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Single Idea 10140

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle ]

Full Idea

Wright says the Fregean arithmetic can be broken down into two steps: first, Hume's Law may be derived from Law V; and then, arithmetic may be derived from Hume's Law without any help from Law V.

Gist of Idea

We derive Hume's Law from Law V, then discard the latter in deriving arithmetic

Source

report of Crispin Wright (Frege's Concept of Numbers as Objects [1983]) by Kit Fine - The Limits of Abstraction I.4

Book Ref

Fine,Kit: 'The Limits of Abstraction' [OUP 2008], p.41

A Reaction

This sounds odd if Law V is false, but presumably Hume's Law ends up as free-standing. It seems doubtful whether the resulting theory would count as logic.

The 19 ideas with the same theme [view that one-one correspondence is basis of numbers]:

8649 Two numbers are equal if all of their units correspond to one another [Hume]
9956 'The number of Fs' is the extension (a collection of first-level concepts) of the concept 'equinumerous with F' [Frege, by George/Velleman]
13527 Frege's cardinals (equivalences of one-one correspondences) is not permissible in ZFC [Frege, by Wolf,RS]
22292 Hume's Principle fails to implicitly define numbers, because of the Julius Caesar [Frege, by Potter]
17442 Frege thinks number is fundamentally bound up with one-one correspondence [Frege, by Heck]
14425 A number is something which characterises collections of the same size [Russell]
18145 Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock]
18149 There are many criteria for the identity of numbers [Bostock]
18148 Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock]
10140 We derive Hume's Law from Law V, then discard the latter in deriving arithmetic [Wright,C, by Fine,K]
8692 Frege has a good system if his 'number principle' replaces his basic law V [Wright,C, by Friend]
17440 Wright says Hume's Principle is analytic of cardinal numbers, like a definition [Wright,C, by Heck]
13893 It is 1-1 correlation of concepts, and not progression, which distinguishes natural number [Wright,C]
8784 Neo-logicism founds arithmetic on Hume's Principle along with second-order logic [Hale/Wright]
10529 If Hume's Principle can define numbers, we needn't worry about its truth [Fine,K]
10530 Hume's Principle is either adequate for number but fails to define properly, or vice versa [Fine,K]
8266 Simple counting is more basic than spotting that one-to-one correlation makes sets equinumerous [Lowe]
8302 Fs and Gs are identical in number if they one-to-one correlate with one another [Lowe]
10133 Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman]