Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T15:31:08.531Z Has data issue: false hasContentIssue false
  • English
  • Français

The number sense represents (rational) numbers

Published online by Cambridge University Press:  12 April 2021

Sam Clarke  and
Jacob Beck Open the ORCID record for Jacob Beck [Opens in a new window]
Sam Clarke
Affiliation:
Department of Philosophy & Centre for Vision Research, York University, Toronto, ONM3J 1P3, Canada. spclarke@yorku.ca; http://www.sampclarke.netjbeck@yorku.ca; http://www.jacobbeck.org
Jacob Beck
Affiliation:
Department of Philosophy & Centre for Vision Research, York University, Toronto, ONM3J 1P3, Canada. spclarke@yorku.ca; http://www.sampclarke.netjbeck@yorku.ca; http://www.jacobbeck.org
Get access Rights & Permissions [Opens in a new window]

Abstract

On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique – the arguments from congruency, confounds, and imprecision – and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind(s) of number being represented. In response, we propose that the ANS represents not only natural numbers (e.g., 7), but also non-natural rational numbers (e.g., 3.5). It does not represent irrational numbers (e.g., √2), however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research.

Keywords

analog magnitude systemapproximate number systemnumber sensenumerical cognitionnumerosity
Type
Target Article
Information
Behavioral and Brain Sciences , Volume 44 , 2021 , e178
Check for updates
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anobile, G., Arrighi, R., Castaldi, E., & Burr, D. C. (2021). A sensorimotor numerosity system. Trends in Cognitive Sciences, 25(1), 24–36. https://doi.org/10.1016/j.tics.2020.10.009.CrossRefGoogle ScholarPubMed
Anobile, G., Cicchini, G. M., & Burr, D. C. (2016). Number as a primary perceptual attribute: A review. Perception, 45(1–2), 5–31. https://doi.org/10.1177/0301006615602599.CrossRefGoogle ScholarPubMed
Arrighi, R., Togoli, I., & Burr, D. C. (2014). A generalized sense of number. Proceedings of the Royal Society B: Biological Sciences, 281(1797), 20141791–20141791. https://doi.org/10.1098/rspb.2014.1791.CrossRefGoogle ScholarPubMed
Ball, B. (2017). On representational content and format in core numerical cognition. Philosophical Psychology, 30(1–2), 119–139. https://doi.org/10.1080/09515089.2016.1263988.CrossRefGoogle Scholar
Barth, H., Baron, A., Spelke, E., & Carey, S. (2009). Children's multiplicative transformations of discrete and continuous quantities. Journal of Experimental Child Psychology, 103(4), 441–454. https://doi.org/10.1016/j.jecp.2009.01.014.CrossRefGoogle ScholarPubMed
Barth, H., Kanwisher, N., & Spelke, E. (2003). The construction of large number representations in adults. Cognition, 86(3), 201–221. https://doi.org/10.1016/S0010-0277(02)00178-6.CrossRefGoogle ScholarPubMed
Barth, H., La Mont, K., Lipton, J., Dehaene, S., Kanwisher, N., & Spelke, E. (2006). Non-symbolic arithmetic in adults and young children. Cognition, 98(3), 199–222. https://doi.org/10.1016/j.cognition.2004.09.011.CrossRefGoogle ScholarPubMed
Barth, H., La Mont, K., Lipton, J., & Spelke, E. S. (2005). Abstract number and arithmetic in preschool children. Proceedings of the National Academy of Sciences, 102(39), 14116–14121. https://doi.org/10.1073/pnas.0505512102.CrossRefGoogle ScholarPubMed
Beck, J. (2013). Why we can't say what animals think. Philosophical Psychology, 26(4), 520–546. https://doi.org/10.1080/09515089.2012.670922.CrossRefGoogle Scholar
Beck, J. (2015). Analogue magnitude representations: A philosophical Introduction. The British Journal for the Philosophy of Science, 66(4), 829–855. https://doi.org/10.1093/bjps/axu014.CrossRefGoogle Scholar
Beck, J. (2018). Analog mental representation. WIREs Cognitive Science, 9(6), e1479. doi.org/10.1002/wcs.1479.CrossRefGoogle ScholarPubMed
