Reflections from Principia Mathematica

Abstract

Reviews 171 c:\users\ken\documents\type3402\rj 3402 050 red.docx 2015-02-04 9:19 PM REFLECTIONS FROM PRINCIPIA MATHEMATICA Russell Wahl English and Philosophy / Idaho State U. Pocatello, id 83209 usa wahlruss@isu.edu Alexandre Guay, ed. Autour des Principia Mathematica de Russell et Whitehead. (Collection Histoire et Philosophie des Sciences.) Dijon: Editions Universitaires de Dijon, 2012. Pp. 168. isbn 978-2-36441-001-4. €20.00 (pb). his collection, by several distinguished French philosophers, is intended to be a work on problems which are suggested by or perhaps stem from Whitehead and Russell’s Principia Mathematica. It turns out to be a somewhat eclectic collection. Some of the papers explicate points in Principia, some delve into issues from Principia or perhaps suggested by something in it, and q= 172 Reviews c:\users\ken\documents\type3402\rj 3402 050 red.docx 2015-02-04 9:19 PM some into issues suggested by later works of Russell’s. The collection was put together for the centenary of Principia, with the idea of focusing on “its posterity , the doors it has opened, the questions it has brought up, and the techniques it has initiated” (p. 5). The essays are all interesting, but in some of them the main focuses are only introduced and are not developed very much, appearing to explicate points already made in detail elsewhere and not adding a great deal that is new. The first essay, for example, gives an overview of the development of Russell’s logic, the theory of classes and the account of relations, the antinomies , the theory of descriptions and even the ramified theory of types. While the review is useful, there doesn’t seem to be here anything more than what would be found in the previous works of Denis Vernant, Jules Vuillemin, or Philippe de Rouilhan, on which the author draws. Hints of further issues are to be found in the discussion, but they are not developed because of the sheer number of points being reviewed. Denis Vernant, in the fifth essay, gives a thorough review of Russell’s account of similarity of relations and his use of this in relation-arithmetic, mostly from Chapter 6 of Introduction to Mathematical Philosophy. Vernant points out that Russell uses the terms “analogy” and “exactly analogous” in several places and thinks of these in terms of his definition of structure. At the end of the essay Vernant begins with Russell’s formal account of structure and seeks to give an account of analogy, and specifically metaphor, using the tools of similarity of structure. Again, the discussion of similarity relations is good, but does not really go beyond previous work, and the application to metaphor, while interesting, is very brief. The sixth essay, by Nadine Gessler and Denis Miéville, discusses Lesniewski ”s criticisms of Principia and gives some hint of Lesniewski’s own positions. Lesniewski’s work is very interesting and has probably not been given its due. These authors briefly mention Lesniewski’s concerns about classes, the assertion sign and definitions, and then give a short presentation of Lesniewski’s own views. This last part is again just sketched out. The authors do point out difficulties Lesniewski had with parts of Principia, including the sloppiness about use and mention, the requirement that definitions be non-creative and the theory of types. They also point out that Lesniewski had a view of logic (reminiscent of Arnauld and Nicod) as “the art of thinking”, and they discuss his suspicion of sets as abstract entities rather than mereological wholes of the kind which can be grounded in experience. Lesniewski apparently thought of mathematical logic as limiting and not encompassing the whole of thinking. This is curious, because often Russell and Whitehead are accused of expanding logic in order to capture all mathematics as part of logic. Gessler and Miéville accent the differences between Lesniewski ’s position and Principia, but in many ways Lesniewski’s views are actually Reviews 173 c:\users\ken\documents\type3402\rj 3402 050 red.docx 2015-02-04 9:19 PM closer to those of Whitehead and Russell than to, e.g., the views...