Hume’s Principle (HP) states that the cardinal number of the concept F is identical with the cardinal number of G if and only if F and G can be put into one-to-one correspondence. The Schwartzkopff-Rosen Principle (SR Principle) is a modification of HP in terms of metaphysical grounding: it states that if the number of F is identical with the number of G, then this identity is (strictly, fully, and immediately) grounded by the fact that F and G can be paired one-to-one (Rosen 2010, 117; Schwartzkopff Grazer Philosophische Studien, 82(1), 353–373, 2011, 362). HP is central to the neo-logicist program in the philosophy of mathematics (Wright 1983; Hale and Wright 2001); in this paper we submit that, even if the neo-logicists wish to venture into the metaphysics of grounding, they can avoid the SR Principle. In Section 1 we introduce neo-logicism. In Sections 2 and 3 we examine the SR Principle. We then formulate an account of arithmetical facts which does not rest on the SR Principle; we finally argue that the neo-logicists should avoid the SR Principle in favour of this alternative proposal.
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Grounding is strict if the notion of grounding does not allow a fact to ground itself; a ground for a fact F is full if there is no set of facts Γ to which that fact belongs such that Γ grounds F (while that fact does not ground F on its own); a ground for F is immediate if there is no ground “in between” that fact and F (Fine 2012, 51).
We will assume that grounding is a relation between facts; we will adopt the following conventions: we will enclose sentences in square brackets as term-forming device for facts, and we will use the symbol ‘<’ for the grounding relation.
‘F ≈ G’ abbreviates the second-order formula:$$ \exists R \ (\forall x \ (F(x) \rightarrow \exists ! y \ (G (y) \land R(x,y))) \land \forall x \ (G(x) \rightarrow \exists ! y \ (F(y) \land R(y, x)))), $$
where ‘∃x! ϕ(x)’ is defined as: ∃x(ϕ(x) ∧∀y (ϕ(y) → x = y)).
The notion of grounding has become increasingly popular in the literature (Schaffer 2009; Rosen 2011; Audi 2012; Fine 2012; for intro, see Clark and Liggins 2012; Correia and Schnieder 2012b). However, its application to the philosophy of mathematics has so far been limited; notable exceptions include Cameron 2008; Rosen 2011, 2016; Schwartzkopff 2011; Donaldson 2017; Carrara and De Florio 2018; De Florio 2018; Wigglesworth 2018.
A binary relation R is an equivalence relation iff it is reflexive, symmetric, and transitive.
It is worth noting that the right-hand side of HP, which functions as the definiens of the meaning of numerical identities, is formulated in purely (second-order) logical terms; cf. fn. 4.
Wright (1983, 153).
Donaldson (2017, 783).
Øystein Linnebo has pointed out to us that the neo-Fregeans would probably find it more congenial to rely on this alternative formulation of the SR Principle:$$ \forall F, G (F \approx G \rightarrow ([F \approx G] < [\# (F) = \# (G)])) $$
Even though we are sympathetic with this proposal, we will stick to the standard formulation for convenience.
Rosen and Donaldson refer to properties rather than to concepts; we will refer to concepts to be closer to Frege’s analysis in (1970), and to neo-logicism more in general.
This exotic example is due to Donaldson (2017, 784).
Horsten and Leigeb (2009, 218).
We are grateful to an anonymous reviewer of Philosophia for suggesting a more precise formulation of the Aristotelian view.
Schwartzkopff (2011, 353-4).
Notice that, on this formulation, the view is that mathematical facts are grounded by facts concerning pluralities of objects, not concepts or properties; cf. fn. 10.
Donaldson (2017, 785).
A form of pluralism of the kind adumbrated here is actually defended in Rosen (2016). The neo-logicists, by contrast, would be happy to concede that some facts concerning concepts are necessary; it is worth noting, for example, that it is a logical truth (of second-order logic with identity) that the concept x≠x, whose extension is (necessarily) empty, is equinumerous with itself.
Including the fact that the number of Welsh cities is six among the grounds of [#Flamingo species = 6] raises however some further difficulties; these difficulties are adumbrated in Donaldson (2017, 789-90, fn. 25).
We are grateful to an anonymous reviewer of Philosophia for many insightful comments on the contents of this Section.
Reck and Price (2000, 356).
This schema would lead to many undesirable consequences if one makes standard assumptions about universally quantified truths being grounded in each one of their instances. Rosen could reply by rehearsing the view that at least some universally quantified truths are not grounded by, but rather ground, their instances; cf. Roski (2018).
Rosen (2011, 121).
Rosen (2011, 285).
However, even if our account is not committed to the SR Principle, it does not exclude it either; in particular, one can argue that SR Principle still gives the correct account of contingent facts involving cardinal numbers.
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De Florio, C., Zanetti, L. On the Schwartzkopff-Rosen Principle. Philosophia 48, 405–419 (2020). https://doi.org/10.1007/s11406-019-00068-6
- Hume’s Principle
- Schwartzkopff-Rosen Principle