more from Bertrand Russell

Single Idea 14442

[catalogued under 6. Mathematics / A. Nature of Mathematics / 2. Geometry]

Full Idea

We wish to say that when two straight lines cross each other they have a point in common, but if the series of points on a line were similar to the series of ratios, the two lines might cross in a 'gap' and have no point in common.

Gist of Idea

If straight lines were like ratios they might intersect at a 'gap', and have no point in common


Bertrand Russell (Introduction to Mathematical Philosophy [1919], X)

Book Reference

Russell,Bertrand: 'Introduction to Mathematical Philosophy' [George Allen and Unwin 1975], p.101

A Reaction

You can make a Dedekind Cut in the line of ratios (the rationals), so there must be gaps. I love this idea. We take for granted intersection at a point, but physical lines may not coincide. That abstract lines might fail also is lovely!

Related Idea

Idea 18188 The line of rationals has gaps, but set theory provided an ordered continuum [Maddy]