Relationship between mathematics and physics

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).{{cite book|author=Arthur Mazer|title=The Ellipse: A Historical and Mathematical Journey|url=https://books.google.com/books?id=twWkDe1Y9YQC&pg=SA5-PA28|date=26 September 2011|publisher=John Wiley & Sons|isbn=978-1-118-21143-4|pages=5|bibcode=2010ehmj.book.....M}} 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).{{Citation |last=Shields |first=Christopher |title=Aristotle |date=2023 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/aristotle/ |access-date=2024-11-11 |edition=Winter 2023 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}}

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).E. J. Post, [http://www22.pair.com/csdc/pdf/philos.pdf A History of Physics as an Exercise in Philosophy, p. 76.]Arkady Plotnitsky, [https://books.google.com/books?id=dmdUp97S4AYC&pg=PA177 Niels Bohr and Complementarity: An Introduction, p. 177]. The creation and development of calculus were strongly linked to the needs of physics:{{cite book|author=Roger G. Newton|title=The Truth of Science: Physical Theories and Reality|url=https://archive.org/details/truthofscienceph00newt|url-access=registration|year=1997|publisher=Harvard University Press|isbn=978-0-674-91092-8|pages=[https://archive.org/details/truthofscienceph00newt/page/125 125]–126}} 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.Eoin P. O'Neill (editor), [https://books.google.com/books?id=h8TaAAAAMAAJ What Did You Do Today, Professor?: Fifteen Illuminating Responses from Trinity College Dublin, p. 62]. 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.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 During this period there was little distinction between physics and mathematics;{{cite book|author1=Timothy Gowers|author-link=Timothy Gowers|author2=June Barrow-Green|author3=Imre Leader|title=The Princeton Companion to Mathematics|url=https://books.google.com/books?id=ZOfUsvemJDMC&pg=PA7|date=18 July 2010|publisher=Princeton University Press|isbn=978-1-4008-3039-8|pages=7}} as an example, Newton regarded geometry as a branch of mechanics.{{cite journal|author=David E. Rowe|author-link=David E. Rowe|title=Euclidean Geometry and Physical Space|journal=The Mathematical Intelligencer|year=2008|volume=28|issue=2|pages=51–59|doi=10.1007/BF02987157|s2cid=56161170}}

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.{{cite journal | last=Read | first=Charlotte | title=Alfred Korzybski: His contributions and their historical development | journal=The Polish Review | publisher=University of Illinois Press | volume=13 | issue=2 | year=1968 | issn=00322970 | jstor=25776770 | pages=5–13 | url=http://www.jstor.org/stable/25776770 | access-date=2025-01-01}} 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.{{Cite web |title="Riemann, Georg Friedrich Bernhard" Complete Dictionary of Scientific Biography |url=https://www.encyclopedia.com/people/science-and-technology/mathematics-biographies/bernhard-riemann#2830903674 |access-date=2025-01-08 |website=www.encyclopedia.com}}

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.{{Citation |last=Bourdeau |first=Michel |title=Auguste Comte |date=2023 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/comte/ |access-date=2024-11-08 |edition=Spring 2023 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}} 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.{{Cite journal |last=Gorban |first=A. N. |date=2018-04-28 |title=Hilbert's sixth problem: the endless road to rigour |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |language=en |volume=376 |issue=2118 |pages=20170238 |doi=10.1098/rsta.2017.0238 |issn=1364-503X |pmc=5869544 |pmid=29555808}}

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.{{Cite magazine |last=Lamb |first=Evelyn |date=2018-04-24 |title=The coevolution of physics and math |url=https://www.symmetrymagazine.org/article/the-coevolution-of-physics-and-math |magazine=symmetry magazine |language=en}}

