THE PRINCIPLES OF MATHEMATICS

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1 THE PRINCIPLES OF MATHEMATICS PREFACE by Bertrand Russell, M. A., Late Fellow of Trinity College, Cambridge Vol. I First published in 1903 by Cambridge University Press. This online edition (version 0.13: 24 Jan 2019) is based on that edition, with various typographical corrections. Missing here is the Introduction to the 1937 second edition, which is not yet in the public domain. Rather than publishing a second volume, Russell and his co-author A. N. Whitehead published the three volumes of Principia Mathematica in Original page numbers are marked in the margins. The page citations in the Table of Contents and Index refer to these numbers. The present work has two main objects. One of these, the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles, is undertaken in Parts II VII of this Volume, and will be established by strict symbolic reasoning in Volume II. The demonstration of this thesis has, if I am not mistaken, all the certainty and precision of which mathematical demonstrations are capable. As the thesis is very recent among mathematicians, and is almost universally denied by philosophers, I have undertaken, in this volume, to defend its various parts, as occasion arose, against such adverse theories as appeared most widely held or most difficult to disprove. I have also endeavoured to present, in language as untechnical as possible, the more important stages in the deductions by which the thesis is established. The other object of this work, which occupies Part I, is the explanation of the fundamental concepts which mathematics accepts as indefinable. This is a purely philosophical task, and I cannot flatter myself that I have done more than indicate a vast field of inquiry, and give a sample of the methods by which the inquiry may be conducted. The discussion of indefinables which forms the chief part of philosophical v

2 ii Bertrand Russell The Principles of Mathematics iii vi logic is the endeavour to see clearly, and to make others see clearly, the entities concerned, in order that the mind may have that kind of acquaintance with them which it has with redness or the taste of a pineapple. Where, as in the present case, the indefinables are obtained primarily as the necessary residue in a process of analysis, it is often easier to know that there must be such entities than actually to perceive them; there is a process analogous to that which resulted in the discovery of Neptune, with the difference that the final stage the search with a mental telescope for the entity which has been inferred is often the most difficult part of the undertaking. In the case of classes, I must confess, I have failed to perceive any concept fulfilling the conditions requisite for the notion of class. And the contradiction discussed in Chapter x proves that something is amiss, but what this is I have hitherto failed to discover. The second volume, in which I have had the great good fortune to secure the collaboration of Mr A. N. Whitehead, will be addressed exclusively to mathematicians; it will contain chains of deductions, from the premisses of symbolic logic through Arithmetic, finite and infinite, to Geometry, in an order similar to that adopted in the present volume; it will also contain various original developments, in which the method of Professor Peano, as supplemented by the Logic of Relations, has shown itself a powerful instrument of mathematical investigation. The present volume, which may be regarded either as a commentary upon, or as an introduction to, the second volume, is addressed in equal measure to the philosopher and to the mathematician; but some parts will be more interesting to the one, others to the other. I should advise mathematicians, unless they are specially interested in Symbolic Logic, to begin with Part IV, and only refer to earlier parts as occasion arises. The following portions are more specially philosophical: Part I (omitting Chapter ii); Part II, Chapters xi, xv, xvi, xvii; Part III; Part IV, 207, Chapters xxvi, xxvii, xxxi; Part V, Chapters xli, xlii, xliii; Part VI, Chapters l, li, lii; Part VII, Chapters liii, liv, lv, lvii, lviii; and the two Appendices, which belong to Part I, and should be read in connection with it. Professor Frege s work, which largely anticipates my own, was for the most part unknown to me when the printing of the present work began; I had seen his Grundgesetze der Arithmetik, but, owing to the great difficulty of his symbolism, I had failed to grasp its importance or to understand its contents. The only method, at so late a stage, of doing justice to his work, was to devote an Appendix to it; and in some points the views contained in the Appendix differ from those in Chapter vi, especially in 71, 73, 74. On questions discussed in these sections, I discovered errors after passing the sheets for the press; these errors, of which the chief are the denial of the null-class, and the identification of a term with the class whose only member it is, are rectified in the Appendices. The subjects treated are so difficult that I feel little confidence in my present opinions, and regard any conclusions which may be advocated as essentially hypotheses. A few words as to the origin of the present work may serve to show the importance of the questions discussed. About six years ago, I began an investigation into the philosophy of Dynamics. I was met by the difficulty that, when a particle is subject to several forces, no one of the component accelerations actually occurs, but only the resultant acceleration, of which they are not parts; this fact rendered illusory such causation of particulars by particulars as is affirmed, at first sight, by the law of gravitation. It appeared also that the difficulty in regard to absolute motion is insoluble on a relational theory of space. From these two questions I was led to a re-examination vii

