I am reading more and more commentary around the value/lack thereof of Mathematics applied to Finance. You can find it in the comments on this very blog. The argument usually goes that the field has gone overboard on complexity, witness the synthetic CDO's that triggered the contagion of the subprime crisis. Other answer that mathematics is just about models and models are worth what they are worth no more no less. Basically it seems the debate is about ego between those that idolize models and those that intuitively distrust them.
Biology reminds me of Economics a lot. Except over there the debate is not that polarized. It is clear that mathematics is a support tool in the complex field of Biology. People use it for what it is good at and don't take it to the logical extremes. The field is still more medicine field practitioners than differential geometry theorists. And that is probably a good thing. It is however providing new insights in the emerging field of system biology.
Mathematic models are powerful predictive tools in Physics. That geometric considerations can unify theoretical models from gravity to quantum models boggles the mind, at least mine. And that is probably the problem with physics and physics/math graduates, it is so powerful you get intoxicated. One can easily forget the underlying reality and take the predictive mathematical models for the only reality that matters. Until they get contradicted by observation that is. But that doesn't happen often, hasn't really happened since the gravity and quantum revolutions.
It is easy, as a modern physicist, to get lulled in taking the mathematics to a non-sensical extreme as seems to be the case in modern theoretical physics where thousands of models cannot be distinguished by experiments. It results in a certain arrogance of the hard sciences.
But in the case of living system, and soft sciences, one cannot afford such certitudes, one doesn't have that luxury. You remain subject to and humbled by reality and nature. Continuously. How come all these PhD's didn't see the crisis coming laments the Queen of England this week?
In the case of biology, inherent complexity linked to the number of variables, what we know about them, and their feedback interaction gives rise to a "living system" that becomes quickly intractable, non-linear behavior being a staple dynamic signature of such complex systems. So in the end, the search for "one true model" feels as elusive and futile a pursuit in Economics as it is in Biology.