Jean-charles Pelland

The study of numerical cognition has undergone tremendous progress in recent years, accumulating scores of data on cognitive systems that could be involved in the uniquely human ability to practice formal arithmetic. Among the important questions tackled by this burgeoning domain of research is what happens to the limited cognitive systems that we share with many animal species to allow us to develop arithmetically-viable numerical content. While answers to this question have varied, most have attributed a constitutive role to culturally-inherited extracranial cognitive support in their explanation of how numerical content emerges from our innate cognitive machinery. The idea here is that we need to look at our interaction with external support for cognition like fingers, numerals, and number words, to explain what allows us to go beyond the size and precision limitations of the cognitive systems we are born with. In this paper, I challenge this externalist answer to the origins of ...

AbstractOne of the most important questions in the young field of numerical cognition studies is how humans bridge the gap between the quantity-related content produced by our evolutionarily ancient brains and the precise numerical content associated with numeration systems like Indo-Arabic numerals. This gap problem is the main focus of this paper. The aim here is to evaluate the extent to which cultural factors can help explain how we come to think about numbers beyond the subitizing range. To do this, I summarize Clark’s (Analysis 58:7–19, 1998) notion of a difference maker in explaining complex causation that criss-crosses between mind and world and apply it to Menary’s (In: Metzinger T, Windt JM (eds) Open MIND. MIND Group, Frankfurt, 2015a) discussion of mathematical cognition as a case of enculturation. I argue that while Menary’s views on enculturation can help explain what makes the difference between numerate and anumerate cultures, it cannot help specify what makes the difference between numerate and anumerate individuals. I argue that features of enculturation do not provide an account of innovation capable of explaining how individuals manage to improve and modify the practices of their cultural niche. This is because Menary’s construal of the role of enculturation in the development of mathematical cognition focuses mostly on the inheritance and transmission of practices, not on their origins, which involve individual-level understanding, rather than population-level practices and pressures. The upshot is that culture provides the necessary background conditions against which individuals can innovate. This role is crucial in the development of numerical abilities—crucial, but explanatorily limited.

Propulsee par le developpement de nouvelles technologies et methodes d'investigation dans divers domaines de recherche, l'etude de la cognition numerique progresse a un rythme fulgurant depuis quelques annees. Des domaines aussi varies que la psychologie developpementale, l'anthropologie, la linguistique, la neuropsychologie, l'ethologie, et la philosophie contribuent tous a une explosion de donnees qui nous permettent de croire que l'on pourrait bientot percer le mystere de comment notre systeme nerveux pourrait nous permettre de representer des entites objectives comme les nombres naturels. Une des plus importantes decouvertes tirees de ce progres est celle de ce qu'il est maintenant coutume d'appeler le « Approximate Number System » (ANS), un ensemble de neurones qui nous permet de determiner la quantite approximative d'objets dans des collections auxquelles nous portons notre attention. Un autre systeme, le « Object-File System » (OFS), serait qua...

L'objectif du present texte est de tenter de construire un modele de l'acquisition des concepts mathematiques sans l'aide du langage en s'inspirant des theses intuitionnistes de L.E.J. Brouwer et en les appliquant a des theories plus modernes de l'acquisition et de la representation des concepts mathematiques, notamment, la theorie du sens des nombres de Stanislas Dehaene. Pour ce faire, nous initierons le lecteur a la pensee de Brouwer dans les deux premiers chapitres et developperons dans le troisieme chapitre une nouvelle analyse de l'Intuition Primordiale de Brouwer dans laquelle il est possible d'identifier chaque element implique dans l'acquisition des concepts mathematiques chez Brouwer et le role joue par chacun. Le chapitre quatre exposera la theorie de Dehaene selon laquelle nos capacites mathematiques sont le resultat de deux systemes cognitifs de base, soit le systeme de repertoire d'objets et le systeme de representation approximative...

Saul Kripke’s (1982) sceptical take on Wittgenstein’s rule-following paradox challenges us to find facts that can justify one interpretation of a symbol’s past use over another. While Ruth Millikan (1990) has answered this challenge by appealing to biological purposes, her answer has been criticized for failing to account for the normativity of rules like addition, which require explicit representations. In this paper, I offer a defense of Millikan. I claim that we can explain how we build intentions to add from the content of core cognition modules like the approximate number system, and argue that Millikan’s answer is better equipped to explain the origins of rules than communitarian approaches like that endorsed by Kusch (2006). I then explore the worth of pluralism about rules and try to find common ground between expressed and unexpressed rules in terms of expectations of how the world is supposed to behave.