3 iv Bertrand Russell The Principles of Mathematics v of the principles of Geometry, thence to the philosophy of continuity and infinity, and thence, with a view to discovering the meaning of the word any, to Symbolic Logic. The final outcome, as regards the philosophy of Dynamics, is perhaps rather slender; the reason of this is, that almost all the problems of Dynamics appear to me empirical, and therefore outside the scope of such a work as the present. Many very interesting questions have had to be omitted, especially in Parts VI and VII, as not relevant to my purpose, which, for fear of misunderstandings, it may be well to explain at this stage. When actual objects are counted, or when Geometry and Dynamics are applied to actual space or actual matter, or when, in any other way, mathematical reasoning is applied to what exists, the reasoning employed has a form not dependent upon the objects to which it is applied being just those objects that they are, but only upon their having certain general properties. In pure mathematics, actual objects in the world of existence will never be in question, but only hypothetical objects having those general properties upon which depends whatever deduction is being considered; and these general properties will always be expressible in terms of the fundamental concepts which I have called logical constants. Thus when space or motion is spoken of in pure mathematics, it is not actual space or actual motion, as we know them in experience, that are spoken of, but any entity possessing those abstract general properties of space or motion that are employed in the reasonings of geometry or dynamics. The question whether these properties belong, as a matter of fact, to actual space or actual motion, is irrelevant to pure mathematics, and therefore to the present work, being, in my opinion, a purely empirical question, to be investigated in the laboratory or the observatory. Indirectly, it is true, the discussions connected with pure mathematics have a very important bearing upon such empirical questions, since mathematical space and motion are held by many, perhaps most, philosophers to be self-contradictory, and therefore necessarily different from actual space and motion, whereas, if the views advocated in the following pages be valid, no such self-contradictions are to be found in mathematical space and motion. But extra-mathematical considerations of this kind have been almost wholly excluded from the present work. On fundamental questions of philosophy, my position, in all its chief features, is derived from Mr G. E. Moore. I have accepted from him the non-existential nature of propositions (except such as happen to assert existence) and their independence of any knowing mind; also the pluralism which regards the world, both that of existents and that of entities, as composed of an infinite number of mutually independent entities, with relations which are ultimate, and not reducible to adjectives of their terms or of the whole which these compose. Before learning these views from him, I found myself completely unable to construct any philosophy of arithmetic, whereas their acceptance brought about an immediate liberation from a large number of difficulties which I believe to be otherwise insuperable. The doctrines just mentioned are, in my opinion, quite indispensable to any even tolerably satisfactory philosophy of mathematics, as I hope the following pages will show. But I must leave it to my readers to judge how far the reasoning assumes these doctrines, and how far it supports them. Formally, my premisses are simply assumed; but the fact that they allow mathematics to be true, which most current philosophies do not, is surely a powerful argument in their favour. In Mathematics, my chief obligations, as is indeed evident, are to Georg Cantor and Professor Peano. If I had become acquainted sooner with the work of Professor Frege, I should viii

