Abstract
Nearly all psychologists think that cardinality is the basis of number knowledge. When they test infants’ sensitivity to number, they look for evidence that the infants grasp the cardinality of groups of physical objects. And when they test older children’s understanding of the meaning of number words, they look for evidence that the children can, for example, “Give [the experimenter] three pencils” or can “Point to the picture of four balloons.” But when people think about the positive integers, do they single them out by means of the numbers’ cardinality, by means of the ordinal relations that hold among them, or in some other way? This chapter reviews recent research in cognitive psychology that compares people’s judgments about the integers’ cardinal and ordinal properties. It also presents new experimental evidence suggesting that, at least for adults, the integers’ cardinality is less central than their number-theoretic and arithmetic properties.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This issue might be put by asking whether people think of the first few naturals as cardinal or ordinal numbers. However, “cardinal number” and “ordinal number” have special meanings in set theory, and these meanings don’t provide the intended contrast. In their usual development (e.g., [8]), the ordinals and cardinals do not differ in the finite range, with which we will be concerned in this chapter. (Both ordinal and cardinal numbers include transfinite numbers—for which they do differ—and so extend beyond the natural numbers.)
- 2.
You might wonder about the use of sets in this construction: Do young children have a notion of set that’s comparable to mathematicians’ sets? Attributing to children at this age a concept of set in the full-blooded sense would be fatal to Carey’s claim that Quinian bootstrapping produces new primitive concepts (e.g., the concept FIVE) that can’t in principle be expressed in terms of the child’s pre-bootstrap vocabulary. We know how to express FIVE in terms of sets (see, e.g., [8]). So the notion of set implicit in representations like {object 1 , object 2 } is presumably more restrictive than ordinary sets.
- 3.
Many quantitative contexts, including some just mentioned, involve measurement of continuous dimensions (e.g., temperature, time, and length) rather than counting. Current research in developmental psychology suggests that infants are about equally sensitive to the number of objects in a collection (for n > 3) and to the continuous extent (e.g., area) of a single object when it is presented alone (see [9] for an overview). This is presumably because the same kind of psychophysical mechanism handles both types of information (see Section 3.1). However, infants’ accuracy for the number of objects in a collection (again, for n > 3) is better than that for the continuous extent of the objects in the same collection [5]. Of course, children’s knowledge of continuous extent, like their knowledge of number, has to undergo further changes before it can support adult uses of measurement (see, e.g., [24]). Some intricate issues exist about children’s understanding of continuous quantity that are the topic of current research [39], but because this chapter focuses on knowledge of natural numbers, this brief summary will have to do.
- 4.
The same is true in education theory, as Sinclair points out in her chapter in this volume. See that chapter for an alternative that is more in line with the structural perspective.
References
W.P. Banks, M. Fujii and F. Kayra-Stuart, “Semantic congruity effects in comparative judgments of magnitudes of digits,” Journal of Experimental Psychology: Human Perception and Performance, vol. 2 (1976) 435–447.
P.B. Buckley and C.B. Gillman, “Comparisons of digits and dot patterns,” Journal of Experimental Psychology, vol. 103 (1974) 1131–1136.
S. Carey, The origin of concepts, Oxford University Press, New York, NY, 2009.
L. Carroll, Alice’s adventures in wonderland, Macmillan and Co., London, 1866.
S. Cordes and E.M. Brannon, “Attending to one of many: When infants are surprisingly poor at discriminating an item’s size,” Frontiers in Psychology, vol. 2 (2011) doi: 10.3389/fpsyg.2011.00065.
K. Davidson, K. Eng and D. Barner, “Does learning to count involve a semantic induction?,” Cognition, vol. 123 (2012) 162–173.
S. Dehaene, The number sense: how mathematical knowledge is embedded in our brains, Oxford University Press, New York, 1997.
K. Devlin, The joy of sets, 2nd ed., Springer, Berlin, 1993.
L. Feigenson, “The equality of quantity,” Trends in Cognitive Sciences, vol. 11 (2007) 185–187.
M.S. Franklin, J. Jonides and E.E. Smith, “Processing of order information for numbers and months,” Memory & Cognition, vol. 37 (2009) 644–654.
G. Frege, The foundations of arithmetic: A logico-mathematical enquiry into the concept of number, 2nd rev. ed., Northwestern University Press, Evanston, IL, 1980.
K.C. Fuson, Children’s counting and concepts of number, Springer-Verlag, New York, 1988.
R. Gelman and C.R. Gallistel, The child’s understanding of number, Harvard University Press, Cambridge, Mass., 1978.
C. Hanlon, “The emergence of set-relational quantifiers in early childhood”, in: The development of language and language researchers: Essays in honor of Roger Brown, F.S. Kessel, ed., Lawrence Erlbaum Associates Inc, Hillsdale, NJ, 1988, pp. 65–78.
P.M. Hartnett, “The development of mathematical insight: From one, two, three to infinity,” Dissertation Abstracts International, vol. 52 (1992) 3921.
R.G. Heck Jr, “Cardinality, counting, and equinumerosity,” Notre Dame Journal of Formal Logic, vol. 41 (2000) 187–209.
R.G. Heck Jr, “Frege’s theorem: An overview”, in: Frege’s theorem, Oxford University Press, Oxford, UK, 2011, pp. 1–39.
