On Mathematics Education and the Financial Crisis
I was doing a mathematical finance search on knol and came across this article. I wholeheartedly agree with your characterization of the financial market 2008 crisis and ensuing recession as having its foundation in mathematical education.
First, because you are not an expert as you acknowledge, you miss the meaning of what a derivatives contract is in Finance. It is not what is commonly meant in mathematics. Please see my knol at http://knol.google.com/k/introduction-to-basis-instruments-contracts-bics-for-mathematics-finance-and#
Your comments on the dangers of exponential growth would have been more enriched by more awareness of Malthusian theory on population growth. See http://en.wikipedia.org/wiki/Thomas_Robert_Malthus
However, you hit something important on the instability of differentiation. I developed BICs theory after working for a number of years on wall street as a quant responsible for the pricing and hedging of complex derivatives. One of the issues I realized is that the mathematics taught at the best universities and used in risk management was really inadequate for effective risk management. As I worked to get at the core of what was the problem, it gradually emerged that continuous time continuous space mathematics that is at the heart of derivatives pricing and hedging and which comes from physics is really not suitable for quantitave financial risk management. (See my knol on derivatives greeks The Cases for BICs – Greek Derivatives, Derivatives’ Greeks, BICs or http://knol.google.com/k/the-wider-scope-of-bics). When you move from spacial continuity, the issues with traditional concepts of differentiation change. The instability problems you point out become more easily handled. (Note that Schwarz’s notion of generalized functions are also a way of addressing those problems, but still in a continuous space framework and towards addressing the needs of physicists)
The mathematics of continuity of the space and their stochastic part (stochastic differential equations, patial integro differential equations(PIDE),Stochastic Partial Differential Equations (SPDEs), etc.. ) which were developed for physicists, mostly in the first part of the 20th century are very seductive as they yield very surprisingly beautiful formulas that summarize a priori complex descriptions in ways that often seem magical. These were extremely useful to get a hint of reality in a pre-computing age. They are now a hindrance to managing an ever more complex world.
It led me to question the use of the traditional concept of probability for risk management purposes. It does not matter to me if I know that a coin toss has a 50/50 chance of head or tail, if we play a game if tail I lose all my net worth and head I win ten times as much. Losing all I have is way too risky a chance to take. Standard risk management would tell you that if the the expected benefit obtained by averaging each payoff with their likelihood is positive, I should play regardless of what my overall situation is. (This simplification is not to ignore the contributions of compensatory economic theories such as utility theory or prospect theory)
In developing BICs theory starting in 2001, I have laid down a resilient mathematical framework that solves a significant number of hardcore mathematical issues (See my knol “the wider scope of BICs” http://knol.google.com/k/the-wider-scope-of-bics). while addressing the issues I have outlined.
One of the major issues I had anticipated might arise, but not for so long is the extent to which the establishment is reticent to accept new ideas, even after a crisis which I thought was unavoidable. Even people whom I had thought would be more receptive or at least willing to engage have been disappointing. There seems to be a latent view that only certain kinds of people can have ideas worth listening to and even when they are proven right, if they do not have loud mouths, they are still dismissed as out of the mainstream cranks…
While I may agree that 99.99% of self proclaimed original or revolutionary thinkers out there are totally nuts and their so called ‘ideas’ not worth the material they are printed on, there is that 0.001% who largely make up for the failed 99.99% and there has to be a better system for giving such individuals a chance to emerge. One of my early teachers has made a fortune with a book on that theme: “The Black Swan”. And that is why I am so grateful for the Knol effort and the tools and level of freedom of expression it provides, even for those with less means to get heard. So this one thumb up is for the Knol platform creators.
More can be found on my knols and blog and their links and in my book on BICs, BICs 4 Derivatives Volume I : Theory