Beck, J. (2019). Perception is analog: The argument from Weber's Law. The Journal of Philosophy, 116(6), 319–349. https://doi.org/10.5840/jphil2019116621.CrossRefGoogle Scholar
Bhatia, P., Delem, M., Léone, J., Boisin, E., Cheylus, A., Gardes, M.-L., & Prado, J. (2020). The ratio processing system and its role in fraction understanding: Evidence from a match-to-sample task in children and adults with and without dyscalculia. Quarterly Journal of Experimental Psychology, 73(12), 2158–2176. doi:10.1177/1747021820940631.CrossRefGoogle ScholarPubMed
Binzak, J. V., & Hubbard, E. M. (2020). No calculation necessary: Accessing magnitude through decimals and fractions. Cognition, 199, 104219. doi: 10.1016/j.cognition.2020.104219.CrossRefGoogle ScholarPubMed
Buijsman, S. (2021). The representations of the approximate number system. Philosophical Psychology, 34(2), 300–317. https://doi.org/10.1080/09515089.2020.1866755.CrossRefGoogle Scholar
Burge, T. (1982). Other bodies. In Woodfield, A. (Ed.), Thought and object (pp.97–120). Oxford University Press.Google Scholar
Burge, T. (2005). Truth, thought, reason: Essays on frege. Oxford University Press.CrossRefGoogle Scholar
Burge, T. (2010). The origins of objectivity. Oxford University Press.CrossRefGoogle Scholar
Burr, D., & Ross, J. (2008). A visual sense of number. Current Biology, 18(6), 425–428. doi: 10.1016/j.cub.2008.02.052.CrossRefGoogle ScholarPubMed
Burr, D., Anobile, G., Togoli, I., Domenici, N., & Arrighi, R. (2019). Motor adaptation affects perception of time and numerosity. Journal of Vision, 19(10), 164b–164b. https://doi.org/10.1167/19.10.164b.CrossRefGoogle Scholar
Carey, S. (2009). The origin of concepts. Oxford University Press.CrossRefGoogle Scholar
Carey, S., & Barner, D. (2019). Ontogenetic origins of human Integer representations. Trends in Cognitive Sciences, 23(10), P823–835. https://doi.org/10.1016/j.tics.2019.07.004.CrossRefGoogle ScholarPubMed
Casasanto, D., & Boroditsky, L. (2008). Time in the mind: Using space to think about time. Cognition, 106, 579–593. https://doi.org/10.1016/j.cognition.2007.03.004.CrossRefGoogle ScholarPubMed
Cheyette, S. J., & Piantadosi, S. T. (2020). A unified account of numerosity perception. Nature Human Behaviour, 4, 1265–1272. https://doi.org/10.1038/s41562-020-00946-0.CrossRefGoogle ScholarPubMed
Church, R. M. & Meck, W. H. (1984). The numerical attribute of stimuli. In Roitblat, H. L., Bever, T. G., & Terrace, H. S. (Eds.), Animal Cognition (pp. 445–464). Erlbaum.Google Scholar
Cicchini, G. M., Anobile, G., & Burr, D. C. (2016). Spontaneous perception of numerosity in humans. Nature Communications, 7, 12536. https://doi.org/10.1038/ncomms12536.CrossRefGoogle ScholarPubMed
Clarke, S. (forthcoming). Beyond the icon: Core cognition and the bounds of perception. Mind & Language. https://doi.org/10.1111/mila.12315.Google Scholar
Clarke, S. (2021). Cognitive penetration and informational encapsulation: Have we been failing the module? Philosophical Studies, 178, 2599–2620. https://doi.org/10.1007/s11098-020-01565-1.CrossRefGoogle Scholar
Cohen Kadosh, R., & Henik, A. (2006). A common representation for semantic and physical properties. Experimental Psychology, 53(2), 87–94. https://doi.org/10.1027/1618-3169.53.2.87.CrossRefGoogle ScholarPubMed
Content, A., Velde, M., & Adriano, A. (2017). Approximate number sense theory or approximate theory of magnitude? Behavioral and Brain Sciences, 40, E168. doi:10.1017/S0140525X16002089.CrossRefGoogle ScholarPubMed
Cordes, S., Gelman, R., Gallistel, R., & Whalen, J. (2001). Variability signatures distinguish verbal from nonverbal counting for both large and small numbers. Psychonomic Bulletin & Review, 8(4), 698–707. doi:10.3758/BF03196206.CrossRefGoogle ScholarPubMed
Dakin, S. C., Tibber, M. S., Greenwood, J. A., Kingdom, F. A. A., & Morgan, M. J. (2011). A common visual metric for approximate number and density. Proceedings of the National Academy of Sciences, 108(49), 19552–19557. https://doi.org/10.1073/pnas.1113195108.CrossRefGoogle ScholarPubMed
Dehaene, S. (2011). The number sense: How the mind creates mathematics, revised and updated edition: How the mind creates mathematics. Oxford University Press.Google Scholar
Dehaene, S., & Changeux, J. P. (1993). Development of elementary numerical abilities: A neuronal model. Journal of Cognitive Neuroscience 5(4), 390–407. doi:10.1162/jocn.1993.5.4.390.CrossRefGoogle ScholarPubMed
Dehaene, S., Izard, V., & Piazza, M. (2005). Control over non-numerical parameters in numerosity experiments. http://www.unicog.org/docs/DocumentationDotsGeneration.doc%3E.Google Scholar
Denison, S., & Xu, F. (2010). Twelve- to 14-month-old infants can predict single-event probability with large set sizes. Developmental Science, 13(5), 798–803. https://doi.org/10.1111/j.1467-7687.2009.00943.x.CrossRefGoogle ScholarPubMed
Denison, S., & Xu, F. (2014). The origins of probabilistic inference in human infants. Cognition, 130(3), 335–347. https://doi.org/10.1016/j.cognition.2013.12.001.CrossRefGoogle ScholarPubMed
DeSimone, K., Kim, M., & Murray, R. F. (2020). Number adaptation can be dissociated from density adaptation. Psychological Science, 31(11), 1470–1474. doi:10.1177/0956797620956986.CrossRefGoogle ScholarPubMed
DeWind, N. K., Adams, G. K., Platt, M. L., & Brannon, E. M. (2015). Modeling the approximate number system to quantify the contribution of visual stimulus features. Cognition, 142, 247–265. https://doi.org/10.1016/j.cognition.2015.05.016.CrossRefGoogle ScholarPubMed
Drucker, C. B., Rossa, M. A., & Brannon, E. M. (2016). Comparison of discrete ratios by rhesus macaques (Macaca mulatta). Animal Cognition, 19(1), 75–89. https://doi.org/10.1007/s10071-015-0914-9.CrossRefGoogle Scholar
Duhem, P. (1914). La théorie physique son objet et sa structure (2nd Ed.). Paris: Chevalier et Rivière. English Translation Philip P. Wiener, The Aim and Structure of Physical Theory. Princeton University Press, 1954.Google Scholar
Dummett, M. (1981). Frege: Philosophy of language (2nd ed.). Duckworth, and Harvard University Press.Google Scholar
Durgin, F. H. (2008). Texture density adaptation and visual number revisited. Current Biology, 18(18), R855–R856. doi: 10.1016/j.cub.2008.07.053.CrossRefGoogle ScholarPubMed
Eckert, J., Call, J., Hermes, J., Herrmann, E., & Rakoczy, H. (2018). Intuitive statistical inferences in chimpanzees and humans follow Weber's Law. Cognition, 180, 99–107. https://doi.org/10.1016/j.cognition.2018.07.004.CrossRefGoogle ScholarPubMed
Evans, G. (1982). The varieties of reference. Oxford University Press.Google Scholar
Feigenson, L., Dehaene, S., & Spelke, E. (2004). Core systems of number. Trends in Cognitive Sciences, 8(7), 307–314. https://doi.org/10.1016/j.tics.2004.05.002.CrossRefGoogle ScholarPubMed
Fodor, J. (1983). The modularity of mind. MIT Press.CrossRefGoogle Scholar
Fodor, J. (1998). Concepts: Where cognitive science went wrong. Oxford: OUP.CrossRefGoogle Scholar
Fontanari, L., Gonzalez, M., Vallortigara, G., & Girotto, V. (2014). Probabilistic cognition in two indigenous Mayan groups. Proceedings of the National Academy of Sciences, 111(48), 17075–17080. https://doi.org/10.1073/pnas.1410583111.CrossRefGoogle ScholarPubMed
Fornaciai, M., Cicchini, G. M., & Burr, D. C. (2016). Adaptation to number operates on perceived rather than physical numerosity. Cognition, 151, 63–67. https://doi.org/10.1016/j.cognition.2016.03.006.CrossRefGoogle ScholarPubMed
Fornaciai, M., & Park, J. (2018). Early numerosity encoding in visual cortex is not sufficient for the representation of numerical magnitude. Journal of Cognitive Neuroscience, 30(12), 1788–1802. https://doi.org/10.1162/jocn_a_01320.CrossRefGoogle Scholar
