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**Wittgenstein’s annotations to Hardy’s course of pure mathematics. An investigation of Wittgenstein’s non-extensionalist understanding of the real numbers.**
*(English)*
Zbl 1529.01002

Nordic Wittgenstein Studies 7. Cham: Springer (ISBN 978-3-030-48480-4/hbk; 978-3-030-48483-5/pbk; 978-3-030-48481-1/ebook). xx, 322 p. (2020).

This book is devoted to L.Wittgenstein’s annotation to G. H. Hardy’s classic textbook on analysis [A course of pure mathematics. Cambridge: At the University Press (1908; JFM 39.0340.01)] which can be found in the margins of Wittgenstein’s copy of this book. They were written 1942–43. These annotations are neither numerous nor extensive. The authors motivate their publication with the fact that “They directly refer to assertions of a real mathematician”, in a book “that was extremely influential in Great Britain for a very long time” (p.vii). The edition is intended to round out our understanding of how Wittgenstein’s ideas on the philosophy of mathematics “are relevant to what mathematicians think, not only those in Cambridge during his lifetime, but also some in our time” (pp.vii–viii), referring in particular to Wittgenstein’s ideas on real numbers.

Part I is entitled “Overview”. The introductory Chapter 1 surveys Wittgenstein’s thinking about mathematics during the main stages of his philosophical development up to the emergence of the Philosophical Investigations (1937–1944), the context of his comments on Hardy’s book. The relationship between Wittgenstein and Hardy is also discussed. Chapter 2 examines Wittgenstein’s non-extensionalist point of view in the philosophy of real numbers. Wittgenstein does not see the irrational numbers as an extension of the rational numbers but thinks that the domain of rational numbers is imbedded into the domain of irrational numbers, i.e., “the structuring of one kind of numbers in terms of another” (p.29).

Part II is devoted to the annotations. Chapter 3 analyzes and comments on the annotations on pp.2–9, focusing on irrational numbers. Chapter 4 provides an excursus on the status of the law of the excluded middle in mathematics from Wittgenstein’s and the authors’ perspectives. The authors relate this law and its applications to Wittgenstein’s notion of a language game. Chapter 5 gives analyses and comments on the annotations on pp. 10–30, mostly on the continuum of real numbers, Chapter 6 on the annotations on pp.40–37 and 117–121 on functions and limits.

Part III contains longer essays by the authors. In Chapter 7, the second author analyzes Wittgenstein’s position on G.Cantor’s diagonal method (pp.124–191). In Chapter 8, the first author contributes a non-extensionalist discussion of its generality (pp.193–258).

In Part IV, the annotations are edited with facsimiles of the annotated pages, the German text of Wittgenstein’s comments and an English translation. The volume closes with an index of topics and names.

Although Wittgenstein’s annotations as the subject matter of this book may seem to be rather marginal, the book goes far beyond their edition, providing access to important topics in the philosophy of mathematics seen from the special Wittgensteinian perspective.

Part I is entitled “Overview”. The introductory Chapter 1 surveys Wittgenstein’s thinking about mathematics during the main stages of his philosophical development up to the emergence of the Philosophical Investigations (1937–1944), the context of his comments on Hardy’s book. The relationship between Wittgenstein and Hardy is also discussed. Chapter 2 examines Wittgenstein’s non-extensionalist point of view in the philosophy of real numbers. Wittgenstein does not see the irrational numbers as an extension of the rational numbers but thinks that the domain of rational numbers is imbedded into the domain of irrational numbers, i.e., “the structuring of one kind of numbers in terms of another” (p.29).

Part II is devoted to the annotations. Chapter 3 analyzes and comments on the annotations on pp.2–9, focusing on irrational numbers. Chapter 4 provides an excursus on the status of the law of the excluded middle in mathematics from Wittgenstein’s and the authors’ perspectives. The authors relate this law and its applications to Wittgenstein’s notion of a language game. Chapter 5 gives analyses and comments on the annotations on pp. 10–30, mostly on the continuum of real numbers, Chapter 6 on the annotations on pp.40–37 and 117–121 on functions and limits.

Part III contains longer essays by the authors. In Chapter 7, the second author analyzes Wittgenstein’s position on G.Cantor’s diagonal method (pp.124–191). In Chapter 8, the first author contributes a non-extensionalist discussion of its generality (pp.193–258).

In Part IV, the annotations are edited with facsimiles of the annotated pages, the German text of Wittgenstein’s comments and an English translation. The volume closes with an index of topics and names.

Although Wittgenstein’s annotations as the subject matter of this book may seem to be rather marginal, the book goes far beyond their edition, providing access to important topics in the philosophy of mathematics seen from the special Wittgensteinian perspective.

Reviewer: Volker Peckhaus (Paderborn)

### MSC:

01-02 | Research exposition (monographs, survey articles) pertaining to history and biography |

01A60 | History of mathematics in the 20th century |

00A30 | Philosophy of mathematics |