1 Briefly, as is well known, in Die Grundlagen der Arithmetik (Evanston, IL: Northwestern University Press 1980), Frege himself held that numbers are objects assigned to (first-level) concepts (or properties), where each number is correlated with a numerical quantifier, i.e., a particular sort of (second-level) property of concepts. Since every concept has an extension (the class of things that fall under the concept) which is the image of the concept at the level of objects, corresponding to every quantifier there is a first-level numerical concept of equinumerous classes. Such concepts are very much like what proponents of what I'm calling ‘the Fregean thesis’ identify with the numbers. Now, those who adhere more strictly to the Fregean tradition with its inviolable distinction between concept and object, like Maddy, identify the numbers with the extensions of these first-level concepts; cf. Maddy, Penelope ‘Proper Classes,’ Journal of Symbolic Logic 48 (1983) 113–39CrossRefGoogle Scholar, and ‘Sets and Numbers,’ Nous 15 (1981) 495-513; also, Frege himself explicitly adopts this postion in the Grundgesetze der Arithmetik, vol. 1 (Hildesheim: Olms Verlag 1962). Others, who don't accept the inviolability of this distinction, take the numbers to be those very concepts themselves (or closely related ones). Cf., e.g., Bealer, George Quality and Concept (Oxford: Oxford University Press 1982)CrossRefGoogle Scholar, esp. 85-9 and ch. 6; also Christopher Menzel, ‘A Complete, Type-free “Second-order” Logic and Its Philosophical Foundations,’ Report No. CSLI-86-40, Center for the Study of Language and Information, Stanford University, 1986, esp. 1-6 and n. 19. Both views, despite their differences, fall under what I'm calling the Fregean thesis.

I will occasionally modify Austin's translations of the Grundlagen slightly in this paper.

2 Yourgrau, Palle ‘Sets, Aggregates, and Numbers,’ Canadian Journal of Philosophy 15 (1985) 581–92CrossRefGoogle Scholar

3 I base this in particular on the discussion of numbering sets on pages 588-9. We'll have more to say about numbering below. It is important to note that Yourgrau uses a variety of terms to express his idea of a set being assigned a number e.g., he speaks simply of a set ‘being numbered,’ of it ‘having a number,’ of a number ‘directly applying’ to a set, of a number ‘belonging to’ a set, of a number being ‘attributed’ to a set, and so on. As far as I can tell these locutions all express the same idea, which I will be picking out always by means of the term ‘assign’ and its cognates.

4 Kessler, Glenn ‘Frege, Mill, and the Foundations of Arithmetic,’ The Journal of Philosophy 76 (1979) 66Google Scholar

5 This might actually be redundant for Frege, since, as Yourgrau notes (587), Frege seemed to view all physical objects as, at least conceptually, aggregates of other objects. Cf., e.g., the quote from Baumann at the beginning of §22 in the Grundlagen (also quoted below), and Frege, Gottlob ‘On Schoenflies: Die Logischen Paradoxien der Mengenlehre,’ in Posthumous Writings, Hermes, Hans Kambartel, Friedrich and Kaulbach, Friedrich eds. and trans. Long, Peter and White, Roger (Chicago, lL: University of Chicago Press 1979), 176–83, esp. 181Google Scholar.

6 Considerations from relativistic physics aside.

7 It might appear that Frege is drawing a conclusion similar to the one Yourgrau attributes to him when, after discussing his example of the pile of cards, he writes that ‘an object to which I can ascribe different numbers with equal right is not the proper bearer (eigentliche Träger) of number’ (29). This remark, however, is really an aside prompted by the suggestion that aggregates can exemplify several numbers simultaneously, and is clearly not the central point of the passage, which comes in the paragraph following the one from which this quote is taken.

8 Frege's choice of examples here is perhaps not the happiest since for him colors are (at least in general) properties of surfaces of physical objects (cf. 29). Better would have been, e.g., mass or shape, which are properties of entire physical objects.

9 One might object here that, although there are several numbers we might be able to assign to the pile, nonetheless we can't assign just any number to it, and to that extent we’ve isolated certain intrinsic number properties of the pile and not mere Cambridge properties. I think Frege would disagree, given his apparent views about the aggregative nature of all concrete objects (cf. n. 5 above). We can, for example, construct a concept that partitions the pile into top halves and bottom halves of cards, then one that partitions it into left and right halves of top and bottom halves of cards, and so on, ad infinitum. By combining such concepts in various ways we can cook up concepts that will yield n-celled partitions for virtually all finite n. Thus Frege: ‘Anyone who did not know what we call a complete pack would probably discover in the pile any other number you like before hitting on two’ (29). In principle, the same is true of any physical object.

