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]

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 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 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]

Connections between the two fields sometimes only require identifing similar concepts by different names, as shown in the 1975 Wu–Yang dictionary, [25] that related concepts of gauge theory with differential geometry. [26] : 332 

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. 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.[ citation needed ] 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. [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]

Role of rigor in physics

Rigor is indispensable in pure mathematics. [34] 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. [31] [35] [36] [37]

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. [38] : ix  [39] : 2 

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. [50] 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. [51] [52] 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. [53]

See also

Related Research Articles

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.

M-theory is a theory in physics that unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1995. Witten's announcement initiated a flurry of research activity known as the second superstring revolution. Prior to Witten's announcement, string theorists had identified five versions of superstring theory. Although these theories initially appeared to be very different, work by many physicists showed that the theories were related in intricate and nontrivial ways. Physicists found that apparently distinct theories could be unified by mathematical transformations called S-duality and T-duality. Witten's conjecture was based in part on the existence of these dualities and in part on the relationship of the string theories to a field theory called eleven-dimensional supergravity.

<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.

<span class="mw-page-title-main">Paul Dirac</span> English theoretical physicist (1902–1984)

Paul Adrien Maurice Dirac was an English mathematical and theoretical physicist who is considered to be one of the founders of quantum mechanics. Dirac laid the foundations for both quantum electrodynamics and quantum field theory. He was the Lucasian Professor of Mathematics at the University of Cambridge, a professor of physics at Florida State University and a 1933 Nobel Prize in Physics recipient.

<span class="mw-page-title-main">Quantum mechanics</span> Description of physical properties at the atomic and subatomic scale

Quantum mechanics is a fundamental theory that describes the behavior of nature at and below the scale of atoms. It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science.

<span class="mw-page-title-main">Quantum field theory</span> Theoretical framework

In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. The current standard model of particle physics is based on quantum field theory.

<span class="mw-page-title-main">Richard Feynman</span> American theoretical physicist (1918–1988)

Richard Phillips Feynman was an American theoretical physicist. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, and in particle physics, for which he proposed the parton model. For his contributions to the development of quantum electrodynamics, Feynman received the Nobel Prize in Physics in 1965 jointly with Julian Schwinger and Shin'ichirō Tomonaga.

<span class="mw-page-title-main">Space</span> Framework of distances and directions

Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework.

Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship with other human activities.

<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.

<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 theoretical physics, Euclidean quantum gravity is a version of quantum gravity. It seeks to use the Wick rotation to describe the force of gravity according to the principles of quantum mechanics.

<span class="mw-page-title-main">History of quantum field theory</span>

In particle physics, the history of quantum field theory starts with its creation by Paul Dirac, when he attempted to quantize the electromagnetic field in the late 1920s. Major advances in the theory were made in the 1940s and 1950s, leading to the introduction of renormalized quantum electrodynamics (QED). The field theory behind QED was so accurate and successful in predictions that efforts were made to apply the same basic concepts for the other forces of nature. Beginning in 1954, the parallel was found by way of gauge theory, leading by the late 1970s, to quantum field models of strong nuclear force and weak nuclear force, united in the modern Standard Model of particle physics.

Quantum mechanics is the study of matter and its interactions with energy on the scale of atomic and subatomic particles. By contrast, classical physics explains matter and energy only on a scale familiar to human experience, including the behavior of astronomical bodies such as the moon. Classical physics is still used in much of modern science and technology. However, towards the end of the 19th century, scientists discovered phenomena in both the large (macro) and the small (micro) worlds that classical physics could not explain. The desire to resolve inconsistencies between observed phenomena and classical theory led to a revolution in physics, a shift in the original scientific paradigm: the development of quantum mechanics.

<span class="mw-page-title-main">Feynman checkerboard</span> Fermion path integral approach in 1+1 dimensions

The Feynman checkerboard, or relativistic chessboard model, was Richard Feynman's sum-over-paths formulation of the kernel for a free spin-⁠1/2⁠ particle moving in one spatial dimension. It provides a representation of solutions of the Dirac equation in (1+1)-dimensional spacetime as discrete sums.

<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">Branches of physics</span> Overview of the branches of physics

Physics is a scientific discipline that seeks to construct and experimentally test theories of the physical universe. These theories vary in their scope and can be organized into several distinct branches, which are outlined in this article.

Basil J. Hiley, is a British quantum physicist and professor emeritus of the University of London.

References

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  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.
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  10. ↑ Kyle Forinash, William Rumsey, Chris Lang, Galileo's Mathematical Language of Nature Archived 2013-09-27 at the Wayback Machine .
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  14. ↑ Arkady Plotnitsky, Niels Bohr and Complementarity: An Introduction, p. 177.
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  25. ↑ 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.
  26. ↑ 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.
  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.
  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.
  29. ↑ Russell, Paul, ed. (2016). The Oxford Handbook of Hume. Oxford University Press. pp. 34, 94. ISBN   978-0-19-049392-9.
  30. ↑ Fundamentals of Physics - Volume 2 - Page 627, by David Halliday, Robert Resnick, Jearl Walker (1993)
  31. 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"
  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.
  33. ↑ Roland Omnès (2005) Converging Realities: Toward a Common Philosophy of Physics and Mathematics p. 38 and p. 215
  34. ↑ Steven Weinberg, To Explain the World: The Discovery of Modern Science, pp. 9–10.
  35. ↑ 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
  36. ↑ Mark Steiner (1992), "Mathematical Rigor in Physics". https://www.taylorfrancis.com/chapters/edit/10.4324/9780203979105-13/mathematical-rigor-physics-mark-steiner
  37. ↑ P.W. Bridgman (1959), "How Much Rigor is Possible in Physics?" https://doi.org/10.1016/S0049-237X(09)70030-8
  38. ↑ Mitra, Asoke N.; Dyson, Freeman J., eds. (2000). Quantum field theory: a twentieth century profile. New Delhi: Hindustan Book Agency [u.a.] ISBN   978-81-85931-25-8.
  39. ↑ Zeidler, Eberhard (2006). 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.
  40. ↑ Albert Einstein, Geometry and Experience.
  41. ↑ Pierre Bergé, Des rythmes au chaos.
  42. ↑ 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.
  43. ↑ 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.
  44. ↑ "FQXi Community Trick or Truth: the Mysterious Connection Between Physics and Mathematics". Archived from the original on 14 December 2021. Retrieved 16 April 2015.
  45. ↑ 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.
  46. ↑ 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.
  47. ↑ 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.
  48. ↑ "Stephen Hawking. "Godel and the End of the Universe"". Archived from the original on 2020-05-29. Retrieved 2015-06-12.
  49. ↑ Mario Livio (August 2011). "Why math works?". Scientific American: 80–83.
  50. ↑ Karam; Pospiech; & Pietrocola (2010). "Mathematics in physics lessons: developing structural skills"
  51. ↑ Stakhov "Dirac’s Principle of Mathematical Beauty, Mathematics of Harmony"
  52. ↑ 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.
  53. ↑ https://bridge.math.oregonstate.edu/papers/ampere.pdf

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