da “LeScienze.it” 12.12.2009
What Do Transitive Inference and Class Inclusion Have in Common? Categorical (Co)Products and Cognitive Development
1 Neuroscience Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki, Japan, 2 School of Computer Science and Engineering, The University of New South Wales, Sydney, New South Wales, Australia, 3 School of Psychology, Griffith University, Brisbane, Queensland, Australia
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.
Author Summary Top
Children acquire various reasoning skills during a remarkably similar period of development. Yet, the reasons for these similarities are a mystery. Two examples are Transitive Inference and Class Inclusion, which develop around five years of age. Older children understand that if John is taller than Mary, and Mary is taller than Sue, then John is also taller than Sue. This form of reasoning is called transitive inference. Older children also understand that there are more fruits than apples. This inference is called class inclusion. We explain why these and a variety of other abilities show the same development using a branch of mathematics called category theory. Category theory reveals that they have related underlying structure. So, despite their apparent superficial differences these reasoning abilities have similar profiles of development because they involve related sorts of processes.