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]