Wednesday, 31 August 2011

Critical Math

A Mathematician Reads the Newspaper, John Allen Paulos

Ah, unrequited love. I love maths, but maths doesnt love me. Still, it likes me enough for me to learn a lot from books like this. And I, like most people, do need to learn a lot about maths, because not knowing about it can lead you to make all sorts of mistakes and fall into all kinds of misunderstandings.
So we need more people like the mathematician John Allen Paulos, who knows a lot about maths and can express what he knows simply and entertainingly. This book is one of those that divide your life into BR and AR Before Reading and After Reading because it changes the way you look at the world. Take politics and important questions like the way we vote and the way power blocs work. Paulos examines all sorts of paradoxes and contradictions in both and you should come out of that section understanding the imperfections and dangers of democracy a lot better, as well as knowing that its possible to create a set of four dice, A, B, C, and D, in which A beats B, B beats C, C beats D, and D beats A.
Impossible? No, its very simple
Pauloss answers are, respectively, no, not necessarily, and no, not necessarily. What is true of a general population is not necessarily true of its extremes:
As an illustration, assume that two population groups vary along some dimension height, for example. Although it is not essential to the argument, make the further assumption that the two groups heights vary in a normal or bell-shaped manner. Then even if the average height of one group is only slightly greater than the average height of the other, people from the taller group will constitute a large majority among the very tall (the right tail of the curve). Likewise, people from the shorter group will constitute a large majority among the very short (the left tail of the curve). This is true even though the bulk of the people from both groups are of roughly average stature. Thus if group A has a mean height of 58 and group B has a mean height of 57, then (depending on the exact variability of the heights) perhaps 90 percent or more of the those over 62 will be from group A. In general, any differences between two groups will always be greatly accentuated at the extremes.Discrimination undoubtedly exists, but where it exists and how much of an effect it has are not questions that can always be answered in simple ways. Paulos even describes how taking measures against discrimination can make its supposed effects worse.
Look before you leap, in other ways, and look with mathematically trained eyes. It will help you in all sorts of ways, from not being taken in by fallacious political arguments to not being ripped off. Suppose, Paulos asks, a pile of potatoes is left out in the sun. Its 99% water and weighs 100 pounds. A day later, its 98% water. How much does it weigh now?
If you cant work out the answer then you might be on your way to losing a lot of money if someone who does know it looks after your money or investments. Paulos explains the answer which, surprisingly (or not), is 50 pounds very clearly and simply, the way he explains the answers of all the other little puzzles he drops into the text as he discusses gossip, celebrity, cooking, bargains, infectious disease, and a myriad of other subjects that maths can either illuminate or obfuscate, depending on how well you understand it and the logic that underlies it.
once you know how. Or take the much vexed question of discrimination. Women are 50% of the population and blacks are (depending where you live) 5% and you should therefore expect them to be 50% and 5%, respectively, of MPs or bishops or disc-jockeys or senior managers in confectionery factories, shouldn't you? And if they aren't, that's clear proof of discrimination, isn't it?

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