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Series:   Elements in the Philosophy of Mathematics

Number Concepts

An Interdisciplinary Inquiry

Published online by Cambridge University Press:  29 January 2024

Richard Samuels  and
Eric Snyder
Richard Samuels
Affiliation:
Ohio State University
Eric Snyder
Affiliation:
Ashoka University

Summary

This Element, written for researchers and students in philosophy and the behavioral sciences, reviews and critically assesses extant work on number concepts in developmental psychology and cognitive science. It has four main aims. First, it characterizes the core commitments of mainstream number cognition research, including the commitment to representationalism, the hypothesis that there exist certain number-specific cognitive systems, and the key milestones in the development of number cognition. Second, it provides a taxonomy of influential views within mainstream number cognition research, along with the central challenges these views face. Third, it identifies and critically assesses a series of core philosophical assumptions often adopted by number cognition researchers. Finally, the Element articulates and defends a novel version of pluralism about number concepts.
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Keywords

number conceptsnumber cognitionrepresentationalismcore systemsconceptual pluralism
Type
Element
Information
Series: Elements in the Philosophy of Mathematics
Online ISBN: 9781009052337
Publisher: Cambridge University Press
Print publication: 15 February 2024

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References

Barwise, J. and Cooper, R. (1981). Generalized quantifiers and natural language. Linguistics and Philosophy, 1:413–458.Google Scholar
Beck, J. (2019). Perception is analog: The argument from weber’s law. The Journal of Philosophy, 116(6):319–349.Google Scholar
Benacerraf, P. (1965). What numbers could not be. The Philosophical Review, 74(1):47–73.CrossRefGoogle Scholar
Benacerraf, P. (1973). Mathematical truth. Journal of Philosophy, 70(19):661–679.Google Scholar
Bermúdez, J. and Cahen, A. (2020). Nonconceptual mental content. In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Summer 2020 edition.Google Scholar
Bloom, P. and Wynn, K. (1997). Linguistic cues in the acquisition of number words. Journal of Child language, 24(3):511–533.Google Scholar
Burge, T. (2010). Origins of Objectivity. Oxford University Press.Google Scholar
Butterworth, B. (2005). The development of arithmetical abilities. Journal of Child Psychology and Psychiatry, 46(1):3–18.Google Scholar
Carey, S. (2000). The origin of concepts. Journal of Cognition and Development, 1(1):37–41.Google Scholar
Carey, S. (2009a). The Origin of Concepts. Oxford University Press.Google Scholar
Carey, S. (2009b). Where our number concepts come from. The Journal of Philosophy, 106(4):220–254.Google Scholar
Carey, S. (2011). Précis of the origin of concepts. Behavioral and Brain Sciences, 34(3):113–124.Google Scholar
Carey, S. and Barner, D. (2019). Ontogenetic origins of human integer representations. Trends in Cognitive Sciences, 23(10):823–835.CrossRefGoogle ScholarPubMed
Cheung, P., Rubenson, M., and Barner, D. (2017). To infinity and beyond: Children generalize the successor function to all possible numbers years after learning to count. Cognitive Psychology, 92:22–36.CrossRefGoogle ScholarPubMed
Chierchia, G. (1998a). Plurality of mass nouns and the notion of “semantic parameter.” In Rothstein, S., ed., Events and Grammar. Springer.Google Scholar
Chierchia, G. (1998b). Reference to kinds across languages. Natural Language Semantics, 6:339–405.CrossRefGoogle Scholar
