. ↑ Sober, Elliott (1993). "Mathematics and Indispensability". The Philosophical Review 102 (1): 35–57. doi:10.2307/2185652. https://www.jstor.org/stable/2185652. Retrieved 2025-06-24.
41. ↑ Quine, W. (1953). "Two Dogmas of Empiricism" In From a Logical Point of View. Cambridge: Harvard University Press.
42. ↑ Steven Weinberg, To Explain the World: The Discovery of Modern Science, pp. 9–10.
43. ↑ Kevin Davey. "Is Mathematical Rigor Necessary in Physics?", The British Journal for the Philosophy of Science, Vol. 54, No. 3 (Sep., 2003), pp. 439–463 https://www.jstor.org/stable/3541794
44. ↑ Mark Steiner (1992), "Mathematical Rigor in Physics". https://www.taylorfrancis.com/chapters/edit/10.4324/9780203979105-13/mathematical-rigor-physics-mark-steiner
45. ↑ P.W. Bridgman (1959), "How Much Rigor is Possible in Physics?" https://doi.org/10.1016/S0049-237X(09)70030-8
46. ↑ Mitra, Asoke N., ed (2000). Quantum field theory: a twentieth century profile. New Delhi: Hindustan Book Agency [u.a.]. ISBN 978-81-85931-25-8.
47. ↑ Zeidler, Eberhard (2006) (in en). Quantum Field Theory I: Basics in Mathematics and Physics. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-540-34764-4. ISBN 978-3-540-34762-0. http://link.springer.com/10.1007/978-3-540-34764-4.
48. ↑ Albert Einstein, Geometry and Experience.
49. ↑ Pierre Bergé, Des rythmes au chaos.
50. ↑ Gary Carl Hatfield (1990). The Natural and the Normative: Theories of Spatial Perception from Kant to Helmholtz. MIT Press. p. 223. ISBN 978-0-262-08086-6. https://books.google.com/books?id=JikeeDbYeUQC&pg=PA223.
51. ↑ Gila Hanna; Hans Niels Jahnke; Helmut Pulte (4 December 2009). Explanation and Proof in Mathematics: Philosophical and Educational Perspectives. Springer Science & Business Media. pp. 29–30. ISBN 978-1-4419-0576-5. https://books.google.com/books?id=3bLHye8kSAwC&pg=PA29.
52. ↑ "FQXi Community Trick or Truth: the Mysterious Connection Between Physics and Mathematics". http://fqxi.org/community/essay/rules.
53. ↑ James Van Cleve Professor of Philosophy Brown University (16 July 1999). Problems from Kant. Oxford University Press, USA. p. 22. ISBN 978-0-19-534701-2. https://books.google.com/books?id=6WHAgt-Mg1AC&pg=PA22.
54. ↑ Ludwig Wittgenstein; R. G. Bosanquet; Cora Diamond (15 October 1989). Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939. University of Chicago Press. p. 96. ISBN 978-0-226-90426-9. https://books.google.com/books?id=d4YUZVq1JSEC&pg=PA96.
55. ↑ Pudlák, Pavel (2013). Logical Foundations of Mathematics and Computational Complexity: A Gentle Introduction. Springer Science & Business Media. p. 659. ISBN 978-3-319-00119-7. https://books.google.com/books?id=obxDAAAAQBAJ&pg=PA659.
56. ↑ "Stephen Hawking. "Godel and the End of the Universe"". http://www.hawking.org.uk/godel-and-the-end-of-physics.html.
57. ↑ Mario Livio (August 2011). "Why math works?". Scientific American: 80–83. http://www.scientificamerican.com/article/why-math-works/.
58. ↑ Karam; Pospiech; & Pietrocola (2010). "Mathematics in physics lessons: developing structural skills "
59. ↑ Stakhov "Dirac's Principle of Mathematical Beauty, Mathematics of Harmony"
60. ↑ Richard Lesh; Peter L. Galbraith; Christopher R. Haines; Andrew Hurford (2009). Modeling Students' Mathematical Modeling Competencies: ICTMA 13. Springer. p. 14. ISBN 978-1-4419-0561-1. https://books.google.com/books?id=Jj5tfi2594kC&pg=PA14.
61. ↑ "Why is Amp`ere's law so hard? A look at middle-division physics". https://bridge.math.oregonstate.edu/papers/ampere.pdf.

## 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. doi:10.1070/pu1999v042n12abeh000673. Bibcode: 1999PhyU...42.1205A.
* Arnold, V. I. (1998). "On teaching mathematics". Russian Mathematical Surveys 53 (1): 229-236. doi:10.1070/RM1998v053n01ABEH000005. Bibcode: 1998RuMaS..53..229A. http://pauli.uni-muenster.de/~munsteg/arnold.html. 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. doi:10.1098/rsta.2009.0227. PMID 20123740. Bibcode: 2010RSPTA.368..913A.
* 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". Synthese 127 (3): 265-277. doi:10.1023/A:1010309227321. http://www.colyvan.com/papers/miracle.pdf. 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. http://www.damtp.cam.ac.uk/events/strings02/dirac/speach.html. 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 (First electronic ed.). University of Alberta Mathematical Sciences Society. http://www.math.ualberta.ca/mss/misc/A Mathematician's Apology.pdf. Retrieved 30 May 2014.
* Hitchin, Nigel (2007). "Interaction between mathematics and physics". ARBOR Ciencia, Pensamiento y Cultura 725. http://arbor.revistas.csic.es/index.php/arbor/article/viewFile/115/116. Retrieved 31 May 2014.
* Harvey, Alex (2012). "The Reasonable Effectiveness of Mathematics in the Physical Sciences". General Relativity and Gravitation 43 (2011): 3057–3064. doi:10.1007/s10714-011-1248-9. Bibcode: 2011GReGr..43.3657H.
* 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. New York: The Science Press. http://www3.nd.edu/~powers/ame.60611/poincare.pdf.
* Schlager, Neil; Lauer, Josh, eds (2000). "The Intimate Relation between Mathematics and Physics". Science and Its Times: Understanding the Social Significance of Scientific Discovery. 7: 1950 to Present. Gale Group. pp. 226-229. ISBN 978-0-7876-3939-6. https://archive.org/details/scienceitstimesu0000unse/page/226.
* 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". Proceedings of the International Conference of Mathematicians. Berkeley, California. pp. 267-303. http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0267.0306.ocr.pdf. 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. doi:10.1002/cpa.3160130102. Bibcode: 1960CPAM...13....1W. http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html. Retrieved 2014-05-27.

## External links

* The Applicability of Mathematics in Physics, The Internet Encyclopedia of Philosophy
* 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

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