Examples of intuition from fields other than Physics to solve math problems 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?
 A: 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).
A: 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.


 
 
 
 
 


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.

A: 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.
A: 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.
A: 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.
A: 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.
A: 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.  
A: 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