Connections between the two fields sometimes only require identifying similar concepts by different names, as shown in the 1975 Wu–Yang dictionary,{{Cite journal |last1=Wu |first1=Tai Tsun |last2=Yang |first2=Chen Ning |date=1975-12-15 |title=Concept of nonintegrable phase factors and global formulation of gauge fields |url=https://link.aps.org/doi/10.1103/PhysRevD.12.3845 |journal=Physical Review D |language=en |volume=12 |issue=12 |pages=3845–3857 |doi=10.1103/PhysRevD.12.3845 |issn=0556-2821}} that related concepts of gauge theory with differential geometry.{{Cite book |last=Zeidler |first=Eberhard |url=https://books.google.com/books?id=Pk9yyC239scC&q=dictionary |title=Quantum Field Theory II: Quantum Electrodynamics: A Bridge between Mathematicians and Physicists |date=2008-09-03 |publisher=Springer Science & Business Media |isbn=978-3-540-85377-0 |language=en}}{{rp|332}}

{{see also|Deductive reasoning|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.{{Cite book |last=Rasch |first=Georg |url=https://books.google.com/books?id=aB9qLgEACAAJ |title=Probabilistic Models for Some Intelligence and Attainment Tests |publisher=Danish Institute for Educational Research |year=1960 |isbn=9780598554512 |pages=37}} For example, Newton built a physical model around definitions like his second law of motion $\backslash mathbf\; F=m\backslash mathbf\; a$ 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.{{Cite book |last=Feynman |first=Richard P. |url=https://www.feynmanlectures.caltech.edu/I_12.html |title=The Feynman lectures on physics. Volume 1: Mainly mechanics, radiation, and heat |date=2011 |publisher=Basic Books |isbn=978-0-465-02493-3 |edition=The new millennium edition, paperback first published |location=New York |chapter=Characteristics of Force}} Mathematics deals with entities whose properties can be known with certainty.{{Cite book |last=Ernest |first=Paul |title=The philosophy of mathematics education |date=2003 |publisher=Routledge |isbn=978-1-85000-667-1 |edition=Reprint |orig-date=1991 |series=Studies in mathematics education |location=New York}} 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".{{cite book | editor-last=Russell | editor-first=Paul | title=The Oxford Handbook of Hume | publisher=Oxford University Press | date=2016 | isbn=978-0-19-049392-9 |pages=34, 94}} 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."Fundamentals of Physics - Volume 2 - Page 627, by David Halliday, Robert Resnick, Jearl Walker (1993) The ultimate goal in research in pure mathematics are rigorous proofs, while in physics heuristic arguments may sometimes suffice in leading-edge research.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" In short, the methods and goals of physicists and mathematicians are different.{{Cite journal |last=Redish |first=Edward F. |last2=Kuo |first2=Eric |date=2015-07-01 |title=Language of Physics, Language of Math: Disciplinary Culture and Dynamic Epistemology |journal=Science & Education |volume=24 |issue=5 |pages=561–590 |doi=10.1007/s11191-015-9749-7 |issn=1573-1901|doi-access=free |arxiv=1409.6272 }} Nonetheless, according to Roland Omnès, the axioms of mathematics are not mere conventions, but have physical origins.Roland Omnès (2005) Converging Realities: Toward a Common Philosophy of Physics and Mathematics p. 38 and p. 215

{{see also|Theoretical physics|Mathematical physics}}

Rigor is indispensable in pure mathematics.Steven Weinberg, To Explain the World: The Discovery of Modern Science, pp. 9–10. 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.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/3541794Mark Steiner (1992), "Mathematical Rigor in Physics". https://www.taylorfrancis.com/chapters/edit/10.4324/9780203979105-13/mathematical-rigor-physics-mark-steinerP.W. Bridgman (1959), "How Much Rigor is Possible in Physics?" https://doi.org/10.1016/S0049-237X(09)70030-8

