Can mathematics get from other sciences what it got from physics? Throughout history, physics has b

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,