Relationship between mathematics and physics

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A cycloidal pendulum is isochronous, a fact discovered and proved by Christiaan Huygens under certain mathematical assumptions. CyloidPendulum.png
A cycloidal pendulum is isochronous, a fact discovered and proved by Christiaan Huygens under certain mathematical assumptions.
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. Conic Sections.svg
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.

Contents

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]

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]

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. [20] 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. [21]

The mathematical rigor of Dirac's delta function was in doubt until the works of Laurent Schwartz on the theory of distributions. [22]

As time progressed, the mathematics used in physics has become increasingly sophisticated, as in the case of superstring theory. [23] Unconventional connections between the two fields are found all the time as in 1975 Wu–Yang dictionary, that related concepts of gauge theory with differential geometry. [24]

Physics is not mathematics

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. For example, Newton built a physical model around definitions like his second law of motion 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. [25] Mathematics deals with entities whose properties can be known with certainty. [26] According to David Hume, only in logic and mathematics statements can be proved (being known with total certainty). While in the physical world one can never know the properties of its beings in an absolute or complete way, leading 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." [27] The ultimate goal in research in pure mathematics are rigorous proofs, while in physics heuristic arguments may sometimes suffice in leading-edge research. [28] In short, the methods and goals of physicists and mathematicians are different. [29]

Role of rigor

Rigor is indispensable in pure mathematics. [30] 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. [28] [31]

Philosophical problems

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

Education

In recent times the two disciplines have most often been taught separately, despite all the interrelations between physics and mathematics. [42] 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. [43] [44] 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. [45]

See also

Related Research Articles

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Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory, algebra, geometry, analysis, and set theory.

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<span class="mw-page-title-main">Physics</span> Scientific field of study

Physics is the scientific study of matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. Physics is one of the most fundamental scientific disciplines. A scientist who specializes in the field of physics is called a physicist.

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries the gravitational force. Thus, string theory is a theory of quantum gravity.

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<span class="mw-page-title-main">Henri Poincaré</span> French mathematician, physicist and engineer (1854–1912)

Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The Last Universalist", since he excelled in all fields of the discipline as it existed during his lifetime. He has further been called the "Gauss of modern mathematics". Due to his success in science, influence and philosophy, he has been called "the philosopher par excellence of modern science."

<span class="mw-page-title-main">Edward Witten</span> American theoretical physicist

Edward Witten is an American theoretical physicist known for his contributions to string theory, topological quantum field theory, and various areas of mathematics. He is a professor emeritus in the school of natural sciences at the Institute for Advanced Study in Princeton. Witten is a researcher in string theory, quantum gravity, supersymmetric quantum field theories, and other areas of mathematical physics. Witten's work has also significantly impacted pure mathematics. In 1990, he became the first physicist to be awarded a Fields Medal by the International Mathematical Union, for his mathematical insights in physics, such as his 1981 proof of the positive energy theorem in general relativity, and his interpretation of the Jones invariants of knots as Feynman integrals. He is considered the practical founder of M-theory.

In philosophy, the philosophy of physics deals with conceptual and interpretational issues in physics, many of which overlap with research done by certain kinds of theoretical physicists. Historically, philosophers of physics have engaged with questions such as the nature of space, time, matter and the laws that govern their interactions, as well as the epistemological and ontological basis of the theories used by practicing physicists. The discipline draws upon insights from various areas of philosophy, including metaphysics, epistemology, and philosophy of science, while also engaging with the latest developments in theoretical and experimental physics.

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences" is a 1960 article written by the physicist Eugene Wigner, published in Communication in Pure and Applied Mathematics. In it, Wigner observes that a theoretical physics's mathematical structure often points the way to further advances in that theory and to empirical predictions. Mathematical theories often have predictive power in describing nature.

<span class="mw-page-title-main">Mathematical physics</span> Application of mathematical methods to problems in physics

Mathematical physics refers to the development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics, known as physical mathematics.

<span class="mw-page-title-main">Pure mathematics</span> Mathematics independent of applications

Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles.

In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.

<span class="mw-page-title-main">Greek mathematics</span> Mathematics of Ancient Greeks

Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly from the 5th century BC to the 6th century AD, around the shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire region, from Anatolia to Italy and North Africa, but were united by Greek culture and the Greek language. The development of mathematics as a theoretical discipline and the use of deductive reasoning in proofs is an important difference between Greek mathematics and those of preceding civilizations.

