Home > Uncategorized > Math and the caveman imagination

Math and the caveman imagination

November 30, 2015

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.

Categories: Uncategorized
  1. November 30, 2015 at 6:58 am

    I believe evolutionary imperatives which have the potential to shape mathematical thinking may account for mathematical pluralism (in a way that it can similarly account for cultural mathematics: different cultures creating different mathematics not up to some kind o isomorphism).
    In particular, if/since the evolution is an ongoing process, it’d be quite fun to imagine how today’s mathematics will evolve in the future based on the different evolutionary imperatives. Such an evolution could even be more politically inspired (feminist mathematics or lgbti mathematics, for example).
    All in all, I believe this is an important argument for mathematical pluralism that there are more than one mathematics.

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  2. Scott S
    November 30, 2015 at 7:49 am

    Possibly it is because for a creative mind Math is fundamentally predictable in outcomes. In that, by design, Math almost always leads to a repeatable result. The predictability factor is anathema to creativity and devastatingly “hard” to get around.

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    • November 30, 2015 at 8:03 am

      In my experience, creativity is all about recognizing & riffing on patterns.

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  3. November 30, 2015 at 8:02 am

    Paleofantasy by Marlene Zuk is an excellent book if you want a bunch of examples of ways humans and animals have evolved in the very recent past. The “fantasy” of the title is the fallacy that humans (or any other animal) were ever perfectly suited for their environments: evolution is random and ongoing.

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  4. davidwlocke
    November 30, 2015 at 8:14 am

    Who sells math? If anyone did, who would buy it? Who would market it?

    As for how or why, neither of those questions can be answered by machine intelligence. Machine intelligence will do what it can with what it is exposed to, but it will not explain a thing. Selling math is like selling feature-based software. Another feature, another math that fortunately connects to other math, but only if you know that that other math exists. Math textbooks tell us the use cases that will appear on a test. But, are they really like the weekly newsletter explaining something about a product and the use cases associated with it?

    When post-Euclid mathematicians decided that text rather than a figure was the way to go, they lost me. I will take to my death the explanation given by my linear algebra professor when asked what an eigen was? The formula was on the board. It was not informative to me. He said, pointing to that formula, this! I checked out of the class at that moment. I withdrew later, but I was done. It took years, but now I think I know what an eigen whatever is.Once I know, I’ll build the formula myself. We were programmers taking the class because we had to, not because we were interested. But then, now I’m sold.

    Thankfully, there are plenty of resources now. Back in the early 80’s math hell stayed hell.Now, I can answer my own questions. I sell it to myself. Ah, a motivation.

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  5. November 30, 2015 at 8:33 am

    How can anyone even be sure that it’s a *biological* question in the first place?

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  6. mathematrucker
    November 30, 2015 at 12:14 pm

    I love these “we don’t know jack” guest posts.

    Every mathematician can point to conversations that shaped their mathematical development. I will never forget a visit to Galen Shorack’s office in early 1982 to discuss majoring in the then-fledgling statistics program at the University of Washington. While we were chatting about the coursework involved, he described the undergraduate analysis sequence as “where you learn to be a mathematician.”

    This broad statement made the course sound so awesome and mysterious to me, its ultimate and unintended (unless Prof. Shorack was cleverly trying to get rid of me…) consequence when I took the course later that year, was to convert me into a math major.

    He made another, similarly broad statement that morning that stuck with me permanently. It may be a famous quote from the past, but anyway, he’s the person who said it to me: mathematicians “build a world and then live in it.”

    Worlds are hard. In the same way that it always has, to this day basic survival (not being eaten by a lion, not walking off a cliff, etc.) forces us to familiarize ourselves with the world of rings and wands first before we can even begin to think about building different ones and attempting to live in them. That these other worlds sometimes happen to be useful for things like avoiding lions, catching bison and traveling to moons is a nice side effect that may forever remain mysterious.

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  7. November 30, 2015 at 4:45 pm

    I think math is hard simply because it is abstract (by abstract, I mean anything that can’t be clearly pictured/visualized by the brain). It operates through symbols, relations, and recursion, and interacting with the immediate outside world with such mental tools is difficult, because these tools are so many steps removed from that world.
    The far more interesting/baffling question to me is, why is learning to speak and comprehend LANGUAGE so EASY for humans (youngsters in particular)?

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    • Paul Fabel
      November 30, 2015 at 9:55 pm

      Math is hard, but not necessarily because of abstraction. For example Category Theory, (among the most abstract branches of mathematics) conjures up images of lots of dots (the objects) connected by lots of arrows (the morphisms), creating something like a graph,

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  8. December 1, 2015 at 12:47 am

    The reason we do math is because our world is comprehensible. It behaves in a predictable way, but doing the prediction requires focus and attention to formalism. These are hard. With narrative, is quite possible to enjoy a story on emotional and immediate terms. If you’ve studied literature, you also know there is something called “close reading” where you focus on the formalism, the words the author uses, what they say and what they imply, how they achieve the emotional effects, how they drive the narrative. This, like doing math, reqiures focus and attention to formalism, and like math, it is hard work.

