He shot, he scores

The Mathematical Association of America's 2011 Euler Prize was recently awarded to Timothy Gowers, editor of the excellent Princeton Companion to Mathematics (2008). Here is the award committee's citation and Gowers' response (Source: 2011 JMM Prize Booklet):

Committee's Citation

The Princeton Companion to Mathematics is the kind of book that comes along only rarely—a vast compendium of mathematical information in the form of essays by experts on a wide variety of fields, in most cases bringing the reader up to date on developments in recent decades in a way that nonexperts can understand and appreciate. The Committee recommending this award realizes that it is the work of many: Professor Gowers and his two associate editors (June Barrow-Green and Imre Leader) as well as 133 distinguished contributors, a list that includes George Andrews, Sir Michael Atiyah, Béla Bollobás, Alain Connes, Ingrid Daubechies, Persi Diaconis, Jordan Ellenberg, Andrew Granville, János Kollár, Peter D. Lax, Barry Mazur, Dusa McDuff, Karen Hunger Parshall, Carl Pomerance, Peter Sarnak, and Terence Tao, to name but a few. Gowers singles out two of the many contributors for special recognition for their especially valuable help both in writing text and in editing work by others: Jordan Ellenberg and Terence Tao. The Committee, in recommending the award, singled out Professor Gowers because of his extraordinary achievement in putting this whole volume together (over 1000 pages of text) and also for writing a beautiful 76-page introduction as well as 68 of the 288 individual entries. The organization is thematic, with sections on the origins of modern mathematics, mathematical concepts, the various branches of the subject, the big problems, biographical essays, and, though the subjects are mainly confi ned to what we call “pure” mathematics, a section on the influence of mathematics on other fields.

That the level of exposition in this volume is so impressive will come as no surprise to anyone familiar with Professor Gowers’ superb but diminutive volume (a sharp contrast in length at roughly 150 pages), Mathematics: A Very Short Introduction (Oxford University Press, 2002). Both books display an exceptional talent for mathematical exposition.

The Companion has something for everyone who has any interest in mathematics. Many sections can be read with great benefit and considerable pleasure by mathematical amateurs and students. Overwhelmed as we are in the twenty-first century by the enormous size of mathematics, the professional mathematician can benefit from finding out what colleagues are doing in branches of mathematics that did not exist when many of us were in school. Anyone who wonders about “mirror symmetry,” “quantum groups, “vertex operator algebras,” “automorphic forms,” or “Ricci flow,” topics referred to in the current mathematical literature or even in the newspapers, can find help here.

Gowers in his preface points out that in deciding what to include he “simply aims to present for the reader a large and representative sample of the ideas that mathematicians are grappling with at the beginning of the twenty-first century, and to do so in as attractive and accessible a way as possible.” The book, he is quick to add, is not an encyclopedia and “does not have a serious online competitor: rather than competing with the existing Web sites, it complements them.”

Tim Gowers' Response

For a long time I have felt that there was a gap in the market for mathematics books that are much less formal than textbooks and monographs, but aimed at an audience that already knows a substantial amount of mathematics. The Princeton Companion to Mathematics was an attempt to do something about this. I am honoured and delighted that the effort that went into the book has been rewarded with the 2011 Euler Book Award, and in a more general way I am also very pleased that the committee has chosen to recognise a book that demands more of
the reader than most popular mathematics books.

The Princeton Companion to Mathematics was very much a collective undertaking. It could not have been finished without the hard work of June Barrow-Green and Imre Leader, my associate editors, and without the work of the large number of contributors, who were willing not just to send us their contributions, but also to cooperate in a long editing process. The editors also received huge support from Sam Clark of T&T Productions, who converted the authors’ files into a unified format and did a large amount of copyediting. We also received just the right balance of pressure and encouragement from Anne Saverese, the reference editor at Princeton University Press.

The response to the book has been very positive, which suggests to me that the gap in the market that I thought I had identifi ed was real. I hope that people are not just buying the book but also reading it, and that one of its central aims, to improve communication amongst mathematicians by helping them to understand what other mathematicians are doing, is to some extent being fulfilled.

A Base-ic Solution

The "obvious" and wrong answer to yesterday's question is 80%. It's wrong because it ignores the base rate information that we have been given, namely that 85% of the city's cabs are Green and 15% are Blue. Again, Bayes' Theorem is the standard mathematical technique to unravel this problem. But here's a simple, common-sense solution:

Suppose the city has a total of 100 cabs. This means 85 are Green and 15 are Blue. The witness correctly identifies the colour of a cab 80% of the time and wrongly identifies the colour of a cab 20% of the time. In other words, of the 85 Green cabs, the witness would, on average, identify 68 of them (80% of 85) as Green and 17 of them (20% of 85) as "Blue". Likewise, of the 15 Blue cabs, the witness would, on average, identify 12 of them (80% of 15) as Blue and 3 of them (20% of 85) as "Green". So overall, out of these 100 cabs, the witness "sees" 29 of them as Blue (17 "false" Blues plus 12 true Blues). Therefore, the probability that the cab involved in the accident was Blue rather than Green is 12/29, or approximately 41%.

Back to Base-ics

Another base rate problem (from Chapter 10 of Judgment under Uncertainty: Heuristics and Biases - "Evidential impact of base rates" by Tversky and Kahneman):

A cab was involved in a hit and run accident at night. Two cab companies, the Green and the Blue, operate in the city. You are given the following data:

(a) 85% of the cabs in the city are Green and 15% are Blue.

(b) A witness identified the cab as Blue. The court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colours 80% of the time and failed 20% of the time.

What is the probability that the cab involved in the accident was Blue rather than Green?

Fooled by Positiveness

The following question comes to this blog from The New England Journal of Medicine (1978), via Randomness (1998) by Deborah J. Bennett, via Fooled by Randomness (2005, 2nd edition) by Nassim Nicholas Taleb:

If a test to detect a disease whose prevalence is one in a thousand has a false positive rate of 5 percent, what is the chance that a person found to have a positive result actually has the disease, assuming you know nothing about the person’s symptoms or signs?

Almost half of the respondents (consisting of "20 house officers, 20 fourth-year medical students and 20 attending physicians, selected in 67 consecutive hallway encounters at four Harvard Medical School teaching hospitals") answered 95%. Only 11 participants got the correct answer: approximately 2%.

This isn't a trick question, but it is a tricky question because most people fail to take into account the prevelance of the disease (i.e., it afflicts, on average, one in every thousand people). In more technical language, we are dealing with conditional probabilities and not just marginal (i.e., non-conditional or simple) probabilities. The standard mathematical technique to deal with such problems is Bayes' Theorem (named after its discoverer, the Reverend Thomas Bayes). But that requires a whole lesson, or series of lessons, on its own.

Here's a simple, common-sense approach to the problem:

To begin with, assume that the test yields no false negatives. Suppose we test a randomly selected group of 1000 people. Based on the given information, we would expect just one of these people to have the disease and therefore give a true positive test. We would expect 5% of the remaining 999, or roughly 1000, healthy people to also test positive, i.e., about 50 false positive tests. In other words, out of the 51 positive tests, only one would be a true positive. Therefore the chance that a person found to have a positive result actually has the disease is 1/51, or approximately 2%.