On the Brink of Paradox

On the Brink of Paradox is a surprisingly delightful read. The book contains an amazing amount of accessible material even to college mathematics majors, while offering value or review for mathematicians, but never in a stale way. The material is pulled together from some beautiful classics of mathematical recreation and is an outgrowth of the author's columns in the Spanish edition of Scientific American.

 

Part of the surprise of this delight, however, was that the author is a professor not in the mathematics department at MIT, but in its philosophy department. However, to suggest that one is getting any kind of watering down would be way off the mark, and it takes relatively few pages to get into the heart of real analysis.

 

For example, Chapter 1 begins with the classical chestnut of the Hilbert Hotel, a purely theoretical structure that has rooms numbered 0, 1, 2, … ad infinitum. Each room is occupied by one person. If the full hotel were finite, it could not accommodate any more guests without "doubling up" or ejecting somebody from the hotel. Along now comes person X for the infinite Hilbert Hotel, and no problem: Each person in a room is asked to move to the next numbered room. This opens up the first room (zero) and guest X settles down there. From there, it is easy to extend to one million added people, or even countably infinitely many more.

 

How can this be? The author shows the dichotomy of the two historical approaches for comparing set sizes: by set inclusion or by one-to-one correspondence. From the viewpoint of the first, the naïve view would have been that this could not be done. From the modern era since Cantor and Dedekind, this is no longer a problem. With that, Rayo shows why set inclusion is not the preferred way of looking at this, thus setting off a review of functions, injections, bijections, cardinality of sets, number sequences, the power set of a set, equivalence relations, partial orderings, Cantor's diagonalization argument, and more. This is all done in a manner so pleasant that one barely notices all the mathematical machinery and power brought into play. The results are reinforced with several exercises in each section that either extends an existing argument or prove some result that has been only cited, but even then, generally with an amount of help proportional to the level of difficulty.

 

In theory, there is a lot to learn for an undergraduate, and a fair amount of the material may be familiar to the mathematician. However, there is enough independence of chapters to allow a number of ways to read the book. I wanted to skip ahead and jump to the Grandfather Paradox, for example, rather than dig sequentially. In fact, Chapter 3 introduces paradoxes, notably Omega Sequence Paradoxes, which includes a range that the author rates in a "Paradoxicality Grade" from 1 (boring) to 10 (exciting, education). Zeno's very familiar example garners just a 2, given how easily this is addressed with modern limits. On the other hand, I thought the assignment of only a 3 to Thomson's Lamp Paradox was a bit stingy; I would have given it a 4 or even a 5. The paradox here is attributed to a philosophy professor at MIT.. Rayo's version goes like this:

 

The problem is that for any moment the lamp is at one state before midnight, at a later moment before midnight it is in the other state. Good head-scratching meat, good logic, but not so much math. This puzzle made me think of my first exposure to a numerical variant of it in the 1992 MAA book Mathematical Cranks, by Underwood Dudley. In that version, one puts two numbers in an urn at each moment and pulls out just one earlier number, leading to asking what is in the urn at midnight. From one view, the number of numbers in the urn keeps growing as midnight approaches, but from another (the one that is actually correct), there will be no numbers left in the urn, as one can prove directly that no number can be left, despite the fact that, along the way, the size of the urn appears to keep growing.

 

Because Rayo's book can be read at different levels, it can be used to learn of many interesting ideas and historical controversies in mathematics, paradoxes, logic, and philosophy as the reader sees fit. I don't see as many non-mathematicians as mathematicians reading this because of the mathematical maturity, but it is very worthwhile to the library of anybody with even a modest interest in these topics.

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Dr. Michael W. Ecker had a 45-year teaching career and was Associate Professor of Mathematics until his retirement July 2016 from the Pennsylvania State University's Wilkes-Barre Campus, as well as a computer journalist and Editor of Recreational and Educational Computing (1986-2007).