I have to confess at school I was not very good at maths. I managed to fail my O-level when I was 16 though a year later I did a retake and was able to pass. I don’t really blame my teachers because I probably wasn’t ready for it. I didn’t really see the relevance and I didn’t really have a natural aptitude for it. Now in later life I have started to appreciate its importance. The problem is that I lack that foundation. Even so I think that maths is not entirely a hierarchical subject and anyone can appreciate its beauty and the insight it gives on our world. It is essential to have an understanding of mathematics. I don’t just mean to avoid financial mismanagement and running up consumer debt. It goes much deeper than that.
For me probability is the most illuminating area of mathematics. An example is the way people think about risk. We seem to have a poor assessment of risk. After a plane crash many people refuse to fly but have no problem taking the car, which is far more dangerous.
We are often fooled by randomness. If there are a lot of cases of cancer near a mobile phone mast, this does not necessarily mean that there is a problem with the mast. When we think of bad things occurring randomly, we seem often to expect that they will be evenly distributed. But a cluster is perfectly ordinary; in fact what would really be bizarre would be a world where there were no clusters. It is like the gambler’s fallacy where people think if it has been heads four times in a row, it must be time for tails.
The history of how probability theory came about is vital for understanding our contemporary world. In Against the Gods Peter Bernstein shows how a series of groundbreaking discoveries in mathematics helped to spark the affluence of the Western capitalist world. We can have investments and stock markets and the economic growth that they permit because we understand the concepts of probability and risk management. Many of those who worked on this in the 16th and 17th centuries were compulsive gamblers, but also mathematicians and fascinated by the mechanics of gambling. One of them was Pascal who thought about the existence of God and thought about how to manage the risk implicit in that bet. This is of course the famous Pascal’s Wager. The important point for Pascal is that risk is not just about probabilities but also consequences. It goes like this:
If you believe in God and you are wrong, you lose nothing (if death is the absolute end), whereas if you correctly believe in God, you gain everything (eternal salvation). But if you correctly disbelieve in God, you gain nothing (death ends all), whereas if you wrongly disbelieve in God, you lose everything (eternal damnation).
A sceptical perspective is provided by Nassim Nicholas Taleb author of The Black Swan. He describes himself as a “sceptical empiricist” and a “philosopher of probability”. For Taleb too many experts overestimate the value of of rational explanations of past data and underestimate the prevalence of inexplicable randomness in that data. The title of the book refers to the fact that we tend to believe all swans are white because we’ve probably never seen a black swan. Taleb distinguishes between two domains of chance – Mediocristan and. Extremistan. For example people’s weights lie in Mediocristan. Some people are fat others are thin but even a very fat person is only a few times heavier than a very thin person and it will still only represent an insignificant percentage of the total weight of everyone. They will converge at a normal (Gaussian) distribution. This is what Taleb calls “mild randomness”. Now, for Extremistan, instead of weight, think about income Think of Bill Gates or Warren Buffett. Their wealth is a million times greater than that of the average person. This is “wild randomness” The exceptional in this case is not unimportant. In the summer of 1982, large American banks suddenly lost close to all their past earnings made cumulatively in the history of American banking. They were fooled by a small sample; they’d had a number of consecutive good years and had forgotten about the small probability of something really bad happening. This kind of thinking is rife in financial markets, although they do not have the monopoly. According to Taleb no statistician produced a model that predicted the subprime crisis. His conclusion is that we can trust experts and statistics with events that come in a normal distribution but we must be prepared to admit our ignorance in these events which have an extremely low probability but with exceptionally severe consequences.
Further listening and reading
More or Less This BBC programme is presented by Tim Harford, author of the best selling book The Undercover Economist and it deals with the world of numbers. You can listen to all the past programmes They have included stories about the numbers behind the immigration debate, measuring poverty, how much alcohol is too much, whether the financial mathematicians known as “quants” are to blame for the current credit crisis and Eurovision voting.
Who’s Counting? In this website a prestigious mathematician analyses the news from a mathematical perspective.
Nassim Taleb talks about the challenges of coping with uncertainty, predicting events, and understanding history.