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Cardinals, Ordinals, and the Prospects for a Fregean Foundation

Published online by Cambridge University Press:  03 July 2018

Eric Snyder ,
Stewart Shapiro  and
Richard Samuels
Eric Snyder*
Affiliation:
Ohio State University
Stewart Shapiro*
Affiliation:
Ohio State University
Richard Samuels*
Affiliation:
Ohio State University
*
[email protected]
[email protected]
[email protected]
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Abstract

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’.

Type
Papers
Information
Royal Institute of Philosophy Supplements , Volume 82: Metaphysics , July 2018 , pp. 77 - 107
Copyright
Copyright © The Royal Institute of Philosophy and the contributors 2018 

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References

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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.

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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.

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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’.

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  1. (i)
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    2. b.

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    3. c.

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32 Frege, Grundlagen der Arithmetik.

33 For example, Wright (Frege's Conception of Numbers as Objects (Aberdeen University Press, 1983)Google Scholar) is plausibly understood as defending the former view, while Hofweber (Ontology and the Ambitions of Metaphysics (Oxford University Press, 2016)Google Scholar) and Moltmann (‘Reference to Numbers in Natural Language’, Philosophical Studies 162 (2013): 499–536CrossRefGoogle Scholar) 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), 172–200CrossRefGoogle Scholar; Snyder, ‘Numbers and Cardinalities: What's Really Wrong with the Easy Argument?’.

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54 Ibid., §61.

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56 Thanks to Gil Sagi for these suggestions.

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