The Sciences

Minus numbers

Cosmic VarianceBy Sean CarrollAug 3, 2005 5:28 PM


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Bad Blood by literary critic Lorna Sage is a memoir of her postwar childhood in a poor Welsh village. It's full of memorable and quotable passages; this is just one of the paragraphs that struck me.

Hanmer school left its mark on my mental life, though. For instance, one day in a grammar school maths lesson I got into a crying jag over the notion of minus numbers. Minus one threw out my universe, it couldn't exist, I couldn't understand it. This, I realised tearfully, under coaxing from an amused (and mildly amazed) teacher, was because I thought numbers were things. In fact, cabbages. We'd been taught in Miss Myra's class to do addition and subtraction by imagining more cabbages and fewer cabbages. Every time I did mental arithmetic I was juggling ghostly vegetables in my head. And when I tried to think of minus one I was trying to imagine an anti-cabbage, an anti-matter cabbage, which was as hard as conceiving of an alternative universe.

The power of abstraction that allows us to contemplate negative numbers shouldn't be taken for granted; it's downright miraculous. And the "alternative universe" comparison is spot on -- the difference between imagining the existence of negative numbers and imagining the existence of extra dimensions of space is one of degree, not of kind. To indulge in some pop evolutionary psychology (a bad habit, I admit), I can't help but wonder whether our faculty of abstract mathematical reasoning is somehow related to the development of grammar. One of the more intriguing parts of Steven Pinker's The Language Instinct is where he suggests that the important difference between humans and other species resides in grammar, and in particular in the subjunctive mood. We can speak in counterfactuals, and make statements of the form "If X had been the case, Y would have happened instead of Z." An incredibly useful skill, allowing human beings to contract with each other in arbitrarily complicated ways, and therefore opening up the possibility of laws and morality and all that. Best of all, it allows for math. It doesn't seem such a great leap from speaking about situations that are not the case to speaking about quantities that can't exist, abstracting from a certain set of cabbages to the general notion of "numbers." And once you're there, it's a short distance to negative numbers, and imaginary numbers aren't far behind. Pretty soon you're talking confidently about the Riemann hypothesis and category theory, and people know not to invite you to cocktail parties.

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