10 Things are of course more subtle than this. We are following Frege's covert assumption here that the cards and the decks are the same aggregate. However, there are several recent accounts of the semantics of plurals that offer much more fine-grained treatments of such expressions. Cf. esp. Link, Godehard ‘Plural,’ forthcoming in Stechow, Arnim von and Wunderlich, D. eds., Handbuch der Semantik. Link's treatment of aggregates is foreshadowed by Kessler, 67-8Google Scholar. Cf. also Burge, Tyler ‘A Theory of Aggregates,’ Nous 11 (1977) 97–117CrossRefGoogle Scholar.

The parallels with later problems of coreferential singular terms cannot be missed; it is worth speculating on possible conceptual connections between the development of RA and Frege's later work. Here, of course, it is clear that an appeal to oblique contexts could do no work.

11 To push this contrast a bit farther, the reason different ways of conceiving a given aggregate, ‘the decks’ and ‘the cards,’ say, can yield different partitions (hence different numerical assignments) is that the concepts involved in the those expressions are in general not coextensive and hence determine different parts; decks are not cards. By contrast, because of the singular nature of sets, the concepts involved in different ways of conceiving the same set must be coextensive; they must determine precisely the same parts.

12 There is of course nothing genuinely circular about this since there are a variety of well known logical and set theoretic ways of parsing the notion of having n members that don't presuppose the number n.

13 A ‘course-grained’ view of properties which equates (necessarily) coextensive properties would guarantee that there is a unique such property; cf., e.g., the modal logic T1 in Bealer, 58-64. However, on a ‘fine-grained’ view of properties such as that embodied in Bealer's T2 (Bealer, 64-7), there may well not be a unique property shared by all and only equinumerous sets. One may then need an account according to which there is, e.g., a simplest such property which can be identified with the appropriate number; cf. Menzel, n. 19. It should also be noted that the paradoxes raise the possibility of ‘large’ sets which, though equinumerous, nonetheless can't be said to share a common property; cf. Menzel, Christopher ‘On the Iterative Explanation of the Paradoxes,’ Philosophical Studies 49 (1986) 37–61CrossRefGoogle Scholar.

14 And indeed Yourgrau mentions Benacerraf in passing in the paragraph from which the quote is taken.

15 Benacerraf, Paul ‘What Numbers Could Not Be,’ Philosophical Review 74 (1965) 47–73CrossRefGoogle Scholar

16 Though in the first case we should need as a part of our formal theory some sort of pedal axiom of infinity to the effect that there are at least a countable number of feet in the universe; so much for logicism. The second account would be formally feasible, of course, because of the isomorphism that exists between the natural numbers and the even numbers.

17 This seems to be Yourgrau's point in the following passage (588): ‘If we are allowed in the case of sets to construe the number question as “really“: How many (elements)? then we could just as well construe Frege's famous question about the deck of cards as: How many (cards)?’

‘Frege's famous question’ (How many?), by the way, never occurs in the Grundlagen, nor, as near as I can tell, in any of his published writings.

18 If the Fregean thesis is correct, of course, the function F here is just the identity function.

19 Except perhaps where P is a somewhat gerrymandered property like having n members and being self-identical that involves a Frege number (where P involves Q just in case it is not possible to conceive P without conceiving Q; cf.Chisholm, Roderick The First Person [Minneapolis, MN: University of Minnesota Press 1981], 124Google Scholar). In such a case the Frege number is still more fundamental conceptually; see also the following argument.

20 Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers trans. with an introduction by Philip E. B. Jourdain (New York, NY: Dover 1955). I, of course, disavow the psychologistic connotations here regarding the nature of cardinality, though something like abstraction might well be part of coming to grasp the concept of cardinality in the first place. Cf. Piaget, Jean The Child's Conception of Number, trans. by Gattegno, C. and Hodgson, F. M. (New York, NY: Humanities Press 1952)Google Scholar.

21 I would like to thank Penelope Maddy for comments on an earlier draft, and Hugh McCann for helpful discussion.