V. Izard, A. Streri and E.S. Spelke, “Toward exact number: Young children use one-to-one correspondence to measure set identity but not numerical equality,” Cognitive Psychology, vol. 72 (2014) 27–53.
D.H. Krantz, R.D. Luce, P. Suppes and A. Tversky, Foundations of measurement, Academic Press, New York, 1971.
M. Le Corre, “Children acquire the later-greater principle after the cardinal principle,” British Journal of Developmental Psychology, vol. 32 (2014) 163–177.
Ø. Linnebo, “The individuation of the natural numbers”, in: New waves in philosophy of mathematics, O. Bueno, Ø. Linnebo, eds., Palgrave, Houndmills, UK, 2009, pp. 220–238.
I.M. Lyons and S.L. Beilock, “Ordinality and the nature of symbolic numbers,” The Journal of Neuroscience, vol. 33 (2013) 17052–17061.
J. MacFarlane, “Double vision: two questions about the neo-Fregean program,” Synthese, vol. 170 (2009) 443–456.
K.F. Miller, “Measurement as a tool for thought: The role of measuring procedures in children’s understanding of quantitative invariance,” Developmental Psychology, vol. 25 (1989) 589–600.
K.S. Mix, J. Huttenlocher and S.C. Levine, Quantitative development in infancy and early childhood, Oxford University Press, Oxford, UK, 2002.
R.S. Moyer and S.T. Dumais, “Mental comparison,” Psychology of Learning and Motivation, vol. 12 (1978) 117–155.
R.S. Moyer and T.K. Landauer, “Time required for judgements of numerical inequality,” Nature, vol. 215 (1967) 1519–1520.
C. Parsons, Mathematical thought and its objects, Cambridge University Press, New York, 2008.
S.T. Piantadosi, J.B. Tenenbaum and N.D. Goodman, “Bootstrapping in a language of thought: A formal model of numerical concept learning,” Cognition, vol. 123 (2012) 199–217.
M.D. Resnik, Mathematics as a science of patterns, Oxford University Press, Oxford, UK, 1997.
G. Rey, “Innate and learned: Carey, Mad Dog nativism, and the poverty of stimuli and analogies (yet again),” Mind & Language, vol. 29 (2014) 109–132.
L.J. Rips, “Bootstrapping: How not to learn”, in: Encyclopedia of the sciences of learning, N.M. Seel, ed., Springer, Berlin, 2012, pp. 473–477.
L.J. Rips, “How many is a zillion? Sources of number distortion,” Journal of Experimental Psychology: Learning, Memory, and Cognition, vol. 39 (2013) 1257–1264.
L.J. Rips, Lines of thought: Central concepts in cognitive psychology, Oxford University Press, Oxford, UK, 2011.
L.J. Rips, J. Asmuth and A. Bloomfield, “Can statistical learning bootstrap the integers?” Cognition, vol. 128 (2013) 320–330.
L.J. Rips, J. Asmuth and A. Bloomfield, “Do children learn the integers by induction?” Cognition, vol. 106 (2008) 940–951.
L.J. Rips, J. Asmuth and A. Bloomfield, “Giving the boot to the bootstrap: How not to learn the natural numbers,” Cognition, vol. 101 (2006) B51-B60.
L.J. Rips, A. Bloomfield and J. Asmuth, “From numerical concepts to concepts of number,” Behavioral and Brain Sciences, vol. 31 (2008) 623–642.
L.J. Rips and S.J. Hespos, “Divisions of the physical world: Concepts of objects and substances,” Psychological Bulletin, vol. 141 (2015).
F.S. Roberts, Measurement theory with applications to decisionmaking, utility, and the social sciences, Addison-Wesley, Reading, Mass., 1979.
B. Russell, Introduction to mathematical philosophy, Allen & Unwin, London, 1919.
B.W. Sarnecka and C.E. Wright, “The idea of an exact number: Children’s understanding of cardinality and equinumerosity,” Cognitive Science, vol. 37 (2013) 1493–1506.
S. Shapiro, Philosophy of mathematics: Structure and ontology, Oxford University Press, New York, 1997.
E. Turconi, J.I.D. Campbell and X. Seron, “Numerical order and quantity processing in number comparison,” Cognition, vol. 98 (2006) 273–285.
H.M. Wellman and K.F. Miller, “Thinking about nothing: Development of concepts of zero,” British Journal of Developmental Psychology, vol. 4 (1986) 31–42.
K. Wynn, “Children’s acquisition of the number words and the counting system,” Cognitive Psychology, vol. 24 (1992) 220–251.
E.N. Zalta, “Frege’s theorem and foundations for arithmetic”, in: Stanford encyclopedia of philosophy, E.N. Zalta, ed., 2014.
Acknowledgement
Thanks to Ernest Davis, Jacob Dink, and Nicolas Leonard for comments on an earlier version of this chapter and to John Glines, Jane Ko, and Gabrielle McCarthy for their help with the experiments described in Section 4.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Rips, L.J. (2015). Beliefs about the nature of numbers. In: Davis, E., Davis, P. (eds) Mathematics, Substance and Surmise. Springer, Cham. https://doi.org/10.1007/978-3-319-21473-3_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-21473-3_16
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-21472-6
Online ISBN: 978-3-319-21473-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)