Relationship between mathematics and physics
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. Generally considered a relationship of great intimacy, mathematics has already been described as "an essential tool for physics" and physics has already been described as "a rich source of inspiration and insight in mathematics".
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. 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", and two millennia later were also expressed by Galileo Galilei: "The book of nature is written in the language of mathematics".
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). 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, it has already been appointed that from the nineteenth century, mathematics started to become increasingly independent from physics). The creation and development of calculus were strongly linked to the needs of physics: 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. During this period there was little distinction between physics and mathematics, as an example: Newton regarded geometry as a branch of mechanics. As time progressed, increasingly sophisticated mathematics started to be used in physics. The current situation is that the mathematical knowledge used in physics is becoming increasingly sophisticated, as in the case of superstring theory.
Some of the problems considered 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).
- 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.
- Pure mathematics
- Applied mathematics
- Theoretical physics
- Mathematical physics
- Non-Euclidean geometry
- Fourier series
- Conic section
- Kepler's laws of planetary motion
- Saving the phenomena
- The Unreasonable Effectiveness of Mathematics in the Natural Sciences
- Zeno's paradoxes
- 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.
- Olaf Uhden, Ricardo Karam, Maurício Pietrocola, Gesche Pospiech, Modelling Mathematical Reasoning in Physics Education.
- 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.
- Sanjay Moreshwar Wagh; Dilip Abasaheb Deshpande (27 September 2012). Essentials of Physics. PHI Learning Pvt. Ltd. p. 3. ISBN 978-81-203-4642-0.
- Michael Atiyah, On the Work of Edward Witten, pp. 31–35.
- Aristotle on physics and mathematics.
- 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.
- Ibrahim Al-Rasasi, “All is number”.
- Aharon Kantorovich (1 July 1993). Scientific Discovery: Logic and Tinkering. SUNY Press. p. 59. ISBN 978-0-7914-1478-1.
- Kyle Forinash, William Rumsey, Chris Lang, Galileo's Mathematical Language of Nature.
- Arthur Mazer (26 September 2011). The Ellipse: A Historical and Mathematical Journey. John Wiley & Sons. p. 5. ISBN 978-1-118-21143-4.
- E. J. Post, A History of Physics as an Exercise in Philosophy, p. 76.
- Arkady Plotnitsky, Niels Bohr and Complementarity: An Introduction, p. 177.
- Roger G. Newton (1997). The Truth of Science: Physical Theories and Reality. Harvard University Press. pp. 125–126. ISBN 978-0-674-91092-8.
- Eoin P. O'Neill (editor), What Did You Do Today, Professor?: Fifteen Illuminating Responses from Trinity College Dublin, p. 62.
- Timothy Gowers; June Barrow-Green; Imre Leader (18 July 2010). The Princeton Companion to Mathematics. Princeton University Press. p. 7. ISBN 1-4008-3039-7.
- David E. Rowe, Euclidean Geometry and Physical Space.
- Albert Einstein, Geometry and Experience.
- Pierre Bergé, Des rythmes au chaos.
- Arnold, V. I. (1999). "Mathematics and physics: mother and daughter or sisters?". Physics-Uspekhi 42 (12). Retrieved 30 May 2014.
- Arnold, V. I. (1998). Translated by A. V. Goryunov. "On teaching mathematics". Russian Mathematical Surveys 53 (1): 229–236. Bibcode:1998RuMaS..53..229A. doi:10.1070/RM1998v053n01ABEH000005. Retrieved 29 May 2014.
- Atiyah, M.; Dijkgraaf, R.; Hitchin, N. (1 February 2010). "Geometry and physics". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368 (1914): 913–926. Bibcode:2010RSPTA.368..913A. doi:10.1098/rsta.2009.0227. Retrieved 29 May 2014.
- Boniolo, Giovanni; Budinich, Paolo; Trobok, Majda, eds. (2005). The Role of Mathematics in Physical Sciences: Interdisciplinary and Philosophical Aspects. Dordrecht: Springer. ISBN 9781402031069.
- Colyvan, Mark (2001). "The Miracle of Applied Mathematics" (pdf). Synthese 127: 265–277. Retrieved 30 May 2014.
- Dirac, Paul (1938–1939, Part II). "The Relation between Mathematics and Physics". Proceedings of the Royal Society (Edinburgh) 59: 122–129. Retrieved 30 March 2014. Check date values in:
- Feynman, Richard P. (1992). "The Relation of Mathematics to Physics". The Character of Physical Law (Reprint ed.). London: Penguin Books. pp. 35–58. ISBN 978-0140175059.
- Hardy, G. H. (2005). A Mathematician's Apology (First electronic ed.). University of Alberta Mathematical Sciences Society. Retrieved 30 May 2014.
- Hitchin, Nigel (2007). "Interaction between mathematics and physics". ARBOR Ciencia, Pensamiento y Cultura 725. Retrieved 31 May 2014.
- Harvey, Alex (2012). "The Reasonable Effectiveness of Mathematics in the Physical Sciences". Relativity and Gravitation, , () 43 (2011): 3057–3064. arXiv:1212.5854v1. doi:10.1007/s10714-011-1248-9.
- John von Neumann, The Mathematician (part 1) (part 2).
- Henri Poincaré, The Value of Science.
- Schlager, Neil; Lauer, Josh, eds. (2000). "The Intimate Relation between Mathematics and Physics". Science and Its Times: Understanding the Social Significance of Scientific Discovery. 7: 1950 to Present. Gale Group. pp. 226–229. ISBN 0-7876-3939-7.
- Vafa, Cumrun (2000). "On the Future of Mathematics/Physics Interaction". Mathematics: Frontiers and Perspectives. USA: AMS. pp. 321–328. ISBN 0-8218-2070-2.
- Edward Witten, Physics and Geometry.
- Eugene Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences.