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Society of Mind

18.8 mathematics made hard

That theory is worthless. It isn't even wrong! —Wolfgang Pauli

Scientists and philosophers are always searching for simplicity. They're happiest when each new thing can be defined in terms of things that have already been defined. If we can keep doing this, then everything can be defined in successive layers and levels. This is how mathematicians usually define numbers. They begin by defining Zero — or, rather, they assume that Zero needs no definition. Then they define One as the successor of Zero, Two as the successor of One, and so on. But why prefer such slender chains? Why not prefer each thing to be connected to as many other things as possible? The answer is a sort of paradox. As scientists, we like to make our theories as delicate and fragile as possible. We like to arrange things so that if the slightest thing goes wrong, everything will collapse at once!

Why do scientists use such shaky strategies? So that when anything goes wrong, they'll be the first to notice it. Scientists adore that flimsiness because it helps them find the precious proofs they love, with each next step in perfect mesh with every single previous one. Even when the process fails, it only means that we have made a new discovery! Especially in the world of mathematics, it is just as bad to be nearly right as it is to be totally wrong. In a sense, that's just what mathematics is — the quest for absolute consistency.

But that isn't good psychology. In real life, our minds must always tolerate beliefs that later turn out to be wrong. It's also bad the way we let teachers shape our children's mathematics into slender, shaky tower chains instead of robust, cross-connected webs. A chain can break at any link, a tower can topple at the slightest shove. And that's what happens in a mathematics class to a child's mind whose attention turns just for a moment to watch a pretty cloud.

Teachers try to convince their students that equations and formulas are more expressive than ordinary words. But it takes years to become proficient at using the language of mathematics, and until then, formulas and equations are in most respects even less trustworthy than commonsense reasoning. Accordingly, the investment principle works against the mathematics teacher, because even though the potential usefulness of formal mathematics is great, it is also so remote that most children will continue to use only their customary methods in ordinary life, outside of school. It is not enough to tell them, Someday you will find this useful, or even, Learn this and I will love you. Unless the new ideas become connected to the rest of the child's world, that knowledge can't be put to work.

The ordinary goals of ordinary citizens are not the same as those of professional mathematicians and philosophers — who like to put things into forms with as few connections as possible. For children know from everyday experience that the more cross-connected their common-sense ideas are, the more useful they're likely to be. Why do so many schoolchildren learn to fear mathematics? Perhaps in part because we try to teach the children those formal definitions, which were designed to lead to meaning-networks as sparse and thin as possible. We shouldn't assume that making careful, narrow definitions will always help children get things straight. It can also make it easier for them to get things scrambled up. Instead, we ought to help them build more robust networks in their heads.