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This is a chaser for the examples of using physical intuition to solve math problems question.

Physical intuition seems to be used relatively frequently for solving math problems as well as stating new interesting ones. What would be examples of interesting Math questions or frameworks or problem solutions derived from field other than Physics?

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    $\begingroup$ The question linked to concerns "physical intuition", not "intuition that derives from the subject called Physics". I think the answers given here belong perfectly well on that question, so I am voting to close as a duplicate $\endgroup$ – Yemon Choi Sep 17 '14 at 2:15
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    $\begingroup$ @Yemon Choi: If someone applies ideas from biology, chemistry, economics, or linguistics to solve a problem in mathematics, must it be through something that would be called physical intuition? $\endgroup$ – Douglas Zare Sep 17 '14 at 2:43
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    $\begingroup$ The other question says "For the purposes of this question let a 'physical intuition' be an intuition that is derived from your everyday experience of physical reality. Your intuitions about how the spin of a ball affects its subsequent bounce would be considered physical intuitions." So no, not everything falls under that heading, but the answers given below seem to do so, and so I think the scope of this question should be clarified in order to avoid yet another big list of tangentially-related cool-sounding stuff $\endgroup$ – Yemon Choi Sep 17 '14 at 2:54
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    $\begingroup$ I don't see the harm in letting the question stay open to see what happens, especially if the main danger is cool stuff. $\endgroup$ – usul Sep 17 '14 at 2:58
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    $\begingroup$ mathoverflow.net/questions/14782/… $\endgroup$ – Margaret Friedland Sep 17 '14 at 17:03
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Douglas Zare's comment mentioning linguistics brings to mind the important example of the Chomsky hierarchy. In the 50s, in the field of linguistics, Noam Chomsky introduced the notion of formal grammars and identified certain levels of complexity of formal grammars (regular, context-free, etc, these forming the above-mentioned hierarchy). These ideas then became important (even fundamental) in theoretical computer science as corresponding to hierarchies of formal languages, with associated automata that recognized them (e.g. Turing Machines). So I think this counts as an intuition and a framework from outside of math that came to play an important role within it.

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  • $\begingroup$ That's a good example. +1. $\endgroup$ – Joël Sep 17 '14 at 3:56
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The Brownian motion is certainly a fundamental piece of modern probability theory. In a strict sense it does not answer the question, because it became to interest mathematicians after, and because, Einstein's 1905 article on the subject. And of course Einstein was a physicist and his interest was for the brownian motion in physics, for example of molecules in a gas (even though the Brownian motion was first observed by a botanist observing pollen grains in water -- a secondary bronian motion created by the more fundamental molecular one studied later by physicists).

On the other hand, there is an other source for the mathematical interest in Brownian motion, older: financial modeling, in particular of the price of stocks and other financial assets. The fundamental work was done by Bachelier, a student of Poincaré, in his thesis in 1900, five years before Einstein's article. It contains a few basic results in common with Einstein's treatment, but then move in another direction. This paper was essentially forgotten for many years, until it was rediscovered by Samuelson, who built on these ideas to develop the modern framework of mathematical finance, which bolstered a big development in the theory of stochastic differential equations.

On the same kind of idea, the Black-Scholes formula for option pricing is, with Merton's point of view and proof, a purely mathematical, non-trivial theorem about stochastic differential equation. Yet its first proof, by Blach and Scholes, giving the same result under much more stringent hypotheses, was heavily based on financial intuition, especially the theory of marker equilibrium. So that's an example where the concrete intuition in the field of finance led to the discovery of a formula that was soon after reproved mathematically.

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Shannon's entropy has been useful to obtain results in combinatorics (like upper bounds on the size of $A\cup B = C \cup D$-free set systems). One doesn't have to interpret the expressions as entropy, but thinking of it that way is helpful. Of course like many things one could debate if information theory is part of physics (or mathematics).

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    $\begingroup$ I think the physicists will want to claim entropy as their baby: arising originally in classical thermodynamics, later re-interpreted in statistical physics, and from there entering into probability and information theory (which didn't exist before Shannon). Shannon himself was a mathematician and an electrical engineer. $\endgroup$ – Robert Israel Sep 17 '14 at 4:50
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An example of crystallography $\to$ mathematics?

This crystallographer discovered a $38$-face convex polyhedron that tiles $\mathbb{R}^3$:

Engel, Peter. Geometric crystallography. An axiomatic introduction to crystallography (1942). Also: Dordrecht et al.: Reidel, 1986. p.220.


          Fig. 10: Engel
The above drawing is from:

Grünbaum, Branko, and G. C. Shephard. "Tilings with congruent tiles." Bulletin of the American Mathematical Society, 3.3 (1980): 951-973.

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    $\begingroup$ Of course one might view crystallography as a subfield of physics... $\endgroup$ – Joseph O'Rourke Sep 17 '14 at 0:37
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    $\begingroup$ He may have been a crystallographer, but when he discovered this he was doing mathematics, unless this particular crystal exists in nature (which I strongly doubt). $\endgroup$ – Robert Israel Sep 17 '14 at 4:53
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The Lorenz equations were formulated by the meteorologist Edward Lorenz, and published in 1963 in Journal of the Atmospheric Sciences. This led to the mathematical study of what is now called chaos theory.

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    $\begingroup$ I think this is way too simplistic an account of the history of dynamical systems theory. The entire story of the 3-body problem and the discovery of homoclinic orbits by Poincaré, not to mention Fatou, Julia and the origins of complex dynamics are completely missing. Maybe a fair assessment is that Lorentz's work made dynamical systems theory more marketable and indicated the possibility of (further) connections to physical sciences (maybe climate is inherently unpredictable?!). It also helped certain terminology (e.g. "chaos" and "butterfly effect") make its way into pop culture. $\endgroup$ – Pablo Lessa Sep 17 '14 at 14:48
  • $\begingroup$ I didn't mean to give an account of the history of dynamical systems theory, just to point out that this was a case where a significant contribution to mathematics came from meteorology. $\endgroup$ – Robert Israel Sep 17 '14 at 15:34
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    $\begingroup$ Plus don't forget the contribution of Mary Cartwright who studied chaotic dynamical systems (from the world of radar) long before 1963: en.wikipedia.org/wiki/Mary_Cartwright $\endgroup$ – Dan Piponi Sep 17 '14 at 17:06
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    $\begingroup$ Lorenz equations are basically truncated Navier stokes, which are physics models of fluid flows. So this is basically physics ? $\endgroup$ – Piyush Grover Sep 17 '14 at 17:14
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    $\begingroup$ It is from a physical model, yes, but in this case a model that specifically had to do with meteorology: it's a toy model of the kind of convection that occurs in the atmosphere. $\endgroup$ – Robert Israel Sep 17 '14 at 19:17
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Much of R. A. Fisher's pioneering work on statistics, including analysis of variance, was motivated by his interest in evolution and genetics. Indeed, the term "variance" itself was first introduced by Fisher in his 1918 paper titled The Correlation Between Relatives on the Supposition of Mendelian Inheritance.

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This may be a stretch, in that it is not really about "solving math problems", but according to Amir Aczel's book on the subject, the Bourbaki collective (and in particular its emphasis on structure as an organizing principle in mathematics) was heavily influenced by the ideas of Structuralism that were current among philosophers (e.g. Saussure), anthropologists (e.g. Lévi-Strauss) and psychologists (e.g. Piaget) of the day.

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    $\begingroup$ The one time I seriously looked into this question, I concluded that structuralism in philosophy, anthropology and psychology had zero impact on Bourbaki, but that Bourbaki had a very slight impact on structuralism. Besides, reviews that I read of Aczel's book usually say 1) that this book is not reliable and 2) that the thesis of the book was that Bourbaki influenced structuralism, not the other way round. So I think this answer is incorrect. $\endgroup$ – Olivier Sep 17 '14 at 19:29
  • $\begingroup$ @Olivier - It has been a while since I read the Aczel book, but the impression I got from it was that Bourbaki was part of the Structuralist movement, and that the influence went in both directions simultaneously. I may be misremembering -- and even if I am remembering correctly, I can't vouch for the accuracy of Aczel's account. It did seem to me at the time that Aczel was exaggerating the connection between Bourbaki and the Structuralists to make a better story. $\endgroup$ – mweiss Sep 18 '14 at 15:22
  • $\begingroup$ That Aczel was exaggerating is a fixture of the reviews I have read. As for the thesis, I did look seriously into this question (albeit just once) and concluded that there was no influence of structuralism on mathematics. Of course, proving a negative is hard. $\endgroup$ – Olivier Sep 18 '14 at 19:36
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Will attempt an answer here (note that some answers actually mention physics-related examples despite the question, but i personaly like this :))

One example is information geometry and relations to (for example) (artificial) neural networks which helped solve various open problems in statistics and differential geometry.

The point here is the interplay between information-theory (Shannon et al), statistics (Fisher et al), differential geometry (Riemann, et al) and biology/physics (neural networks/boltzmann machines et al). In essense what is stated is that the natural metric on the neural manifold is the correlation (mutual information) between (neural) points.

Another facet is that it made clear why methods like Maximum Likelihood/MAP, et al, work and how are related.

Some refs:

http://www.ncbi.nlm.nih.gov/pubmed/19623488

http://www.maths.manchester.ac.uk/~kd/PREPRINTS/DodsonCSE.pdf

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  • $\begingroup$ How does it help to solve open problems in differential geometry? Your references only suggest applications in statistical modelling. $\endgroup$ – Ben McKay Sep 20 '14 at 11:17
  • $\begingroup$ @BenMcKay, hmm, i'm positive it has solved some open problems in statistics (eg the optimal estiomator problem of Fisher)(ref. Methods of Information Geometry). i think it has provided insights into differential geometry as well (eg dual connections, dually-flat spaces, metric from a general divergence etc..). i will certainly need to find some refs on this point, though, thanks for the notice $\endgroup$ – Nikos M. Sep 21 '14 at 9:58

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