Spring 2017: Talk at “Numbers in Mind: The Philosophy of Numerical Cognition”

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.

Spring 2015: Semantics for Exact-Vague Quantifiers at Szklarska Poręba Workshop

Szklarska Poręba Workshop

The 16th Workshop  on the Roots of Pragmasemantics will be held on the top of the Szrenica mountain in the Giant Mountains on the border of Poland and the Czech Republic on 20-23 February 2015. The two main themes of this year’s convention are (1) “Mental Representation of Semantic and Pragmatic Lexical Knowledge” and (2) “The Role of Linguistics in the Cognitive Sciences”.

Karolina Krzyżanowska will present our joint paper:

Exact numerals as vague quantifiers

Abstract: When we say that the population of Norway is 5 million people, we do not usually mean that there are exactly 5 million people living in this country. On the contrary, if the country happened to have a population of no more and no less than 5 million, we would need to add “exactly” to convey this information. Why do phrases like “5 million,” “two hundreds” or even, in some contexts, numerals denoting smaller numbers like “forty” tend to be interpreted as approximations (cf., e.g., Krifka 2009)? Is it only a matter of pragmatics or is the denotation of these numerals vague? Is the sentence “Norway has a population of 5 million” true if the exact number is 5,109,059? Or is it false but assertable? Drawing from recent developments in cognitive science (e.g. Dehaene 2011; Carey 2009), we will argue that our preference for vague interpretation of numerals might be due to the approximate number system (ANS) being the primary source of our mental representation of numbers, and hence using exact numerals as vague quantifiers is not only a matter of convention. ANS forms one of the core cognitive systems responsible for our representations of quantitative information. Unlike the verbal representations of discrete quantities, ANS-related representations are believed to be analouge and intrinsically imprecise. In the proposed study, we investigate how the practice of use of exact numerals as vague quantifiers correlates with the structure of ANS.

Carey, S. (2009), The Origin of Concepts, Oxford University Press, Oxford.
Dehaene, S. (2011), The number sense: How the mind creates mathematics, revised and updated edn, Oxford University Press, Oxford.
Krifka, M. (2009), Approximate interpretations of number words. the case for strategic communication, in E. Hinrichs and J. Nerbonne, eds, ‘Theory and evidence in semantics’, CSLI Publications, Stanford, pp. 109–132.