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Relationship between mathematics and physics

The relationship between mathematics and physics has intrigued philosophers, mathematicians, and physicists since antiquity. Often described as intimately linked, mathematics is considered “an essential tool for physics,” while physics serves as “a rich source of inspiration” for mathematics. Early reflections, such as those by Aristotle in his Physics, explore differences between their methods. The Pythagoreans famously believed “Numbers rule the world,” a view echoed by Galileo Galilei, who stated “The book of nature is written in the language of mathematics.” These ideas continue to inspire debates on the role of mathematical rigor and the effectiveness of mathematics in describing nature.

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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).10 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).11

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).1213 The creation and development of calculus were strongly linked to the needs of physics:14 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.15 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.16 During this period there was little distinction between physics and mathematics;17 as an example, Newton regarded geometry as a branch of mechanics.18

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.19 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.20

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.21 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.22

In 1930, Paul Dirac invented the Dirac delta function 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.23

Connections between the two fields sometimes only require identifying similar concepts by different names, as shown in the 1975 Wu–Yang dictionary,24 that related concepts of gauge theory with differential geometry.25: 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.26 For example, Newton built a physical model around definitions like his second law of motion F = m a {\displaystyle \mathbf {F} =m\mathbf {a} } 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.27 Mathematics deals with entities whose properties can be known with certainty.28 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".29 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."30 The ultimate goal in research in pure mathematics are rigorous proofs, while in physics heuristic arguments may sometimes suffice in leading-edge research.31 In short, the methods and goals of physicists and mathematicians are different.32 Nonetheless, according to Roland Omnès, the axioms of mathematics are not mere conventions, but have physical origins.33

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".3435 While the phrase generated controversy and even parodies, Arnold has defended it.36 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.37 In the framework of Quine's epistemological holism, our beliefs, even in mathematics, are subjected to the "tribunal of experience", just like in physics.38

Role of rigor in physics

See also: Theoretical physics and Mathematical physics

Rigor is indispensable in pure mathematics.39 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.40414243

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.44: ix 45: 2 

Some mathematicians, such as Arthur Jaffe and Frank Quinn, argue that non-rigorous mathematical work can sometimes bring benefits too.46

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).47
  • 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.48
  • What is the geometry of physical space?49
  • What is the origin of the axioms of mathematics?50
  • How does the already existing mathematics influence in the creation and development of physical theories?51
  • Is arithmetic analytic or synthetic? (from Kant, see Analytic–synthetic distinction)52
  • 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)53
  • Do Gödel's incompleteness theorems imply that physical theories will always be incomplete? (from Stephen Hawking)5455
  • Is mathematics invented or discovered? (millennia-old question, raised among others by Mario Livio)56

Education

In recent times the two disciplines have most often been taught separately, despite all the interrelations between physics and mathematics.57 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.5859 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.60

See also

Further reading

References

  1. Uhden, Olaf; Karam, Ricardo; Pietrocola, Maurício; Pospiech, Gesche (20 October 2011). "Modelling Mathematical Reasoning in Physics Education". Science & Education. 21 (4): 485–506. Bibcode:2012Sc&Ed..21..485U. doi:10.1007/s11191-011-9396-6. S2CID 122869677. /wiki/Bibcode_(identifier) ↩

  2. 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. 978-1-84816-693-6 ↩

  3. Sanjay Moreshwar Wagh; Dilip Abasaheb Deshpande (27 September 2012). Essentials of Physics. PHI Learning Pvt. Ltd. p. 3. ISBN 978-81-203-4642-0. 978-81-203-4642-0 ↩

  4. Atiyah, Michael (1990). On the Work of Edward Witten (PDF). International Congress of Mathematicians. Japan. pp. 31–35. Archived from the original (PDF) on 2017-03-01. /wiki/Michael_Atiyah ↩

  5. Lear, Jonathan (1990). Aristotle: the desire to understand (Repr. ed.). Cambridge [u.a.]: Cambridge Univ. Press. p. 232. ISBN 9780521347624. 9780521347624 ↩

  6. 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. 978-3-540-43727-7 ↩

  7. Al-Rasasi, Ibrahim (21 June 2004). "All is number" (PDF). King Fahd University of Petroleum and Minerals. Archived from the original (PDF) on 28 December 2014. Retrieved 13 June 2015. https://web.archive.org/web/20141228132248/http://faculty.kfupm.edu.sa/math/irasasi/Allisnumber.pdf ↩

  8. Aharon Kantorovich (1 July 1993). Scientific Discovery: Logic and Tinkering. SUNY Press. p. 59. ISBN 978-0-7914-1478-1. 978-0-7914-1478-1 ↩

  9. Kyle Forinash, William Rumsey, Chris Lang, Galileo's Mathematical Language of Nature Archived 2013-09-27 at the Wayback Machine. http://homepages.ius.edu/kforinas/K/pdf/Galileo.pdf ↩

  10. Arthur Mazer (26 September 2011). The Ellipse: A Historical and Mathematical Journey. John Wiley & Sons. p. 5. Bibcode:2010ehmj.book.....M. ISBN 978-1-118-21143-4. 978-1-118-21143-4 ↩

