Geometry, Logicism and ‘If-Thenism’

Abstract

In the two previous chapters, we have dealt in some detail with Russell’s view of projective and metric geometry. But we have not yet explained how he viewed the articulation between geometry and logic. Russell is well known for having claimed that geometry (projective geometry at least) is a part of pure mathematics. But how did he proceed to reduce geometry to logic? As we began to see in Chapter 2, the standard answer is ‘if-thenism’: to regard a given theory as a part of logic, it is sufficient to show that the said theory is consistent (has a model) and can be axiomatized.¹ This interpretation confronts us with a problem however. Russell included projective geometry within pure mathematics, but he did not believe that metrical geometry was a logical science. For all that, he admitted that metrical geometry was consistent and could be axiomatized — in at least two different ways: via the projective definition of metric and also in the direct Leibnizian way. Axiomatization and consistency cannot then be the sole criteria Russell used to characterize the sphere of logical science. More is needed — but what exactly?