Mathematical analysis of the space problem. Bilingual edition German-French in 2 volumes. Commented and edited by Éric Audureau and Julien Bernard (Q2800864)

This two-volume book is a reprint of \textit{H. Weyl}'s [Mathematische Analyse des Raumproblems. Vorlesungen gehalten in Barcelona und Madrid. Berlin: J (1923; JFM 49.0494.02)] with a translation into French on the facing pages, accompanied by an introductory essay and 348 end notes elucidating points in the text.NEWLINENEWLINEThe original book was written towards the end of Weyl's writing his better-known book \textit{Raum Zeit Materie}, between, one might say, the fourth and the final fifth edition of that work, and it reworks to good effect some of the key ideas. One way of approaching Weyl's concerns is to recall the Kleinian view of geometry, according to which a geometry is a space equipped with the action of a group. For Klein, the paradigm space was a projective space of some dimension, and the groups typically familiar transformation groups. Spaces with a metric, such as non-Euclidean (hyperbolic) space fitted in to this framework in an attractive and informative way. It is clear that these spaces are homogeneous, and so ill-suited to the spacetimes of general relativity with their varying metrics. What Weyl did in the book under review was to show how to create a comprehensive geometrical structure (later regarded as fibre bundles and principal bundles for certain Lie groups) into which the Kleinian approach could be extended by letting the groups act infinitesimally.NEWLINENEWLINEThe translation in this edition is fluent, the notes helpful, and the republication of this work should prove to be very useful; Weyl's questions have not gone away, and his approach should be part of an education in differential geometry and general relativity.NEWLINENEWLINEThe most interesting part of the book are the two introductions; the one by Audureau is the more overtly philosophical and wide-ranging, the one by Bernard only a little less philosophical but closer to the mathematics. This overlapping scene-setting reflects Weyl's interests. It seemed to Weyl that the only defensible geometry from an epistemological point of view was an infinitesimal one, just as it seemed to him that a realist perspective was unduly naive. With Einstein's work, new questions were raised about what concepts belong to physics and what to mathematics (insofar as one admits a distinction); to this debate Weyl raised questions about the nature of knowledge. He was influenced by Husserl at this stage in his life, and Kantian ideas, not always welcome to him, inevitably play a role. This book by Weyl, with these two introductions, show just how clearly the theory of general relativity is not only differential geometry and remarkable observations but a challenge to how we think of having knowledge of the external world.