Debts, Fortunes and False Roots

Negative numbers are the clearest case in mathematics of a thing being used long before it was believed in, and the gap is not a few decades. It is about a millennium.

Chinese mathematicians were calculating with negative quantities early, using counting rods of two colours to keep track of which side of a subtraction a quantity sat on. Brahmagupta, in 628, wrote systematic rules for them in the language of debts and fortunes: a debt times a debt is a fortune, a fortune times a debt is a debt, and so on. Note what the framing does. It gives the quantities something to be. A debt is a real thing in the world, so arithmetic about debts is arithmetic about something, and the sign conventions stop looking arbitrary.

Europe refused for centuries after that, and it is worth being fair about why.

If a number is a quantity of something, then a number less than nothing is incoherent. You can have three sheep and you can have no sheep and there is no state of affairs in which you have minus three sheep. That is not a stupid objection. It is a defence of a definition, and the definition was the thing that eventually had to give.

So the mathematicians did what mathematicians generally do under those conditions: they used the quantities and declined to endorse them. Cardano's Ars Magna of 1545 allows negative solutions to equations and calls them fictitious, reserving true for the positive ones. Descartes called them false on the grounds that they represented numbers less than nothing, and went on using them. Both men wanted the answers and neither wanted the ontology, which is a perfectly workable professional position and the one this entire book is about.

What eventually settled it was not a philosophical argument. It was the number line, the idea of negatives as directions rather than amounts, plus the sheer accumulating usefulness of coordinates, of accounting, of anything with two directions in it. The meaning arrived to justify a practice that was already universal.

Which brings up the question that every child asks and almost no textbook answers properly: why does a negative times a negative come out positive?

The usual answers are stories. Turning round twice leaves you facing the way you started. Taking away a debt makes you better off. Both are fine as intuitions and neither is a reason, because you could tell an equally vivid story for the opposite convention if that were the rule.

The reason is that arithmetic has to stay consistent. Suppose you want the distributive law to keep holding, so that a times the sum of b and c equals a times b plus a times c. Write out what that forces for minus one times minus one, and you find there is no choice: any other value makes multiplication contradict addition somewhere.

That is a different kind of answer from a story, and a better one. It says the rule was not designed, chosen or voted on. It is a consequence, and the only alternative to it is breaking something already agreed. A good deal of mathematics has that structure, and this is the smallest clean example of it.