Relationship between mathematics and physics explained

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.[1] Generally considered a relationship of great intimacy,[2] mathematics has been described as "an essential tool for physics"[3] and physics has been described as "a rich source of inspiration and insight in mathematics".[4]

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.[5] 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",[6] [7] and two millennia later were also expressed by Galileo Galilei: "The book of nature is written in the language of mathematics".[8] [9]

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] 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).[11] [12] The creation and development of calculus were strongly linked to the needs of physics:[13] 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.[14] During this period there was little distinction between physics and mathematics;[15] as an example, Newton regarded geometry as a branch of mechanics.[16] As time progressed, the mathematics used in physics has become increasingly sophisticated, as in the case of superstring theory.[17] 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.[18]

Physics is not math

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. For example, Newton built a physical model around definitions like

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.[19] Mathematics deals with entities whose properties can be known with certainty.[20] 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."[21]

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.[32] 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.[33] [34]

See also

Further reading

External links

Notes and References

  1. Uhden. Olaf. Karam. Ricardo. Pietrocola. Maurício. Pospiech. Gesche. Modelling Mathematical Reasoning in Physics Education. Science & Education. 20 October 2011. 21. 4. 485–506. 10.1007/s11191-011-9396-6. 2012Sc&Ed..21..485U . 122869677.
  2. Book: Francis Bailly. Giuseppe Longo. Mathematics and the Natural Sciences: The Physical Singularity of Life. 2011. World Scientific. 978-1-84816-693-6. 149.
  3. Book: Sanjay Moreshwar Wagh. Dilip Abasaheb Deshpande. Essentials of Physics. 27 September 2012. PHI Learning Pvt. Ltd.. 978-81-203-4642-0. 3.
  4. On the Work of Edward Witten . Atiyah . Michael . Michael Atiyah . 1990 . International Congress of Mathematicians . 31–35 . Japan . dead . https://web.archive.org/web/20170301004342/http://www.mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0031.0036.ocr.pdf . 2017-03-01 .
  5. Book: Lear. Jonathan. Aristotle: the desire to understand. 1990. Cambridge Univ. Press. Cambridge [u.a.]. 9780521347624. 232. Repr..
  6. Book: Gerard Assayag. Hans G. Feichtinger. José-Francisco Rodrigues. Mathematics and Music: A Diderot Mathematical Forum. 10 July 2002. Springer. 978-3-540-43727-7. 216.
  7. Web site: Ibrahim . Al-Rasasi . All is number . King Fahd University of Petroleum and Minerals . 21 June 2004 . 13 June 2015 . 28 December 2014 . https://web.archive.org/web/20141228132248/http://faculty.kfupm.edu.sa/math/irasasi/Allisnumber.pdf . dead .
  8. Book: Aharon Kantorovich. Scientific Discovery: Logic and Tinkering. 1 July 1993. SUNY Press. 978-0-7914-1478-1. 59.
  9. Kyle Forinash, William Rumsey, Chris Lang, Galileo's Mathematical Language of Nature .
  10. Book: Arthur Mazer. The Ellipse: A Historical and Mathematical Journey. 26 September 2011. John Wiley & Sons. 978-1-118-21143-4. 5. 2010ehmj.book.....M.
  11. E. J. Post, A History of Physics as an Exercise in Philosophy, p. 76.
  12. Arkady Plotnitsky, Niels Bohr and Complementarity: An Introduction, p. 177.
  13. Book: Roger G. Newton. The Truth of Science: Physical Theories and Reality. registration. 1997. Harvard University Press. 978-0-674-91092-8. 125–126.
  14. Eoin P. O'Neill (editor), What Did You Do Today, Professor?: Fifteen Illuminating Responses from Trinity College Dublin, p. 62.
  15. Book: Timothy Gowers. Timothy Gowers. June Barrow-Green. Imre Leader. The Princeton Companion to Mathematics. 18 July 2010. Princeton University Press. 978-1-4008-3039-8. 7.
  16. David E. Rowe. David E. Rowe. Euclidean Geometry and Physical Space. The Mathematical Intelligencer. 2008. 28. 2. 51–59. 10.1007/BF02987157. 56161170.
  17. Web site: String theories . . Particle Central . Four Peaks Technologies . 13 June 2015 .
  18. Book: Zeidler, Eberhard . Quantum Field Theory II: Quantum Electrodynamics: A Bridge between Mathematicians and Physicists . 2008-09-03 . Springer Science & Business Media . 978-3-540-85377-0 . en.
  19. Book: Feynman, Richard P. . The Feynman lectures on physics. Volume 1: Mainly mechanics, radiation, and heat . 2011 . Basic Books . 978-0-465-02493-3 . The new millennium edition, paperback first published . New York . Characteristics of Force.
  20. The Philosophy of Mathematics Education, by Paul Ernest (2002)
  21. Fundamentals of Physics - Volume 2 - Page 627, by David Halliday, Robert Resnick, Jearl Walker (1993)
  22. [Albert Einstein]
  23. Pierre Bergé, Des rythmes au chaos.
  24. Book: Gary Carl Hatfield. The Natural and the Normative: Theories of Spatial Perception from Kant to Helmholtz. 1990. MIT Press. 978-0-262-08086-6. 223.
  25. Book: Gila Hanna. Gila Hanna. Hans Niels Jahnke. Helmut Pulte. Explanation and Proof in Mathematics: Philosophical and Educational Perspectives. 4 December 2009. Springer Science & Business Media. 978-1-4419-0576-5. 29–30.
  26. Web site: FQXi Community Trick or Truth: the Mysterious Connection Between Physics and Mathematics . 16 April 2015 . 14 December 2021 . https://web.archive.org/web/20211214170940/https://fqxi.org/community/essay/rules . dead .
  27. Book: James Van Cleve Professor of Philosophy Brown University. Problems from Kant. 16 July 1999. Oxford University Press, USA. 978-0-19-534701-2. 22.
  28. Book: Ludwig Wittgenstein. R. G. Bosanquet. Cora Diamond. Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939. 15 October 1989. University of Chicago Press. 978-0-226-90426-9. 96.
  29. Book: Pudlák, Pavel. Logical Foundations of Mathematics and Computational Complexity: A Gentle Introduction. 2013. Springer Science & Business Media. 978-3-319-00119-7. 659.
  30. Web site: Stephen Hawking. "Godel and the End of the Universe" . 2015-06-12 . 2020-05-29 . https://web.archive.org/web/20200529232552/http://www.hawking.org.uk/godel-and-the-end-of-physics.html . dead .
  31. Mario Livio . Mario Livio. Why math works? . Scientific American . 80–83 . August 2011 .
  32. Karam; Pospiech; & Pietrocola (2010). "Mathematics in physics lessons: developing structural skills"
  33. Stakhov "Dirac’s Principle of Mathematical Beauty, Mathematics of Harmony"
  34. Book: Richard Lesh. Peter L. Galbraith. Christopher R. Haines. Andrew Hurford. Modeling Students' Mathematical Modeling Competencies: ICTMA 13. 2009. Springer. 978-1-4419-0561-1. 14.

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