4 vi Bertrand Russell The Principles of Mathematics vii ix have owed a great deal to him, but as it is I arrived independently at many results which he had already established. At every stage of my work, I have been assisted more than I can express by the suggestions, the criticisms, and the generous encouragement of Mr A. N. Whitehead; he also has kindly read my proofs, and greatly improved the final expression of a very large number of passages. Many useful hints I owe also to Mr W. E. Johnson; and in the more philosophical parts of the book I owe much to Mr G. E. Moore besides the general position which underlies the whole. In the endeavour to cover so wide a field, it has been impossible to acquire an exhaustive knowledge of the literature. There are doubtless many important works with which I am unacquainted; but where the labour of thinking and writing necessarily absorbs so much time, such ignorance, however regrettable, seems not wholly avoidable. Many words will be found, in the course of discussion, to be defined in senses apparently departing widely from common usage. Such departures, I must ask the reader to believe, are never wanton, but have been made with great reluctance. In philosophical matters, they have been necessitated mainly by two causes. First, it often happens that two cognate notions are both to be considered, and that language has two names for the one, but none for the other. It is then highly convenient to distinguish between the two names commonly used as synonyms, keeping one for the usual, the other for the hitherto nameless sense. The other cause arises from philosophical disagreement with received views. Where two qualities are commonly supposed inseparably conjoined, but are here regarded as separable, the name which has applied to their combination will usually have to be restricted to one or other. For example, propositions are commonly regarded as (1) true or false, (2) mental. Holding, as I do, that what is true or false is not in general mental, I require a name for the true or false as such, and this name can scarcely be other than proposition. In such a case, the departure from usage is in no degree arbitrary. As regards mathematical terms, the necessity for establishing the existence-theorem in each case i.e. the proof that there are entities of the kind in question has led to many definitions which appear widely different from the notions usually attached to the terms in question. Instances of this are the definitions of cardinal, ordinal and complex numbers. In the two former of these, and in many other cases, the definition as a class, derived from the principle of abstraction, is mainly recommended by the fact that it leaves no doubt as to the existence-theorem. But in many instances of such apparent departure from usage, it may be doubted whether more has been done than to give precision to a notion which had hitherto been more or less vague. For publishing a work containing so many unsolved difficulties, my apology is, that investigation revealed no near prospect of adequately resolving the contradiction discussed in Chapter x, or of acquiring a better insight into the nature of classes. The repeated discovery of errors in solutions which for a time had satisfied me caused these problems to appear such as would have been only concealed by any seemingly satisfactory theories which a slightly longer reflection might have produced; it seemed better, therefore, merely to state the difficulties, than to wait until I had become persuaded of the truth of some almost certainly erroneous doctrine. My thanks are due to the Syndics of the University Press, and to their Secretary, Mr R. T. Wright, for their kindness and courtesy in regard to the present volume. London December, 1902.

5 viii Bertrand Russell The Principles of Mathematics ix TABLE OF CONTENTS page Preface v PART I THE INDEFINABLES OF MATHEMATICS CHAPTER I DEFINITION OF PURE MATHEMATICS 1. Definition of pure mathematics The principles of mathematics are no longer controversial Pure mathematics uses only a few notions, and these are logical constants All pure mathematics follows formally from twenty premisses Asserts formal implications And employs variables Which may have any value without exception Mathematics deals with types of relations Applied mathematics is defined by the occurrence of constants which are not logical Relation of mathematics to logic CHAPTER II SYMBOLIC LOGIC 11. Definition and scope of symbolic logic The indefinables of symbolic logic Symbolic logic consists of three parts A. The Propositional Calculus. 14. Definition Distinction between implication and formal implication Implication indefinable Two indefinables and ten primitive propositions in this calculus The ten primitive propositions Disjunction and negation defined B. The Calculus of Classes. 20. Three new indefinables The relation of an individual to its class Propositional functions The notion of such that Two new primitive propositions Relation to propositional calculus Identity C. The Calculus of Relations. 27. The logic of relations essential to mathematics New primitive propositions Relative products Relations with assigned domains D. Peano s Symbolic Logic. 31. Mathematical and philosophical definitions Peano s indefinables Elementary definitions Peano s primitive propositions

6 x Bertrand Russell The Principles of Mathematics xi 35. Negation and disjunction Existence and the null-class CHAPTER III IMPLICATION AND FORMAL IMPLICATION 37. Meaning of implication Asserted and unasserted propositions Inference does not require two premisses Formal implication is to be interpreted extensionally The variable in a formal implication has an unrestricted field A formal implication is a single propositional function, not a relation of two Assertions Conditions that a term in an implication may be varied Formal implication involved in rules of inference CHAPTER IV PROPER NAMES, ADJECTIVES AND VERBS 46. Proper names, adjectives and verbs distinguished Terms Things and concepts Concepts as such and as terms Conceptual diversity Meaning and the subject-predicate logic Verbs and truth All verbs, except perhaps is, express relations Relations per se and relating relations Relations are not particularized by their terms CHAPTER V DENOTING 56. Definition of denoting Connection with subject-predicate propositions Denoting concepts obtained from predicates Extensional account of all, every, any, a and some Intensional account of the same Illustrations The difference between all, every, etc. lies in the objects denoted, not in the way of denoting them The notion of the and definition The notion of the and identity Summary CHAPTER VI CLASSES 66. Combination of intensional and extensional standpoints required Meaning of class Intensional and extensional genesis of classes Distinctions overlooked by Peano The class as one and as many The notion of and All men is not analyzable into all and men There are null class-concepts, but there is no null class The class as one, except when it has one term, is distinct from the class as many Every, any, a and some each denote one object, but an ambiguous one The relation of a term to its class The relation of inclusion between classes

