Math and the caveman imagination
This is a guest post by Ernie Davis Professor of Computer Science at NYU. Ernie has a BS in Math from MIT (1977) and a PhD in Computer Science from Yale (1984). He does research in artificial intelligence, knowledge representation, and automated commonsense reasoning. He and his father, Philip Davis, are editors of Mathematics, Substance and Surmise: Views on the Ontology and Meaning of Mathematics, published just last week by Springer.
We hear often that our cognitive limitations and our social and psychological flaws are due to our evolutionary heritage. Supposedly, the characteristics of minds and our psyches reflect the conditions in the primordial savannah or caves and therefore are not a good fit to the very different conditions of the 21st century.
The conditions of our primordial ancestors have been blamed for political conservativism, for religious belief , for vengefulness, and especially – since the subject is so fraught and so enjoyable – for gender differences, particularly in sexual fidelity. These kinds of theories have been extensively criticized, most notably by Steven Jay Gould, as being often “just-so” stories. You find a feature of the human mind that you dislike, or one that you think is an ineradicable part of human nature, and you make up a story about why it was good for the cavemen. You find a feature that some people have and others don’t, like political conservatism, and you explain that the stupid bad guys have inherited it from the cavemen, but that the smart good guys have overcome it. I gave my own opinions of the theories about conservatism and religion here.
This week, our ancestors are the fall guys for the fact that we find math difficult. In this week’s New Yorker, Brian Greene is quoted as saying, “[Math] is not what our brains evolved to do. Our brains evolved to survive in the wilderness. You didn’t have to take exponentials or use imaginary numbers in order to avoid that lion or that tiger or to catch that bison for dinner. So the brain is not wired, literally, to do the kinds of things that we now want it to do.”
The problem with this explanation is that it doesn’t explain. The question is not “Why is math hard in an absolute sense?” That’s hardly even a meaningful question. The question is “Why is math (for many people)particularly hard and unpleasant?”; that is to say, harder than a lot of other cognitive tasks. Saying that math is hard because it was useless for avoiding lions and catching bison doesn’t answer the question, because there are many other tasks that were equally useless but are easy and pleasant for people: reading novels, singing songs, looking at pictures, pretending, telling jokes, talking nonsense, dreaming. Nor can the comparative hardness of math be explained in terms of inherent computational complexity; if our experience with artificial intelligence is any indication, doing basic mathematics is much easier computationally than understanding stories. Until we have a much better understanding of how the mind carries out these various cognitive tasks, no explanation of why one task is harder than another can possibly hold much water.*
Conversely, our cognitive apparatus has all kinds of characteristics that, one has to suppose, were unhelpful for primitive people: our working memory is limited in size, our long-term memory is error-prone, we are susceptible to all manner of cognitive illusions and psychological illnesses, we are easily distracted and misled, we are lousy at three-dimensional mental rotation, our languages have any number of bizarre features. We find it harder to communicate distance and direction than bees; we find it harder to navigate long distances than migratory birds. Granted, imaginary numbers would have been useless in primitive life, but other forms of math which would probably have been useful, such as three-dimensional geometry, are also difficult.
Also, our distant ancestors should not be underestimated. The quotation from Greene seem to reflect Hobbes’ view that primitive life was “poor, nasty, brutish, and short”. These are, after all, the people from whom we inherit number systems, art, and language. They did not spend all their time escaping from lions and hunting bison.
Our ancestors on the savannah saw parabolic motion whenever they threw a stone; they experienced spherical geometry whenever they looked up at the starry sky. They never encountered a magic wand or a magic ring. Nonetheless, most people find it easier and much more enjoyable to read and remember and discuss four volumes of intricate tales about magic rings or seven about magic wands than to read a few dozen pages with basic information about parabolas; and even most mathematicians find spherical geometry unappealing and difficult. Why? We have absolutely no idea.
* “I well remember something that Francis Crick said to me many years ago, … ‘Why do you evolutionists always try to identify the value of something before you know how it’s made?’ At the time I dismissed this comment … Now, having wrestled with the question of adaptation for many years, I understand the wisdom of Crick’s remark. If all structures had a `why’ framed in terms of adaptation, then my original dismissal would be justified for we would know that “whys” exist whether or not we had elucidated the “how”. But I am now convinced that many structures … have no direct adaptational ‘why’. And we discover this by studying pathways of genetics and development — or, as Crick so rightly said to me, by first understanding how a structure is built. In other words, we must first establish ‘how’ in order to know whether or now we should be asking ‘why’ at all.” — Steven Jay Gould, “Male Nipples and Clitoral Ripples”, in Bully for Brontosaurus 1991.