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The story of symbolic algebra

Gene Expression
By Razib Khan
Feb 21, 2006 1:07 AMNov 5, 2019 9:13 AM


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Chad is not happy with my previous post where I consider that we shouldn't expect that everyone should be able to pass algebra conditional upon a deep understanding of the subject. First, let me state that my post was in part operating outside what I will call the "Cohen narrative." Rather, I wanted to interject the opinion that variation is a contingent fact of human history (otherwise, we wouldn't have been shaped by natural selection). I was attempting to offer that the alternatives are not black and white in that everyone should learn algebra or that everyone need not learn algebra. Granted, many of the observers qualified that any educated person needed to know algebra. I simply suggest that not everyone is educable to the same extent. If basic literacy and arithmetic are the standard for being educated, then everyone is probably educable. If algebra and geometry are the standard for being educated, I suspect a large minority are not educable. If basic diffential and integral calculus is the standard for being educated (18^th century math) than only a small minority are educable, and excludingMatthew Yglesias (Harvard, philosophy, 2003 magna cum laude). If the ability to learn multiple languages with a reasonable level of fluency after initiation of puberty is the standard for being educable, then I must withdraw my name from the pool based upon induction. This goes to another part of Chad's comment, where he states: it seems a little too Steven Pinker-- "our ape-like ancestors on the savannah didn't need algebra, so we never evolved the brain module for it.... Well, I don't have that much of a problem with Steven Pinker, though I disagree on a lot of details with him. But in the case of algebra Chad's point is pretty orthogonal, if you may excuse a mathematical analogy, to what I'm getting at. I don't think higher cognitive functions are really "hard wired" as tightly integrated modules which were subject to selection during our Pleistocene past. A bit off topic, I suspect humans are still subject to selection on cognitive traits, and that our phenotypic variation is in part a reflection of this. So the "evolution on the savannah" narrative is one I will forgo. But, as I alluded to in my reference to The Number Sense, it seems to me that abstract mathematics is a cultural innovation which is contingent upon a wide range of congitive faculties. When I state abstract mathematics, I am being precise because analog numeracy is a capacity which rats and pigeons exhibit. We do possess a gestalt ability to "count" a set of objects arrayed before us. But, this ability reaches its limit somewhere around 10 objects for the vast majority of humans. Though we can assess rough proportions and have a general sense of "amount," if you threw 58 marbles at front of any normal person they couldn't spit back a number with any level of accuracy. They could tell you about how many marbles they were seeing with rough consistency, but they would have to count sequentially in their mind's eye or verbally. Counting precisely (as opposed to comprehending the rough gist of proportions and amounts) requires extending our learning facilities, and coopts language and meta-representational capacities. I offered the example of individuals who suffer brain damange which imobilizes both their fingers and their ability to meta-represent sequences of numbers to suggest that even this cultural innovation is in part rooted in a pre-existent cognitive architecture. This brings me back to variation:

though humans are a cultural animal with an innate aptitude toward plasticity of behavior and thought, we exhibit universal and intraspecies cognitive biases

. The French cognitive anthropologist Dan Sperber has been developing a theory over the past 25 years which posits that cultural ideas inhabit a cognitive fitness landscape where particular motifs and tendencies are strong favored and spread as mental epidemics. Sperber takes the basic thrust of memes and adds a large dosage of cognitive science and hypothesizes that the probability of a particular path of cultural development is conditional upon the weights of various initial mental parameters. This is premised upon the common evolutionary psychological idea that humans share a unified cognitive substrate, so universal preferences will result in particularly favored cultural conformations. To use a chemical analogy, the various conformations of cyclohexane are characterized by different levels of strain. Though the chair conformation is energetically favored, that does not imply that the other forms are not ever extent or possible, rather, the steric parameter is simply a strong biasing factor. Similarly, an evolutionary adaptive landscape might have a primary peak, but temporal and spatial vicissitudes may result in stabilization of a population's gene frequencies at a local fitness peak which is "sub-optimal." What does this have to do with algebra? We need to move behind the common universal set of cognitive biases towards an appreciation of human variation in said biases. I suspect that abstract mathematics draws upon a wide range of cognitive faculties. I also suspect that virtuosity in this phenotype is contingent upon a particular combination of aptitudes, as well as the necessary developmental and social nurturing (ie, aptitude is irrelevant outside of a cultural matrix which allows the expression of a given trait). The various cognitive faculties and developmental and social factors can be thought of as random variables with particular states, and these states together contribute to a distribution of mathematical aptitude. In contrast, I believe that language is a relatively tightly integrated module (if not necessarily localized to one region of the brain), and so humans without basic language fluency are pathological. I believe that arithmetic is "close enough the code" to our innate numeracy, (which is just one of the various faculties that contribute to mathematical aptitude) that the number of random variables are limited, and that they aren't so random (so they skew toward a distribution which leaves almost all humans within the range of teachability). As mathematics becomes more "decoupled" from our cognitive substrate and necessarily coopts a far wider range of our hardware and necessitates more and more software input, the higher the probability becomes that you are shifting up the distribution of mathematical aptitude so that a greater and greater proprotion of the population falls below the threshold needed for educability in a particular domain. Now, I will tack to Chad's example of new techniques in literary presentation. I don't know much about literature, though I am trying to read Jane Austen, so I am a righteous philistine. But, to go back to a mathematical analogy, I would hold that literature is of a different magnitude of a rather universal vector, our love of story telling. This is a strong cognitive bias which draws upon both our linguistic and social intelligence, both of which we are generally endowed with in spades. While I can not do subpar algebraic topology, I can write down a subpar story. In other words, the seed for this aptitude is there, I can conceive of the lands I have not seen. In contrast, I have a really difficult time even conceiving of the wilds of algebraic topology, "algebraic topology" are two words whose general outlines I can sea, but whose face is hidden. In short, mathematics is a vector that is exploring an entirely new space. I do not believe this space is always intuitive to people who attain fluency, even virtuosity, in mathematics. When I took introductory linear algebra one of the most bizarre moments was when I made a joke about "6-space being confused for 5-space." I will spare you the details because they escape me, but I don't know too many people who can conceive of dimensionalities beyond 3 (a physicist friend tells me that some physicists claim they can imagine higher dimensionalities). To go into the strange lands of abstract mathematics we need to depend on axioms and the systems we derive from them, but we are taking a cultural boat into uncharted waters, a vessel whose planks are constructed from a wide range of materials. So finally, let me respond to the most hurtful charge Chad has cast my way:

I'm not really comfortable with the claim that the relatively late development of an idea indicates that it's somehow counter to our brain chemistry, and thus ok for people to not understand it.

First, to marginal issues, I think my prose above elucidates my position that it is not brain chemistry alone which I am speaking to. Of course, there are issues relating to necessary and sufficient conditions, and a particular brain chemistry, roughly interpreted, might be necessary to algebraic fluency. But the primary issue is that I am not engaging in the naturalistic fallacy, anymore than I would assert that because the chair conformation is energenetically favored it is the form of cyclohexane we must utilize. Rather, I am suggesting that to actually get everyone in our society to understand algebra, or be educated in a modern sense, will be as feasible as trapping noble gases in bucky balls.

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