more from B Hale / C Wright

### Single Idea 8784

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

Full Idea

The result of joining Hume's Principle to second-order logic is a consistent system which is a foundation for arithmetic, in the sense that all the fundamental laws of arithmetic are derivable within it as theorems. This seems a vindication of logicism.

Clarification

Hume's Principle involves one-to-one correlation

Gist of Idea

Neo-logicism founds arithmetic on Hume's Principle along with second-order logic

Source

B Hale / C Wright (Logicism in the 21st Century [2007], 1)

Book Reference

'Oxf Handbk of Philosophy of Maths and Logic', ed/tr. Shapiro,Stewart [OUP 2007], p.169

A Reaction

The controversial part seems to be second-order logic, which Quine (for example) vigorously challenged. The contention against most attempts to improve Frege's logicism is that they thereby cease to be properly logical.