Franconeri, S. L., Bemis, D. K., & Alvarez, G. A. (2009). Number estimation relies on a set of segmented objects. Cognition, 113(1), 1–13. https://doi.org/10.1016/j.cognition.2009.07.002.CrossRefGoogle ScholarPubMed
Frege, G. (1884). Die Grundlagen der Arithmetik: eine logisch mathematische Untersuchung über den Begriff der Zahl, Breslau: W. Koebner; translated as The foundations of arithmetic: A logico-mathematical enquiry into the concept of number, by J.L. Austin, Oxford: Blackwell, second revised edition, 1974.Google Scholar
Frege, G. (1892). Über Sinn und Bedeutung. In Zeitschrift für Philosophie und philosophische Kritik (Vol. 100, pp. 25–50); translated as ‘On sense and reference’ by M. Black in Geach and Black (eds. and trans.), 1980, 56–78.Google Scholar
Gallistel, R., & Gelman, R. (2000). Non-verbal numerical cognition: From reals to integers. Trends in Cognitive Sciences, 4(2), 59–65. doi: 10.1016/s1364-6613(99)01424-2CrossRefGoogle ScholarPubMed
Gebuis, T., Cohen Kadosh, R., & Gevers, W. (2016). Sensory-integration system rather than approximate number system underlies numerosity processing: A critical review. Acta Psychologica, 171, 17–35. https://doi.org/10.1016/j.actpsy.2016.09.003.CrossRefGoogle ScholarPubMed
Gebuis, T., Herfs, I. K., Kenemans, J. L., de Haan, E. H. F., & van der Smagt, M. J. (2009). The development of automated access to number knowledge in children: An ERP study. European Journal of Neuroscience 30, 1999–2008. doi:10.1111/j.1460-9568.2009.06994.x.CrossRefGoogle Scholar
Gebuis, T., Kenemans, J. L., de Haan, E. H. F., & van der Smagt, M. J. (2010). Conflict processing of symbolic and non-symbolic numerosity. Neuropsychologia, 48(2), 394–401. https://doi.org/10.1016/j.neuropsychologia.2009.09.027.CrossRefGoogle ScholarPubMed
Gebuis, T., & Reynvoet, B. (2012a). Continuous visual properties explain neural responses to nonsymbolic number. Psychophysiology, 49(11), 1649–1659. https://doi.org/10.1111/j.1469-8986.2012.01461.x.CrossRefGoogle Scholar
Gebuis, T., & Reynvoet, B. (2012b). The interplay between nonsymbolic number and its continuous visual properties. Journal of Experimental Psychology: General, 141(4), 642–648. https://doi.org/10.1037/a0026218.CrossRefGoogle Scholar
Gebuis, T, & Reynvoet, B (2012c). The role of visual information in numerosity estimation. PLoS ONE, 7(5), e37426. https://doi.org/10.1371/journal.pone.0037426.CrossRefGoogle Scholar
Gergely, G, & Csibra, G. (2003). Teleological reasoning in infancy: The naïve theory of rational action. Trends in Cognitive Sciences, 7(7), 287–292. doi:10.1016/s1364-6613(03)00128-1.CrossRefGoogle Scholar
Girotto, V., & Gonzalez, M. (2008). Children's understanding of posterior probability. Cognition, 106(1), 325–344. https://doi.org/10.1016/j.cognition.2007.02.005.CrossRefGoogle ScholarPubMed
Gordon, P. (2005). Numerical cognition without words: Evidence from Amazonia. Science (New York, N.Y.), 306(5695), 496–499. https://doi.org/10.1126/science.1094492.CrossRefGoogle Scholar
Green, E. J. (2018). Psychosemantics and the rich/thin debate. Philosophical Topics, 31(1), 153–186. https://doi.org/10.1111/phpe.12097Google Scholar
Gweon, H., Tenenbaum, J. B., & Schulz, L. E. (2010). Infants consider both the sample and the sampling process in inductive generalization. Proceedings of the National Academy of Sciences, 107(20), 9066–9071. https://doi.org/10.1073/pnas.1003095107.CrossRefGoogle ScholarPubMed
Halberda, J. (2016). Epistemic limitations and precise estimates in analog magnitude representation. In D. Barner and A. S. Baron (Eds.), Oxford Series in cognitive development. Core knowledge and conceptual change (pp. 171–190). Oxford University Press.CrossRefGoogle Scholar
Halberda, J. (2019). Perceptual input is not conceptual content. Trends in Cognitive Sciences, 23(8), 636–638. https://doi.org/10.1016/j.tics.2019.05.007.CrossRefGoogle Scholar
