Is the crisis basically a financial crisis caused by irresponsible financiers, or lies the root of the crisis deeper? Can we get out of it? Or is the crisis here to stay? Let’s seek some answers, in mathematics by mathematics.
The Danger of Exponential Growth
The exponential function
y(t) = exp(t) is maybe the most perfect and ego-centered function of all functions, defined by the differential equation
dy/dt = y(t) with initial condition y(0) = 1,
stating that the rate of growth or derivative dy/dt of y(t) with respext to time t is equal to y(t) itself. How perfect to be equal to the derivative of oneself!
The rate of growth dy/dt of the exponential function has the same form of percentual growth as we expect of the GNP of our economy or our money on a savings account, which we before the crisis expected to be around 3% a year.
The exponential function looks harmless, but is not because sooner or later it becomes very very large. Exponential growth is dangerous, and unstable. Anything growing exponentially will explode in finite time.
We recall the wheat and chessboard fable
with a mathematician suggesting to the King as a prize for inventing the game of chess, to receive as many grains of wheat as would result from putting 1 grain on the first square and doubling on each successive square. The King, who was not a mathematician, thought this was pretty modest, but changed opinion when realizing that total number of grains would be 18,446,744,073,709,551,615 corresponding to 37 cubic-kilometers of grain:
Exponential growth from doubling every unit step.
states that the computing power doubles every 18 months, which according to Kurzweil
already in the 21st century century will lead to an explosion of intelligence or technological singularity
based on infinite computing power shaping an infinitely fast development of technology engaging the whole Universe in a gigantic intelligent computation:
Of course, there will be no technological singularity because exponential growth is exponentially unstable and exponentially unstable processes cannot be controled. Anything growing exponentially, will break down
because of instability. The only exception is a nuclear bomb explosion which is so fast and viloent that
break-down or blow-up is the same as the explosion itself:
The Myth of Western Leadership
General Motors and Ford are now facing bankruptcy. The once leading US car industry has not been able to reform to meet the challenges of the 21st century and is no longer competitive.
A main reason for the lack of vitality is that engineering education has resisted reform and is based on mathematics programs formed 100 years ago, long before the computer revolution. A clash of civilizations beween new smart technology based on advanced computation
and old-fashioned dumb technology based on primitive pre-computer computation, is now developing in science, education and industry. The winner has to be inventive and cannot solely rely on a once glorious tradition:
The once successful T-Ford.
Traditional engineering education based on traditional mathematics is not the sole cause of the crisis, but
The Danger of Derivatives
In a capitalistic market system its possible to make business from differences in resources, technology, knowledge or demand, with small business thriving from small differences and big business on big differences. You can run a small consulting business of you know a little more than most people. If you happen to have a big oil well on your property, then you can start a big business.
The financial markets of derivates thrives on differences in the form of the fluctuations or rates of change or derivatives with respect to time of e.g. stocks on the stock market in the form of options. A successful day trader can run a small business buying and selling small amounts of stocks on a daily basis, but upscaling is risky:
- small business thriving on small differences can be (fairly) stable
- big business thriving on small differences is very unstable (cf. Instability of Capitalism).
The booming financial market of derivatives of the first decade of the 21st century can be described as big options business based on small differences or derivatives, and the inherent instability of such a system has contributed to the financial crisis.
In mathematical terms the financial crisis is thus a consequence of the instability of the operation of differentiation or computing derivatives: Small fluctuations dy of a function y(t) are turned into large variations
in the derivative dy/dt because the fluctuation dy is divided by the small time increment dt.
In retrospect, it is thus possible to explain the financial crisis as being mathematically inevitable, but this was not understood because the mathematical concept of derivate was not properly understood. What is needed is focus on analysis of stability aspects in mathematical finance
in particular, and on reform of mathematics education in general. Unless we are prepared to meet repeated crises.
To sum up, we are led to put blame traditional mathematics education for
- incorrect understanding of exponential growth
- old dumb technology
- the financial crisis
threatening to throw developed industrial countries like the US and Sweden into permanent recession. A reformation of mathematics education correcting these mistakes seems to be necessary, but strong forces oppose reform and the future of the (car) industry in the US and Sweden is not bright. It could have been different, with a different math education.
What Can Be Done?
The educational system is looking back and does not want (is not capable of) reform. What is needed is first debate on the role of mathematics in our information society and then political initiatives towards reform of mathematics/science/engineering education. This alone will not bring us out of the crisis but it is a necessary part of a rescue plan. And it is feasible if only the road blocks to debate reform could be removed. But the system is controled by a powerful lobby on a seemingly sinking ship of traditional mathematics education, and independently thinking politicians are rare… Can Obama stimulate Change?