Abstract
A core commitment of Bob Hale and Crispin Wright’s neologicism is their invocation of Frege’s Constraint—roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. According to these neologicists, if legitimate, Frege’s Constraint adjudicates in favor of their preferred foundation—Hume’s Principle—and against alternatives, such as the Dedekind–Peano axioms. In this paper, we consider a recent argument for legitimating Frege’s Constraint due to Hale, according to which the primary empirical application of the naturals is transitive counting, or answering ‘how many’-questions using numerals. We make two claims regarding Hale’s argument. First, it fails to legitimate Frege’s Constraint in virtue of resting on unsupported and highly contentious assumptions. Secondly, even if sound, Hale’s argument would vindicate a version of Frege’s Constraint which fails to adjudicate in favor of Hume’s Principle over alternative characterizations of the naturals.
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Notes
On Wright (2000)’s gloss, as well as Hale (2016)’s, Frege’s constraint is limited specifically to empirical applications. Of course, one could reformulate Frege’s constraint so as to include non-empirical applications as well. However, because our concern here is with Hale’s argument, we will ignore this latter possibility.
We use this label to distinguish the view from other versions of neologicism, such as Tennant (1997)’s Constructive Logicism.
For the purposes of this paper, we will adopt the convention of naming concepts in CAPITALS.
It has been suggested to us that ‘natural number’ is “systematically ambiguous”, in that the DP axioms and HP characterize “different things”. In particular, the DP axioms characterize the “natural numbers in sequence”, presumably the natural number structure, while HP characterizes “individual natural numbers”; and this is somehow reflected in ‘natural number’ or the corresponding concept(s). However, the structuralist does define the “individual natural numbers”—they are places in the natural number structure—and the abstractionist neologicist does characterize the “natural numbers in sequence”—they are the finite cardinal numbers ordered in the usual way. The present paper concerns the priority between these two accounts, an issue posed by the abstractionists themselves. Of course, the DP axioms and HP are not the only formal characterizations of the natural numbers (either “individually” or “in sequence”). For example, a third characterization is to be found in Tennant’s Constructive Logicism; and a fourth, ordinal-based characterization, is due to Linnebo (2009). We will return to both these proposals in subsequent work.
See Heck (2011).
In fact, one must further suppose that the conditions described by TCP1 and TCP2 respectively exhaust the relevant alternatives for concept possession. However, since this assumption will play no role in our discussion, we set it to one side.
As one anonymous reviewer notes, one might think it is more felicitous to express claims about essentiality as involving a relationship between kinds or nominalized properties –e.g. “Being striped is essential to being a zebra.” That said, as our presentation of the argument makes clear, Hale uses other constructions to express essentialist claims. For present purposes, however, nothing of philosophical significance turns on this. For example, the claim in TCC1 can be readily reformulated as follows: “One’s being able to transitively count is essential to one’s possessing natural number concepts.” Mutatis mutandis for other claims in the argument.
Hale (2016, p. 340): “If [TCC1] is right, then the fact that the natural numbers can be used to count collections of things is no mere accidental feature, but is essential to them. And if that is so, then a satisfactory definition of the natural numbers–a characterisation of what they essentially are–should reflect or incorporate that fact.”
It is possible, we suppose, that someone might subscribe to POP for all concepts, not just natural number concepts. Since this view is both implausible and unnecessary for Hale’s argument to go through, we don’t discuss it here.
See MacBride (2003) and Ebert and Rossberg (forthcoming).
There is philosophical precedent for the distinction, however. Indeed, Heck (2000) speculates that children, relatively early in their cognitive development, might be what we are calling nominal transitive counters: “Such children may well understand the numerals as mere tags, having no independent significance. For them, ‘There are four hats on the table’ really does mean something like: I ended with ‘four’ when I counted the hats. But they seem to have no grasp at all of the point of such ‘ascriptions’ of number.”
This is, of course, a point on which Hale, qua modal essentialist, is wholly in agreement. See Hale (2013), especially Chapter 6.
At least in terms of the development of number concept acquisition, which we return to in §5.
Wright (2000, p. 327) is committed to a similar conclusion. He notes that the structuralist, who bases her account of the natural numbers on the DP axioms, will "be open to the charge of changing the subject: whatever the detail of her epistemological story about the simplest truths of arithmetic, the content of the knowledge thereby explained will not be that of the knowledge we actually have.”
Many thanks to an anonymous referee for this suggestion.
Some may find certain of these examples more acceptable than others. To be clear, the judgments reported here are those of Moltmann (2013) and Snyder (2017), and they appear to be shared by many native English speakers, though this is ultimately an empirical question. What’s important for our purposes is that there is a contrast, witnessed in a variety of contexts, between the number of-terms and terms like the number four.
To be clear, not all mathematical modifiers are unacceptable with the number of-terms. For example, as Moltmann (2013) observes, there is no difference in acceptability between ‘the even number {of ducks/four}’.
Balcerak-Jackson and Penka (2017) come a similar conclusion, though based on different considerations.
Many thanks to an anonymous referee for this observation.
In view of this, it is unsurprising that ability to intransitively count comes prior to an ability to transitively count. That is, children learn to count intransitively well before learning how to count transitively. See Carey (2009), especially Chapter 4.