Halberda, J., Mazzocco, M. M. M., & Feigenson, L. (2008). Individual differences in non-verbal number acuity correlate with maths achievement. Nature, 455(7213), 665–668. https://doi.org/10.1038/nature07246.CrossRefGoogle ScholarPubMed
He, L., Zhang, J., Zhou, T., & Chen, L. (2009). Connectedness affects dot numerosity judgment: Implications for configural processing. Psychonomic Bulletin & Review, 16(3), 509–517. https://doi.org/10.3758/PBR.16.3.509.CrossRefGoogle ScholarPubMed
Henik, A., & Tzelgov, J. (1982). Is three greater than five: The relation between physical and semantic size in comparison tasks. Memory & Cognition, 10(4), 389–395. https://doi.org/10.3758/BF03202431.CrossRefGoogle ScholarPubMed
Irie, N., Hiraiwa-Hasegawa, M., & Kutsukake, N. (2019). Unique numerical competence of Asian elephants on the relative numerosity judgment task. Journal of Ethology, 37(1), 111–115. https://doi.org/10.1007/s10164-018-0563-y.CrossRefGoogle Scholar
Izard, V., Sann, C., Spelke, E. S., & Streri, A. (2009). Newborn infants perceive abstract numbers. Proceedings of the National Academy of Sciences, 106(25), 10382–10385. https://doi.org/10.1073/pnas.0812142106.CrossRefGoogle ScholarPubMed
Johnson, A. (2004). Attention: Theory and practice. Thousand Oaks, Calif: Sage Publications.CrossRefGoogle Scholar
Kayhan, E., Gredebäck, G., & Lindskog, M. (2018). Infants distinguish between two events based on their relative likelihood. Child Development, 89(6), e507–e519. https://doi.org/10.1111/cdev.12970.CrossRefGoogle ScholarPubMed
Kirjakovski, A., & Matsumoto, E. (2016). Numerosity underestimation in sets with illusory contours. Vision Research, 122, 34–42. https://doi.org/10.1016/j.visres.2016.03.005.CrossRefGoogle ScholarPubMed
Kominsky, J., & Carey, S. (2018). Early-developing causal perception is sensitive to multiple physical constraints. In Rogers, T., Rau, M., Zhu, J., & Kalish, C. (Eds.), Proceedings of the 40th annual conference of the cognitive science society (pp. 622–627). Cognitive Science Society.Google Scholar
Laurence, S., & Margolis, E. (2005). Number and natural language. In Carruthers, P., Laurence, S., & Stich, S. (Eds.), The Innate Mind: Structure and Contents (pp. 216–235). Oxford University Press. https://doi.org/10.1093/acprof:oso/9780195179675.003.0013CrossRefGoogle Scholar
Leibovich, T., & Henik, A. (2013). Magnitude processing in non-symbolic stimuli. Frontiers in Psychology, 4. https://doi.org/10.3389/fpsyg.2013.00375.CrossRefGoogle ScholarPubMed
Leibovich, T., & Henik, A. (2014). Comparing performance in discrete and continuous comparison tasks. Quarterly Journal of Experimental Psychology, 67(5), 899–917. https://doi.org/10.1080/17470218.2013.837940.CrossRefGoogle ScholarPubMed
Leibovich, T., Katzin, N., Harel, M., & Henik, A. (2017). From “sense of number” to “sense of magnitude” – The role of continuous magnitudes in numerical cognition. Behavioral and Brain Sciences, 40, e164. https://doi.org/10.1017/S0140525X16000960.CrossRefGoogle Scholar
Lemer, C., Dehaene, S., Spelke, E., & Cohen, L. (2003). Approximate quantities and exact number words: Dissociable systems. Neuropsychologia, 41(14), 1942–1958. https://doi.org/10.1016/S0028-3932(03)00123-4.CrossRefGoogle ScholarPubMed
Lewis, D. (1971). Analog and digital. Noûs, 5(3), 321–327. doi:10.2307/2214671.CrossRefGoogle Scholar
Lewis, M. R., Matthews, P. G., & Hubbard, E. M. (2016). Chapter 6 – Neurocognitive architectures and the nonsymbolic foundations of fractions understanding. In Berch, D. B., Geary, D. C., & Koepke, K. M. (Eds.), Development of mathematical cognition (pp. 141–164). Academic Press. https://doi.org/10.1016/B978-0-12-801871-2.00006-X.CrossRefGoogle Scholar
Lipton, J. S., & Spelke, E. S. (2003). Origins of number sense: Large-number discrimination in human infants. Psychological Science, 14(5), 396–401. https://doi.org/10.1111/1467-9280.01453.CrossRefGoogle ScholarPubMed
