Close Encounters with Mathematical Cranks of the Third Kind Mathematical Cranks. By Underwood Dudley. Mathematical Association of America, Washington, D.C., 1992. 372 pp. A Budget of Trisections. By Underwood Dudley. Springer-Verlag, New York, 1987. 169 pp. ROBERT L. FOOTE Well, what's the excuse for this book? winds up in the wastebasket (fre- I guess that what's in it isn't in any quently after serving as departmental other book. entertainment), occasionally a well- Like what? meaning mathematician makes the Things about mathematical cranks mistake of responding to a crank, and eccentrics. No one pays much pointing out his errors, hoping to set attention to them. him straight. Thus may begin a series So this is a book about nuts? of communications between crank and No, no. They're not nuts. Well, a few are, but most aren't. mathematician (mostly from crank to J see. So why would anyone want to mathematician), usually ending when read it? the mathematician is forced to be Well, cranks are interesting. rude, or the crank becomes angry and What would you say if 1 said that goes in search of a more "tolerant" cranks are odd and anyone who's mathematician. The moral of the story interested in them is probably a little is: Don't get involved with a crank. peculiar too? While frequently giving this advice What? But. to others, Underwood Dudley, profes- —Mathematical Cranks, p. v sor of mathematics at DePauw Uni- versity in Greencastle, Indiana, has or better or for worse, pseudo- made it his professional hobby to mathematics does not enjoy the collect the works of modern mathe- popularity among cranks that matical cranks, to correspond with F them, and even to meet them. In these other subjects seem to have. It just doesn't have the same appeal as two books, he shares his fascinations psychic claims, astrology, or UFOs. and frustrations with people bent on Nevertheless, mathematical cranks do pseudo-mathematics. exist, as mathematics departments at What types of things do mathe- many major universities are well matical cranks work on? Most of aware. When I w a s a graduate student Dudley's examples can be placed into at the University of Michigan, it was two broad categories: (l) real mathe- not uncommon for the departmental matics that has been twisted, has been mailboxes to be stuffed periodically misunderstood, or has had its signif- with the local crank's most recent icance horribly exaggerated, and (2) proof of the rationality of n or impenetrable nonsense. yfl. Into the first category goes any While most crank mathematics work based on an actual mathematical 182 SKEPTICAL INQUIRER, Vol. 18 problem whose author understands, impossibility. Cranks, on the other at least at some level, the basic hand, don't understand, and appar- mathematics behind it, but who then ently never get tired. goes on to display a lack of logical reasoning, often caused by a zeal to In more than 12,000 working hours achieve the desired result. Although I have in the course of 40 years found this solution. I am not a most of modern mathematics is safely mathematician but a retired civil beyond the reach of crankery (I don't servant, now 69 years of age. (A imagine there are many cranks work- Budget of Trisections, p. 19) ing on problems in vector bundles over homogeneous spaces or topological The proof that these constructions supercompactifications), everyone are impossible is one of mathematics' knows some mathematics, and there triumphs. Modern mathematics is full are a number of problems whose of insightful connections between its statements can be understood by varied branches—in this case between anyone with a high-school mathe- geometry and algebra. In brief, given matical background. an initial collection of points in a plane A Budget of Trisections, written in where coordinates have been estab- the spirit of Augustus De Morgan's lished, the points that can be obtained A Budget of Paradoxes (see article by from them by compass and straight- Milton Rothman, SI, Spring 1992), is edge constructions are those whose devoted entirely to attempts to trisect coordinates are related to the coordi- the angle (given an angle, divide it into nates of the original points by solu- three equal parts) using only compass tions of linear and quadratic and straightedge. This is one of the equations. Thus the coordinates of the three classical construction problems constructed points are formed of Greek geometry, each of which has through combinations of addition, attracted its share of cranks. The other subtraction, multiplication, division, two are squaring the circle (given a and taking square roots. Higher roots circle, construct a square having the are possible, but only in powers of 2, same area), and duplicating the cube