Sleight of Mind 75 Ingenious Paradoxes in Mathematics, Physics, and Philosophy

Merriam-Webster defines a paradox as: “a tenet contrary to received opinion; a statement that is seemingly contradictory or opposed to common sense and yet is perhaps true...” Paradoxes are probably ancient, and a recurring source of humor, but they also reveal problems, sometimes even limits, to the way we reason about things. 

 

Reason is ancient (many animals show signs of reasoning) but so far as we know, only humans make vast systems out of it. This is not merely an intellectual exercise: in his Magic, Reason and Experience : Studies in the Origins and Development of Greek Science, Geoffrey Lloyd proposes that reactions to mistakes, delusions, and fraud in science, medicine, politics, and religion led to the system we inherited from Aristotle; no doubt the development of analogous systems in ancient India and China were similarly motivated. Reason has acquired such a formidable reputation that polemicists often claim to be “logical” when seeking credibility for their arguments or themselves. Yet reason is difficult, forbidding,  and unpredictable enough that many laymen are averse to following or checking the reasoning of philosophers, scientists, mathematicians, and bureaucrats.

 

But there is Pythagoras’ problem: if reductio ad absurdum is a club to wield against quacks, what happens when a transparent geometric construction leads to incommensurable lengths?

 

Matt Cook initially takes a lighter view of paradoxes, launching his book with Gilbert & Sullivan’s Pirates of Penzance, and readers may remember the lines,

 

 

He then leads us through an arboretum of 75 paradoxes, helpfully partitioned into thirteen chapters: there are paradoxes of infinity, paradoxes of probability, paradoxes of geometry, paradoxes of social choice, paradoxes of game theory, and so on. Each paradox is introduced with a little background: Zeno’s paradox of The Arrow is briefly described (an arrow in flight is at one particular place, so it cannot be moving to another place, so it is immobile), and then he presents a naive Claim 1 (Zeno is right: motion is impossible) and a correct Claim 2 (the arrow can be at one place and still move).

 

Curiously, incommensurables do not appear in this book, but the other foundational paradox - the Liar (this sentence is false) - gets a lot of play, along with a number of allied conundrums, most notably Bertrand Russell’s paradox (what is \( \left\{ S\; : \; S \notin S \right\} \)?), and Cook describes some of the efforts to resolve the paradox. The Liar is classified as a paradox of self-reference, and the Incompleteness Theorem is one of these, but while the Incompleteness Theorem satisfies Merrian-Webster’s definition of paradox (or at least it did when it was published), it is a particular kind of paradox.

 

Unlike the Liar, incommensurables and incompleteness are not contradictions; they are merely outlandish, and perhaps should be considered monsters. In his Mathematical Intuition vs. Mathematical Monsters, Solomon Feferman wrote that “I dont know who first applied the word ‘monster’ to examples of counter-intuitive ‘pathological’ functions and figures of the sort that began to emerge in the latter part of the nineteenth century” before quoting Henri Poincaré’s notorious line, “Logic sometimes breeds monsters.” And Cook’s arboretum is full of monsters, and not just from logic.

 

Common sense, as Albert Einstein may or may not have said to the editor of Life magazine, is nothing more than the deposit of prejudices laid down in the mind prior to the age of eighteen. Twentieth century physics is the abode of many monsters, and Cook describes a few, such as the one labeled the Relativity of Simultaneity. Alice is standing in the middle of a train car that has lights at both ends and is barreling down a straight track as it passes a station. When Alice is aligned with the station’s sign, she flips a switch, and sees the lights go on simultaneously. But Bob, standing next to the sign, sees the tail light come on first, and then the head light. General relativity is a rather complex subject, so Cook does not address the question of who is “right” in this monster, but that issue is inescapable in the Twin Paradox, which concerns two twins, one of whom takes an extended trip on a space ship going nearly the speed of light and returns ... not as aged as the twin who stayed home. That required some general relativistic explanation, and Cook makes a go of it in the typical qualitative way and mentions that particle physicists have found that speeding unstable particles decay more slowly, confirming that the monstrous situation is real.

 

Some of the paradoxes are popular errors, notably the proof by induction that all horses are of the same color, a proof that may look convincing until a student tries to prove that all horses are brown. There are two proofs that \( 0 = 1 \), and Cook includes the Monty Hall Paradox. That paradox is fun because it featured credentialed experts being very wrong in public (a bit of historical context that Cook doesn’t mention, alas). In Monty Hall’s game show, Let’s Make a Deal, a preposterously dressed contestant would be offered a choice of three boxes, one containing a brand new luxury sedan, and the other two containing goats. The contestant would choose a box, but before it is opened, Hall would open one of the other two boxes and voilá! A goat. Then Hall would ask the contestant if (s)he wants to keep the chosen box, or switch to the unchosen and still closed box. So, what should the contestant do? The columnist Marilyn Mach, a.k.a. Marilyn vos Savant, correctly concluded that the unchosen, unopened box had a 2/3 probability of containing the sedan.

 

This book is fun to read, and in the end, Cook does glance at the stakes of the issue, but only glancingly. Whether mathematics is independent of Humanity or a cultural construct is an ancient debate, and Cook presents the familiar argument that mathematics works, which suggests that it is real, but he does not go to the significance of paradoxes. Cook cites Ayn Rand frequently in this book, and Rand is notable for her optimism

about the powers of Reason. But Reason leads us to strange places. For example, in his encounter with the Banach-Tarski Paradox (a solid ball can be dismantled and reassembled into two solid balls, each the same volume as the original), Richard Feynman, in his Surely You’re Joking, Mr. Feynman, observed that one cannot actually reassemble an orange this way. The apparatus of points in space, and the assertion of the existence of an undefined disassembly process courtesy of the Axiom of Choice, seems a bit like a word game. In fact, one is reminded of Richard Hamming’s remark that, “If the design of this airplane depends on the difference between the Riemann and Lebesgue integral, I don’t want to fly it.”

 

And there’s the rub. Aristotle and Euclid may, in theory, have sold us on the metaphor of a logical argument as a mechanically checkable chain whose soundness depends solely on the soundness of each link, but the primate inside us relies on its intuition. Reason is one of our primary tools for running a complex technical society, but contradictions suggest that there are limits to what Reason can give us, while monsters and recurring errors are warnings from our inner primate.

 

This is a fun book, and a useful one with many important paradoxes and many that the reader will not have heard of. It does not provide philosophical or historical context - that would have made the book at least twice as long - but it is useful in bringing these problematica into one place.

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Greg McColm (mccolm@usf.edu) is an associate professor of mathematics & statistics at the University of South Florida. Originally working in logic and computer science, he shifted to geometry with applications to crystallography and nanoscience.