Ebert P (2008) A puzzle about ontological commitments. Philosophia Mathematica, 16 (2), pp. 209-226. https://doi.org/10.1093/philmat/nkm050
The aim of this paper is to scrutinise the necessary conditions for a specific mathematical principle to be ontologically committing and, as such, to identify the source of its ontological commitments. The principle in ques- tion is Hume’s Principle – a statement that embedded in second-order logic allows for a deduction of the second-order Peano axioms. This principle is at the heart of the so-called Neo-Fregean programme as defended by Bob Hale and Crispin Wright. Once it is clear what the source of the commitment to infinitely many ob jects of Hume’s Principle is, we should be able to re-evaluate the debate between the Neo-Fregeans – who defend Hume’s Principle as an analytic principle – and the so-called epistemic rejectionists – who deny its analytic status. The conclusions can then be generalised to other abstraction principles, principle that share a similar form to Hume’s Principle. In the first section, I will clarify what epistemic rejectionism is committed to and provide a theoretical basis for the position by introducing the notion of presumptuousness as the underlying criterion on the basis of which Hume’s Principle is to be rejected as an analytic principle. Then, in section 2 and 3, I will review certain formal results which prima facie put pressure on epistemic rejectionism. In section 4, I will propose a short thought-experiment to highlight the problem for epistemic rejectionism posed by the formal results and then suggest various responses on behalf of the epistemic rejectionist. The upshot will be to elicit a new and very basic disagreement between epistemic rejectionism and the Neo-Fregeans which will provide a new angle to properly assess and re-evaluate the current debate.
Abstractionism; ontological commitments; Neo-Fregeanism; apriori; analyticity; Hume's Principle
Philosophia Mathematica: Volume 16, Issue 2