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  9. Sophonax
    December 1, 2015 at 4:19 pm

    “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.”

    You assume that because a given behavior is useless, its cognitive substrate is useless as well. But this is completely unwarranted: of course cognitive mechanisms will sometimes be used for other purposes than the ones they evolved for (this is why evolutionary psychologists came up with the distinction between the proper and the actual domain of a cognitive adaptation), but this doesn’t mean they are not adaptations in the first place.

    As an example, even if we assume that reading novels is useless, it is easy to see why it is an attractive activity for the evolved human mind: it hijacks systems designed to gather information, especially information about the social world, and these systems are usually very adaptive (most novels deal with violence and love, definitely fitness-relevant topics).

    Also, what is the Gould footnote supposed to accomplish, apart from being a buzzwordy authority argument?

    Liked by 1 person

    • December 2, 2015 at 11:13 am

      Andrew Sundstrom points out to me this same caveman theory in an otherwise fine article that a few weeks ago I pointed out to Cathy and she posted on mathbabe:
      https://mathbabe.org/2015/11/16/debiasing-techniques-in-science/

      Our brains evolved long ago on the African savannah, where jumping to plausible conclusions about the location of ripe fruit or the presence of a predator was a matter of survival. But a smart strategy for evading lions does not necessarily translate well to a modern laboratory, where tenure may be riding on the analysis of terabytes of multidimensional data. In today’s environment, our talent for jumping to conclusions makes it all too easy to find false patterns in randomness, to ignore alternative explanations for a result or to accept ‘reasonable’ outcomes without question — that is, to ceaselessly lead ourselves astray without realizing it.

      Apparently, this half-baked idea is very pervasive.

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    • December 2, 2015 at 11:23 am

      “You assume that because a given behavior is useless, its cognitive substrate is useless as well. ” On the contrary, that is my point. Our abilities are not directly a function of what is adaptive. Certainly one can come up with an explanation of why novel reading is useful; explaining why listening to music is so pleasurable is harder, but probably can be done as well. My point is that we have no theory that _distinguishes_ between novel reading and learning math in that regard. We don’t have an explanation for why novel reading much _more_ pleasurable than learning three-dimensional geometry. So saying “we find math hard because it wasn’t directly adaptive for the caveman” is not much of an explanation, because there are a lot of things we do enjoy that were not adaptive, and there is math that would have been adaptive for the caveman that we find hard.

      As for the quote from Gould/Crick, I thought it was a particularly well-written statement of the principle. I recommend the essay, and the collection, and everything else that Gould wrote.

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      • Sophonax
        December 3, 2015 at 12:59 am

        Fair point. Greene’s explanation is short and thus not very rigorous indeed.
        However, his idea is not totally irrelevant. It doesn’t constitute a full explanation on its own, but it is helpful nonetheless. We are wondering why people find math hard. The fact that computing exponentials wasn’t a feature on people’s life during the Pleistocene is a relevant bit of information. As your point about reading novels, etc, makes clear, it is not a sufficient explanation, because we know of activities that didn’t constitute a selection pressure but that we nonetheless do easily; but this doesn’t mean the idea is worthless. As an analogy, pointing out that birds have wings is a good step towards explaining why they can fly, even if ostriches have wings but can’t fly.

        “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.”

        As it turns out, cognitive scientists have their ideas on the issue. Because predicting the motion of a stone is so useful, we do it automatically without needing/being able to consciously access the underlying computations. So we are very good at doing certain kinds of math but just can’t harness these skills to solve the same problems explicitly. As for magic rings, they rely on the trick of minimal counter-intuitiveness. They fit well into our intuitive ontology (it is easy to grasp the concept of a ring), but have a counter-intuitive feature (e.g. making people invisible) that make them attention-grabbing. On the other hand, some arcane scientific concepts are so remote from our commonsense understanding of the world that we don’t have any intuitive concept to anchor them to.

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        • davidwlocke
          December 3, 2015 at 6:23 am

          We don’t learn anything from an intuitive grasp. Things are harder to learn when we have nothing to anchor them to. The typical pathway to spherical geometry is the triangle inequality, not looking up at the sky. We learn along that pathway. Things are harder to learn when that pathway has not been adequately delineated.

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        • Scott S
          December 3, 2015 at 6:36 am

          “A triangle is a triangle but a magical ring is what ever you wish it to be” The magical ring engages the imagination and creativity of the reader where as the triangle doesn’t quite seem to do the same IMO

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