more from E Reck / M Price

### Single Idea 10178

#### [catalogued under 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism]

Full Idea

It is tempting to take a modal turn, and quantify over all possible objects, because if there are only a finite number of actual objects, then there are no models (of the right sort) for Peano Arithmetic, and arithmetic is vacuously true.

Gist of Idea

Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous

Source

E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5)

A Reaction

[compressed; Geoffrey Hellman is the chief champion of this view] The article asks whether we are not still left with the puzzle of whether infinitely many objects are possible, instead of existent.