On applying conceptual engineering to number cognition
“Number in Mind: The Philosophy of Numerical Cognition”
February 18, 2017, School of Advanced Studies, University of London
In my paper I show that applying the method of conceptual engineering to interpretation of the research in number cognition can open up new perspectives on what natural numbers are and how humans construct them.
Conceptual engineering is understood as a methodological toolkit comprising conceptual analysis, explication, explanation etc. In the paper the “engineering” is applied to early math knowledge and to “full-blooded” arithmetical concepts known today, to their both syntactical and semantical aspects and on both local and structural levels.
In consequence, conceptual gaps between the contemporary mature concept and the first conceptual content get disclosed. Most importantly it becomes visible that regularity of distances between subsequent elements of the natural number progression does not descend from subitizing or ANS nor from the commonly used Linguistic Number System. This regularity is in the basis of recursivity of the number progression. Its presence is visible in tallying and at the linear number line from Cartesian coordinate system.
The solution to this problem that the paper proposes is based on a version of enculturation. Enculturation is understood as the idea that cultural factors, such as language, influence activation of new cognitive modules that add to the core specific ones in providing content to a specific concept. I advance hypothesis that the innate sensibility to rhythm (beat-induction) might be such a new cognitive module participating in number concept creation. This explains our intuition that the use of tallying is related to the number concept creation. This view is also coherent with Chomskyan claim that only humans count (as opposed to other animals) because the only purely human aspect of language is recursivity hidden in various aspects of its structure.
On the basis of my arguments showing that the concept gets input from many different sources, I opt for pluralism in arithmetic. For practical reasons I propose to adapt an “extended” version of Frege’s constraint according to which the criterion of choice of the foundational principle for a mathematical theory should be such that the main feature enables definition of all other important features of natural numbers. This “extended” constraint can be adapted to developmental cognitive sciences by replacing the research of the “true” way in which children acquire the conceptual content, by the research of the optimal way that will foster children learning of what we today understand as natural numbers.