Minds and mathematics

9 Jan 2012mathematics

In 1998, the mathematical physicist David Ruelle wrote an odd little article, entitled Conversations on Mathematics with a Visitor from Outer Space, which appeared as a chapter of the very interesting looking book Mathematics: Frontiers and Perspectives, published by the IMU in 2000. I’ve not read the whole volume, although I think I’ve read at least one of the other chapters as a preprint. Most of the 30 authors seem to have written more or less “straight” articles, but Ruelle’s is different. He wanted to explore the constraints imposed on human mathematics by the structure and capabilities of the human brain. Perhaps constraints is too strong a word–“predispositions” might be better: human mathematics is necessarily tied to human brains, and human brains have evolved to their current state to solve problems that are quite different from the problems encountered in mathematics. One might thus expect human mathematics to have followed the “fault lines” in the Platonic edifice of Mathematics that are most easily appreciated using the mental tools that evolution has given us.

To explore this idea, Ruelle introduces the conceit of a visitor from outer space, a “galatic mathematician” pursuing doctoral studies investigating the nature of human mathematics. She describes some of the characteristics of the human mental apparatus that are relevant: limited short-term memory, and hence inability directly to execute complex algorithms; predisposition to see symmetry and pattern; human brains operate very slowly, even compared to human computers; the importance of geometrical and visual reasoning; and so on. Ruelle comments that despite our limitations, humans have managed to do some fairly complex mathematics, but the question is always: how much further could we go if we could transcend the limitations of our history?

Ruelle’s visitor tells him that it would be impossible for her to explain Galactic Mathematics, since galactic mathematicians have already undergone this transcendence: “an ancient civilization would have fixed such intellectual inadequacies by assisted coevolution, biological engineering, and so on”. Humanity must necessarily do the same if they are to go beyond Human Mathematics.

This reminded me very strongly of an after-dinner talk by Freeman Dyson at a conference at Caltech I attended some years ago. The conference was about the exploration of MarsAt the time, I was a PhD student in Oxford, working on the ill-fated Mars Climate Orbiter mission. This was the infamous “metric vs. imperial” screw-up, which put paid to that PhD project quite handily. I had better luck second time out, a long time after this., and Dyson’s chosen topic was, naturally enough, the human exploration of space. The main point of his talk, as I remember it, was that he didn’t think that humans would get all that far into space. At least, not humans as we are now. Too much effort expended on life support, too much danger from radiation or extended periods of weightlessness. Would it not make much more sense, he asked, for us to modify ourselves for life in outer space? Not overnight, of course, since such a process would involve a long period of experimentation and gradual adaptation, but eventually the idea would be to have people able to survive life in a weightless environment, with better resistance to ionising radiation, and perhaps even limited ability to survive in vacuum.

And if we might do that to travel and live in outer space, why not follow a similar process to allow us to explore the further reaches of mathematics? If arithmetic was no longer a question of rote learning but just came naturally, if complex symbolic manipulations were “obvious” instead of a lottery of sign errors, transpositions and copying slip-ups, if complex logical arguments flowed effortlessly at first sight, what might mathematics feel like to us, and how might it feel to be freed from the bondage of our (relatively) plodding squishy brains? What sort of mathematics would be produced by an entity for whom, for instance, the classification of finite groups could be held in the mind’s eye like a gem to be rotated and admired from every angle?

Of course, it’s pretty useless to speculate what it would be like to be such an “extended” mathematician: not so different from the archetypal example of imagining what it might be like to be a bat, although perhaps more like the bat imagining life as a human. Nevertheless, thinking about “extended” or modified human intellects is tied to another topic I’ve been idly thinking about recently, mostly as a follow-on from the Stanford AI course. That’s the question of “common sense”. I’ll write about that another time.