Sleight of Mind

Matt Cook is an economist, a composer, a storyteller (as the author of thrillers), and he performs as a magician. Several magical tricks rely on creating an intuitive expectation and then come up with a totally different result. This creates amazement and unbelief in the audience. This is also very much the effect of a paradox. Given that Cook is not a professional mathematician himself, it comes as a surprise to find rather much abstraction and mathematics in this book.

Logical paradoxes are often found in popular science books discussing mathematics, games, and puzzles. Many of these "popular" paradoxes you can also find in this book but there are many more. Although this book is written for a general public, it is not leisure reading, since the discussion of the paradoxes goes in depth and that requires precise definitions and sometimes it touches upon the foundations of logic, mathematics, probability, or whatever topic the paradox is about.

The different topics are arranged in different chapters and the format is always similar. There is a general introduction to the subject, and that involves the definition of the concepts that are required for the discussion of the paradoxes to follow. These are precise but the selected terms and the associated technicalities are restricted to a minimum. Only what is essential is defined and only as precise as needed. For example in the chapter on probability it is defined what a probability space is, and that involves a sample space, a sigma-algebra, and a probability function, which are described by words, rather than formulas. Of course it is also explained how a random variable and its density function are defined and the Bayes theorem is introduced (this inevitably results in a formula). So, there are some formulas, but they are suppressed as much as possible, describing the definitions mostly in words and by using examples. I guess this is intended not to shy away the non-mathematician, but if you are a mathematician, then, given the intended rigour, it feels a bit awkward and verbose. Of course some formulas cannot be avoided, for example to illustrate what is in the Principia Mathematica of Whitehead and Russell a formula here and there is unavoidable.

When Cook comes to the many examples of paradoxes, it assumes an attentive reader because the lack of formulas requires sometimes complicated sentences that are often almost philosophical. Also here, a returning format is used. First the paradox is formulated, wherever possible, mentioning its origin. Cook usually tells a story to make the paradox concrete for the reader, rather than formulating it in its mathematical or abstract form. Then the opposing explanations (often there are only two) are formulated. The main discussion then explains why one is wrong and the other is correct. Sometimes there are more possibilities and more than one explanation is possible depending on how some components are defined or interpreted, which happens when the problem is ill-posed or under-defined.

Let me give some examples that illustrate the types of paradoxes and the depth of the discussion. A first chapter is dealing with infinity, which is not the simplest one to start with, but it is also the underlying concept in some subsequent chapters. It is clearly a concept that has caused a lot of confusion throughout the history of mathematics and logic. First we are instructed about bijections and countable sets, Cantor's diagonalization process, the cardinals $\aleph_k$, and the continuity hypothesis. Then the paradoxes can be explained: Hilbert's Hotel, Stewart's HyperWebster Dictionary, and many more. After introducing some additional group theory also the Banach-Tarsky theorem is explained in some detail. Not really a proof, but still the reader is given some idea of why this seemingly impossible result holds. Zeno's paradoxes of motion are of course somewhat related to the concept infinity, and so these are discussed making use of what was obtained in the previous chapter. Thomson's lamp is also related. If a lamp is alternately switched on and off at time instances $1−2^{−n}$, then deciding whether at time $t=1$ the lamp will be on or off is impossible.

With chapter four, probability is introduced. The Simpson paradox and the Monty Hall problem are probably the best known but there are others that allow much more variations and require much more discussion. In the chapter on voting systems we are introduced to social choice theory and Arrow's impossibility theorem. This is not completely unrelated to the topic of game theory which plays a role in, for example, price setting in a economic system. The Braess paradox is the unexpected result that by adding an extra road to a traffic system, the traffic may be slowed down.

With self-reference we are back to the foundations of mathematics with axiomatic set theory, and, among others, the paradoxes of Russell (the set of al sets that are not a member of themselves) and the liar (I am always lying). Inevitably this leads to Gödel's incompleteness theorems, a theory of types, the ZFC axiomatic system, etc. Also the unexpected hanging is a tough paradox discussed here. Somewhat in the same style is the chapter on induction, where some elements of formal logic are introduced.

A chapter involving geometry has curves, areas, and volumes with fractal dimension. There is not really a paradox here, but the fact that a dimension can be a fraction and need not be integer is considered to be paradoxical. But there are other simpler geometric examples. In many calculus books, we find the hard-to-believe fact that we can create an infinitely large overhang by stacking bricks if brick $k$ (numbered from top to bottom) overhangs the underlying one by $1/(k+1)$. This is an example where the mathematical fact that $\sum_{k=}^\infty 1/k$ diverges is replaced by a "story" of stacking overhanging bricks. Some typical mathematical beginners errors can also give some unexpected results, dividing by zero for example, or summing divergent series.

Finally Matt Cook has invited some colleagues to discuss paradoxes from physics. With statistical mechanics, the reader learns about entropy, Maxwell's Demon, and other classics such as the Brownian Ratchet driven by Brownian motion, and the Feynman's sprinkler problem. The unexpected results of special relativity are well known, and quantum physics is still difficult to understand in all its consequences and different interpretations are still discussed today.

In the final chapter the age-old question whether mathematics is discovered or invented is tackled. As one might expect, the answer is not exclusive for one or the other.

Mind, the paradoxes that are mentioned in this survey, are only few and exemplary for the many examples that can be found in this book (there are over 75). I can imagine that for readers who are totally mathematically illiterate, some steps may be hard, if these use terminology or arguments that are taken for granted. Nevertheless also those are considered potential readers because there is a short addendum introducing some very elementary mathematical notation. Cook also added a rather extensive bibliography, but many of the references are papers where the paradoxes were originally formulated, or papers discussing the solution. Thus not really the popularizing kind of literature for further reading. The index though is well stuffed and useful, since there is sometimes cross referencing across the chapters.

I could spot a typo in the discussion of the Banach-Tarsky theorem. When discussing successions of irrational spherical rotations left, right, up, down, denoted as L,R,U,D, strings of these letters are formed to denote points on a sphere. Uniqueness requires eliminating the succession of opposite rotations (free group). Thus UD, DU, LR, or RL are not allowed in a string. However in the table page 25 appears the string DUL which is not allowed.

Adhemar Bultheel
Book details

Several paradoxes are analysed in depth. Some are well known others are less familiar: Zeno's paradoxes, Monty Hall problem, Banach-Tarsky theorem, paradoxes related to voting systems, self reference, but also statistical mechanics, special relativity and quantum physics and many more pass the review. The finale is a discussion of the ultimate question: Is mathematics invented or discovered?

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9780262043465 (hbk)
USD 34.95 (hbk)