From E. O. Wilson (via Wonkblog, via PK), the suggested answer is “not much” – that, when it comes to research breakthroughs in the science, there is much more to it than mathematics:
Everyone sometimes daydreams like a scientist. Ramped up and disciplined, fantasies are the fountainhead of all creative thinking. Newton dreamed, Darwin dreamed, you dream. The images evoked are at first vague. They may shift in form and fade in and out. They grow a bit firmer when sketched as diagrams on pads of paper, and they take on life as real examples are sought and found.
Pioneers in science only rarely make discoveries by extracting ideas from pure mathematics. Most of the stereotypical photographs of scientists studying rows of equations on a blackboard are instructors explaining discoveries already made. Real progress comes in the field writing notes, at the office amid a litter of doodled paper, in the hallway struggling to explain something to a friend, or eating lunch alone. Eureka moments require hard work. And focus.
I’d love to compile (or see an already-compiled version of) a list of some of the most significant research contributions in economics in the last, say, 20 years, accompanied by a discussion of the degree of required mathematical rigor.