The certainty principle

Nonplussed! Mathematical Proof of Implausible Ideas - The Pythagorean Theorem
June 22, 2007

Much of the joy of mathematics derives from the certainty of its assertions. Mathematical proof is incontrovertible. Once we know what numbers are and what multiplication is, we can be sure that seven eights make 56. We start from basic facts or definitions, argue logically and arrive at totally dependable, if sometimes non-­obvious, conclusions. That is what both these books are about. What they have in common is that they discuss mathematically derivable conclusions, are from the same publisher at about the same time and should both be accessible to a wide readership. Beyond that, they are very ­different.

Nonplussed! is a collection of lovely paradoxes: facts that are provable logically but are nevertheless seriously counterintuitive. Would you believe, for example, that a top-quality tennis player playing an equally good opponent has a smaller chance of winning the game when serving at 40-30 or at 30-15 than at its start? Of course not. But if the players are equally good, and good enough to win just over 90 per cent of their service points, then this is the case. The proof uses elementary calculation with probabilities at what in the English educational system is AS level and requires some basic facility with algebra.

Here is another one. Take three cubes and turn them into non-standard dice. Die A has a face showing 1 and five faces showing 4, die B has a face showing 6 and five faces showing 3, and die C has three faces showing 2, the other three showing 5. Being apparently more courteous than wily, you let your opponent choose first, then you choose one of the two remaining dice, and you both roll. If your opponent chooses A then you choose C, your probability of winning is 7/12; if your opponent chooses B then you choose A, your probability of winning is 25/36; if your opponent chooses C then you choose B, your probability of winning is 7/12. That is, A is better than B, which is better than C, which in turn is better than A. Your courtesy and self-interest coincide.
On February 13, 1970, The Times published a letter from the famous Cambridge mathematical astronomer Raymond A. Lyttleton. He wrote: “You may be willing to allow me, on this doubly unlucky date of Friday the Thirteenth of February, to remind any superstitious among your readers that the 13th day of the month falls more frequently on a Friday than upon any other day of the week.” There is no deep mathematics in this. Simple but careful counting yields that in each 400-year cycle of the Gregorian calendar the number of times that the 13th of a month is a Monday is 685, a Tuesday 685, a Wednesday 687, a Thursday 684, a Friday 688, a Saturday 684, and a Sunday 687.

About half of Julian Havil’s nicely chosen paradoxes come from applications of probability theory. The remainder range widely over other topics. In mechanics, for example, we have the roller that is so designed as to be able to roll uphill. In geometry there is Torricelli’s trumpet, a trumpet-like vase that has finite volume but infinite surface area, so that it is impossible to paint, and yet
if you filled it with paint and poured the surplus away then the inner surface would be painted.
In geometry again, spheres in high dimensional space exhibit quite unexpected behaviour. If you can imagine an n-dimensional sphere whose radius is one unit and compute its n-dimensional volume, you will find the following surprising facts: the ordinary three-dimensional sphere has volume 4p/3, which is very nearly 4.2; the four-­dimensional sphere has volume p2/2, which is just over 4.9; the five-dimensional sphere has volume 8p2/15, which is just over 5.2; but the volume of the six-dimensional sphere is p3/6, which is less than 5.2, and from there on, as the dimension grows, the volume gets smaller and smaller.

The mathematics required for an understanding of this sphere paradox is at first-year undergraduate level, and a few more require A-level mathematics, but many of them require no more than a clear head and an open and inquiring mind. This is an exciting book. It should be in every sixth-form and college library. It would be the right gift for mathematicians and anyone who uses mathematics — economists, business analysts and many others — and indeed for anyone who would claim to be educated.

The theme of Eli Maor’s book is the theorem that, in a right-angled triangle, the square on the hypotenuse is the sum of the squares on the two shorter sides. Until the “new math” displaced geometrical teaching in schools a generation or two ago, this was well known. It was widely learnt as Proposition 47 of Book I of Euclid’s Elements of Geometry , written in Greece well over 2,000 years ago, and which dominated Western European teaching of mathematics from late mediaeval times until about 1960. Knowledge of it can be detected, however, in ­earlier civilisations, which is why the subtitle of The Pythagorean Theorem is “A4,000-Year History”.

It is not only the subtitle that describes the book as a history; the author also writes in his preface that the book is aimed at the reader with an interest in the history of mathematics. It should also appeal to most well-educated people — but most is not all. I doubt if it will appeal to professional historians of mathematics. For them, it is likely to be too unspecific and unreliable in point of detail.

Too often the author tries to cover our ignorance about the past with conjecture — “there is only one plausible explanation: they must have known an algorithm, which 1,500 years later...”; “In all likelihood they used a geometric proof based on ...”. Writing of this kind recalls, on the one hand, Sherlock Holmes and his confidence that he has eliminated all other possibilities than the truth, and, on the other, Mark Twain’s cheeky observation (having just calculated that a million years ago the lower Mississippi ­River was well over a million miles long and that in 742 years it would be no more than a mile and a half): “There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact” ( Life on the Mississippi ). Rather, it would be better to acknowledge that there are some details about the distant past that we are never likely to know and should be content not to know until real evidence emerges.

Too often, also, words are used inappropriately. For example, Mersenne and Viète are described as “freelance mathematicians”, which in the context of their times is meaningless, and the first English translation (1570) of Euclid’s Elements is said to have been “attributed to” Sir Henry Billingsley, whereas in fact his name as translator is plain to see on the title page (and in 1570 he had not yet been knighted).

The book is a history only insofar as it has a timeline. Rather, it is a story based on a theme with variations and guided by its timeline. The variations are many and interesting. For example, chapter eight tells us about the 371 proofs of Pythagoras’ Theorem collected into a book by Elisha Scott Loomis, an American schoolteacher, and later chapters explain how the theorem forms the basis of the geometry that underlies the theory of relativity.

The story is not history, but it is a good one. As a popular account of important ideas and their development, the book should be read by anyone with a good education. It deserves to succeed.

Peter M. Neumann is fellow and tutor at The Queen’s College, Oxford.

Nonplussed! Mathematical Proof of Implausible Ideas

Author - Julian Havil
Publisher - Princeton University Press
Pages - 196
Price - £15.95
ISBN - 9780691120560

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