Lourenco, S. F., & Longo, M. R. (2010). General magnitude representation in human infants. Psychological Science, 21(6), 873–881. https://doi.org/10.1177/0956797610370158.CrossRefGoogle ScholarPubMed
Lucero, C., Brookshire, G., Sava-Segal, C., Bottini, R., Goldin-Meadow, S., Vogel, E., & Casasanto, D. (2020). Unconscious number discrimination in the human visual system. Cerebral Cortex, 30(11), 5821–5829. doi:10.1093/cercor/bhaa155.CrossRefGoogle ScholarPubMed
Maley, C. J. (2011). Analog and digital, continuous and discrete. Philosophical Studies, 155(1), 117–131. https://doi.org/10.1007/s11098-010-9562-8.CrossRefGoogle Scholar
Mandelbaum, E. (2013). Numerical architecture. Topics in Cognitive Science, 5(2), 367–386. https://doi.org/10.1111/tops.12014.CrossRefGoogle ScholarPubMed
Margolis, E. (2020). The small number system. Philosophy of Science, 87(1), 113–134. https://doi.org/10.1086/706087.CrossRefGoogle Scholar
Margolis, E., & Laurence, S. (2008). How to learn the natural numbers: Inductive inference and the acquisition of number concepts. Cognition, 106(2), 924–939. https://doi.org/10.1016/j.cognition.2007.03.003.CrossRefGoogle ScholarPubMed
Marr, D. (1982). Vision: A computational investigation into the human representation and processing of visual information. MIT Press.Google Scholar
Marshall, O. R. (2017). The psychology and philosophy of natural numbers. Philosophia Mathematica, 26(1), 40–58. https://doi.org/10.1093/philmat/nkx002.CrossRefGoogle Scholar
Martin, B., Wiener, M., & van Wassenhove, V. (2017). A Bayesian perspective on accumulation in the magnitude system. Scientific Reports, 7(1), 630. https://doi.org/10.1038/s41598-017-00680-0.CrossRefGoogle ScholarPubMed
Matthews, P. G., & Chesney, D. L. (2015). Fractions as percepts? Exploring cross-format distance effects for fractional magnitudes. Cognitive Psychology, 78, 28–56. https://doi.org/10.1016/j.cogpsych.2015.01.006.CrossRefGoogle ScholarPubMed
Matthews, P. G., Lewis, M. R., & Hubbard, E. M. (2016). Individual differences in nonsymbolic ratio processing predict symbolic math performance. Psychological Science, 27(2), 191–202. doi:10.1177/0956797615617799.CrossRefGoogle ScholarPubMed
McCrink, K., & Spelke, E. S. (2010). Core multiplication in childhood. Cognition, 116(2), 204–216. https://doi.org/10.1016/j.cognition.2010.05.003.CrossRefGoogle ScholarPubMed
McCrink, K., & Spelke, E. S. (2016). Non-symbolic division in childhood. Journal of Experimental Child Psychology, 142, 66–82. https://doi.org/10.1016/j.jecp.2015.09.015.CrossRefGoogle ScholarPubMed
McCrink, K., Spelke, E. S., Dehaene, S., & Pica, P. (2012). Non-symbolic halving in an Amazonian indigene group. Developmental Science, 16(3), 451–462. https://doi.org/10.1111/desc.12037.CrossRefGoogle Scholar
McCrink, K., & Wynn, K. (2007). Ratio abstraction by 6-month-old infants. Psychological Science, 18(8), 740–745. https://doi.org/10.1111/j.1467-9280.2007.01969.x.CrossRefGoogle ScholarPubMed
Morgan, M. J., Raphael, S., Tibber, M. S., & Dakin, S. C. (2014). A texture-processing model of the “visual sense of number.” Proceedings of the Royal Society B: Biological Sciences, 281(1790), 20141137. https://doi.org/10.1098/rspb.2014.1137.CrossRefGoogle Scholar
Neander, K. (2017). A mark of the mental: A defence of informational teleosemantics. MIT Press.CrossRefGoogle Scholar
Nieder, A. (2016). The neuronal code for number. Nature Reviews Neuroscience, 17(6), 366–382. https://doi.org/10.1038/nrn.2016.40.CrossRefGoogle ScholarPubMed
Nieder, A. (2017). Number faculty is rooted in our biological heritage. Trends in Cognitive Sciences, 21(6), 403–404. https://doi.org/10.1016/j.tics.2017.03.014.CrossRefGoogle ScholarPubMed
Nieder, A. (2020). Neural constraints on human number concepts. Current Opinion in Neurobiology, 60, 28–36. https://doi.org/10.1016/j.conb.2019.10.003.CrossRefGoogle ScholarPubMed
