I have a story in this week’s New Scientist about robot mathematicians. The idea, really, is to see if we can work out what maths is, and how we do it, though it’s tempting to speculate about how a robot mathematician might create new mathematics. One of the aspects I didn’t get to go into is how intertwined are our bodies and our mathematical concepts. ET, for instance, might do maths differently from us.
When researching the feature, I talked at length with Rafael Nunez of the University of California, San Diego. Nunez, working with George Lakoff of the University of California, Berkeley, was first to put forward the idea that mathematics is “embodied” (their book is here).
The original forms of measuring and accounting all have to do with bodily forms, they pointed out: the foot, the yard (a stride length), hands for measuring horse size. Even concepts such as future and past are related to our bodies. Western cultures speak of “looking forward” to a future that is “ahead” of us, for example, while the past is “behind us”.
That research took place in the early 1990s. Now, thanks to further discoveries, they are more convinced than ever that mathematics is nothing more than a useful way of making sense of the world – and that it is certainly not innate.
For a start, plenty of cultures have no mathematical concepts to speak of, Nunez says. Then there is the Andean culture that he has been studying. They have time the other way round from us. Because they can’t see the future, they think of it as lying behind them. Our maths is all to do with our bodies. “Our mathematics comes out of things like our bipedalism, and having a visual field on only one side of the body,” Nunez told me. “But the fact that this culture is so different to ours shows it is not hard-wired or genetically determined.”
Our bodies make maths possible, but it has only ever been a tool, invented only if it can help with something else humans are trying to do – like snowboarding or windsurfing, Nunez says. The mathematical concept of transfinite numbers are an example. They were invented in Germany at the end of the 19th century as an extension of natural numbers. How, Nunez asks, is that any different from the invention of snowboarding, invented in the 20th century as a new form of winter thrill?
What’s really interesting is that, just as snowboarding would be different if we had differently-shaped bodies, so would maths. The fact that we have ten fingers is probably what leads us to prefer the decimal system, for instance. That has interesting implications for the popular notion that we would recognise alien communications by their mathematical form. Martin Rees put this idea forward most recently, also in New Scientist.
The thing is, if your maths depends on the kind of body you have, aliens with different bodies would have different maths. So we wouldn’t necessarily even notice that signal from space. Does that mean that SETI should go back to the drawing board? Even better, does it mean that a signal based on physical constants – like the Wow! Signal – is the most likely to be seen?