Chomsky, N. (1987). Language and Problems of Knowledge the Managua Lectures. MIT Press.Google Scholar
Chomsky, N. (2014). The Minimalist Program. MIT press.Google Scholar
Clarke, S. and Beck, J. (2021). The number sense represents (rational) numbers. Behavioral and Brain Sciences, 44:1–33.Google Scholar
Cole, J. (2013). Towards an institutional account of the objectivity, necessity, and atemporality of mathematics. Philosophia Mathematica, 21(1):9–36.Google Scholar
Cowie, F. (1998). What’s within? Nativism Reconsidered. Oxford University Press.Google Scholar
Davidson, K., Eng, K., and Barner, D. (2012). Does learning to count involve a semantic induction? Cognition, 123(1):162–173.Google Scholar
Davies, M. (2015). Knowledge–explicit, implicit and tacit: Philosophical aspects. International Encyclopedia of the Social & Behavioral Sciences, 13:74–90.Google Scholar
de Cruz, H. (2016). Numerical cognition and mathematical realism. Philosophers’ Imprint, 16:1–13.Google Scholar
Dehaene, S. (2011). The Number Sense: How the Mind Creates Mathematics. Oxford University Press.Google Scholar
Dummett, M. (1991a). Frege: Philosophy of Mathematics. Harvard University Press.Google Scholar
Dummett, M. (1991b). The Logical Basis of Metaphysics. Harvard University Press.Google Scholar
Egan, F. (2012). Representationalism. In Margolis, E., Samuels, R., and Stich, S., editors, The Oxford Handbook of Philosophy and Cognitive Science, pages 249–272. Oxford University Press.Google Scholar
Field, H. (1989). Fictionalism, epistemology, and modality. Realism, Mathematics and Modality, pages 1–52. Oxford: Basil Blackwell.Google Scholar
Fodor, J. A. (1975). The Language of Thought, volume 5. Harvard University Press.Google Scholar
Fodor, J. A. (1981). The present status of the innateness controversy. In Fodor, J., editor, RePresentations: Philosophical Essays on the Foundations of Cognitive Science, pages 257–316. MIT Press.Google Scholar
Fodor, J. A. (1990). A theory of content i. In Fodor, J. A., editor, A Theory of Content. MIT Press.Google Scholar
Fodor, J. A. (1992). A theory of the child’s theory of mind. Cognition, 44:283–296.CrossRefGoogle ScholarPubMed
Fodor, J. A. (1998). Concepts: Where Cognitive Science Went Wrong. Oxford University Press.CrossRefGoogle Scholar
Frege, G. (1884). Grundlagen der Arithmetik. Wilhelm Koebner.Google Scholar
Frege, G. (1903). Grundgesetze der Arithmetik II. Olms.Google Scholar
Fuson, K. C. (2012). Children’s Counting and Concepts of Number. Springer Science & Business Media.Google Scholar
Gallistel, C., Gelman, R., and Cordes, S. (2006). The cultural and evolutionary history of the real numbers. In Levinson, S. C. & Jaisson, P. editor, Evolution and Culture, page 247. MIT Press.Google Scholar
Gallistel, C. R. (2021). The approximate number system represents magnitude and precision. Behavioral and Brain Sciences, 44:e187–e187.Google Scholar
Gallistel, C. R. and Gelman, R. (1992). Preverbal and verbal counting and computation. Cognition, 44(1–2):43–74.Google Scholar
Gallistel, C. R. and Gelman, R. (2000). Non-verbal numerical cognition: From reals to integers. Trends in Cognitive Sciences, 4(2):59–65.Google Scholar
Geach, P. T. (1972). Logic Matters. University of California Press.Google Scholar
Gelman, R. and Gallistel, C. R. (1986). The Child’s Understanding of Number. Harvard University Press.Google Scholar
Gelman, S. A., Leslie, S.-J., Gelman, R., and Leslie, A. (2019). Do children recall numbers as generic? A strong test of the generics-as-default hypothesis. Language Learning and Development, 15(3):217–231.Google Scholar
