Can mathematics get from other sciences what it got from physics? Throughout history, physics has been an unparalleled source of '' inspiration'' for discovering/inventing mathematical ideas, which is due to its ability to describe the physical world. But can this connection be made as profoundly with other fields of science like biology? Can other fields deepen our understanding of mathematics and generate new discoveries/inventions in it? Has this already happened? How so?
 A: You might find this interesting, it is written by an applied mathematician who is very well know in the finance industry. http://keplerianfinance.com/2013/05/what-is-keplerian-finance/
A: Yes! This might apply to more of computer science and algorithms but look at genetic algorithm which comes straight from Biology. It is a heuristic that mimics the process of natural selection to arrive at an optimal solution or configuration. Basically we define a metric and then generate random solutions and keep the ones that score high on our metric. Then we breed them while also generating random samples. More information here.
A: Of course from economics, as already said, think of Game Theory or Utility Theory.
In the 1940s-1950s a new field called Mathematical Psychology emerged, basic questions of measurement theory(*) (under what conditions and to what extend could qualitative data quantified, and what operations are permissable on that gathered data), of artificial intelligence, learning (for which Markov models where applied). Artificial intelligence is nowadays a part of its own, from connectionistic models, data mining, pattern matching etc all borrow ideas from mathematics. 
Also in the 1940s-1950s information theory emerged, which stems some ideas and analogies from thermodynmaics, but essentially has its origin in coding and transmission, so was motivated by electrical engineering.
Last but not least computer science, modelling programs, domain theory, database theory, verification borrowes much from logic and computability theory and also created much theory for themselves (non-hausdorff topology, reasoning about programs by calculus). But also Formal Language Theory, for example finite machines could be interpreted as monoids which opens a whole new door to the mathematical investigation of languages (see Eilenberg, J.E.Pin and many others who created a whole new mathematical theory around that). By the way, formal language theory has its origins also in psychological considerations (see Chomsky, who asked questions about a universal grammar "hard-wired in our brains", to said it simply). Also you might search for Donald Knuth, who wrote a multi-volume compendium on mathematics as applied to computer science, where much of combinatorics is applied to the analysis of algorithms.
And let me add, one of the oldest books, Euclids elements, where not directly motivated by solving physical problems (other then that our physical world permit counting), indeed the platonian view was that mathematical ideas exist outside the physical reality and so mathematics could in theory be done without any appeal to reality.
There are also problems from music, apart form such physical question at what is sound, vibrations, but more concerned with the structure of music itself and not its medium, which motivate new mathematical ideas, look for a guy called Guerino Mazzola who uses Category Theory in Music.
(*) by the way this is not measure theory which is applied in probability and integration,
A: This not exactly a science, but it is an outside subject that has helped a lot in the development of mathematics. It was and still is a source of inspiration. I'm talking about gambling or any game which depends on luck. 
Probability, during its early days, was not concerned with describing mathematics, or any other subject, but gambling. Nowadays probability has a special place in Quantum Mechanics, Economics, Biology, etc. And of course, probability has its special place in mathematics, being a fundamental subject of study for any mathematician. 
A: Yes, and I think that psychology has a great potential here. Perhaps not many scientist are devoted to this combination, but a "Newton" might be born somewhere in the jungle of category theory?
Human mind as such has probably absolutely nothing to do with mathematics and my intuition here is directed to the human way of thinking and making decisions. And since there initially could be no better objects to study than the mathematicians themselves, experts of thinking as they are, perhaps some day a light will start to twinkle and enlightening some darkness in human mind?
The benefit would be to extract and classify the patterns of human thinking and their consequences, possibilities and drawbacks. Not in vague philosophical ways or with statistics, but as mathematical models of categories of thinking. 
When analyzing the constructive way of thinking, mathematically, the less valuable and even delusive patterns may appear more clear for psychologists and psychiatrists. But also, the study would lead mathematics forward if restraints in our thinking become clear and new examples from reality enriches the theories.
I'm especially interested in qualitative mathematical-psychological theories. The quantitative theories from the 19's and the 20's seems uninteresting and elusive.
I think that mathematics is just as much about inner world as about outer world and might be considered as an extension or product of those two worlds together.

I recently found topics like this
Characterizing projective geometry of binocular visual space by Möbius transformation
Which seems interesting both from the perspective of psychology and of mathematics.
