Russell's unknown logicism. A study in the history and philosophy of mathematics (Q1941180)

The stated aim of this important study of neglected parts of the Russellian \textit{opus} is to show that ``the widespread view that logicism is a continuation of arithmetization ought to be rejected.'' (p.\ 7) This, in itself, could be read off a letter Whitehead wrote to Russell on 14 September 1909, in which he writes ``The modern arithmeticisation of mathematics is an entire mistake.'' However, the author goes into great length to show where Russell rejects that view and what takes the place of arithmetic in those instances. The two examples are Russell's rejection of ``the definition of geometrical space as a numerical manifold'' and of ``Dedekind's cut-construction of real numbers.'' (p.\ 7) The first three chapters are devoted to Russell's views on geometry, the next three deal with his views on quantity, and the last chapter tries to explain the tension between the universality of logic and the topic-specificity of his choices of formal set-ups in the cases of geometry and of the real numbers. In the chapters on geometry, after presenting a short history of projective geometry, a closer look at Chapter VI of [\textit{B. Russell}, The principles of mathematics. Vol I. Cambridge: University Press (1903; JFM 34.0062.14)] (to be referred to as PoM VI) and at [\textit{A. N. Whitehead}, The axioms of projective geometry. Cambridge: University Press (1906; JFM 37.0559.01)] (which ``is nothing else than an extension of Russell's PoM VI theory'') reveals that Russell, having rejected the ``Cartesian approach'' of treating projective geometry in its coordinatized variant, prefers \textit{M. Pieri}'s incidence-based approach from [Mem.\ Real.\ Accad.\ Sci.\ Torino (2) 48, 1--62 (1898; JFM 29.0407.01)] to \textit{M. Pasch}'s order-based approach from [Vorlesungen über neuere Geometrie. Leipzig: Teubner (1882; JFM 14.0498.01)]. The reason for this choice constitutes one of the main topics of the author's research. Russell appears to have considered projective geometry as belonging to logic, and felt that Pasch's approach, with its ``ordinal'' nature, did not as adequately reflect ``the specificity of geometrical thinking'' (p.\ 48) as Pieri's. Deeply rooted in the 19th-century understanding of geometry, and in particular influenced by von Staudt's work, when asking for the most appropriate way of formalizing metric geometry (Euclidean, hyperbolic, elliptic), Russell had the option of (i) accepting Klein's projective definition of a metric, (ii) Leibniz's approach, which starts with a congruence relation (``distance relation'' for Russell) and defines collinearity in terms of the locus of all points equidistant from two fixed points, or (iii) introducing a new logically undefinable relation of metrical distance, understood not as the distance in (ii), but as a ``magnitude of divisibility'', a Russellian term that moved metric geometry out of ``pure mathematics'' (``there is a genuinely distinct science of metrical geometry, but, since it introduces a new undefinable, it does not belong to pure mathematics in the sense in which we have used the words in this work.'' (PoM, p.\ 428)). The reasons for opting for (iii), which defeats the grand plan of showing that mathematics can be reduced to logic, are carefully analyzed. In the case of (ii), Russell, relying on \textit{M. Pieri} [Mem.\ Real.\ Accad.\ Sci.\ Torino (2) 49, 173--222 (1899; JFM 30.0426.02)], finds that ``the method is so complicated as to be not practically desirable'' (PoM, p.\ 411). Had he known of the very natural congruence-based axiom systems for Euclidean geometry presented in [\textit{R. Schnabel}, Euklidische Geometrie. Kiel: Univ. Kiel (Habil.) (1978)] (see also [\textit{R. Schnabel} and the reviewer, Expo.\ Math.\ 3, 285--288 (1985; Zbl 0566.51018)]) and in [\textit{G. Richter} and \textit{R. Schnabel}, J. Geom.\ 58, No. 1--2, 164--178 (1997; Zbl 0879.51010)] he might have opted otherwise. And here is precisely where the author finds that Russell's approach lacks ``universality'', that his choices depend on the actual mathematical practice of the time, on having had the chance to read Pieri's works, and many other accidental, historical features. Chapter 3 is devoted to showing that Russell could not have meant that geometry belongs to logic by ``if-thenism'' (a thesis proposed by \textit{A. Musgrave} [Br. J. Philos.\ Sci. 28, No. 2, 99--127 (1977; Zbl 1271.03020)]), i.e., by the simple observation that any finitely axiomatized theory can be said to belong to logic as it is concerned with deductions of statements (theorems) from the conjunction of all of its axioms. The second half of the book looks at the analysis of quantity in PoM III and in \textit{Principia mathematica}, at the ``application constraint'' from \textit{Principia mathematica} VI ``according to which the definition of rational and real numbers should explain how these numbers are used to measure quantities'' (p.\ 134) (the arithmetization approach does not satisfy that constraint as ``it leaves the whole theory of applied mathematics (measurement, etc.) unproved'', as noted by Whitehead in the letter to Russell cited above). In the final chapter the author summarizes what sets his reading of logicism apart from the standard one: ``My main claim is that Russell used logic as a frame for discussing the various ways of expanding logic as a tool for reaching a reflective agreement on the way mathematics should be defined.'' (p.\ 210) Reviewer's remark: The main bibliographic omission, which could have provided a fruitful exchange of views, is [\textit{S. Donati}, I fondamenti della matematica nel logicismo di Bertrand Russell. Florence: Firenze Atheneum (2003; Zbl 1112.03300)].