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.{{Cite book |title=Quantum field theory: a twentieth century profile |date=2000 |publisher=Hindustan Book Agency [u.a.] |isbn=978-81-85931-25-8 |editor-last=Mitra |editor-first=Asoke N. |location=New Delhi |editor-last2=Dyson |editor-first2=Freeman J.}}{{rp|ix}}{{Cite book |last=Zeidler |first=Eberhard |url=http://link.springer.com/10.1007/978-3-540-34764-4 |title=Quantum Field Theory I: Basics in Mathematics and Physics |date=2006 |publisher=Springer Berlin Heidelberg |isbn=978-3-540-34762-0 |location=Berlin, Heidelberg |language=en |doi=10.1007/978-3-540-34764-4}}{{rp|2}}

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).Albert Einstein, [https://mathshistory.st-andrews.ac.uk/Extras/Einstein_geometry/ Geometry and Experience].
• 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.Pierre Bergé, [https://books.google.com/books?id=umFTtQAACAAJ&q=%22Des+rythmes+au+chaos%22 Des rythmes au chaos].
• What is the geometry of physical space?{{cite book|author=Gary Carl Hatfield|title=The Natural and the Normative: Theories of Spatial Perception from Kant to Helmholtz|url=https://books.google.com/books?id=JikeeDbYeUQC&pg=PA223|year=1990|publisher=MIT Press|isbn=978-0-262-08086-6|page=223}}
• What is the origin of the axioms of mathematics?{{cite book|author1=Gila Hanna|author1-link=Gila Hanna|author2=Hans Niels Jahnke|author3=Helmut Pulte|title=Explanation and Proof in Mathematics: Philosophical and Educational Perspectives|url=https://books.google.com/books?id=3bLHye8kSAwC&pg=PA29|date=4 December 2009|publisher=Springer Science & Business Media|isbn=978-1-4419-0576-5|pages=29–30}}
• How does the already existing mathematics influence in the creation and development of physical theories?{{cite web |url=http://fqxi.org/community/essay/rules |title=FQXi Community Trick or Truth: the Mysterious Connection Between Physics and Mathematics |access-date=16 April 2015 |archive-date=14 December 2021 |archive-url=https://web.archive.org/web/20211214170940/https://fqxi.org/community/essay/rules |url-status=dead }}
• Is arithmetic analytic or synthetic? (from Kant, see Analytic–synthetic distinction){{cite book|author=James Van Cleve Professor of Philosophy Brown University|title=Problems from Kant|url=https://books.google.com/books?id=6WHAgt-Mg1AC&pg=PA22|date=16 July 1999|publisher=Oxford University Press, USA|isbn=978-0-19-534701-2|pages=22}}
• 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){{cite book|author1=Ludwig Wittgenstein|author2=R. G. Bosanquet|author3=Cora Diamond|title=Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939|url=https://books.google.com/books?id=d4YUZVq1JSEC&pg=PA96|date=15 October 1989|publisher=University of Chicago Press|isbn=978-0-226-90426-9|page=96}}
• Do Gödel's incompleteness theorems imply that physical theories will always be incomplete? (from Stephen Hawking){{cite book|first=Pavel|last=Pudlák|title=Logical Foundations of Mathematics and Computational Complexity: A Gentle Introduction|url=https://books.google.com/books?id=obxDAAAAQBAJ&pg=PA659|year=2013|publisher=Springer Science & Business Media|isbn=978-3-319-00119-7|page=659}}{{Cite web |url=http://www.hawking.org.uk/godel-and-the-end-of-physics.html |title=Stephen Hawking. "Godel and the End of the Universe" |access-date=2015-06-12 |archive-date=2020-05-29 |archive-url=https://web.archive.org/web/20200529232552/http://www.hawking.org.uk/godel-and-the-end-of-physics.html |url-status=dead }}
• Is mathematics invented or discovered? (millennia-old question, raised among others by Mario Livio){{Cite journal | author = Mario Livio | author-link = Mario Livio| title = Why math works? | journal = Scientific American | pages = 80–83 | date = August 2011 | url = http://www.scientificamerican.com/article/why-math-works/}}

