Animal Cognition, Species Invariantism, and Mathematical Realism

What can we infer from numerical cognition about mathematical realism? In this paper, I will consider one aspect of numerical cognition that has received little attention in the literature: the remarkable similarities of numerical cognitive capacities across many animal species. This Invariantism in Numerical Cognition (INC) indicates that mathematics and morality are disanalogous in an important respect: proto-moral beliefs differ substantially between animal species, whereas proto-mathematical beliefs (at least in the animals studied) seem to show more similarities. This makes moral beliefs more susceptible to a contingency challenge from evolution compared to mathematical beliefs, and indicates that mathematical beliefs might be less vulnerable to evolutionary debunking arguments. I will then examine to what extent INC can be used to flesh out a positive case for mathematical realism. Finally, I will review two forms of mathematical realism that are promising in the light of the evolutionary evidence about numerical cognition, ante rem structuralism and Millean empiricism

Numerical cognition in bees and other insects

The ability to perceive the number of objects has been known to exist in vertebrates for a few decades, but recent behavioral investigations have demonstrated that several invertebrate species can also be placed on the continuum of numerical abilities shared with birds, mammals, and reptiles. In this review article, we present the main experimental studies that have examined the ability of insects to use numerical information. These studies have made use of a wide range of methodologies, and for this reason it is striking that a common finding is the inability of the tested animals to discriminate numerical quantities greater than four. Furthermore, the finding that bees can not only transfer learnt numerical discrimination to novel objects, but also to novel numerosities, is strongly suggestive of a true, albeit limited, ability to count. Later in the review, we evaluate the available evidence to narrow down the possible mechanisms that the animals might be using to solve the number-based experimental tasks presented to them. We conclude by suggesting avenues of further research that take into account variables such as the animals' age and experience, as well as complementary cognitive systems such as attention and the time sense.This publication was funded by the German Research Foundation (DFG) and the University of Wuerzburg in the funding program Open Access Publishing. Shaowu Zhang was supported by the ARC-CoE in Vision Science

Constructing a concept of number

Numbers are concepts whose content, structure, and organization are influenced by the material forms used to represent and manipulate them. Indeed, as argued here, it is the inclusion of multiple forms (distributed objects, fingers, single- and two-dimensional forms like pebbles and abaci, and written notations) that is the mechanism of numerical elaboration. Further, variety in employed forms explains at least part of the synchronic and diachronic variability that exists between and within cultural number systems. Material forms also impart characteristics like linearity that may persist in the form of knowledge and behaviors, ultimately yielding numerical concepts that are irreducible to and functionally independent of any particular form. Material devices used to represent and manipulate numbers also interact with language in ways that reinforce or contrast different aspects of numerical cognition. Not only does this interaction potentially explain some of the unique aspects of numerical language, it suggests that the two are complementary but ultimately distinct means of accessing numerical intuitions and insights. The potential inclusion of materiality in contemporary research in numerical cognition is advocated, both for its explanatory power, as well as its influence on psychological, behavioral, and linguistic aspects of numerical cognition

Did language give us numbers? : Symbolic thinking and the emergence of systematic numerical cognition

What role does language play in the development of numerical cognition? In the present paper I argue that the evolution of symbolic thinking (as a basis for language) laid the grounds for the emergence of a systematic concept of number. This concept is grounded in the notion of an infinite sequence and encompasses number assignments that can focus on cardinal aspects ("three pencils"), ordinal aspects ("the third runner"), and even nominal aspects ("bus #3"). I show that these number assignments are based on a specific association of relational structures, and that it is the human language faculty that provides a cognitive paradigm for such an association, suggesting that language played a pivotal role in the evolution of systematic numerical cognition

Finding a voice for numerical cognition

oai:journals.psychopen.eu:article/5671No abstract available

Early numerical cognition and mathematical processes

In this paper I study the development of arithmetical cognition with the focus on metaphorical thinking. In an approach developing on Lakoff and Núñez (2000), I propose one particular conceptual metaphor, the Process → Object Metaphor (POM), as a key element in understanding the development of mathematical thinking.; En este artículo estudio el desarrollo de la cognición aritmética enfocado en el pensamiento metafórico. En una propuesta que se desarrolla desde la de Lakoff y Núñez (2000), propongo una metáfora particular conceptual, la Metáfora Proceso → Objeto (POM), como un elemento clave para comprender el desarrollo del pensamiento matemátic

Early numerical cognition and mathematical processes

In this paper I study the development of arithmetical cognition with the focus on metaphorical thinking. In an approach developing on Lakoff and Núñez (2000), I propose one particular conceptual metaphor, the Process → Object Metaphor (POM), as a key element in understanding the development of mathematical thinking.; En este artículo estudio el desarrollo de la cognición aritmética enfocado en el pensamiento metafórico. En una propuesta que se desarrolla desde la de Lakoff y Núñez (2000), propongo una metáfora particular conceptual, la Metáfora Proceso → Objeto (POM), como un elemento clave para comprender el desarrollo del pensamiento matemátic

Integrating numerical cognition research and mathematics education to strengthen the teaching and learning of early number

BACKGROUND: Research into numerical cognition has contributed to a large body of knowledge on how children learn and perform mathematics. This knowledge has the potential to inform mathematics education. Unfortunately, numerical cognition research and mathematics education remain disconnected from one another, lacking the proper infrastructure to allow for productive and meaningful exchange between disciplines. The present study was designed to address this gap. AIM: This study reports on the design, implementation, and effects of a 16-week (25-hour) mathematics Professional Development (PD) model for Kindergarten to Grade 3 educators and their students. A central goal of the PD was to better integrate numerical cognition research and mathematics education. SAMPLE: A total of 45 K-3 educators and 180 of their students participated. METHODS: To test the reproducibility and replicability of the model, the study was carried out across two different sites, over a two-year period, and involved a combination of two different study designs: a quasi-experimental pre-post-research design and a within-group crossover intervention design. RESULT: The results of the first implementation (Year 1), indicated that compared to a control group, both teachers and students benefited from the intervention. Teachers demonstrated gains on both a self-report measure and a test of numerical cognition knowledge, while students demonstrated gains in number line estimation, arithmetic, and numeration. In Year 2, teachers in the intervention group demonstrated greater improvements than the control group on the self-report measure, but not the test of numerical cognition knowledge. At the student level, there was some evidence of gains in numeration. CONCLUSION: The current PD model is a promising approach to better integrate research and practice. However, more research is needed to determine in which school contexts the model is most effective

Consumers' immediate memory for prices

In this article, the authors examine the cognitive mechanics involved in keeping prices in short-terme memory for subsequent recall. Consumers code and store prices verbally, visually, and in terms of their magnitude. The encoding used influences immediate recall performance. The memorability of prices depends on their verbal length, usualness and overall magnitude. They find that the performance of consumers recall prices better than what previous digit span studies with simple numbers have suggested.consumer behavior; numerical cognition; price memory