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Cardinals, Ordinals, and the Prospects for a Fregean Foundation

  • Eric Snyder (a1), Stewart Shapiro (a1) and Richard Samuels (a1)


There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is ‘more basic’ or ‘more fundamental’ than the others. This paper addresses two related issues. First, we review some of these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege's foundationalist epistemology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally ‘legitimate’.


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1 It is possible to start with one instead of zero.

2 Wright, C., Frege's Conception of Numbers as Objects (Aberdeen University Press, 1983)

3 In modern notation, equinumerosity is defined as follows, where ‘∃!x’ translates as ‘there is exactly one x such that …’.

$$\eqalign{\forall F,G\{ F & \approx G \leftrightarrow \exists R[(\forall x(F\left( x \right){\rm} \to \exists !y.{\rm} R\left( {x,y} \right) \wedge G\left( y \right))) \wedge (\forall x.(G\left( x \right){\rm} \cr & \to \exists !y.R\left( {x,y} \right) \wedge F\left( y \right)))]\}}$$

The first conjunct on the right-hand side of the biconditional states that R is many-to-one, while the second states that R is one-to-many. Thus, taken together, they state that a bijection holds between the Fs and the Gs.

4 Heck, R., Frege's Theorem (Clarendon Press, 2011).

5 Frege, G., Grundlagen der Arithmetik (1884).

6 Linnebo, Ø., ‘The Individuation of the Natural Numbers’ in Bueno, O. and Linnebo, Ø. (eds), New Waves in Philosophy of Mathematics (Palgrave-MacMillan, 2009), 220238.

7 If left unrestricted, 2L-N is inconsistent, falling to the Burali-Forti paradox. Consequently, Linnebo restricts it to concrete relations among systems of numerals.

8 Here we characterize the (finite) ordinal numbers as applying to individual objects, with respect to a given ordering. It is common in mathematics, however, to define an ordinal to be the order-type of a well-ordering. This may be because well-orderings are at least one natural way to extend the typical finite orderings used in ordinary ordinal discourse, into the transfinite. In the official foundation for mathematics, Zermelo-Fraenkel set theory, ordinals are identified with pure, transitive sets that are well-ordered under the membership relation. These are typically called von Neumann ordinals. And cardinal numbers are identified with certain of the von Neumann ordinals, those that are not equinumerous with any smaller von Neumann ordinal. We will return to these foundational matters in the final section below.

Notice, incidentally, that in the sense of 2L-N, an ordinal is defined in terms of an object with respect to an ordering. So there must be such an object in order to get an ordinal at all. So the smallest ordinal, in that sense, is one (or ‘first’). There is no zero ordinal. But there is a zero ordinal, in the mathematical sense. It is the order-type of an empty well-ordering, codified by the the empty set in set theory.

9 Resnik, M., Mathematics as a Science of Patterns (Oxford University Press, 1997).

10 Shapiro, S., Philosophy of Mathematics: Structure and Ontology (Oxford University Press, 1997).

11 G. Frege, Grundlagen der Arithmetik.

12 Hale, B. and Wright, C., The Reason's Proper Study: Towards a Neo-Fregean Philosophy of Mathematics (Oxford University Press, 2001).

13 Tennant, N., Anti-Realism and Logic: Truth as Eternal (Clarendon Library of Logic and Philosophy, Oxford University Press, 1987); Tennant, N., ‘On the Necessary Existence of Numbers’, Nous, 31 (1997): 307336.

14 Though they characterize the natural numbers in different ways.

15 Linnebo, ‘The Individuation of the Natural Numbers’.

16 G. Frege, Grundlagen der Arithmetik.

17 See Rothstein, , Semantics for Counting and Measuring (Cambridge University Press, 2017), and Snyder, , ‘Numbers and Cadinalities: What's Really Wrong with the Easy Argument for Numbers’, Linguistics and Philosophy 70 (2017): 373400.

18 Russell, B., Introduction to Mathematical Philosophy (Dover Publications, 1919), 1617, emphasis added.

19 Benacerraf, P., ‘What Numbers Could Not Be’, The Philosophical Review 74(1) (1965): 4773.

20 Intransitive counting is the mere reciting of numerals, in order, without correlating them with objects.

21 G. Frege, Grundlagen der Arithmetik.

22 Hale, B., ‘Definitions of Numbers and their Applications’, in Ebert, P. and Rossberg, M. (eds), Abstractionism (Oxford University Press, 2016).

23 See also Wright, C., ‘Neo-Fregean Foundations for Real Analysis: Some Reflections on Frege's Constraint’, Notre Dame Journal of Formal Logic 41 (2000): 317334.

24 E. Snyder, R. Samuels and S. Shapiro, ‘Neologicism, Frege's Constraint, and the Frege-Heck Condition’, Nous (forthcoming).

25 Dummett, M., Frege: Philosophy of Mathematics (Duckworth, 1991), 293, emphasis added.

26 It is curious that earlier in the same book, Dummett (Frege: Philosophy of Mathematics, 53), gives pride of place to the notion of cardinal:

…what is constitutive of the number 3 is not its position in any progression whatever, or even in some particular progression, …but something more fundamental than any of these: the fact that, if certain objects are counted ‘One, two, three’ or, equally, ‘Nought, one, two’, then there are 3 of them. The point is so simple that it needs a sophisticated intellect to overlook it; and it shows Frege to have been right, as against Dedekind, to have made the use of the natural numbers as finite cardinals intrinsic to their characterisation.

Perhaps the proper exegetical conclusion to draw is that, for Dummett, the notion of ordinal is more fundamental than that of cardinal, but that the natural numbers are, after all, cardinal numbers.

