Reason's Nearest Kin : Philosophies of Arithmetic From Kant to Carnap

## Abstract This book is a critical examination of the astonishing progress made in the philosophical study of the properties of the natural numbers from the 1880s to the 1930s. It reassesses the brilliant innovations of Frege, Russell, Wittgenstein, and others, which transformed philosophy as well as the understanding of mathematics. The book argues that through the problem of arithmetic participates in the larger puzzle of the relationship between thought, language, experience, and the world, we can distinguish accounts that look to each of these to supply the content we require: those that involve the structure of our experience of the world; those that explicitly involve our grasp of a ‘third realm’ of abstract objects distinct from the concrete objects of the empirical world and the ideas of the author's private Gedankenwelt; those that appeal to something non-physical that is nevertheless an aspect of reality in harmony with which the physical aspect of the world is configured; and finally those that involve only our grasp of language. Title 4 Copyright 5 Preface 6 Contents 8 Introduction 12 0.1 Arithmetic 12 0.2 The a priori 15 0.3 Empiricism 17 0.4 Psychologism 20 0.5 Pure formalism 21 0.6 Trivial formalism 23 0.7 Reflexive formalism 26 0.8 Arithmetic and reason 28 1 Kant 31 1.1 Intuitions and concepts 32 1.2 Geometrical propositions 35 1.3 Arithmetical propositions 36 1.4 The Transcendental Deduction 38 1.5 Analytic and synthetic 41 1.6 The principle of analytic judgements 42 1.7 Geometry is not analytic 46 1.8 Arithmetic is not analytic 48 1.9 The principle of synthetic judgements 50 1.10 Geometry as synthetic 53 1.11 Arithmetic as synthetic 61 1.12 Arithmetic and sensibility 63 2 Grundlagen 66 2.1 Axiomatization 67 2.2 Arithmetic independent of sensibility 71 2.3 The Begriffsschrift 73 2.4 Frege’s conception of analyticity 76 2.5 Numerically definite quantifiers 80 2.6 The numerical equivalence 83 2.7 Frege’s explicit definition 86 2.8 The context principle again 89 2.9 The analyticity of the numerical equivalence 90 3 Dedekind 92 3.1 Dedekind’s recursion theorem 92 3.2 Frege and Dedekind 94 3.3 Axiomatic structuralism 96 3.4 Existence 98 3.5 Uniqueness 102 3.6 Implicationism 106 3.7 Systems 108 3.8 Dedekind on existence 110 3.9 Dedekind on uniqueness 113 4 Frege’s account of classes 116 4.1 The Julius Caesar problem yet again 116 4.2 The context principle in Grundgesetze 120 4.3 Russell’s paradox 123 4.4 Numbers as concepts 126 4.5 The status of the numerical equivalence 128 5 Russell’s account of classes 130 5.1 Propositions 130 5.2 The old theory of denoting 133 5.3 The new theory of denoting 136 5.4 The substitutional theory 139 5.5 Russell’s propositional paradox 142 5.6 Frege’s hierarchy of senses 145 5.7 Mathematical logic as based on the theory of types 147 5.8 Elementary propositions 150 5.9 The hierarchy of propositional functions in *12 151 5.10 The hierarchy of propositional functions in the Introduction 152 5.11 Typical ambiguity 155 5.12 Cumulative types 157 5.13 The hierarchy of classes 158 5.14 Numbers 161 5.15 The axiom of reducibility 163 5.16 Propositional functions and reducibility 165 5.17 The regressive method 168 5.18 The Introduction to Mathematical Philosophy 171 6 TheTractatus 175 6.1 Sign and symbol 175 6.2 The hierarchy of types 177 6.3 The doctrine of inexpressibility 179 6.4 Operations and functions 182 6.5 Sense 185 6.6 The rejection of class-theoretic foundations for mathematics 187 6.7 Number as the exponent of an operation 188 6.8 The adjectival strategy 190 6.9 Equations 192 6.10 Numerical identities 195 6.11 Generalization 196 6.12 The axiom of infinity 198 6.13 A transcendental argument 200 6.14 Another transcendental argument 203 7 The second edition of Principia 206 7.1 Logical atomism and empiricism 206 7.2 The hierarchy of propositional functions 208 7.3 Mathematical induction 210 7.4 The definition of identity 212 8 Ramsey 217 8.1 Propositions 217 8.2 Predicating functions 219 8.3 Extending Wittgenstein’s account of identity 224 8.4 Propositional functions in extension 227 8.5 Wittgenstein’s objections 229 8.6 The axiom of infinity 232 9 Hilbert’s programme 234 9.1 Formal consistency 234 9.2 Real arithmetic 239 9.3 Schematic arithmetic 243 9.4 Ideal arithmetic 248 9.5 Metamathematics 250 9.6 Hilbert’s programme 252 10 Godel 254 10.1 Incompleteness 255 10.2 Formal theories 257 10.3 The unprovability of outer consistency 259 10.4 The demise of Hilbert’s programme 261 10.5 The unprovability of consistency 265 10.6 Axiomatic formalism 269 11 Carnap 272 11.1 Language and symbolism 272 11.2 The rejection of the Tractatus 274 11.3 Conventionalism 276 11.4 Completeness 279 11.5 Consistency 280 11.6 Semantics 282 11.7 Pragmatics 285 Conclusion 289 Bibliography 301 Index 310 How do we account for the truth of arithmetic? And if it does not depend for its truth on the way the world is, what constrains the world to conform to arithmetic? Reason's Nearest Kin is a critical examination of the astonishing progress made towards answering these questions from the late nineteenth to the mid-twentieth century. In the space of fifty years Frege, Dedekind, Russell, Wittgenstein, Ramsey, Hilbert, and Carnap developed accounts of the content of arithmetic that were brilliantly original both technically and philosophically. Michael Potter's innovative study presents them all as finding that content in various aspects of the complex linkage between experience, language, thought, and the world. Potter's reading places them all in Kant's shadow since it was his attempt to ground arithmetic in the spatio-temporal structure of reality that they were reacting against; but it places us in Gödel's shadow since his incompleteness theorems supply us with a measure of the richness of the content they were trying to explain. This stimulating reassessment of some of the classic texts in the philosophy of mathematics reveals many unexpected connections and illuminating comparisons, and offers a wealth of ideas for future work in the subject. How Do We Account For The Truth Of Arithmetic? And If It Does Not Depend For Its Truth On The Way The World Is, What Constrains The World To Conform To Arithmetic? Reason's Nearest Kin Is A Critical Examination Of The Astonishing Progress Made Towards Answering These Questions From The Late Nineteenth To The Mid-twentieth Century. In The Space Of Fifty Years Frege, Dedekind, Russell, Wittgenstein, Ramsey, Hilbert, And Carnap Developed Accounts Of The Content Of Arithmetic That Were Brilliantly Original Both Technically And Philosophically. Michael Potter's Innovative Study Presents Them All As Finding That Content In Various Aspects Of The Complex Linkage Between Experience, Language, Thought, And The World.--jacket. Kant -- Grundlagen -- Dedekind -- Frege's Account Of Classes -- Russell's Account Of Classes -- The Tractatus -- The Second Edition Of Principia -- Ramsey -- Hilbert's Programme -- Gödel -- Carnap. Michael Potter. Includes Bibliographical References (p. [290]-298) And Index. Reason's Nearest Kin is a critical examination of the most exciting period there has been in the philosophical study of the properties of the natural numbers, from the 1880s to the 1930s. Reassessing the brilliant innovations of Frege, Russell, Wittgenstein, and others, which transformed philosophy as well as our understanding of mathematics, Michael Potter places arithmetic at the interface between experience, language, thought, and the world. - ;How do we account for the truth of arithmetic? And if it does not depend for its truth on the way the world is, what constrains the world to conform to arithmetic? Reason's Nearest Kin is a critical examination of the astonishing progress made towards answering these questions from the late nineteenth to the mid-twentieth century. In the space of fifty years Frege, Dedekind, Russell, Wittgenstein, Ramsey, Hilbert, and Carnap developed accounts of the content of arithmetic that were brilliantly original both technically and philosophically
This is a critical examination of the astonishing progress made in the philosophical study of the properties of the natural numbers from the 1880s to the 1930s. Reassessing the brilliant innovations of Frege, Russell, Wittgenstein, and others, which transformed philosophy as well as our understanding of mathematics, Michael Potter places arithmetic at the interface between experience, language, thought, and the world. How do we account for the truth of arithmetic? And if it does not depend for its truth on the way the world is, what constrains the world to conform to arithmetic? This text examines the progress made towards answering these questions