In recent times the two disciplines have most often been taught separately, despite all the interrelations between physics and mathematics.Karam; Pospiech; & Pietrocola (2010). "[http://www.univ-reims.fr/site/evenement/girep-icpe-mptl-2010-reims-international-conference/gallery_files/site/1/90/4401/22908/29476/30505.pdf Mathematics in physics lessons: developing structural skills]" 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.Stakhov "[http://www2.fisica.unlp.edu.ar/materias/algebralineal/documentos/mathharm.pdf Dirac’s Principle of Mathematical Beauty, Mathematics of Harmony]"{{cite book|author1=Richard Lesh|author2=Peter L. Galbraith|author3=Christopher R. Haines|author4=Andrew Hurford|title=Modeling Students' Mathematical Modeling Competencies: ICTMA 13|url=https://books.google.com/books?id=Jj5tfi2594kC&pg=PA14|year=2009|publisher=Springer|isbn=978-1-4419-0561-1|page=14}} 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.https://bridge.math.oregonstate.edu/papers/ampere.pdf

{{div col|colwidth=28em}}

• Non-Euclidean geometry
• Fourier series
• Conic section
• Kepler's laws of planetary motion
• Saving the phenomena
• {{section link|Positron|History}}
• The Unreasonable Effectiveness of Mathematics in the Natural Sciences
• Mathematical universe hypothesis
• Zeno's paradoxes
• Axiomatic system
• Mathematical model
• Empiricism
• Logicism
• Formalism
• Mathematics of general relativity
• Bourbaki
• Experimental mathematics
• History of Maxwell's equations
• {{section link|Philosophy of mathematics|Platonism}}
• History of astronomy
• Why Johnny Can't Add
• Mathematical formulation of quantum mechanics
• Scientific modelling
• All models are wrong

{{div col end}}

{{reflist}}

• {{cite journal|last=Arnold|first=V. I.|author-link=Vladimir Arnold |title=Mathematics and physics: mother and daughter or sisters?|journal=Physics-Uspekhi|volume=42|issue=12|pages=1205–1217|year=1999|doi=10.1070/pu1999v042n12abeh000673|bibcode = 1999PhyU...42.1205A |s2cid=250835608 }}
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• {{cite journal|last=Hitchin|first=Nigel|author-link=Nigel Hitchin |title=Interaction between mathematics and physics|journal=ARBOR Ciencia, Pensamiento y Cultura|volume=725|url=http://arbor.revistas.csic.es/index.php/arbor/article/viewFile/115/116|year=2007|access-date=31 May 2014}}
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• {{cite journal |first=John von |last=Neumann |author-link=John von Neumann |title=The Mathematician |journal=Works of the Mind|volume=1|number=1|pages=180–196 |year=1947}} ([https://mathshistory.st-andrews.ac.uk/Extras/Von_Neumann_Part_1/ part 1]) ([https://mathshistory.st-andrews.ac.uk/Extras/Von_Neumann_Part_2/ part 2]).
• {{cite book|first=Henri |last=Poincaré |author-link = Henri Poincaré |translator=George Bruce Halsted |title=The Value of Science |publisher=The Science Press|location=New York |year=1907 |url=http://www3.nd.edu/~powers/ame.60611/poincare.pdf}}
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• [http://www.physics.rutgers.edu/~gmoore/PhysicalMathematicsAndFuture.pdf Gregory W. Moore – Physical Mathematics and the Future (July 4, 2014)]
• [http://www.iop.org/publications/iop/2014/file_64122.pdf IOP Institute of Physics – Mathematical Physics: What is it and why do we need it? (September 2014)]
• [https://www.youtube.com/watch?v=B-eh2SD54fM Feynman explaining the differences between mathematics and physics in a video available on YouTube]

Category:Philosophy of physics

Category:Philosophy of mathematics

Category:History of science

Category:Mathematics education

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Category:History of mathematics

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