M13 9PL Research Articles

russell: the Journal of Bertrand Russell Studies n.s. 35 (summer 2015): 71–94 The Bertrand Russell Research Centre, McMaster U. issn 0036–01631; online 1913–8032 c:\users\arlene\documents\rj issues\type3501\rj 3501 061 red.docx 2015-07-10 4:07 PM oeviews A CENTENARY COMPANION TO PRINCIPIA MATHEMATICA Graham Stevens Philosophy / U. of Manchester Manchester m13 9pl, uk graham.p.stevens@manchester.ac.uk Nicholas Griffin and Bernard Linsky, eds. The Palgrave Centenary Companion to Principia Mathematica. Basingstoke and New York: Palgrave Macmillan, 2013. Pp. xxvii, 458. isbn: 978-1-137-34462-5. £65; us$120. famous anecdote of Russell’s neatly summarizes the impact of Principia Mathematica in Russell’s lifetime: I used to know of only six people who had read the later parts of the book. Three of these were Poles, subsequently (I believe) liquidated by Hitler. The other three were Texans, subsequently successfully assimilated. (MPD, p. 86) That the later parts (dealing with purely mathematical concerns) were somewhat neglected, was a disappointment to Russell and Whitehead. However, the same could not be said for the early parts of the work (dealing with philosophical and mathematical logic).The inﬂuence of these parts has been profound . Logicism is a simple enough thesis: mathematics (to a greater or lesser degree , depending on which version of logicism—Frege’s or Russell’s—we are talking about) is part of logic. Every mathematical truth is really just a logical truth.This deceptively simple philosophical claim, however, can only be taken seriously if conﬁrmed by supporting evidence. Logicism is a philosophical thesis that requires logical demonstration. Both Frege and Russell began by stating the philosophical case for logicism, and then attempting to demonstrate the thesis formally in subsequent work. Frege’s attempt famously ﬂoundered after Russell discovered its inconsistency. After the best part of a decade spent trying to remove that inconsistency, Russell and Whitehead produced Principia Mathematica, which was their attempt to demonstrate logicism on, in the words of the editors of this volume, “a truly epic scale” (p. xvi). ^= 72 Reviews c:\users\arlene\documents\rj issues\type3501\rj 3501 061 red.docx 2015-07-10 4:07 PM Much appeared to have changed during those years in which PM was constructed . Russell’s original philosophical statement of logicism in The Principles of Mathematics was a paradigm of elegance, at least with regard to its reduction of number theory to the calculus of classes. But in PM, things look very different.The logic of PM is stratiﬁed into a theory of types, something only tentatively considered in an appendix to the Principles; the classes to which numbers were reduced in the Principles are absent from the “no-classes” theory of PM; the much celebrated theory of quantiﬁcation ﬁrst presented in “On Denoting” is incorporated into PM. Additionally, Russell andWhitehead claim at some points to have rejected any commitment to propositions as entities , replacing them with a new theory of judgment to explain “propositional ” content. After protracted attempts to digest these many changes and innovations to Russell’s logic, most commentators remained unconvinced that the demonstration of logicism was successful. It is a powerful testament to the lasting importance and value of PM that, in the century following its publication, the fact that the orthodox view of the work was that it had failed in its intended purpose posed no obstacle to its dramatic impact on the development of both mathematical logic and analytical philosophy. PM’s enormous inﬂuence in mathematical logic is largely due to subsidiary achievements made in the service of the logicist enterprise, rather than a reﬂection of that enterprise. PM gave the ﬁrst accessible axiomatization of propositional and predicate logic (Frege had previously given an arguably more rigorous axiomatization, but his notation made the work far less accessible) which became the point of reference for subsequent work in the development of metamathematics, culminating in Gödel’s famous incompleteness theorems which proved the incompleteness of any consistent formalization of arithmetic based on PM’s formal system. Furthermore, for generations of logicians, PM was the only book available in which to study mathematical logic. From our current perspective...

Read full 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...

Read full abstract

M13 9PL Research Articles

Articles published on M13 9PL

Proteolysis Targeting Chimeras (PROTACs): An Innovative Strategy for Targeted Protein Degradation and Disease Treatment

A Review of Multi-vehicle Cooperative Control System in Intelligent Transportation

Correspondence.

A Minimally Invasive Approach for Cardiac Electrophysiology Studies in Mice

Quantitative Analysis Between Blackboard Learning Management System and Students’ Learning

The COVID-19 pandemic: lifestyle and cardiovascular risk factors.

Contribution to ‐ Distant Surgical teaching during COVID‐19 ‐ A pilot study on final year medical students

A Centenary Companion to Principia Mathematica

Editorial Introduction

Meeting report on YBPW 2014 (the 3rd Young Bioinformatics PIs Workshop)

Changing the Ties That Bind? The Emerging Roles and Identities of General Practitioners and Managers in the New Clinical Commissioning Groups in the English NHS

Social and Academic Abilities in Children with High‐Functioning Autism Spectrum Disorders by NiritBauminger‐ZvielyThe Guilford Press, 2013, 322pp ISBN 9781462509423

DFT and Falicov-Kimball Model Approach to Cr_{9} Molecular Ring

Uncovering the Mathematics Behind Russell’s Philosophy of Mathematics

When two worlds collide: A twenty‐first century approach to mental health and policing

Structure of Clean and Adsorbate-Covered Single-Crystal Rutile TiO2Surfaces

Lanthanide Single-Molecule Magnets

Dr. Murray, et al reply

P3‐275: Identifying dementia in deaf sign language users

MORAL PROGRESS AND MORAL ENHANCEMENT

Lead the way for us

Editage

Paperpal

R Discovery

Mind the Graph

M13 9PL Research Articles

Articles published on M13 9PL

Proteolysis Targeting Chimeras (PROTACs): An Innovative Strategy for Targeted Protein Degradation and Disease Treatment

A Review of Multi-vehicle Cooperative Control System in Intelligent Transportation

Correspondence.

A Minimally Invasive Approach for Cardiac Electrophysiology Studies in Mice

Quantitative Analysis Between Blackboard Learning Management System and Students’ Learning

The COVID-19 pandemic: lifestyle and cardiovascular risk factors.

Contribution to ‐ Distant Surgical teaching during COVID‐19 ‐ A pilot study on final year medical students

A Centenary Companion to Principia Mathematica

Editorial Introduction

Meeting report on YBPW 2014 (the 3rd Young Bioinformatics PIs Workshop)

Changing the Ties That Bind? The Emerging Roles and Identities of General Practitioners and Managers in the New Clinical Commissioning Groups in the English NHS

Social and Academic Abilities in Children with High‐Functioning Autism Spectrum Disorders by NiritBauminger‐ZvielyThe Guilford Press, 2013, 322pp ISBN 9781462509423

DFT and Falicov-Kimball Model Approach to Cr_{9} Molecular Ring

Uncovering the Mathematics Behind Russell’s Philosophy of Mathematics

When two worlds collide: A twenty‐first century approach to mental health and policing

Structure of Clean and Adsorbate-Covered Single-Crystal Rutile TiO2Surfaces

Lanthanide Single-Molecule Magnets

Dr. Murray, et al reply

P3‐275: Identifying dementia in deaf sign language users

MORAL PROGRESS AND MORAL ENHANCEMENT