### Single Idea 10008

#### [catalogued under 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism]

Full Idea

I argue for an internalist conception of arithmetic. Arithmetic is not about a domain of entities, not even quantified entities. Quantifiers over natural numbers occur in their inferential-role reading in which they merely generalize over the instances.

Clarification

See Idea 10007 for the two roles of quantifiers

Gist of Idea

Arithmetic is not about a domain of entities, as the quantifiers are purely inferential

Source

Thomas Hofweber (Number Determiners, Numbers, Arithmetic [2005], §6.3)

Book Reference

-: 'Philosophical Review 114' [Phil Review 2005], p.219

A Reaction

Hofweber offers the hope that modern semantics can disentangle the confusions in platonist arithmetic. Very interesting. The fear is that after digging into the semantics for twenty years, you find the same old problems re-emerging at a lower level.

Related Idea

Idea 10007
Quantifiers for domains and for inference come apart if there are no entities **[Hofweber]**