One of the most important questions in the young field of numerical cognition studies is how humans bridge the gap between the quantity-related content produced by our evolutionarily ancient brains and the precise numerical content associated with numeration systems like Indo-Arabic numerals. This gap problem is the main focus of this paper. The aim here is to evaluate the extent to which cultural factorse can help explain how we bridge the gapcome to think about numbers beyond the subitizing range. To do this, I apply summarize Clark’s (1998) notion of a difference maker in explaining complex causation that criss-crosses between mind and world and apply it to Menary’s (2015a) discussion of mathematical cognition as a case of enculturation. I argue that while Menary’s views on enculturated enculturation framework can help explain what makes the difference between numerate and anumerate cultures, it cannot help specify what makes the difference between numerate and anumerate individuals. I argue that features of enculturation does not have provide an account of innovation capable of explaining how individuals manage to improve and modify the practices of their cultural niche. This is because Menary’s construal of the role of enculturation in the development of mathematical cognition focuses focuses mostly on the inheritance and transmission of practices, not on their origins, which involve individual-level understanding, rather than population-level practices and pressures. The upshot is that culture provides the necessary background conditions against which individuals can innovate. This role is crucial in the development of numerical abilities – crucial, but explanatorily limited.

The study of numerical cognition has undergone tremendous progress in recent years, accumulating scores of data on cognitive systems that could be involved in the uniquely human ability to practice formal arithmetic. Among the important questions tackled by this burgeoning domain of research is what happens to limited cognitive systems that we share with many animal species to allow us to develop arithmetically-viable numerical content. While answers to this questions have varied, most have appealed to the presence of culturally-inherited extracranial cognitive support to explain how numerical content emerges from our innate cognitive machinery. In this paper I challenge this externalist approach and argue that we should favor explanations that focus on cognitive processing inside the head. To support my claim, I take a look at research into Spontaneous Focusing on Numerosity (SFON) to show that internalism can explain some aspects of the development of numerical cognition without needing to factor in the cultural setting in which this development takes place.

In this paper, I propose to sketch the main lines of some recent theoretical accounts of numerical cognition and evaluate their ability to explain the development of mathematically viable concepts of number (Dehaene 2011; Carey 2009; DeCruz 2008). I will try to show that, despite their differences, these all share a critical flaw in that they explain the development of number concepts by relying in part on numerical symbols. I will argue that explaining how number concepts develop in the brain by relying on the presence of mathematical symbols in the environment is akin to putting the cart before the ox, since such symbols cannot emerge without there being number concepts beforehand.

The study of the cognitive and perceptual systems underlying our numerical abilities has progressed tremendously in the past few decades, yielding scores of data on the potential role played by the so-called Approximate Number System (ANS) and the Object-File System (OFS) in the development of natural number concepts (Dehaene 1997/2011). While there is still disagreement on the relationship between these systems and on the extent to which they produce representations with numerical content, there is overwhelming consensus that, on their own, neither of these systems produces representations with sufficient precision and numerical range to account for the development of natural number concepts.
The question I am interested in in my doctoral thesis is, given these systems' limitations, how do we manage to build representations with mathematically-viable numerical content? That is, how do we bridge the gap between the quantity-related content produced by our evolutionarily ancient brains and the mathematically-viable numerical content associated with numeration systems like Indo-Arabic or Roman numerals? This is what I refer to as the gap problem. It is the main problem that concerns us in this thesis. While many answers have been proposed to this question over the years, virtually all of them rely on culturally-inherited symbols and external artefacts to bridge the gap between natural numbers and the output of our innate cognitive machinery. And yet, as I argue in my thesis, if we want to bridge the gap between natural number concepts and the content produced by systems like the ANS and the OFS, such externalist approaches to cognition are limited in their explanatory power. The main problem with externalist accounts is what I call the origins problem: how can we explain the origins of numerical cognition by appealing to external symbols for numbers, when these symbols in turn depend for internal representations of number for their origins?
To support my claim that externalism is unable to answer this problem, the first two chapters start by summarizing data concerning the main cognitive systems involved in numerical cognition. Then I present the main lines of two of the most influential externalist accounts, Stanislas Dehaene's Number Sense (chapter 3) and Susan Carey's Quinian Bootstrap (chapter 4), in order to illustrate externalist approaches to the gap problem. I show that despite their differences, both accounts attribute a central role to external representations of numbers like number words in explaining how we bridge
the gap, and that this seems problematic given that there are historical cases of development of numerical content without such external support. In chapter 5, I take a closer look at the philosophical motivations behind this externalism in Clark & Chalmer’s classic 1998 paper on the extended mind before exploring the constitutivity of external supports for cognition, as framed by Catarina Dutilh Novaes (2013). This leads me to discuss the relationship between external and internal representations for numbers at both ontogenetic and historical timescales. To ground this discussion, I then discuss data concerning possible origins of the first abstract symbols for numbers in Sumeria (Malafouris 2010) and then explore the limits of anumerate cultures like the Pirahã and the Mundurucu in order to determine whether there is evidence that we can explain the origins of the concept of precise quantity by appealing to the ability to put objects into one-to-one correspondence. I argue that the data do not support this conclusion, and thus that externalists do not have an account of how the first numerical content emerged in a numeral-free environment. In chapter 6, I explore two potential externalist replies to my gap problem based on the potential constructive role of culture. The first is to try to explain the emergence of novel numerical content by appealing to mechanisms of cultural evolution, as described by Helen De Cruz (2007). I argue that this doesn’t help, since such mechanisms are population-level, while the generation of novel content occurs at the level of the individual. I then consider Richard Menary’s (2015a) enculturated approach to numerical cognition to see if extending cognition to include our cultural niche can help the externalist, and find this option wanting as well. Here, I argue that innovation matters in how we want to answer the gap problem and that enculturation is not well equipped to describe the individual-level construction of novel content, due to its focus on population-level processes of innovation. The upshot is that externalist approaches need to be restricted in order to make room for internalist theories of the development of numerical content for an initial segment of the natural numbers.