that is, fourth roots, eighth roots, (given a cube, construct a second cube sixteenth roots, etc. The important with twice the volume), again, using fact is that no cube roots are possible. only compass and straightedge. Over Duplicating the cube would involve the years many futile attempts have constructing a segment of length been made at these constructions. In \fl, the solution of x3 = 2. Similarly, the early 1800s they were proved trisecting an angle is related to the impossible. solution of another cubic equation. Impossible? Ah, you hear the Squaring the circle would involve a cranks say, nothing is impossible. segment of length n, but n is tran- Think of the excitement of doing scendental; that is, it is not the something no one has ever been able solution of any polynomial equation to do, and the fame and fortune that with integer coefficients, let alone surely go with it. What child has never linear or quadratic. The conclusion is thought this? Many people, some bud- that these constructions are impossi- ding mathematicians (the reviewer ble. included), spent some time in their Trisectors, cube duplicators, and youth attempting these constructions circle squarers balk at algebraic dis- until they got tired or finally under- proofs of what they think they have stood the meaning of mathematical proved. If the problem is geometric, Winter 1994 183 they say, any refutation should be books. The note was discovered after geometric. his death, but no sign of any proof, and mathematicians have been trying Those who are skeptical should to find one ever since. offer something more than rhetoric (Perhaps Fermat's Last Theorem or argument in order to disprove has finally been proved. Shortly after geometrical facts. ... If the lines this review was written, Andrew constituting the respective pairs of trisectors of both sectors do not Wiles, professor of mathematics at intersect on the quadrantal arc they Princeton University, announced that should show by the ruler and com- he has a proof, an event noteworthy passes where they do intersect. {A enough to be covered by the media Budget of Trisections, p. 27) (New York Times, June 24, 1993, Al, and June 29, 1993, C1). His 200-page The only thing that will convince this manuscript will no doubt be carefully trisector that his construction is scrutinized by the world's algebraic wrong is a correct trisection, which geometers and their graduate stu- is impossible! dents over the next several months. Cranks who work on these three If no errors are found, the book may classical problems are under the be closed on this famous problem.) delusion that solutions to them would The Four Color Theorem states be of great importance and would that every map can be colored using revolutionize mathematics and sci- at most four colors. Countries in a ence. In fact, all three problems have valid map must be connected (so simple solutions, but by constructions Michigan would not be allowed), and that are not Euclidean; that is, they different colors must be used for use more than compass and straight- adjacent countries. This was the Four edge. For example, it is easy to trisect Color Conjecture up until 1976. Its an angle using a compass and a marked proof caused immense controversy in straightedge (i.e., a ruler). These the mathematical community since it constructions were known to the involved hundreds of hours of com- Greeks, and Dudley includes them. puter calculations, but is now gener- Mathematical Cranks covers a wide ally accepted. variety of pseudo-mathematics topics, The appeal of these problems for including Fermat's Last Theorem, the cranks is immediate. As with the Four Color Theorem, and the two Greek constructions they are easily remaining problems from classical understood, and when working on Greece. them they give one the feeling that Fermat's Last Theorem is inaccu- success is imminent. Dudley observes rately named—it is really a conjecture. that many of the works of cranks It is easy to find positive integers x, contain y, and z that satisfy x2 + y2 = z2. For example, 32 + A2 = 52 and 52 + 12* = pages, in some cases a few and in 132. Fermat noticed that he could not other cases many, of elementary find any positive integers that would and correct reasoning followed by solve x3 + y3 = z3, or for that matter a quick descent into incomprehen- sibility, then a return to the surface x" + y" = z" for any integer exponent with something like "therefore x" + n 3, and he conjectured that there y" = z" has no solutions in positive are none. He evidently thought he had integers" that does not follow from a proof, for he wrote a note to that what has gone before. (Mathe- effect in the margin of one of his matical Cranks, p. 117) 184 SKEPTICAL INQUIRER, Vol.