The n-Category Café

The basic point seems to me to be that the more difficult tasks (the ones that children older than about 5 years are able to perform while children younger than about 5 years are not) involve analysing two pieces of data at once, while the simpler tasks (that even younger children are able to perform) involve analysing only one piece of data at a time (and other pieces are irrelevant even if present). Of course, this is just the sort of thing that may be modelled categorially with products.

Why use category theory at all? One answer may be to get other researchers to focus on the relevant points. While the others are worrying about the level of complexity of each piece of data (such whether it refers to individuals or to sets of individuals), the authors want to consider the relationships between the data.

From the introduction:

This theoretical difficulty is symptomatic of the general problem in cognitive science where the basic components of cognition are unknown. In the absence of such detailed knowledge, cognitive modelers have been forced to assume a particular representational format (e.g., symbolic [6], or subsymbolic [7]). This approach, however, does not lend itself to the current problem, because the elements of Transitive Inference and Class Inclusion tasks (i.e., objects and classes of objects) do not share a common basis. Understandably, then, these sorts of behaviours have tended to be studied in detailed isolation, narrowing the scope for identifying general principles.

What a coincidence. Just over the past few days Todd and I were discussing Piaget and cognitive development , keeping in mind Piaget’s consideration of category theory (however superficial) in his theory of genetic epistemology, lo and behold, you brought up this paper! This should make some interesting reading.

For what its worth (and perhaps others may have thought along these lines), I have always suspected that there is a good reason why CT is able to unify so many diverse areas of mathematics, that reason being we are perhaps genetically endowed with cognitive structures that help us reason categorically with varying degrees of success. To me, a promising line of research in cognitive neuroscience would be one that verifies (perhaps elucidates) such structures. Of course, such structures are not meant to be located in specific parts of the brain; rather, they must exist as dynamic representations in the mind/brain.

As I recall, Levi-Strauss states in no uncertain terms in several places that the sense of ‘structure’ that he uses is pretty much the same as in mathematics. I think Piaget’s introductory book on structuralism has group theory in it somewhere, for example. At a superficial level, structuralists were looking for (approximate) isomorphisms between various concrete structures (e.g. families). Perhaps one could say they tried to model a concrete collection of objects as a single category, with varying degrees of success.

Since category theory, on the other hand, is interested in functors between different kinds of structures, one might have suspected it to be a more suitably flexible framework for making the kind of comparisons anthropologists or cognitive scientists are interested in. So I had assumed that someone had already done something of this sort, convincingly or otherwise.

Re the title…

I am 38 years old and cannot compute coproducts :)

PS: Greetings from HK :)

At the risk of sounding pedantic, I think it is worth distinguishing between:

Hypothesis A: Children compute coproducts (starting at age X or so)

Hypothesis B: Children perform certain tasks (such as transitive inference) that are best explained by cognitive scientists as evidence for the possession of category theoretic structures in their minds.

In cognitive science the distinction between hypothesis A and hypothesis B is quite important - what Chomsky has called the difference between competence and performance. For example, if A is true, we would be entitled to look for brain mechanisms that implement coproducts, while if B is the better hypothesis, it would be premature if not false to look for brain mechanisms.

In my admittedly biased view, good cognitive science has mostly happened when research has proceeded along directions that resemble hypothesis-B, resulting in competence theories. Unfortunately, the authors of this paper have not clarified their views on the matter, though it seems as if they have a hypothesis-A like view.

What I find interesting about the tasks that they test in children is that all of them involve a comparison between one kind of structure represented in one mental state and another structure represented in another mental state (such as the appearance/reality false belief test in the theory of mind segment). A strong case could be made that some category theory like formalism is needed to perform cross structural comparisons, especially if the comparison abilities are systematic and productive. However, arguing for category theory on grounds of systematicity is a competence argument, not a performance argument and the authors lose out on an opportunity as a result of their confusion between the two categories.

To give a linguistic analogy, English speakers are able to perform wh-movements (replace an assertion by a question, for example). Generative grammarians have tried to describe children’s knowledge of wh-movements using constraints on tree structures that represent the grammatical structure of a sentence. However, Chomsky and other generative grammarians never claimed that children actually compute those trees in their heads or that those tree operations are implemented somewhere in our brains.

Another analogy: we can clearly predict the motion of planets using differential equations, but no one believes that planets compute the differential calculus. (I should say ‘almost no one’, for there are people like Ed Fredkin, Norm Morgolus and Steve Wolfram who believe the universe IS a computer).

I find it strange that views that are demonstrably false and/or problematic in physics are so easily accepted when it comes to cognitive science and the philosophy of mind.

My colleague Chris Athorne sent me the following by email, which I’m reposting here with his permission.

I still maintain that I taught Thomas how to “count” before he could speak by giving him a set of cards with numbers of dots marked on them and showing him that he should sort them into correspondence classes, which he was able to do. Cards of the same … er, Card … were not geometrically congruent so he did not have any patterns to offer clues. I told the child psychologist this and she gave me a look over her glasses as though I’d just claimed there were fairies at the bottom of my garden. Which, of course, there are but I thought it prudent not to go there!