KEYWORDS: Numerical cognition; Extended Mind; Approximate Number Sense; Object-File System; Cultural Evolution; Foundations of Mathematics

Saul Kripke's (1982) sceptical take on Wittgenstein's rule-following paradox challenges us to find facts that can justify one interpretation of a symbol's past use over another. While Ruth Millikan (1990) has answered this challenge by appealing to biological purposes, her answer has been criticized for failing to account for the normativity of rules like addition, which require explicit representations. In this paper, I offer a defense of Millikan. I claim that we can explain how we build intentions to add from the content of core cognition modules like the approximate number system, and argue that Millikan's answer is better equipped to explain the origins of rules than communitarian approaches like that endorsed by Kusch (2006). I then explore the worth of pluralism about rules and try to find common ground between expressed and unexpressed rules in terms of expectation on how the world is supposed to behave.

Malgré l’imposant corpus soutenant l’existence du “Approximate Number System” (ANS), certains proposent d’expliquer notre comportement dans des études sur la cognition numérique en se fiant à un système dédié au traitement de grandeurs continues comme la durée, la taille, la luminosité, etc. Selon ces sceptiques de l’ANS, les méthodes expérimentales utilisées pour étudier l’ANS ne nous permettent pas d’identifier ce système comme étant responsable de notre comportement dans de telles études, puisque la quantité d’objets discrets varie toujours avec une autre grandeur continue. Dans cet article, je soutiens qu’un tel scepticisme n’est pas tenable pour des raisons conceptuelles, étant donné l’opposition fondamentale entre le contenu mental associé au discret et celui associé au continu. Pour soutenir ma thèse, je propose un bref résumé de la conception de la relation entre le continu et le discret dans l’histoire de la philosophie, en particulier telle que conçue par le mathématicien intutionniste L.E.J Brouwer, selon qui il est impossible de construire un continu à partir d’éléments discrets, et vice versa.

Le présent texte tente de jeter de la lumière sur une hypothèse problématique que partagent plusieurs théories récemment proposées pour expliquer l’origine du concept de nombre. Bien que le domaine de la cognition numérique soit encore trop jeune pour parler d’idée reçue, il n’en demeure pas moins que l’existence de symboles numériques dans l’environnement joue un rôle central dans les théories les plus citées dans la littérature – dont celles de Stanislas Dehaene, Susan Carey, et Helen DeCruz. Je propose de démontrer que, dans ces trois modèles, le développement de concepts de nombres est le résultat d’une interaction entre l’être humain et des symboles numériques dans son environnement. En se fiant à une telle interaction, ces théories font ainsi appel à une forme de cognition externalisée et d’esprit étendu pour expliquer l’émergence d’une nouvelle catégorie de représentations, celle des nombres. Or, si l’argument présenté ci-dessous tient la route, un tel appel à des symboles numériques pour expliquer l’émergence du concept de nombre est l’équivalent philosophique de placer la charrue devant les bœufs, puisque ces symboles sont parasitiques sur les concepts de nombres dont on tente d’expliquer l’émergence.