Gelman and Gallistel (1986, p. 79–80).
For example Tennant (1997)’s Schema N.
To illustrate, consider again the second HP numeral, and consider a class of two objects, say the moons belonging to Mars. In order to infer from HP that the number of Martian moons is the number referenced by the second HP numeral, the HP Novice needs to make an additional inference, given the numeral’s disjunctive character. Namely, he needs to infer that the number of the concept BEING IDENTICAL TO THE NUMBER ZERO (i.e., THE CONCEPT BEING NON-SELF-IDENTICAL) and the number of the concept BEING IDENTICAL TO EITHER THE NUMBER ZERO OR THE NUMBER ONE are distinct. This is not something he can know from HP alone, and the point generalizes to all HP numerals beyond the first.
This translation is from Ebert & Rossberg (forthcoming), and differs from Wright (2000, p. 324)’s translation (“patch them on from the outside”).
See Snyder et al. (2018) for relevant discussion.
References
Benacerraf, P. (1965). What numbers could not be. The Philosophical Review, 74, 47–73.
Carey, S. (2009). The origin of concepts. Oxford: Oxford University Press.
Dedekind, R. (1888/1963). The nature and meaning of number words. Mineola: Dover.
Dummett, M. D. (1991). Frege: Philosophy of mathematics. Cambridge, MA: Harvard University Press.
Ebert, P., & Rossberg, M. Essays on Frege’s basic laws of arithmetic. Oxford University Press, forthcoming
Frege, G. (1884). The foundations of arithmetic, translated by JL Austin. New York: Philosophical Library.
Frege, G. (1893/1903). Grundgesetze der Arithmetik, vol. 1 and 2. Jena: Pohle. English translation: (Frege, 2013).
Frege, G. (2013). Gottlob Frege: Basic laws of arithmetic. Oxford: Oxford University Press.
Fuson, K. (1988). Children’s counting and concepts of number. New York: Springer.
Gelman, R., & Gallistel, C. R. (1986). The child’s understanding of number. Cambridge: Harvard University Press.
Hale, B. (2013). Necessary beings: An essay on ontology, modality, and the relations between them. Oxford: Oxford University Press.
Hale, B. (2016). Definitions of numbers and their applications. In P. Ebert & M. Rossberg (Eds.), Abstractionism (pp. 326–341). Oxford: Oxford University Press.
Hardy, G. H., & Wright, E. M. (1938). An introduction to the theory of numbers. Oxford: Clarendon.
Heck, R. G. (2000). Cardinality, counting, and equinumerosity. Notre Dame Journal of Formal Logic, 41, 187–209.
Heck, R. (2011). Frege’s theorem. Oxford University Press.
Hellman, G. (1989). Mathematics without numbers: Towards a modal-structural interpretation. Oxford: Clarendon Press.
Jackson, B. B., & Penka, D. (2017). Number word constructions, degree semantics and the metaphysics of degrees. Linguistics and Philosophy, 40, 347–372.
Linnebo, Ø. (2009). The individuation of the natural numbers. In O. Bueno & Ø. Linnebo (Eds.), New waves in philosophy of mathematics (pp. 220–238). London: Palgrave.
MacBride, F. (2003). Speaking with shadows: A study of neo-logicism. The British Journal for the Philosophy of Science, 54(1), 103–163.
Moltmann, F. (2013). Reference to numbers in natural language. Philosophical Studies, 162, 499–536.
Resnik, M. D. (1997). Mathematics as a science of patterns. Oxford: Oxford University Press.
Sarnecka, B., & Carey, S. (2008). How counting represents number: What children must learn and when they learn it. Cognition, 108, 662–674.
Searle, J. R. (1980). Minds, brains, and programs. Behavioral and Brain Sciences, 3, 417–424.
Shapiro, S. (1997). Philosophy of mathematics: Structure and ontology. Oxford: Oxford University Press.
Snyder, E. (2017). Numbers and cardinalities: What’s really wrong with the easy argument for numbers? Linguistics and Philosophy, 40, 373–400.
Snyder, E., Shapiro, S., & Samuels, R. (2018). Cardinals, ordinals, and the prospects of a Fregean foundation. Royal Institute of Philosophy Supplements, 82, 77–107.
Tennant, N. (1987). Anti-realism and logic: Truth as eternal. Oxford: Oxford University Press.
Tennant, N. (1997). On the necessary existence of numbers. Nous, 31, 307–336.
Wright, C. (2000). Neo-fregean foundations for real analysis: Some reflections on Frege’s constraint. Notre Dame Journal of Formal Logic, 41, 317–334.
Zalta, E. (1999). Natural numbers and natural cardinals as abstract objects: A partial reconstruction of Frege’s Grundgesetze in object theory. Journal of Philosophical Logic, 28, 619–660.
Acknowledgements
We would like to thank a number of people for many helpful conversations leading to substantial improvements in this paper, including Bob Hale, Øystein Linnebo, Susan Rothstein, Neil Tennant, various audiences at various conferences, and two anonymous reviewers.
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Snyder, E., Samuels, R. & Shaprio, S. Hale’s argument from transitive counting. Synthese 198, 1905–1933 (2021). https://doi.org/10.1007/s11229-019-02178-w
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DOI: https://doi.org/10.1007/s11229-019-02178-w