### Single Idea 9623

#### [catalogued under 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics]

Full Idea

Field needs the notion of logical consequence in second-order logic, but (since this is not recursively axiomatizable) this is a semantical notion, which involves the idea of 'true in all models', a set-theoretic idea if there ever was one.

Gist of Idea

Field needs a semantical notion of second-order consequence, and that needs sets

Source

comment on Hartry Field (Science without Numbers [1980], Ch.4) by James Robert Brown - Philosophy of Mathematics

Book Reference

Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.54

A Reaction

Brown here summarises a group of critics. Field was arguing for modern nominalism, that actual numbers could (in principle) be written out of the story, as useful fictions. Popper's attempt to dump induction seemed to need induction.