<span class="mw-page-title-main">Theoretical physics</span> Branch of physics

Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena.

<span class="mw-page-title-main">Shahn Majid</span>

Shahn Majid is an English pure mathematician and theoretical physicist, trained at Cambridge University and Harvard University and, since 2001, a professor of mathematics at the School of Mathematical Sciences, Queen Mary, University of London.

<span class="mw-page-title-main">Matilde Marcolli</span> Italian mathematician and physicist

Matilde Marcolli is an Italian and American mathematical physicist. She has conducted research work in areas of mathematics and theoretical physics; obtained the Heinz Maier-Leibnitz-Preis of the Deutsche Forschungsgemeinschaft, and the Sofia Kovalevskaya Award of the Alexander von Humboldt Foundation. Marcolli has authored and edited numerous books in the field. She is currently the Robert F. Christy Professor of Mathematics and Computing and Mathematical Sciences at the California Institute of Technology.

<span class="mw-page-title-main">Sergio Doplicher</span> Italian mathematical physicist

Sergio Doplicher is an Italian mathematical physicist, who mainly dealt with the mathematical foundations of quantum field theory and quantum gravity.

<span class="mw-page-title-main">Giovanni Felder</span> Swiss physicist and mathematician

Giovanni Felder is a Swiss mathematical physicist and mathematician, working at ETH Zurich. He specializes in algebraic and geometric properties of integrable models of statistical mechanics and quantum field theory.

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.
  2. ↑ 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.
  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.
  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.
  5. ↑ 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.
  6. ↑ Lear, Jonathan (1990). Aristotle: the desire to understand (Repr. ed.). Cambridge [u.a.]: Cambridge Univ. Press. p.  232. ISBN   9780521347624.
  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.
  8. ↑ 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.
  9. ↑ Aharon Kantorovich (1 July 1993). Scientific Discovery: Logic and Tinkering. SUNY Press. p. 59. ISBN   978-0-7914-1478-1.
  10. ↑ Kyle Forinash, William Rumsey, Chris Lang, Galileo's Mathematical Language of Nature Archived 2013-09-27 at the Wayback Machine .
  11. ↑ 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.
  12. ↑ 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
  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.
  16. ↑ Eoin P. O'Neill (editor), What Did You Do Today, Professor?: Fifteen Illuminating Responses from Trinity College Dublin, p. 62.
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  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.
  19. ↑ David E. Rowe (2008). "Euclidean Geometry and Physical Space". The Mathematical Intelligencer . 28 (2): 51–59. doi:10.1007/BF02987157. S2CID   56161170.
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  21. ↑ 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.
  22. ↑ "The coevolution of physics and math", Symmetry Magazine, April 24, 2018
  23. ↑ "String theories". Particle Central. Four Peaks Technologies. Retrieved 13 June 2015.
  24. ↑ 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.
  25. ↑ 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.
  26. ↑ Ernest, Paul (2003). The philosophy of mathematics education. Studies in mathematics education (Transferred to digital printing 2005; [Reprint of the ed. 1991] ed.). New York, NY: Routledge. ISBN   978-1-85000-667-1.
  27. ↑ Fundamentals of Physics - Volume 2 - Page 627, by David Halliday, Robert Resnick, Jearl Walker (1993)
  28. 1 2 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"
  29. ↑ 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.
  30. ↑ Steven Weinberg, To Explain the World: The Discovery of Modern Science, pp. 9–10.
  31. ↑ 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
  32. ↑ Albert Einstein, Geometry and Experience.
  33. ↑ Pierre Bergé, Des rythmes au chaos.
  34. ↑ 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.
  35. ↑ 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.
  36. ↑ "FQXi Community Trick or Truth: the Mysterious Connection Between Physics and Mathematics". Archived from the original on 14 December 2021. Retrieved 16 April 2015.
  37. ↑ 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.
  38. ↑ 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.
  39. ↑ 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.
  40. ↑ "Stephen Hawking. "Godel and the End of the Universe"". Archived from the original on 2020-05-29. Retrieved 2015-06-12.
  41. ↑ Mario Livio (August 2011). "Why math works?". Scientific American: 80–83.
  42. ↑ Karam; Pospiech; & Pietrocola (2010). "Mathematics in physics lessons: developing structural skills"
  43. ↑ Stakhov "Dirac’s Principle of Mathematical Beauty, Mathematics of Harmony"
  44. ↑ 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.
  45. ↑ https://bridge.math.oregonstate.edu/papers/ampere.pdf

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