The Cult of Pythagoras: Math and Myths

11. Inventing Mathematics?

181 Through the so-called Platonist outlook, many people construed mathematics in religious ways. They assumed that its principles were eternal truths discovered by special men, geniuses, and they accepted that these truths were valid everywhere and could never change. The laws of geometry and numbers seemed like the laws of God, and therefore mathematics was valued as a preparation to discipline the mind for studies of metaphysics and theology. In the 1730s, Bishop Berkeley complained that some people accepted strange mathematical propositions on the basis of faith instead of reason. Some students learned rules on the basis of authority, “because the Master said it,” like the Pythagoreans. Along with rules of operation, students learned the rules of “Thou shalt not.” Children learn that they cannot subtract a greater number from a lesser. Later they learn that actually it’s possible. But they then learn that they cannot take square roots of negative numbers. Later, they learn that it is possible to take roots of negatives, and they learn about imaginary numbers, and so forth. As they advance in mathematics, they learn that a series of operations that first seemed impossible are actually fine. Even nowadays, some students still learn certain basic rules by obedience. In an interview, when asked why we must use the rule that division by zero is undefined, one ninth-grade student replied: “The question is not ‘Why is this the rule?’ You just have to know the rule. Clever mathematicians make rules and we should memorize them. The problems we want to solve by applying 11 INVENTING MATHEMATICS? 182 I N V E N T I N G M AT H E M AT I C S ? the rules are what we have to understand.” Another student, in the eleventh grade, said: “It is not allowed to divide by zero. In mathematics we have rules, and we operate according to them. These rules often do not seem reasonable. For instance, it is illogical that minus times minus is plus. When studying mathematics, we have to obey the rules and to work with them. There is no point at all in looking for explanations. One just has to accept them.”1 To illustrate how people variously explain rules, consider negative numbers . When multiplying positives and negatives we have + × + = + − × + = − + × − = − − × − = + Years ago, in 1992, a college freshman told me “why minus times minus is plus,” by making + mean “good.” She said: When good things happen to good people, that’s good. When bad things happen to good people, that’s bad. When good things happen to bad people, that’s bad. When bad things happen to bad people, that’s good! I laughed. But we might think that if bad things happen to bad people, that’s not really good. Maybe we imagine that if good things happen to bad people, they might cease to be bad. The same problem affects some popular justifications for why the product oftwonegativesispositive.Oneclaimisthat,inlanguage,twonegativesmake a positive. A sentence such as “I do not disagree” seems to express agreement. But it sounds ambiguous: maybe the speaker is just undecided. Likewise, when someone says, “Don’t bring me no food,” it might mean that he really does not want food. Still, some linguists tried to find certainty or universality in the way that positives and negatives are used in human languages. A famous anecdote says that in the 1950s, the British philosopher of language J. L. Austin, a professor at Oxford University, presented a lecture in which he argued that although double negatives often express positives, in many languages, there exists no language in which two positives express a negative. But then someone in the audience replied in a dismissive voice: “Professor Sidney Morgenbesser is said to have piped up from the back of the room with an I N V E N T I N G M AT H E M AT I C S ? 183 instant, sarcastic, ‘Yeah, yeah.’ This convulsed the audience in laughter and put a blot on the speaker’s career.”2 The larger problem in claims that language entails that minus times minus is plus is that double negatives in language hardly involve multiplication. Two negatives might make a positive, but why not apply that, say, to addition rather than multiplication? If someone says, “I do not disagree,” again, what exactly is being multiplied? Conventions of grammar do not decide mathematical...