Quantity in The Principles of Mathematics

Abstract

Part III of PoM has not been widely studied. Certain aspects of Russell’s doctrine of quantity have been touched on when discussing Russell’s use of the abstraction principle¹ or when examining the genesis of PoM.² But, except Michell (1997; 1999), nobody, to my knowledge, has tried to present in detail Russell’s doctrine. There are some reasons for that. First, even if it is quite short (40 pages), PoM III is terribly complicated. Russell introduced there several idiosyncratic notions (for instance: kind of magnitude, divisibility, stretch, relational magnitude) that are nowhere precisely defined and which almost never occur again in the book. Second, Russell himself, at the beginning of Part III, warned his reader that ‘the whole of this part … is a concession to tradition; for quantity … is not properly a notion belonging to pure mathematics at all’ (p. 158). He is here alluding to the fact that, thanks to the works of Dedekind and Cantor, there was no need, in 1903, to refer to quantity for giving an account of the theory of real numbers and real analysis .³ Owing to these recent mathematical developments, Russell himself seemed to consider PoM III as a dispensable outgrowth, withdrawn into itself and unconnected with the rest of the book. No surprise then if scholars did not rush into this mire on the side.