  11. Shields, Christopher (2023), "Aristotle", in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Winter 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-11-11 https://plato.stanford.edu/entries/aristotle/ ↩

  12. E. J. Post, A History of Physics as an Exercise in Philosophy, p. 76. http://www22.pair.com/csdc/pdf/philos.pdf ↩

  13. Arkady Plotnitsky, Niels Bohr and Complementarity: An Introduction, p. 177. https://books.google.com/books?id=dmdUp97S4AYC&pg=PA177 ↩

  14. Roger G. Newton (1997). The Truth of Science: Physical Theories and Reality. Harvard University Press. pp. 125–126. ISBN 978-0-674-91092-8. 978-0-674-91092-8 ↩

  15. Eoin P. O'Neill (editor), What Did You Do Today, Professor?: Fifteen Illuminating Responses from Trinity College Dublin, p. 62. https://books.google.com/books?id=h8TaAAAAMAAJ ↩

  16. 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 https://doi.org/10.1007/s10838-019-09496-0 ↩

  17. 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. 978-1-4008-3039-8 ↩

  18. David E. Rowe (2008). "Euclidean Geometry and Physical Space". The Mathematical Intelligencer. 28 (2): 51–59. doi:10.1007/BF02987157. S2CID 56161170. /wiki/David_E._Rowe ↩

  19. Read, Charlotte (1968). "Alfred Korzybski: His contributions and their historical development". The Polish Review. 13 (2). University of Illinois Press: 5–13. ISSN 0032-2970. JSTOR 25776770. Retrieved 2025-01-01. http://www.jstor.org/stable/25776770 ↩

  20. ""Riemann, Georg Friedrich Bernhard" Complete Dictionary of Scientific Biography". www.encyclopedia.com. Retrieved 2025-01-08. https://www.encyclopedia.com/people/science-and-technology/mathematics-biographies/bernhard-riemann#2830903674 ↩

  21. Bourdeau, Michel (2023), "Auguste Comte", in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Spring 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-11-08 https://plato.stanford.edu/entries/comte/ ↩

  22. Gorban, A. N. (2018-04-28). "Hilbert's sixth problem: the endless road to rigour". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 376 (2118): 20170238. doi:10.1098/rsta.2017.0238. ISSN 1364-503X. PMC 5869544. PMID 29555808. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5869544 ↩

  23. Lamb, Evelyn (2018-04-24). "The coevolution of physics and math". symmetry magazine. https://www.symmetrymagazine.org/article/the-coevolution-of-physics-and-math ↩

  24. Wu, Tai Tsun; Yang, Chen Ning (1975-12-15). "Concept of nonintegrable phase factors and global formulation of gauge fields". 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 ↩

  25. Zeidler, Eberhard (2008-09-03). Quantum Field Theory II: Quantum Electrodynamics: A Bridge between Mathematicians and Physicists. Springer Science & Business Media. ISBN 978-3-540-85377-0. 978-3-540-85377-0 ↩

  26. Rasch, Georg (1960). Probabilistic Models for Some Intelligence and Attainment Tests. Danish Institute for Educational Research. p. 37. ISBN 9780598554512. {{cite book}}: ISBN / Date incompatibility (help) 9780598554512 ↩

  27. 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. 978-0-465-02493-3 ↩

  28. Ernest, Paul (2003) [1991]. The philosophy of mathematics education. Studies in mathematics education (Reprint ed.). New York: Routledge. ISBN 978-1-85000-667-1. 978-1-85000-667-1 ↩

  29. Russell, Paul, ed. (2016). The Oxford Handbook of Hume. Oxford University Press. pp. 34, 94. ISBN 978-0-19-049392-9. 978-0-19-049392-9 ↩

  30. Fundamentals of Physics - Volume 2 - Page 627, by David Halliday, Robert Resnick, Jearl Walker (1993) ↩

  31. 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" https://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00503-8/S0273-0979-1994-00503-8.pdf ↩

  32. 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. arXiv:1409.6272. doi:10.1007/s11191-015-9749-7. ISSN 1573-1901. https://doi.org/10.1007%2Fs11191-015-9749-7 ↩

  33. Roland Omnès (2005) Converging Realities: Toward a Common Philosophy of Physics and Mathematics p. 38 and p. 215 /wiki/Roland_Omn%C3%A8s ↩

  34. Gusein-Zade, Sabir M.; Varchenko, Alexander N (December 2010), "Obituary: Vladimir Arnold (12 June 1937 – 3 June 2010)" (PDF), Newsletter of the European Mathematical Society, 78: 28–29 /wiki/Sabir_Gusein-Zade ↩

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  36. 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.. /wiki/Vladimir_Arnold ↩

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  38. Sober, Elliott. "Mathematics and Indispensability." The Philosophical Review 102.1 (1993): 35–57. https://doi.org/10.2307/2185652 /wiki/Elliott_Sober ↩

  39. Steven Weinberg, To Explain the World: The Discovery of Modern Science, pp. 9–10. /wiki/Steven_Weinberg ↩

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