Friday, 19 August 2011
Gábor, Gábor, Hej!
Paradoxes in Probability Theory and Mathematical Statistics, Gábor J. Székely
A fascinating book in a number of ways. First the obvious way: probability contains some of the strangest and most counter-intuitive mathematics easily open to amateurs and dabblers. I can actually understand a lot of this book, but it still stretches and even re-shapes my mind and my understanding of the world more far deeply than almost all art has ever done. There are very odd things to be found even in something as simple as the patterns of heads-and-tails in coin-tossing. For example, although HH and HT are equally likely to occur first when you start tossing a fair coin, “more tosses are necessary, on average, for HH than for HT to turn up”. That just doesn’t make sense at first glance. It’s a paradox, in other words, and if you can understand it you’ve taken a step even the most intelligent human beings were once completely unable to take.
Much less subtle, but probably much more important in life, is this:
Consider two random events with probabilities of 99% and 99.99%, respectively. One could say that the two probabilities are nearly the same, both events are almost sure to occur. Nevertheless the difference may become significant in certain cases. Consider, for instance, independent events which may occur on any day of the year with probability p = 99%; then the probability that it will occur every day of year is less than P = 3%, while if p = 99.99%, then P = 97%. (ch. 1, “Classical paradoxes of probability theory”, pp. 54-5)
Then there’s the question of why buses always seem to run “more frequently in the opposite direction”. The mathematics gets much trickier here, but that’s an example of how mathematical analysis, unlike so much of what passes for analysis in the modern humanities, extracts deep meaning from apparently simple things because it’s actually there to be extracted. Mathematics is both the most fundamental and the purest of all subjects, and is something that can unite minds across barriers of language, culture, and politics.This book is actually a good example of that, because it was first published not only in a communist country but in what is, to almost all Europeans, a very strange European language: Hungarian. Hungarian isn’t related to the Indo-European family spoken almost everywhere else. If this book had been written in French or German or Spanish, its original title would look more or less familiar to an English-speaker. But its original title in Hungarian ― Paradoxonok a véletlen matematikában ― looks very odd. Even without being told you could guess from some of the English that the book is a translation, but that adds to its charm and helps prove the universality of mathematics. A Hungarian can speak mathematics to anyone without an accent, and vice versa ― though I suspect that the mathematics here occasionally stutters because of typos. The English certainly does, but then the book was printed under communism, with all the inefficiency and carelessness that entailed. Communism is gone now, Hungarian and maths both continue, but maths will outlast Hungarian too, just as it will outlast all other languages spoken today. Vivat