Can Five-Year-Olds Compute Coproducts?
Posted by John Baez
Charlie Clingen points out the following article:
- Steven Phillips, William H. Wilson and Graeme S. Halford, What do transitive inference and class inclusion have in common? Categorical (co)products and cognitive development, PLoS Computational Biology.
The abstract explains what the authors are up to:
Transitive inference, class inclusion and a variety of other inferential abilities have strikingly similar developmental profiles—all are acquired around the age of five. Yet, little is known about the reasons for this correspondence. Category theory was invented as a formal means of establishing commonalities between various mathematical structures. We use category theory to show that transitive inference and class inclusion involve dual mathematical structures, called product and coproduct. Other inferential tasks with similar developmental profiles, including matrix completion, cardinality, dimensional changed card sorting, balance-scale (weight-distance integration), and Theory of Mind also involve these structures. By contrast, (co)products are not involved in the behaviours exhibited by younger children on these tasks, or simplified versions that are within their ability. These results point to a fundamental cognitive principle under development during childhood that is the capacity to compute (co)products in the categorical sense.
A lot of the paper is taken up with explaining coproducts, which suggests a variant of the old Groucho Marx joke:
Coproducts are so simple, any five-year-old child can understand them. Quick, someone find me a five-year-old child!
But mathematicians who already understand coproducts might find this the most interesting part:
Children acquire various reasoning skills over remarkably similar periods of development. Transitive Inference and Class Inclusion are two behaviours among a suite of inferential abilities that have strikingly similar developmental profiles—all are acquired around the age of five years. For example, older children can infer that if John is taller than Mary, and Mary is taller than Sue, then John is taller than Sue. This form of reasoning is called Transitive Inference. Older children also understand that a grocery store will contain more fruit than apples. That is, the number of items belonging to the superclass is greater than the number of items in any one of its subclasses. This form of reasoning is called Class Inclusion. These two types of inference appear to have little in common. Transitive Inference typically involves physical relationships between objects, while Class Inclusion involves abstract relative sizes of object classes. Nonetheless, explicit tests of these and other inferences for a range of age groups revealed that success was attained from about the median age of five years.
Since Piaget, decades of research have revealed important clues regarding the development of inference, yet little is known about the reasons underlying these correspondences. A common theme in two recent proposals is the computing of relational information. In regard to Relational Complexity theory, the correspondence between commonly acquired cognitive behaviours is based on the maximum arity of relations that must be processed (e.g., tasks acquired after age five involve ternary relations, i.e., relations between three items). In regard to Cognitive Complexity and Control theory, the correspondence is based on the common depth of relation hierarchies. Although a relational approach to cognitive behaviour has a formal basis in relational algebra, certain assumptions must be made about the units of analysis. For tasks as diverse in procedure and content as Transitive Inference and Class Inclusion, it is difficult to see how the analysis of one task leads naturally to the other. For Relational Complexity theory, Transitive Inference is considered to involve the integration of two binary relations between task elements into an ordered triple, or ternary relation; whereas Class Inclusion is regarded as the integration of three binary relations between three sets of elements (one complement and two containments) into a ternary relation. For Cognitive Complexity and Control theory, Transitive Inference involves relations over items; whereas Class Inclusion involves relations over sets of items.
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, or subsymbolic). 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.
Category theory was born out of a desire to establish formal commonalities between various mathematical structures, and has since been applied to the analysis of computational structures in computer science. The seminal insight was a shift from objects as the primary focus of analysis to their transformations. Contrast, for instance, sets defined in terms of (the properties of) the objects they contain—Set Theory—against sets defined in terms of the morphisms that map to or from them—Category Theory. This insight motivates our categorical approach to the analysis of inference, and our way around the current impasse. In cognitive science, several authors have used category theory for a conceptual analysis of space and time, though we know of only one other application that has modeled empirical data. Since our application of category theory to cognitive behaviour is novel, we first introduce the basic category theory constructs needed for our subsequent analysis of Transitive Inference, Class Inclusion, and other paradigms. The analysis begins with a brief introduction of the sort of data our approach is intended to explain, which primarily concerns contrasts between younger and older children relative to age five, and correlations across paradigms. Finally, we extend our categorical approach to more complex levels of inference. Our main point is that, despite the apparent lack of resemblance, all these tasks are formally connected via the categorical (co)product, to be defined below. The significance of this result is that it opens the door to an entirely new (empirical) approach to identifying general principles, particularly in regard to the development of inferential abilities, that are less likely to be revealed by standard modeling methods.
How convincingly do the authors argue that products and coproducts are the unifying theme of the reasoning skills that develop around the age of 5? I’m not sure… they tackle this later in the paper, but I haven’t read it carefully yet.
Re: Can Five-Year-Olds Compute Coproducts?
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: