The paper aims to put certain basic mathematical elements and operations into an empirical perspective, evaluate the empirical status of various analytic operations widely used within psychology and suggest alternatives to procedures criticized as inadequate. Experimentation shows the "manyness" of items to be a perceptual quality for both young children and animals and that natural operations are performed by naive children analogous to those performed by persons tutored in arithmetic. Number, counting, arithmetic operations therefore can make distinctions that are not (...) inevitably arbitrary, and conceptual operations can obviously have a status as natural events with psychology. If the elements and conceptual operations involved in mathematical systems were not inherent in physiological process, various primitive discriminations could not be possible. Also, since some calculi have a natural status in a given empirical circumstance, the axioms of others can not be satisfied. Therefore the psychologist when acting as an empirical scientist seeks a calculus having a structure whose elements are isomorphic with natural units of stimulus and response and whose operations are isomorphic with whatever natural processes are involved. Measurement poses a special problem for the empirical scientist. It concerns but a single class of natural qualities and this only in a limited way. The concept of quantity has a natural counterpart but quantity and measurement are not wholly analogous. Measurement is defined, as H. S. Leonard suggests, as a theoretical activity. Measurement theory has a formal structure but empirical end. Measurement hypothesizes about the position of a particular quality within a definite range of qualities. It therefore is beholden to definite empirical restrictions. Some hypotheses-making systems use terms and relations per se as the context and starting point for dealing with discriminable events. Such procedures are 'transcendent." In empirical science, questions are part of problem-solving activity and their reference is naturally restricted. In providing description and explanation, psychological researchers frequently use calculi in a transcendent way. This results in theories that are only quasi-empirical and "half" true. The roles measurement plays in psychology are discussed. Of particular concern are those cases in which the results of measuring or a theory of measurement are used to define the "real" units, or the "real" relations involved in problematic psychological events, and thence to describe and explain behaviors of interest. Metaphysical or ontological usages of measurement sometimes occur. The implication of these arguments with regard to a view of empirical science is discussed. (shrink)

We see no grounds for insisting that, because the concept natural number is abstract, its foundations must be innate. It is possible to specify domain general learning processes that feed into more abstract concepts of numerical infinity. By neglecting the messiness of children's slow acquisition of arithmetical concepts, Rips et al. present an idealized, unnecessarily insular, view of number development.

Recent studies have shown that deductive reasoning skills are related to mathematical abilities. Nevertheless, so far the links between mathematical abilities and these two forms of deductive inference have not been investigated in a single study. It is also unclear whether these inference forms are related to both basic maths skills and mathematical reasoning, and whether these relationships still hold if the effects of fluid intelligence are controlled. We conducted a study with 87 adult participants. The results showed that transitive (...) reasoning skills were related to performance on a number line task, and conditional inferences were related to arithmetic skills. Additionally, both types of deductive inference were related to mathematical reasoning skills, although transitive and conditional reasoning ability were unrelated. Our results also highlighted the important role that ordering abilities play in mathematical reasoning, extending findings regarding the role of ordering abilities in basic maths skills. These results have implications for the theories of mathematical and deductive reasoning, and they could inspire the development of novel educational interventions. (shrink)

Textual and historical subtleties aside, let's call the idea that numbers are properties of equinumerous sets ‘the Fregean thesis.’ In a recent paper, Palle Yourgrau claims to have found a decisive refutation of this thesis. More surprising still, he claims in addition that the essence of this refutation is found in the Grundlagen itself – the very masterpiece in which Frege first proffered his thesis. My intention in this note is to evaluate these claims, and along the way to shed (...) some light on relevant passages of the Grundlagen. I will argue that Yourgrau does not make his case.The arguments with which we are concerned are found in the last three sections of Yourgrau's paper. A pervasive difficulty in these sections is that it is not clear exactly what Yourgrau is arguing against. The stated object of his attack is the Fregean thesis, a thesis about what numbers are; however, instead of a frontal assault, his strategy is to embark on a foray into the ill-defined issue of what it is that numbers number, where, roughly speaking, a number n numbers an object x just in case n can be legitimately assigned to x. The reason for this shift in emphasis appears to be rooted in a misconception. As we’ll see in more detail shortly, Yourgrau's argument against the Fregean thesis is based on an extension of a well known argument of Frege's found in §§22-3 of the Grundlagen, which Glenn Kessler has tagged the ‘relativity argument,’. According to Yourgrau, this is an argument ‘to the effect that what is literally numbered cannot simply be concrete objects’. This is incorrect. Frege himself clarifies the point of the argument in §21 with the following preface:In language, numbers (i.e., numerals] most commonly appear in adjectival form and attributive construction in the same sort of way as the words "hard" or "heavy" or "red," which have for their meanings properties of external things. It is natural to ask whether we must think of the individual numbers too as such properties, and whether, accordingly, the concept of number can be classed along with that, say, of color. (shrink)

Children often personify non-living objects, such as puppets and stars. This attribution is considered a healthy phenomenon, which can simulate social exchange and enhance children's understanding of social relationships. In this study, we considered that the tendency of children to engage in personification could potentially be observed in abstract entities, such as numbers. We hypothesized that children tend to attribute personalities to numbers, which diminishes during the course of development. By consulting the methodology to measure ordinal linguistic personification (OLP), which (...) is a type of synesthesia, we quantified the frequency with which child and adult populations engage in number personification. Questionnaires were completed by 151 non-synesthetic children (9--12 years old) and 55 non-synesthetic adults. Children showed a higher tendency than adults to engage in number personification, with respect to temporal consistency and the frequency of choosing meaningful answers. Additionally, children tended to assign unique and exclusive descriptions to each number from zero to nine. By synthesizing the series of analyses, we revealed the process in which number personification diminishes throughout development. In the discussion, we examined the possibility that number personification serves as a discrimination clue to aid children's comprehension of the relationships between numbers. (shrink)

In this commentary, we outline an epistemological continuum between earlier and later number concepts, showing how empirical findings support the view that specific and general underpinnings to number develop in parallel in children; and we raise the question, based on cross-syndrome comparisons in infancy, as to whether exact or approximate number abilities underlie these later skills.

According to one of the most powerful paradigms explaining the meaning of the concept of natural number, natural numbers get a large part of their conceptual content from core cognitive abilities. Carey’s bootstrapping provides a model of the role of core cognition in the creation of mature mathematical concepts. In this paper, I conduct conceptual analyses of various theories within this paradigm, concluding that the theories based on the ability to subitize (i.e., to assess anexactquantity of the elements in a (...) collection without counting them), or on the ability to approximate quantities (i.e., to assess anapproximatequantity of the elements in a collection without counting them), or both, fail to provide a conceptual basis for bootstrapping the concept of an exact natural number. In particular, I argue that none of the existing theories explains one of the key characteristics of the natural number structure: the equidistances between successive elements of the natural numbers progression. I suggest that this regularity could be based on another innate cognitive ability, namely sensitivity to the regularity of rhythm. In the final section, I propose a new position within the core cognition paradigm, inspired by structuralist positions in philosophy of mathematics. (shrink)

This paper aims to balance the conceptual reception of Bourdieu's sociology in the United States through a conceptual re-examination of the concept of Habitus. I retrace the intellectual lineage of the Habitus idea, showing it to have roots in Claude Levi-Strauss structural anthropology and in the developmental psychology of Jean Piaget, especially the latter's generalization of the idea of operations from mathematics to the study of practical, bodily-mediated cognition. One important payoff of this exercise is that the common misinterpretation of (...) the Habitus as an objectivist and reductionist element in Bourdieu's thought is dispelled. The Habitus is shown to be instead a useful and flexible way to concep-tualize agency and the ability to transform social structure. Thus ultimately one of Bourdieu's major contributions to social theory consists of his development of a new radical form of cognitive sociology, along with an innovative variety of multilevel sociological explanation in which the interplay of different structural orders is highlighted. In keeping with the usual view, the goal of sociology is to uncover the most deeply buried structures of the different social worlds that make up the social universe, as well as the "mechanisms" that tend to ensure their reproduction or transformation. Merging with psychology, though with a kind of psychology undoubtedly quite different from the most widely accepted image of this science, such an exploration of the cognitive structures that agents bring to bear in their practical knowledge of the social worlds thus structured. Indeed there exists a correspondence between social structures and mental structures, between the objective divisions of the social world . . . and the principles of vision and division that agents apply to them. (shrink)

Picking up on John Forrester’s (1949–2015) disclosure that he felt ‘haunted’ by the suspicion that Thomas Kuhn’s (1922–96) interests had become his own, this essay complexifies our understanding of both of their legacies by presenting two sites for that haunting. The first is located by engaging Forrester’s argument that the connection between Kuhn and psychoanalysis was direct. (This was the supposed source of his historiographical method: ‘climbing into other people’s heads’.) However, recent archival discoveries suggest that that is incorrect. Instead, (...) Kuhn’s influence in this regard was Jean Piaget (1896–1980). And it is Piaget’s thinking that was influenced directly by psychoanalysis. Psychoanalysis then haunts Kuhn’s thinking through Piaget, and thus Piaget haunts Forrester through Kuhn. To better understand this second site of the haunting—which is ultimately the more important one, given the intent of this special issue—Piaget’s early psychoanalytic ideas are uncovered through their interaction with his early biology and subsequent turn to philosophy. But several layers of conflicting contemporary misunderstandings are first excavated. The method of hauntology is also developed, taking advantage of its origins as a critical response to the psychoanalytic discourse. As a result of adopting this approach, a larger than usual number of primary sources have been unearthed and presented as evidence (including new translations from French originals). Where those influences have continued to have an impact, but their sources forgotten, they have thus been returned. They can then all be considered together in deriving new perspectives of Forrester’s cases/Kuhn’s exemplars/Piaget’s stages. (shrink)

The characterization of early token-based accounting using a concrete concept of number, later numerical notations an abstract one, has become well entrenched in the literature. After reviewing its history and assumptions, this article challenges the abstract–concrete distinction, presenting an alternative view of change in Ancient Near Eastern number concepts, wherein numbers are abstract from their inception and materially bound when most elaborated. The alternative draws on the chronological sequence of material counting technologies used in the Ancient Near East—fingers, tallies, tokens, (...) and numerical notations—as reconstructed through archaeological and textual evidence and as interpreted through Material Engagement Theory, an extended-mind framework in which materiality plays an active role (Malafouris 2013). (shrink)

Beyond modularityattempts a synthesis of Fodor's anticonstructivist nativism and Piaget's antinativist constructivism. Contra Fodor, I argue that: (1) the study of cognitive development is essential to cognitive science, (2) the module/central processing dichotomy is too rigid, and (3) the mind does not begin with prespecified modules; rather, development involves a gradual process of “modularization.” Contra Piaget, I argue that: (1) development rarely involves stagelike domain-general change and (2) domainspecific predispositions give development a small but significant kickstart by focusing the infant's (...) attention on proprietary inputs. Development does not stop at efficient learning. A fundamental aspect of human development (“representational redescription”) is the hypothesized process by which information that isina cognitive system becomes progressively explicit knowledgetothat system. Development thus involves two complementary processes of progressive modularization and progressive “explicitation.” Empirical findings on the child as linguist, physicist, mathematician, psychologist, and notator are discussed in support of the theoretical framework. Each chapter concentrates first on the initial state of the infant mind/brain and on subsequent domain-specific learning in infancy and early childhood. It then goes on to explore data on older children's problem solving and theory building, with particular focus on evolving cognitive flexibility. Emphasis is placed throughout on the status of representations underlying different capacities and on the multiple levels at which knowledge is stored and accessible. Finally, consideration is given to the need for more formal developmental models, and a comparison is made between representational redescription and connectionist simulations of development. In conclusion, I consider what is special about human cognition by speculating on the status of representations underlying the structure of behavior in other species. (shrink)