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Hume’s principle: a plea for austerity

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Abstract

According to Hume’s principle, a sentence of the form ⌜The number of Fs = the number of Gs⌝ is true if and only if the Fs are bijectively correlatable to the Gs. Neo-Fregeans maintain that this principle provides an implicit definition of the notion of cardinal number that vindicates a platonist construal of such numerical equations. Based on a clarification of the explanatory status of Hume’s principle, I will provide an argument in favour of a nominalist construal of numerical equations. The neo-Fregean objections to such a construal will be examined and rejected. And the implications of the nominalist construal for the use of numerals and for the understanding of ontological questions for the existence of numbers will be spelled out.

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Notes

  1. Quinean quasi-quotation will be employed wherever necessary (cf. Quine 1981, ch. 6). ‘F’ and ‘G’ will be used as predicate variables, ‘F’ and ‘G’ as the corresponding object variables. Accordingly, phrases like ‘the Fs’, ‘a F’, etc. are short for ‘the object to which F applies’, ‘an object to which F applies’, etc.

  2. cf. Rundle (1979, p. 257).

  3. Suppose, for example, that we are to apply the procedure to given sets of cups and spoons. The assumption in question, then, tells us that it does not matter which spoon is paired with which cup, nor how they are paired (whether, for instance, a spoon is put to the left of a cup or the other way around). Either they will pair off or they won’t.

  4. In the quoted passage, Wright is concerned with the relation between statements that claim that two lines are parallel and the corresponding statements about the identity of the lines’ directions. But his considerations are supposed to be transferable to the case of statements that claim the bijective correlatability of two concepts and the corresponding numerical equations.

  5. This point has also been stressed by Contessa (2016) against Thomasson’s defence of the neo-Fregean project in (2013). It has been accepted by Thomasson in her response (2017).

  6. Again, the practice of transitive counting determines a consistent use of number ascriptions only on the true, though non-trivial assumption that the procedure’s outcome does not depend on which F is counted with which numeral.

  7. In this sense, even a connective such as ‘and’ could be said to refer to something, viz. a truth-function.

  8. Recent discussions of the problem can be found in Greimann (2003), MacBride (2006), Pederson (2009), and Kim (2011). Only MacBride mentions the possibility that (11) may actually be nonsense (2006, pp. 187–8).

  9. It can be granted that there is no trivial for proof for this intuitive principle. Still, the Neo-Fregean’s would certainly accept the principle and classify it as a logical truth.

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Acknowledgements

I would like to thank David Dolby, Severin Schroeder and an anonymous reviewer for their comments on previous drafts. Further to this, I received helpful suggestions from the audience of the 5th conference of the Latin American Association for Analytic Philosophy (ALFAn V) at which I presented an earlier version of this paper.

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  1. Universidad del Norte, Km. 5, Antigua Vía a Puerto Colombia, Área Metropolitana de Barranquilla, Barranquilla, 1569, Colombia

    Kai Michael Büttner

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Büttner, K.M. Hume’s principle: a plea for austerity. Synthese 198, 3759–3781 (2021). https://doi.org/10.1007/s11229-019-02309-3

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