Combining Texts

Ideas for 'Ordinary Objects', 'Elements of Mathematical Logic' and 'Logicism and Ontological Commits. of Arithmetic'

unexpand these ideas     |    start again     |     choose another area for these texts

display all the ideas for this combination of texts

---

2 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers

10027	Mathematics is higher-order modal logic [Hodes]
	Full Idea: I take the view that (agreeing with Aristotle) mathematics only requires the notion of a potential infinity, ...and that mathematics is higher-order modal logic.
	From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984])
	A reaction: Modern 'modal' accounts of mathematics I take to be heirs of 'if-thenism', which seems to have been Russell's development of Frege's original logicism. I'm beginning to think it is right. But what is the subject-matter of arithmetic?

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic

10026	Arithmetic must allow for the possibility of only a finite total of objects [Hodes]
	Full Idea: Arithmetic should be able to face boldly the dreadful chance that in the actual world there are only finitely many objects.
	From: Harold Hodes (Logicism and Ontological Commits. of Arithmetic [1984], p.148)
	A reaction: This seems to be a basic requirement for any account of arithmetic, but it was famously a difficulty for early logicism, evaded by making the existence of an infinity of objects into an axiom of the system.