more from Penelope Maddy

### Single Idea 18188

#### [catalogued under 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory]

Full Idea

The structure of a geometric line by rational points left gaps, which were inconsistent with a continuous line. Set theory provided an ordering that contained no gaps. These reals are constructed from rationals, which come from integers and naturals.

Gist of Idea

The line of rationals has gaps, but set theory provided an ordered continuum

Source

Penelope Maddy (Naturalism in Mathematics [1997], I.2)

Book Reference

Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.27

A Reaction

This completes the reduction of geometry to arithmetic and algebra, which was launch 250 years earlier by Descartes.

Related Idea

Idea 14442
If straight lines were like ratios they might intersect at a 'gap', and have no point in common **[Russell]**