# Russell, Idealism, and the Emergence of Analytic Philosophy

By Peter Hylton | Go to book overview

7 The Logic of Principia Mathematica

The Principles of Mathematics, as we saw in Chapter 5 , contains many admittedly unsolved problems. By far the most serious of these arise from Russell's paradox. We can think of the paradox as posing two distinguishable problems for Russell. One is logical or mathematical: the paradox casts doubt on the coherence of the very notion of a class, yet this notion seems inescapable. Classes were, obviously, central to Russell's logicism. Worse, the notion of a class, or something like it, is presupposed in much nineteenth-century work which had appeared as purely mathematical, and so as free from philosophical doubts. As a result of the rigorization of analysis, reasoning that appealed to arbitrary classes of real numbers was seen as fundamental to mathematics. Russell's paradox threatened the notion of an arbitrary class. This aspect of the problem arises quite independently of Russell's philosophical views. The second aspect of the problem is logical or metaphysical. 1 In the view of Principles there are no ultimate distinctions of ontological category among entities; all are conceived of as object-like. Once the notion of a propositional function is introduced to allow for generality, and to explain the validity of certain patterns of inference, a paradox follows immediately, unless special restrictions are laid down. One has only to see that certain propositional functions cannot be truly applied to themselves and note that since we can say this, there is presumably a propositional function expressed by the words ' . . . is a propositional function which can (not) be truly applied to itself' (see pp. 227-9, above, and pp. 298-9, below). For Russell this problem would, in principle, have remained even if he had given up logicism and abandoned all interest in the philosophical status of mathematics. The contrast between Russell and Frege, which we have already noted,

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Russell, Idealism, and the Emergence of Analytic Philosophy

• Russell, Idealism, and the Emergence of Analytic Philosophy iii
• Preface vii
• Preface to the Paperback Edition x
• Contents xi
• Abbreviations xiii
• Introduction 1
• Part I the Idealist Background 19
• 1: T. H. Green 21
• 2: F. H. Bradley 44
• 3: Russell's Idealist Period 72
• Part II Platonic Atomism 103
• 4: The Underlying Metaphysics 117
• 5: Russell's Principles of Mathematics 167
• 6: 'On Denoting' 237
• Part III Logic, Fact, and Knowledge 277
• 7: The Logic of Principia Mathematica 285
• 8: Judgement, Belief, and Knowledge 328
• Bibliography 393
• Index 401
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