Uncovering the Mathematics Behind Russell’s Philosophy of Mathematics

Abstract

c:\users\kenneth\documents\type3401\rj 3401 193 red.docx 2014-05-14 8:54 PM UNCOVERING THE MATHEMATICS BEHIND RUSSELL’S PHILOSOPHY OF MATHEMATICS Graham Stevens Philosophy / U. of Manchester Manchester m13 9pl, uk graham.p.stevens@manchester.ac.uk Sébastien Gandon. Russell’s Unknown Logicism. (History of Analytic Philosophy series.) Basingstoke, uk, and New York: Palgrave Macmillan, 2012. Pp. xiii + 263. isbn: 978-0-230-57699-5. £55; us$90. n My Philosophical Development, Russell recorded the disappointment that both he and Whitehead felt about the reception of Principia Mathematica by the mathematical community. While the philosophical parts of the book, including those parts dealing with philosophical logic, were of course widely discussed and of tremendous importance in the subsequent development of philosophical logic, the purely mathematical aspects of the work went largely ignored. Russell was perhaps exaggerating when he claimed he used to know of “only six people who had read the later parts of the book” (MPD, p. 86) but it was certainly true that the impact of the book on mathematicians working in areas outside of the philosophical foundations of their subject was minimal . The situation has hardly changed since the publication of those remarks in 1959. Principia Mathematica remains the target of philosophical, not mathematical , attention. In this outstanding new book on Russell’s logicism, however , Sébastien Gandon offers a welcome exception to the rule. While the book is still very much a book on Russell’s philosophy, its central claim is that new light can be shed on that philosophy by examining the hitherto neglected mathematical parts of both Principia Mathematica and The Principles of Mathematics . What parts of Russell’s philosophy are to be better understood in light of these forays into the “terra incognita” (p. 2) of Russell’s treatment of advanced mathematics in Principia and the Principles? Gandon’s insightful suggestion is that the very notion of analysis at the heart of Russell’s logicist project can be grasped in a new and more complete way by reflection on Russell’s development of areas of mathematics such as his theories of geometry and quantity. The suggestion is a compelling one, not least because Gandon does a superb job of arguing the case through his own painstaking analysis of Russell ’s mathematics in the book. To understand why these seemingly remote parts of Russell’s philosophical writings carry such significance, Gandon argues that Russell’s logicist project must be understood both in terms of its contribution to philosophy and to f= Reviews 93 c:\users\kenneth\documents\type3401\rj 3401 193 red.docx 2014-05-14 8:54 PM mathematics. Logicism makes a bold assertion about mathematics—that mathematical truths (or, perhaps, some subset of them—e.g. those that exclude geometry in the case of Frege’s version of logicism) are nothing more than logical truths. As we know, the logicist project does not rest content with bold assertion, it seeks to prove the assertion. The proof will be a demonstration that every mathematical truth can be translated into a logical truth. Frege’s attempt at that demonstration famously failed because it overlooked Russell’s paradox. Russell and Whitehead’s attempted demonstration may have avoided that pitfall, but it had problems of its own (the axioms of reducibility and infinity) that left many unconvinced of its success. Gandon’s project here is not to provide a fresh argument for accepting Russell and Whitehead ’s demonstration, nor even for accepting the truth of logicism. Rather, it is to examine more closely what the claim made by the logicist is and to subsequently draw a subtle but important distinction between the projects of Frege and Russell, as well as the project which Gandon, more tentatively, ascribes to Wittgenstein. To illustrate the differences between the three approaches, Gandon invites us to reflect on what kind of analysis of mathematical reasoning is being proposed by the logicist, and suggests that the three positions mentioned above present themselves as three possible outcomes of that reflection. The logicist is faced with an analysandum—pre-logicized mathematics—to which the analysans must be related in a certain way. One option...