Abstract
Recent years have seen an explosion of empirical data concerning arithmetical cognition. In this paper that data is taken to be philosophically important and an outline for an empirically feasible epistemological theory of arithmetic is presented. The epistemological theory is based on the empirically well-supported hypothesis that our arithmetical ability is built on a protoarithmetical ability to categorize observations in terms of quantities that we have already as infants and share with many nonhuman animals. It is argued here that arithmetical knowledge developed in such a way cannot be totally conceptual in the sense relevant to the philosophy of arithmetic, but neither can arithmetic understood to be empirical. Rather, we need to develop a contextual a priori notion of arithmetical knowledge that preserves the special mathematical characteristics without ignoring the roots of arithmetical cognition. Such a contextual a priori theory is shown not to require any ontologically problematic assumptions, in addition to fitting well within a standard framework of general epistemology.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Here is a case where the empirical scientists often are less than careful about equivocating terminology. To talk about natural numbers is premature at this primitive stage, as many scientists note. Hence the term “numerosity”, referring to a primitive conception of a discrete quantity. In the literature, numerosities are often referred to as cardinalities, but this can also be confusing since “cardinality” has an established meaning as a technical term in set theory. Another common confusion is to call primitive processing of numerosities “arithmetic”. In this paper, arithmetic means a system with explicit rules and number symbols (or number words). For the more primitive processing of numerosities, it is better to talk of proto-arithmetic.
I believe that the many replications of the Wynn experiment quite clearly show that the infants acted based on the quantity of objects they saw. But at the same time, it seems that Wynn and others are hasty in making the conclusion that the infants are adding and subtracting. They could equally well just be keeping track of the quantity of the objects they expect to see, thus holding only one numerosity in their minds. Postulating the ability to do arithmetical (or even proto-arithmetical) operations is not necessary, and in fact quite problematic.
While the nonhuman and infant abilities with numerosities are widely accepted in modern cognitive science and psychology, not everybody subscribes to it. Kelly Mix, in particular, has been a prominent critic of the ability to deal with numerosities in infants, claiming that it is continuous magnitudes rather than discrete numerosities that the infants respond to. I cannot go here into the details (for one part of them, see Mix et al. 2002) but it should be noted that many experiments seem extremely unlikely to fall into such equivocations. For example, experiments are often controlled so that the total visible surface area of objects remains the same when their quantity is changed. One criticism by Mix (2002), however, is philosophically particularly interesting and should be addressed here:
Just as animal researchers are at risk for anthropomorphizing their non-human subject, infant researchers are at risk for overlaying adult reasoning on basic perceptual processes. Longer looking times are commonly interpreted as evidence that infants have formed an abstract representation, compared it to a test stimulus and effectively said to themselves, ‘Hey, that’s different!’ or ‘How surprising!’
While it is indeed important not to postulate excessive cognitive capability to the subjects, what Mix criticizes is the exact thing that proponents of proto-arithmetical ability in infants and animals deny. They do not claim that infants make abstract representations which they discuss within their minds. Rather, this process happens automatically.
The ability is not properly logarithmical, however, as distinguishing between two and four objects is easier than between 8 and 16 objects. See Dehaene (2011, chap. 4).
In addition to numerosities, similar results have been acquired in the study of proportions (Nieder 2011).
Interestingly, in the intraparietal sulcus, which is another part of the brain associated with numerosity, the number signs triggered much less activity. This suggests that while there are clear connections between the number symbols and the numerosities of the ANS, these are different in the two areas of the brain.
See also Brannon and Merritt (2011) for a good overview of the subject. Research on an Amazon tribe with limited numerical lexicon has also established that education in verbal numerical thinking enhances the acuity of the ANS, thus suggesting that the connection between the ANS and language-based numerical thinking goes both ways (Piazza et al. 2013).
It is a matter of debate whether we should accept there being non-linguistic concepts in the first place. Fortunately, the particular definition of “concept” does not change the essential argument here. The key question is whether we accept that there is a relevant difference between dealing with language-dependent mathematical concepts and the proto-arithmetical ability. I am confident that few are ready to deny the existence of such difference.