Núñez, R. E. (2017). Is there really an evolved capacity for number? Trends in Cognitive Sciences, 21(6), 409–424. https://doi.org/10.1016/j.tics.2017.03.005.CrossRefGoogle ScholarPubMed
Nys, J., & Content, A. (2012). Judgement of discrete and continuous quantity in adults: Number counts! The Quarterly Journal of Experimental Psychology, 65(4), 675–690. https://doi.org/10.1080/17470218.2011.619661.CrossRefGoogle ScholarPubMed
Odic, D. (2018). Children's intuitive sense of number develops independently of their perception of area, density, length, and time. Developmental Science, 21(2), e12533. https://doi.org/10.1111/desc.12533.CrossRefGoogle ScholarPubMed
Odic, D., & Starr, A. (2018). An introduction to the approximate number system. Child Development Perspectives, 12(4), 223–229. https://doi.org/10.1111/cdep.12288.CrossRefGoogle ScholarPubMed
Park, J., & Brannon, E. M. (2013). Training the approximate number system improves math proficiency. Psychological Science, 24(10), 2013–2019. https://doi.org/10.1177/0956797613482944.CrossRefGoogle ScholarPubMed
Park, J., DeWind, N., & Brannon, E. (2017). Direct and rapid encoding of numerosity in the visual stream. Behavioral and Brain Sciences, 40, E185. doi:10.1017/S0140525X16002235.CrossRefGoogle ScholarPubMed
Peacocke, C. (1986). Analogue content. Proceedings of the Aristotelian Society, Supplementary Volumes, 60, 1–17. https://www.jstor.org/stable/4106896.CrossRefGoogle Scholar
Peacocke, C. (1992). A study of concepts. MIT Press.Google Scholar
Peacocke, C. (2020). The primacy of metaphysics. Oxford: OUP.Google Scholar
Petzschner, F. H., Glasauer, S., & Stephan, K. E. (2015). A Bayesian perspective on magnitude estimation. Trends in Cognitive Sciences, 19(5), 285–293. https://doi.org/10.1016/j.tics.2015.03.002.CrossRefGoogle ScholarPubMed
Piazza, M. (2011). Neurocognitive start-up tools for symbolic number representations. Trends in Cognitive Sciences, 14(12), 542–551. https://doi.org/10.1016/j.tics.2010.09.008.CrossRefGoogle Scholar
Pica, P., Lemer, C., Izard, V., & Dehaene, S. (2005). Exact and approximate arithmetic in an Amazonian indigene group. Science (New York, N.Y.), 306(5695), 499–503. https://doi.org/10.1126/science.1102085.CrossRefGoogle Scholar
Picon, E., Dramkin, D., & Odic, D. (2019). Visual illusions help reveal the primitives of number perception. Journal of Experimental Psychology: General, 148(10), 1675–1687. https://doi.org/10.1037/xge0000553.CrossRefGoogle ScholarPubMed
Plotnik, J. M., Brubaker, D. L., Dale, R., Tiller, L. N., Mumby, H. S., & Clayton, N. S. (2019). Elephants have a nose for quantity. Proceedings of the National Academy of Sciences, 116(25), 12566–12571. https://doi.org/10.1073/pnas.1818284116.CrossRefGoogle ScholarPubMed
Putnam, H. (1975). The meaning of “meaning.” Minnesota Studies in the Philosophy of Science 7, 131–193. http://hdl.handle.net/11299/185225.Google Scholar
Quine, W. V. (1951). Two dogmas of empiricism. Philosophical Review, 60(1), 20–43. https://doi.org/10.2307/2181906CrossRefGoogle Scholar
Rakoczy, H., Clüver, A., Saucke, L., Stoffregen, N., Gräbener, A., Migura, J., & Call, J. (2014). Apes are intuitive statisticians. Cognition, 131(1), 60–68. https://doi.org/10.1016/j.cognition.2013.12.011.CrossRefGoogle ScholarPubMed
Sarrazin, J.-C., Giraudo, M.-D., Pailhous, J., & Bootsma, R. J. (2004). Dynamics of balancing space and time in memory: Tau and kappa effects revisited. Journal of Experimental Psychology: Human Perception and Performance, 30(3), 411–430. https://doi.org/10.1037/0096-1523.30.3.411.Google ScholarPubMed
Savelkouls, S., & Cordes, S. (2020). The impact of set size on cumulative area judgments. Acta Psychologica, 210, 103163. https://doi.org/10.1016/j.actpsy.2020.103163.CrossRefGoogle ScholarPubMed
Shea, N. (2018). Representation in cognitive science. Oxford University Press.CrossRefGoogle Scholar
