The Puzzle Instinct: The Meaning of Puzzles in Human Life

4 Puzzling Logic

In the fifth century B.C. heated debates broke out frequently in Greece over one of the greatest philosophical puzzles of all time—how human beings think. Prominent in the debates were the philosopher Parmenides (born c. 515 B.C.) and his disciple Zeno of Elea (c. 489–c. 435 B.C.). In Zeno’s time, leading scholars had based mathematics on the method of deductive logic. Zeno challenged this practice with a series of clever arguments. They were designated paradoxes (meaning literally “conflicting with expectation”) by the people who witnessed the fascinating debates (Salmon 1970). In one of his paradoxes, Zeno argued that a runner would never be able to reach a finish line. He argued his case as follows. The runner must first traverse half the distance to the finish line. Then, from that position, the runner would face a new, but similar, task—he must traverse half of the remaining distance between himself and the finish line. But thereupon, the runner would face a new, but again similar, task—he must once more cover half of the new remaining distance between himself and the finish line. Although the successive half distances between himself and the finish line would become increasingly (indeed infinitesimally ) small, the wily Zeno concluded that the runner would come 4 PuzzlingLogic D, P,  O F  M P The paradox is really the pathos of intellectual life and just as only great souls are exposed to passions it is only the great thinker who is exposed to what I call paradoxes, which are nothing else than grandiose thoughts in embryo. —SØren Kierkegaard (1813–1855) very close to the finish line, but would never cross it. The successive distances that the runner must cover form an infinite geometric sequence, each term of which is half the one before: {1 ⁄2, 1 ⁄4, 1 ⁄8, 1 ⁄16, . . .}. The sum of the terms in this sequence—{1 ⁄2 + 1 ⁄4 + 1 ⁄8 + 1 ⁄16 + . . .}—will never reach 1, the whole distance to be covered. Below is a visual portrayal of Zeno’s runner paradox: The runner will never reach the finish line because in order to do so, he must first traverse 1 ⁄2 of the distance to the finish line; and then half of that, i.e., 1 ⁄2 of 1 ⁄2 = 1 ⁄4; and then half of that, i.e., 1 ⁄2 of 1 ⁄4 = 1 ⁄8; and so on, ad infinitum. Hence, the runner can never reach the finish line: With such shrewdly contrived arguments, Zeno called into question the whole deductive-analytical edifice of Greek mathematics and philosophy. As Devlin (1998a: 101) aptly puts it, “Zeno’s puzzles presented challenges to the attempts of the day to provide analytic explanations of space, time, and motion—challenges that the Greeks themselves were not able to meet.” The Sophists—a group of traveling teachers who became famous throughout Greece toward the end of the fifth century B.C.—sided with Zeno, arguing that the very existence of paradoxes meant that it was impossible to deduce anything worthwhile, contrary to what philosophers such as Thales and Socrates believed. Aristotle, on the other hand, dismissed Zeno’s paradoxes as exercises in specious reasoning. The central characteristic of human thinking, Aristotle insisted, was its ability to deduce things from given facts. He then proceeded to give the deductive method a formal structure, which he designated syllogistic: PuzzlingLogic 113 Figure 4.1 Major premise: All mammals are warm-blooded. Minor premise: Whales are mammals. Conclusion: Therefore whales are warm-blooded. The major premise states that a category has (or does not have) a certain characteristic, and the minor premise states that a certain thing is a member of the given category. The conclusion then affirms (or denies) that the thing has that characteristic. As clever as they were, Aristotle stated, Zeno’s paradoxes were inconsequential, since they did not impugn the validity of the syllogism, which is based on the logic of common sense. But Aristotle’s attempt to demolish Zeno’s clever arguments did not eradicate them from the history of logic and mathematics. On the contrary, they became important factors in the development of those two disciplines, and are still being debated heatedly today. As Kasner and Newman (1940: 39) perceptively state, the “history of mathematics, in fact, recounts a poetic vindication of Zeno’s stand.” Aristotelian logic was...