Extreme Quants, I thought, I say a few words about maths and how it changed my life. You can find more about my background here.
Although I find mathematics is a result of the evolution of the "lazy" brain (a theorem proof frees you from testing many cases, the theorem holds for infinite many cases), I never found it boring. But as an algebraist, a kind of an "abstractonaut", I did not see immediately when maths became "real".
In my business as factory automator I first dealt with geometric modeling (later with the kinematics and dynamics of numerically controlled mechanisms, vehicles, ...). Constructing complicated geometric objects from primitives, like boxes, wedges, spheres, cylinders, .. by union and difference operations was a kind of algebraic approach with exact results for a limited zoo of geometric objects.
We came closer to reality by using bounded geometry representations, by, say, NURBS approximating any shape. But whatever the representations of your geometric objects are they have pros and cons from their usage. If you are interested in manufacturing, you see geometries as features, if you want to calculate the weight as a solid ....
This was the time when I learned how important it is to distinguish between models and their representation - to organize them orthogonally.
Physical vs financial modeling and solving
It is said: maths is the "language of nature". It is used to represent (physical) reality. Tested and verified in experiments, it enables predictive models forecasting very precise future behavior of a physical system. Real maths - what a great concept.
Economic and financial modeling is a different story - the maths is not longer "real". Because a forecasted behavior (say, a fair price of an instrument) influences itself when "published". We talk about price dynamics and even turbulences that are unpredictable.
Not so few respond to this by recommending to use simple maths only or even end up with giving it up.
But also the behavior of physical systems is often not so predictable as we think (see the Blank Swan of Metal Forming)
Harnessing the power of feedback loops
In factory automation feedback loops are used to control interacting machine tools, robots, handling and assembly systems, ... the wished system behavior is a result of constant instantiation, calibration and recalibration of models in feedback loops.
Not so different from valuation and risk management in finance. Risk managers can use information about the market dynamic and the impact of their actions to change them.
The decision process must be evident, the information must be inverted into valuation and risk information, recalibration adjusts the risk management models for a better next act. This process goes on and on.
The implementation of such a process needs a special maths - the maths of inverse problems. We, at UnRisk have put a lot of effort and inverse problems expertise in doing calibrations right.
Picture from sehfelder