Wittgenstein's Apprenticeship with Russell (review)

Although everyone knows that Russell had an immense influence upon Wittgenstein's early philosophy, the degree to which Wittgenstein is either adopting or renouncing Russell's views is still largely a matter of dispute. Recent commentators have been in nearly univocal agreement that the Tractatus should be understood as a rejection of Russell's philosophy, and that Wittgenstein was instead more influenced by the "great works of Frege." In his earlier work, Gregory Landini has proposed a more nuanced way to understand Russell than is commonly assumed. In his new book, Landini directs his attention at re-thinking Wittgenstein's early philosophy in light of his revolutionary interpretation of Russell.

Too often Russell is read backwards, as it were; that is, his views in later published works are assimilated back into Principia Mathematica. And this leads commentators to interpret the Tractatus as criticizing views which Russell simply did not hold. Two "dogmas" that are commonly perpetuated about Russell are: first, that Principia features a ramified type-theory of entities; and second, that logical atomism is a theory of reductive empiricism, grounded upon "knowledge by acquaintance." The first chapter of the book seeks to dispel these dogmas. Ascribing to Russell a type-theory of entities is mistaken on Landini's view, since the formal language of Principia features only one type of genuine variable for individuals, and type/order indices are built into the syntax, through the use of structured predicate variables. As to the latter dogma, Landini presents logical atomism as a form of eliminative reductionism, which seeks to "reconceptualize the laws of the old theory without countenancing its objects" (15). The second chapter puts forth a positive reading of Russell's atomism independent of these two dogmas. From this perspective, logical atomism is not a philosophical theory, but rather "a research program for dissolving philosophical conundrums by employing variables with structure—an ontologically austere structural realism" (ix).

Throughout the third chapter, Landini discusses some key insights in Wittgenstein's early philosophy, including the doctrine of showing, and Wittgenstein's self-described Grundgedanke or "fundamental idea"—that "the 'logical constants' are not representatives" (TLP, 4. 0312 ). Landini interprets the Grundgedanke as coextensive with the doctrine of showing. That is, for Wittgenstein, "all and only notions with logico-semantic content," including what Wittgenstein calls "formal" or "internal" concepts and relations, are shown by the use of structured variables (85). This insight, Landini claims, he borrowed from Russell. According to this reading, Wittgenstein is continuing the project of Russell's logical atomism in a radically eliminativist direction, by building semantic notions into variables which show their compositional roles in the syntax of an ideal language.

Landini's fourth and fifth chapters are perhaps the best part of the book. In the fourth chapter, he seeks to expound some key notions in Wittgenstein's logical symbolism, including why Wittgenstein thought logic must consist of tautologies, the question of decidability and its connection to Wittgenstein's early "a-b notation" and the truth-table method. It also includes an enlightening discussion of the N-operator, which underscores its importance as a procedure for determining whether a formula is a tautology. The fifth chapter elucidates some of the tougher aspects of Wittgenstein's philosophy of mathematics, with the goal of dispelling the popular misconception that Wittgenstein rejected Russell's logicist project. A great deal of his discussion concerns Wittgenstein's dissatisfaction with the first edition of Principia—especially the use of the identity notation, as well as the axioms of reducibility and infinity, all of which Wittgenstein explicitly rejected. Landini suggests that Wittgenstein's dissatisfaction results from his commitment to 'showing' all of them by the use of exclusive quantifiers (and in fact, in an appendix he provides a system of polyadic predicate logic without need for the identity sign, through exclusive quantifiers).

The sixth chapter chronicles Russell's attempts to incorporate Wittgenstein's ideas into the second edition of Principia, including how to save mathematical induction and Cantor's power-class...