Dummett (e.g. 1973, p. 228) has made this “Kreisel’s dictum” famous: “The point is not the existence of mathematical objects, but the objectivity of mathematical truth,” although the quote is not known to be found in Kreisel’s own writings. The quotation as attributed to Kreisel can only be found in Putnam (2004, p. 67).
In the philosophy of mathematics, such objectivism without objects is not a new suggestion in mathematical ontology. If we take a structuralist approach to mathematics, the focus can be turned from objects to characteristics that arithmetical structures have. Among structuralists, however, there is great disagreement about the ontological status of mathematical structures, ranging from the objectively existing ante rem structures of Resnik (1981) and Shapiro (1997) to the modal constructions of Hellman (1989).
As before, by strict conventionalism I understand the position that mathematical statements are ultimately mere human conventions not based on any stronger objective basis.
I acknowledge that this understanding of objectivity is somewhat non-standard, but it seems to be the relevant one when it comes to the present arithmetical context. See Footnote 20 for further analysis.
It should be noted that having some grasp of the idea of successor does not necessarily mean understanding the formal consequences of defining natural numbers in terms of a successor function. When a child learns to count beyond the first few number words, she clearly understands something about numerosities forming a succession. Yet at that stage she is unlikely to grasp the full meaning of the successor-based nature of numerosities, e.g., their infinity.
Carey (2009b, chap. 8) also presents an account of children learning the inductive nature of natural numbers that is based on the successor function. While I believe such an account is highly plausible, we must be careful not to postulate the inductive ability before we know when it is actually acquired. Interestingly, Carey recounts experiments with nonhuman animals which fail to make the inductive step, thus giving further evidence that while ANS may be the basis for our arithmetical ability, it is not enough by itself.
See Boghossian and Peacocke (eds.) (2000) for a nice collection of the kind of problems that engage modern philosophers in the study of a priori.
Such a faculty is most often associated with Gödel (1947) in the literature.
This should not be confused with informal mathematical insights, which undoubtedly serve an important purpose in mathematical thought process. Mathematicians often report, for example, that they know a theorem to be true before they can prove it. The psychological processes involved in such cases are an interesting subject, but it cannot be treated here. But as a prima facie explanation, it seems more likely that mathematicians are so familiar with their subject matter that they can recognize patterns and lines of arguments before they are fully articulated—rather than having a special epistemic connection to a world of abstract mathematical ideas.
It should be mentioned that there are also finitist approaches which aim to be non-revisionist mathematically. See Lavine (1994) for an example of non-revisionist set-theoretical ultra-finitism.
Ifrah (1998, p. 298) writes that Mayans had some notion of infinity, but in the arithmetical writings that remain, that is not clear.
The matter of objectivity was already mentioned in Footnote 12 and should now be dealt with in more detail. The sense I have understood “objectivity” in this paper is obviously much weaker than some standard understandings of objectivity in the philosophy of mathematics, including the objective existence of natural numbers or the natural number structure. But in the light of the empirical data reviewed in this paper, that strong kind of objectivity seems too much to require. If there are general physiological features responsible for at least some of the content of arithmetical theories, it seems clear that arithmetical statements have content that is in a relevant sense objective. Although it is not possible here to go deeper into the philosophical question of objectivity, I believe that any arithmetically relevant understanding of “objectivity” should include this sense. That is why many important writings about mathematical objects and objectivity, e.g. Putnam (1980) and Field (1998), do not touch the argument given here. While questions treated in those writings—such as whether the continuum hypothesis has an objective truth-value—clearly are central to the question of mathematical objectivity, so is the question whether that is the case for \(1 + 1 = 2\). To discuss the objectivity of this latter question without taking into account the empirical data reviewed in this paper would seem to be taking a needlessly limiting view of objectivity and thus ending up ignoring important evidence.
References
Agrillo, C., Dadda, M., Serena, G., & Bisazza, A. (2009). Use of number by fish. PLoS One, 4(3), e4786.
Azzouni, J. (2010). Talking about nothing—Numbers, hallucinations and fiction. Oxford: Oxford University Press.
Boghossian, P., & Peacocke, C. (2000). New essays on the a priori. Oxford: Oxford University Press.
Brannon, E., & Merritt, D. (2011). Evolutionary foundations of the approximate number system. In Dehaene (Ed.), Space, time and number in the brain (pp. 107–122). London: Academic Press.
Burge, T. (2007). Postscript to “Belief de re” in foundations of mind. Oxford: Oxford University Press.
Burge, T. (2010). Origins of objectivity. Oxford: Oxford University Press.
Butterworth, B. (2010). Foundational numerical capacities and the origins of dyscalculia. Trends in Cognitive Sciences, 14, 534–541.
Cantlon, J. F., & Brannon, E. M. (2006). Shared system for ordering small and large numbers in monkeys and humans. Psychological Science, 17(5), 402–407.
Carey, S. (2009a). Where our number concepts come from. The Journal of Philosophy, 106(4), 220–254.
Carey, S. (2009b). The origin of concepts. Oxford: Oxford University Press.
Church, R., & Meck, W. (1984). The numerical attribute of stimuli. In H. L. Roitblat, T. G. Bever, & H. S. Terrace (Eds.), Animal cognition (pp. 445–464). Hillsdale: Erlbaum.
Clarke-Doane, J. (2012). Morality and mathematics: The evolutionary challenge. Ethics, 122(2), 313–340.
Dehaene, S. (2011). Number sense (2nd ed.). New York: Oxford University Press.
Dehaene, S., & Brannon, E. (Eds.). (2011). Space, time and number in the brain. London: Academic Press.
Diester, I., & Nieder, A. (2007). Semantic associations between signs and numerical categories in the prefrontal cortex. PLoS Biology, 5, e294.
Dummett, M. (1973). Frege: Philosophy of language. London: Duckworth.
Essenin-Volpin, A. (1961). Le Programme Ultra-intuitionniste des fondements des mathématiques. Infinistic methods (pp. 201–223). New York: Pergamon Press.
Feigenson, L. (2011). Objects, sets, and ensembles. In Dehaene & Brennan (Eds.), Space, time and number in the brain (pp. 13–22). London: Academic Press.
Field, H. (1980). Science without numbers: A defense of nominalism. Princeton, NJ: Princeton University Press.
Field, H. (1998). Mathematical objectivity and mathematical objects. In S. Laurence & C. Macdonald (Eds.), Contemporary readings in the foundations of metaphysics (pp. 387–403). Oxford: Basil Blackwell.
Gödel, K. (1947). What is Cantor’s continuum problem? The American Mathematical Monthly, 54, 515–25.
Gordon, P. (2004). Numerical cognition without words: Evidence from Amazonia. Science, 306, 496–499.
Hellman, G. (1989). Mathematics without numbers. Oxford: Oxford University Press.
Ifrah, G. (1998). The universal history of numbers: From prehistory to the invention of the computer. Great Britain: The Harvill Press.
Kitcher, P. (1983). The nature of mathematical knowledge. New York: Oxford University Press.
Kuhn, T. (1993). Afterwords. In Paul Horwich (Ed.), World changes (pp. 331–332). Cambridge, MA: MIT Press.
Lakoff, G., & Núñez, R. (2000). Where mathematics comes from. New York: Basic Books.
Lavine, S. (1994). Understanding the infinite. Cambridge, MA: Harvard University Press.
Leslie, A. M., et al. (2008). The generative basis of natural number concepts. Trends in Cognitive Science, 12, 213–218.
Lourenco, S., & Longo, M. (2011). Origins and development of generalized magnitude representation. In Dehaene & Brennan (Eds.), Space, time and number in the brain (pp. 225–241). London: Academic Press.
Mechner, F. (1958). Probability relations within response sequences under ratio reinforcement. Journal of Experimental Analysis of Behavior, 1, 109–21.
Mechner, F., & Guevrekian, L. (1962). Effects of deprivation upon counting and timing in rats. Journal of Experimental Analysis of Behavior, 5, 463–66.
