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.
usul
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