During a discussion with a friend over the weekend, the proposition arose that scientific and technological progress is largely limited by mathematics. The transistor relies on quantum mechanics, which couldn't exist without mathematics, general relativity calls for mathematics beyond the undergraduate level, even biology/biochemistry rely on instruments whose function relies on principles that are described mathematically.

With that in mind, I wonder whether there are any areas - most likely various fields of physics - that feel limited by the mathematical tools available. A few months ago, I heard or read somewhere that progress in string theory, for example, is limited by the lack of analytic solutions to some problems, though I can't remember the source, much less vouch for it.

My current research is in computational neuroscience and attempts to describe the neuron using a set of differential equations; if even a small subset of the celullar machinery requires such a complex system of equations to be described, can the tools of today's mathematics possibly fully describe even a single cell to completion? My feeling is that it can't be done today and that some new paradigm might be necessary, in mathematics or otherwise.