Siegler, R. S., Fazio, L. K., Bailey, D. H., & Zhou, X. (2013). Fractions: The new frontier for theories of numerical development. Trends in Cognitive Sciences, 17(1), 13–19. doi: 10.1016/j.tics.2012.11.004.CrossRefGoogle ScholarPubMed
Starr, A., Libertus, M. E., & Brannon, E. M. (2013). Number sense in infancy predicts mathematical abilities in childhood. Proceedings of the National Academy of Sciences, 110(45), 18116–18120. https://doi.org/10.1073/pnas.1302751110.CrossRefGoogle ScholarPubMed
Szkudlarek, E., & Brannon, E. (2021). First and second graders successfully reason about ratio with both dot arrays and Arabic numerals. Child Development, 92(3), 1011–1027. https://doi.org/10.1111/cdev.13470.CrossRefGoogle ScholarPubMed
Tecwyn, E. C., Denison, S., Messer, E. J. E., & Buchsbaum, D. (2017). Intuitive probabilistic inference in capuchin monkeys. Animal Cognition, 20(2), 243–256. https://doi.org/10.1007/s10071-016-1043-9.CrossRefGoogle ScholarPubMed
Téglás, E., Vul, E., Girotto, V., Gonzalez, M., Tenenbaum, J. B., & Bonatti, L. L. (2011). Pure reasoning in 12-month-old infants as probabilistic inference. Science (New York, N.Y.), 332(6033), 1054–1059. https://doi.org/10.1126/science.1196404.CrossRefGoogle ScholarPubMed
Tenenbaum, J. B., Kemp, C., Griffiths, T. L., & Goodman, N. D. (2011). How to grow a mind: Statistics, structure, and abstraction. Science (New York, N.Y.), 331(6022), 1279–1285. https://doi.org/10.1126/science.1192788.CrossRefGoogle Scholar
Tokita, M., Ashitani, Y., & Ishiguchi, A. (2013). Is approximate numerical judgment truly modality-independent? Visual, auditory, and cross-modal comparisons. Attention, Perception, & Psychophysics, 75(8), 1852–1861. https://doi.org/10.3758/s13414-013-0526-x.CrossRefGoogle ScholarPubMed
Tokita, M., & Ishiguchi, A. (2012). Behavioral evidence for format-dependent processes in approximate numerosity representation. Psychonomic Bulletin & Review 19, 285–293. https://doi.org/10.3758/s13423-011-0206-6.CrossRefGoogle ScholarPubMed
Tomlinson, R. C., DeWind, N. K., & Brannon, E. M. (2020). Number sense biases children's area judgments. Cognition, 204, 104352. https://doi.org/10.1016/j.cognition.2020.104352.CrossRefGoogle ScholarPubMed
von Neumann, J. (1958). The computer and the brain. Yale University Press.Google Scholar
Wagner, J. B., & Johnson, S. C. (2011). An association between understanding cardinality and analog magnitude representations in preschoolers. Cognition, 119(1), 10–22. https://doi.org/10.1016/j.cognition.2010.11.014.CrossRefGoogle ScholarPubMed
Wood, J. N., & Spelke, E. S. (2005). Infants’ enumeration of actions: Numerical discrimination and its signature limits. Developmental Science, 8(2), 173–181. https://doi.org/10.1111/j.1467-7687.2005.00404.x.CrossRefGoogle ScholarPubMed
Xu, F., & Denison, S. (2009). Statistical inference and sensitivity to sampling in 11-month-old infants. Cognition, 112(1), 97–104. https://doi.org/10.1016/j.cognition.2009.04.006.CrossRefGoogle ScholarPubMed
Xu, F., & Garcia, V. (2008). Intuitive statistics by 8-month-old infants. Proceedings of the National Academy of Sciences, 105(13), 5012–5015. https://doi.org/10.1073/pnas.0704450105.CrossRefGoogle ScholarPubMed
Xu, F., & Spelke, E. S. (2000). Large number discrimination in 6-month-old infants. Cognition, 74(1), B1–B11. https://doi.org/10.1016/S0010-0277(99)00066-9.CrossRefGoogle ScholarPubMed
Xuan, B., Zhang, D., He, S., & Chen, X. (2007). Larger stimuli are judged to last longer. Journal of Vision, 7(10), 2, 1–5. https://doi.org/10.1167/7.10.2.CrossRefGoogle ScholarPubMed
Yousif, S. R., & Keil, F. C. (2020). Area, not number, dominates estimates of visual quantities. Scientific Reports, 10(1), 13407. https://doi.org/10.1038/s41598-020-68593-z.CrossRefGoogle Scholar
Zimmermann, E. (2018). Small numbers are sensed directly, high numbers constructed from size and density. Cognition, 173, 1–7. https://doi.org/10.1016/j.cognition.2017.12.003.CrossRefGoogle ScholarPubMed