Goodman, N. and Quine, W. V. O. (1947). Steps toward a constructive nominalism. Journal of Symbolic Logic, 12(4):105–122.Google Scholar
Grinstead, J., MacSwan, J., Curtiss, S., and Gelman, R. (1997). The independence of language and number. In Twenty-Second Boston University Conference on Language Development.Google Scholar
Hale, B. and Wright, C. (2001). The Reason’s Proper Study: Towards a Neo-Fregean Philosophy of Mathematics. Oxford University Press.Google Scholar
Harnish, R. M. (2000). Minds, Brains, Computers: An Historical Introduction to the Foundations of Cognitive Science. Wiley-Blackwell.Google Scholar
Hart, W. (1991). Benacerraf’s dilemma. Crítica: Revista Hispanoamericana de Filosofía, 23(68):87–103.Google Scholar
Horn, L. (1972). On the Semantic Properties of Logical Operators. PhD thesis, University of California, Los Angeles.Google Scholar
Hurford, J. R. (1987). Language and Number: The Emergence of a Cognitive System. Basil Blackwell Oxford.Google Scholar
Hyde, D. C., Simon, C. E., Berteletti, I., and Mou, Y. (2017). The relationship between non-verbal systems of number and counting development: A neural signatures approach. Developmental Science, 20(6):e12464.Google Scholar
Kadosh, R. C. and Dowker, A. (2015). The Oxford Handbook of Numerical Cognition. Oxford Library of Psychology.Google Scholar
Kennedy, C. and Syrett, K. (2022). Numerals denote degree quantifiers: Evidence from child language. In Measurements, Numerals and Scales: Essays in Honour of Stephanie Solt, pages 135–162. Springer.Google Scholar
Kim, J. (1981). The role of perception in a priori knowledge: Some remarks. Philosophical Studies: An International Journal for Philosophy in the Analytic Tradition, 40(3):339–354.Google Scholar
Kratzer, A. and Heim, I. (1998). Semantics in Generative Grammar, volume 1185. Blackwell Oxford.Google Scholar
Landman, F. (1989). Groups, i. Linguistics and Philosophy, 12(5):559–605.Google Scholar
Laurence, S. and Margolis, E. (2005). Number and natural. The Innate Mind: Structure and Contents, 1:216.Google Scholar
Laurence, S. and Margolis, E. (2007). Linguistic determinism and the innate basis of number. In The Innate Mind, pages 139–169.Google Scholar
Le Corre, M. and Carey, S. (2007). One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition, 105(2):395–438.Google Scholar
Lee, A. Y., Myers, J., and Rabin, G. O. (2022). The structure of analog representation. Noûs, 57(1):209–237.Google Scholar
Leslie, A. M., Gelman, R., and Gallistel, C. (2008). The generative basis of natural number concepts. Trends in Cognitive Sciences, 12(6):213–218.Google Scholar
Leslie, A. M., Xu, F., Tremoulet, P. D., and Scholl, B. J. (1998). Indexing and the object concept: Developing “what” and “where” systems. Trends in Cognitive Sciences, 2(1):10–18.Google Scholar
Link, G. (1983). The logical analysis of plurals and mass terms: A lattice-theoretic approach. In Bäuerle, R., Schwarze, C., and von Stechow, A., editors, Meaning, Use, and Interpretation of Language, pages 303–323. de Gruyter.Google Scholar
Linnebo, Ø. (2009). The individuation of the natural numbers. In Bueno, O. and Linnebo, Ø. , editors, New Waves in Philosophy of Mathematics, pages 220–238. Palgrave-MacMillan.Google Scholar
Linnebo, Ø. (2017). Philosophy of Mathematics. Princeton University Press.Google Scholar
Linnebo, Ø. and Shapiro, S. (2017). Actual and potential infinity. Noûs, 53(1):160–191.Google Scholar
Macnamara, J. T. (1986). A border dispute: the place of logic in psychology. MIT Press.Google Scholar
Maddy, P. (1990). Realism in Mathematics. Oxford University Press.Google Scholar
Maddy, P. (2018). Psychology and the a priori sciences. In Bangu, S., editor, Naturalizing Logico-Mathematical Knowledge, pages 15–29. Routledge.Google Scholar