Mix, K. (2002). Trying to build on shifting sand: Commentary on Cohen and Marks. Developmental Science, 5, 205–206.
Mix, K., et al. (2002). Multiple cues for quantification in infancy: Is number one of them? Psychological Bulletin, 128, 278–294.
Nieder, A. (2011). The neural code for number. In S. Dehaene & E. Brannon (Eds.), Space, time and number in the brain (pp. 107–122). London: Academic Press.
Nieder, A. (2012). Supramodal numerosity selectivity of neurons in primate prefrontal and posterior parietal cortices. Proceedings of the National Academy of Sciences of the United States of America, 109, 11860–11865.
Nieder, A. (2013). Coding of abstract quantity by ’number neurons’ of the primate brain. Journal of Comparative Psychology A, 199, 1–16.
Nieder, A., et al. (2006). Temporal and spatial enumeration processes in the primate parietal cortex. Science, 313(2006), 1431–1435.
Nieder, A., & Dehaene, S. (2009). Representation of number in the brain. Annual Review of Neuroscience, 32, 185–208.
Nieder, A., Freedman, D., & Miller, E. (2002). Representations of the quantity of visual items in the primate prefrontal cortex. Science, 297, 1708–1711.
Nieder, A., & Miller, E. (2003). Coding of cognitive magnitude. Compressed scaling of numerical information in the primate prefrontal cortex. Neuron, 37, 149–157.
Piazza, M. (2010). Neurocognitive start-up tools for symbolic number representations. Trends in Cognitive Sciences, 14, 542–551.
Piazza, M., et al. (2007). A magnitude code common to numerosities and number symbols in human intraparietal cortex. Neuron, 53, 293–305.
Piazza, M., et al. (2013). Education enhances the acuity of nonverbal approximate number system. Psychological Science, 24, 1037–1043.
Putnam, H. (1968). Is logic empirical? In Boston studies in the philosophy of science, vol. 5, D (pp. 216–241). Dordrecht: Reidel.
Putnam, H. (1980). Models and reality. Journal of Symbolic Logic, 45, 464–482.
Putnam, H. (1983). ‘Two dogmas’ revisited. In Realism and reason: Philosophical papers (Vol. 3, pp. 87–97). Cambridge: Cambridge University Press.
Putnam, H. (2004). Ethics without ontology. Cambridge, MA: Harvard University Press.
Quine, W. V. O. (1951). Two dogmas of empiricism. In From a logical point of view. Cambridge, MA: Harvard University Press.
Resnik, M. (1981). Mathematics as a science of patterns: Ontology and reference. Nous, 15, 529–550.
Rumbaugh, D., Savage-Rumbaugh, S., & Hegel, M. (1987). Summation in the chimpanzee. Journal of Experimental Psychology: Animal Behavior Processes, 13, 107–115.
Ruse, M. (1986). Taking Darwin seriously: A naturalistic approach to philosophy. New York: Prometheus Book.
Shapiro, S. (1997). Philosophy of mathematics: Structure and ontology. New York: Oxford University Press.
Simon, T. J., Hespos, S. J., & Rochat, P. (1995). Do infants understand simple arithmetic? A replication of Wynn (1992). Cognitive Development, 10, 253–269.
Spelke, E. (2011). Natural number and natural geometry. In S. Dehaene & E. Brannon (Eds.), Space, time and number in the brain (pp. 287–318). London: Academic Press.
Street, S. (2006). A Darwinian dilemma for realist theories of value. Philosophical Studies, 127, 109–166.
Wynn, K. (1992). Addition and subtraction by human infants. Nature, 358, 749–751.
Acknowledgments
This work has been made possible by the generous support of the Academy of Finland and the University of Bucharest. I am in great gratitude to the many colleagues who have provided valuable comments on the manuscript at some point, especially Daniel Cohnitz and Dirk Schlimm.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pantsar, M. An empirically feasible approach to the epistemology of arithmetic. Synthese 191, 4201–4229 (2014). https://doi.org/10.1007/s11229-014-0526-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-014-0526-y