27 Linnebo, ‘The Individuation of the Natural Numbers’.

28 Linnebo, ‘The Individuation of the Natural Numbers’.

29 For classic discussions of type-shifting with respect to conjunctions and names, see Partee, B. and Rooth, M., ‘Generalized Conjunction and Type Ambiguity’, in Bauerle, R., Schwarze, C. and von Stechow, A. (eds), Meaning, Use, and Interpretation of Language (De Gruyter, 1983), 361383; Partee, B., ‘Ambiguous Pseudoclefts with Unambiguous Be’, in Bergman, S., Choe, J. and McDonough, J. (eds) Proceedings of the Northwestern Linguistics Society 16 (GLSA, 1986). And, for discussions relevant to number expressions, see e.g. Geurts, B., ‘Take “Five”, in Vogleer, S., and Tasmowski, L. (eds), Non-Definiteness and Plurality (Benjamins, 2006), 311329; Snyder, E., ‘Numbers and Cardinalities: What's Really Wrong with the Easy Argument?’, Linguistics and Philosophy 40 (2017): 373400.

30 See, for example, Matushansky, O., ‘Why Rose is the Rose: On the Use of Definite Articles in Proper Names’, Empirical Issues in Syntax and Semantics, 6 (2006): 285307; and Fara, D. G., ‘Names are predicatesPhilosophical Review 124(1) (2015): 59117.

31 Especially considering that names take on a interesting variety of uses beyond those witnessed in (10). Consider those in (i), for instance.

  1. (i)
    1. a.

      a. He was part of the Obama election team.

    2. b.

      b. Let's Skype tomorrow.

    3. c.

      c. How much Rover is on the road?

32 Frege, Grundlagen der Arithmetik.

33 For example, Wright (Frege's Conception of Numbers as Objects (Aberdeen University Press, 1983)) is plausibly understood as defending the former view, while Hofweber (Ontology and the Ambitions of Metaphysics (Oxford University Press, 2016)) and Moltmann (Reference to Numbers in Natural Language’, Philosophical Studies 162 (2013): 499536) have recently defended the latter.

34 See e.g. Kennedy, C., ‘A Scalar Semantics for Scalar Readings of Number Words’ in Caponigro, I. and Cecchetto, C. (eds) From Grammar to Meaning: the Spontaneous Logicality of Language (Cambridge University Press, 2013), 172200; Snyder, ‘Numbers and Cardinalities: What's Really Wrong with the Easy Argument?’.

35 Fine, K., ‘Guide to Ground’, in Correia, F. and Schnieder, B. (eds) Metaphysical Grounding (Cambridge University Press, 2012), 3780.

36 Sider, T., Writing the Book of the World (Oxford University Press, 2013).

37 Schaffer, J., ‘On What Grounds What’, in Manley, D., Chalmers, D. J. and Wasserman, R. (eds), Metametaphysics: New Essays on the Foundations of Ontology (Oxford University Press, 2009), 347383.

38 Though Sider (Writing the Book of the World) also speaks of entities as being fundamental, or as being more fundamental than others.

39 Dummett, Frege: Philosophy of Mathematics.

40 Frege, Grundlagen der Arithmetik, §2.

41 Frege, Grundlagen der Arithmetik, §17; Leibniz, G.W., et al. , ‘Nouveaux Essais Sur l'Entendement Humain: Avantpropos et Premier Livre’ (Belin, 1885), §9.

42 Bolzano, B., Theory of Science, trans. by George, R. (University of Berkeley, 1837).

43 Frege, G., Grundlagen der Arithmetik, 1884, §3.

44 Aristotle, (Physics, Chapter 3): ‘Men do not think they know a thing unless they have grasped the ‘why’ of it’. So proper foundational knowledge is of-a-piece with what Jaegwon Kim calls ‘explanatory knowledge’ – knowledge why – as opposed to mere descriptive knowledge – knowledge that (Kim, J., ‘Explanatory Knowledge and Metaphysical Dependence’, Philosophical Issues 5 (1994), 51). However, Joshua Schechter (p.c.) suggested that what we call ‘proper foundational knowledge’ may not be a special kind of knowledge at all: it is to know and to have an (or the) explanation for what one knows. That would make the Fregean hierarchy purely metaphysical, and in no way epistemic. There is no need to settle this matter of classification/exegesis here.

45 Jeshion, R., ‘Frege's Notions of Self-Evidence’, Mind 110(440) (2001), 944.

46 Jeshion, R., ‘Frege: Evidence for Self-Evidence’, Mind 113(449) (2004): 131138.

47 Frege, G., Grundgesetze der Arithmetik vol. II (Olms, 1903), 253.

48 Frege, G., Grundgesetze der Arithmetik, vol. I (H. Pohle, 1893), vii.

49 Jeshion, ‘Frege's Notions of Self-Evidence’, 967.

50 Frege, G., ‘Logik in der Mathematik’, Nachgelassene Schriften (1914), 205.

51 Burge, T., ‘Frege on Knowing the Foundation’, Mind 107(426) (1998), 328.

52 Jeshion, ‘Frege's Notions of Self-Evidence’, 969.

53 Frege, Grundgesetze der Arithmetik, vii.

54 Ibid., §61.

55 Whitehead, A.N. and Russell, B., Principia mathematica, 3 (1910/1913).

56 Thanks to Gil Sagi for these suggestions.

57 See Shapiro, S., ‘We Hold These Truths to be Self-Evident: But What Do We Mean by That?’, Review of Symbolic Logic 2 (2009): 175207.

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