Margolis, E. (2020). The small number system. Philosophy of Science, 87(1):113–134.Google Scholar
Margolis, E. and Laurence, S. (2008). How to learn the natural numbers: Inductive inference and the acquisition of number concepts. Cognition, 106(2):924–939.Google Scholar
Margolis, E. and Laurence, S. (2013). In defense of nativism. Philosophical Studies, 165(2):693–718.Google Scholar
Margolis, E. and Laurence, S. (2022). Concepts. In Zalta, E. N. and Nodelman, U., editors, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Fall 2022 edition.Google Scholar
Margolis, E. E. and Laurence, S. E. (1999). Concepts: Core Readings. The MIT Press.Google Scholar
McMurray, B. (2007). Defusing the childhood vocabulary explosion. Science, 317(5838):631–631.Google Scholar
McNally, L., de Swart, H., Aloni, M. et al. (2011). Inflection and derivation: How adjectives and nouns refer to abstract objects. In Proceedings of the 18th Amsterdam Colloquium, pages 425–434. ILLC.Google Scholar
Meck, W. H. and Church, R. M. (1983). A mode control model of counting and timing processes. Journal of Experimental Psychology: Animal Behavior Processes, 9(3):320–356.Google Scholar
Mix, K. S. and Sandhofer, C. M. (2007). Do we need a number sense? In Roberts, M. J., editor, Integrating the Mind: Domain General vs Domain Specific Processes in Higher Cognition, pages 293–326. Psychology Press.Google Scholar
Moltmann, F. (2013). Reference to numbers in natural language. Philosophical Studies, 162:499–536.Google Scholar
O’Shaughnessy, D. M., Gibson, E., and Piantadosi, S. T. (2021). The cultural origins of symbolic number. Psychological Review, 129 (6):1442.Google ScholarPubMed
Partee, B. (1986). Noun phrase interpretation and type-shifting principles. In Groenendijk, J., de Jongh, D., and Stokhof, M., editors, Studies in Discourse Representation Theory and the Theory of Generalized Quantifiers. Foris. pages 115–143.Google Scholar
Piaget, J. and Chomsky, N. (1980). Opening the debate: The psychogenesis of knowledge and its epistemological significance. In Piattelli-Palmarini, M., editor, Language and learning: The debate between Jean Piaget and Noam Chomsky, pages 23–34. Cambridge, MA: Harvard University.Google Scholar
Pinker, S. (2007). The Stuff of Thought: Language as a Window into Human Nature. Penguin.Google Scholar
Ramsey, W. (2022). Implicit mental representation. In Bangu, S., (editor), The Routledge Handbook of Philosophy and Implicit Cognition. Routledge.Google Scholar
Ramsey, W. M. (2007). Representation Reconsidered. Cambridge University Press.Google Scholar
Resnik, M. (1997). Mathematics as a Science of Patterns. Oxford University Press.Google Scholar
Rips, L. J., Bloomfield, A., and Asmuth, J. (2008). From numerical concepts to concepts of number. Behavioral and Brain Sciences, 31(6):623–642.Google Scholar
Russell, B. (1919). Introduction to Mathematical Philosophy. Dover.Google Scholar
Sanford, E. M. and Halberda, J. (2023). Successful discrimination of tiny numerical differences. Journal of Numerical Cognition, 9(1):196–205.Google Scholar
Sarnecka, B. W. and Carey, S. (2008). How counting represents number: What children must learn and when they learn it. Cognition, 108(3):662–674.Google Scholar
Schlimm, D. (2018). Numbers through numerals: The constitutive role of external representations. In Naturalizing Logico-Mathematical Knowledge, pages 195–217. Routledge.Google Scholar
Scholl, B. J. and Leslie, A. M. (1999). Explaining the infant’s object concept: Beyond the perception/cognition dichotomy. In Lepore, E., and Pylyshyn, Z., (editors), What Is Cognitive Science, pages 26–73. Blackwell.Google Scholar
Scontras, G. (2014). The Semantics of Measurement. PhD thesis, Harvard University.Google Scholar
Searle, J. R. (1982). The Chinese room revisited. Behavioral and Brain Sciences, 5(2):345–348.Google Scholar
Shapiro, S. (1997). Philosophy of Mathematics: Structure and Ontology. Oxford University Press.Google Scholar
Shapiro, S. (2000). Thinking about Mathematics: The Philosophy of Mathematics. Oxford University Press.Google Scholar
Shea, N. (2018). Representation in Cognitive Science. Oxford University Press.Google Scholar
Simon, T. J. (1997). Reconceptualizing the origins of number knowledge: A “non-numerical” account. Cognitive Development, 12(3):349–372.Google Scholar
Snyder, E. (2017). Numbers and cardinalities: What’s really wrong with the easy argument? Linguistics and Philosophy, 40:373–400.Google Scholar
Snyder, E. (2021a). Counting, measuring, and the fractional cardinalities puzzle. Linguistics and Philosophy, 44(3):513–550.Google Scholar
Snyder, E. (2021b). Semantics and the Ontology of Number. Cambridge University Press.Google Scholar
Snyder, E., Samuels, R., and Shapiro, S. (2018a). Neologicism, Frege’s constraint, and the Frege-heck condition. Noûs, 54(1):54–77.Google Scholar
Snyder, E., Samuels, R., and Shapiro, S. (2022). Resolving Frege’s other puzzle. Philosophia Mathematica, 30(1):59–87.Google Scholar
Snyder, E., Samuels, R., and Shaprio, S. (2019). Hale’s argument from transitive counting. Synthese, 198(3):1905–1933.Google Scholar
Snyder, E. and Shapiro, S. (2022). Groups, sets, and paradox. Linguistics and Philosophy, 45(6):1277–1313.Google Scholar
Snyder, E., Shapiro, S., and Samuels, R. (2018b). Cardinals, ordinals, and the prospects for a Fregean foundation. Royal Institute of Philosophy Supplements, 82:77–107.Google Scholar
Spelke, E. S. (2000). Core knowledge. American Psychologist, 55(11):1233–1250.Google Scholar
Spelke, E. S. (2003). What makes us smart? Core knowledgeand natural language. In Gentner, D., and Goldin-Meadow, S., editors, Language in Mind: Advances in the Study of Language and Thought, pages 277–311. MIT Press.Google Scholar
Spelke, E. S. (2017). Core knowledge, language, and number. Language Learning and Development, 13(2):147–170.Google Scholar
Spelke, E. S. (2022). What Babies Know: Core Knowledge and Composition. Volume 1. Oxford University Press.Google Scholar
Strawson, P. F. (2002). Individuals. Routledge.Google Scholar
Tennant, N. (1997). The Taming of the True. Oxford University Press.Google Scholar
vanMarle, K. (2018). What happens when a child learns to count? The development of the number concept. In Bangu, S., editor, Naturalizing Logico-Mathematical Knowledge: Approaches from Philosophy, Psychology and Cognitive Science, pages 131–147. Routledge.Google Scholar
Weir, A. (2022). Formalism in the philosophy of Mathematics. In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.Google Scholar
Wright, C. (2000). Neo-Fregean foundations for real analysis: Some reflections on Frege’s constraint. Notre Dame Journal of Formal Logic, 41:317–334.Google Scholar
Wynn, K. (1992). Children’s acquisition of the number words and the counting system. Cognitive Psychology, 24(2):220–251.Google Scholar
Wynn, K. (2018). Origins of numerical knowledge. In Bangu, S., editor, Naturalizing Logico-Mathematical Knowledge, pages 106–130. Routledge.Google Scholar
Yi, B.-u. (2018). Numerical cognition and mathematical knowledge: The plural property view. In Bangu, S., editor, Naturalizing Logico-Mathematical Knowledge, pages 52–88. Routledge.Google Scholar

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