7 xii Bertrand Russell The Principles of Mathematics xiii 78. The contradiction Summary CHAPTER VII PROPOSITIONAL FUNCTIONS 80. Indefinability of such that Where a fixed relation to a fixed term is asserted, a propositional function can be analyzed into a variable subject and a constant assertion But this analysis is impossible in other cases Variation of the concept in a proposition Relation of propositional functions to classes A propositional function is in general not analyzable into a constant and a variable element CHAPTER VIII THE VARIABLE 86. Nature of the variable Relation of the variable to any Formal and restricted variables Formal implication presupposes any Duality of any and some The class-concept propositional function is indefinable Other classes can be defined by means of such that Analysis of the variable CHAPTER IX RELATIONS 94. Characteristics of relations Relations of terms to themselves The domain and the converse domain of a relation Logical sum, logical product and relative product of relations A relation is not a class of couples Relations of a relation to its terms CHAPTER X THE CONTRADICTION 100. Consequences of the contradiction Various statements of the contradiction An analogous generalized argument Variable propositional functions are in general inadmissible The contradiction arises from treating as one a class which is only many Other primâ facie possible solutions appear inadequate Summary of Part I PART II NUMBER CHAPTER XI DEFINITION OF CARDINAL NUMBERS 107. Plan of Part II Mathematical meaning of definition Definition of numbers by abstraction Objections to this definition Nominal definition of numbers

8 xiv Bertrand Russell The Principles of Mathematics xv CHAPTER XII ADDITION AND MULTIPLICATION 112. Only integers to be considered at present Definition of arithmetical addition Dependence upon the logical addition of classes Definition of multiplication Connection of addition, multiplication and exponentiation CHAPTER XIII FINITE AND INFINITE 117. Definition of finite and infinite Definition of α Definition of finite numbers by mathematical induction CHAPTER XIV THEORY OF FINITE NUMBERS 120. Peano s indefinables and primitive propositions Mutual independence of the latter Peano really defines progressions, not finite numbers Proof of Peano s primitive propositions CHAPTER XV ADDITION OF TERMS AND ADDITION OF CLASSES 124. Philosophy and mathematics distinguished Is there a more fundamental sense of number than that defined above? Numbers must be classes Numbers apply to classes as many One is to be asserted, not of terms, but of unit classes Counting not fundamental in arithmetic Numerical conjunction and plurality Addition of terms generates classes primarily, not numbers A term is indefinable, but not the number CHAPTER XVI WHOLE AND PART 133. Single terms may be either simple or complex Whole and part cannot be defined by logical priority Three kinds of relation of whole and part distinguished Two kinds of wholes distinguished A whole is distinct from the numerical conjunction of its parts How far analysis is falsification A class as one is an aggregate CHAPTER XVII INFINITE WHOLES 140. Infinite aggregates must be admitted Infinite unities, if there are any, are unknown to us Are all infinite wholes aggregates of terms? Grounds in favour of this view CHAPTER XVIII RATIOS AND FRACTIONS 144. Definition of ratio Ratios are one-one relations Fractions are concerned with relations of whole and part

9 xvi Bertrand Russell The Principles of Mathematics xvii 147. Fractions depend, not upon number, but upon magnitude of divisibility Summary of Part II PART III QUANTITY CHAPTER XIX THE MEANING OF MAGNITUDE 149. Previous views on the relation of number and quantity Quantity not fundamental in mathematics Meaning of magnitude and quantity Three possible theories of equality to be examined Equality is not identity of number of parts Equality is not an unanalyzable relation of quantities Equality is sameness of magnitude Every particular magnitude is simple The principle of abstraction Summary Note CHAPTER XX THE RANGE OF QUANTITY 159. Divisibility does not belong to all quantities Distance Differential coefficients A magnitude is never divisible, but may be a magnitude of divisibility Every magnitude is unanalyzable CHAPTER XXI NUMBERS AS EXPRESSING MAGNITUDES: MEASUREMENT 164. Definition of measurement Possible grounds for holding all magnitudes to be measurable Intrinsic measurability Of divisibilities And of distances Measure of distance and measure of stretch Distance-theories and stretch-theories of geometry Extensive and intensive magnitudes CHAPTER XXII ZERO 172. Difficulties as to zero Meinong s theory Zero as minimum Zero distance as identity Zero as a null segment Zero and negation Every kind of zero magnitude is in a sense indefinable CHAPTER XXIII INFINITY, THE INFINITESIMAL, AND CONTINUITY 179. Problems of infinity not specially quantitative Statement of the problem in regard to quantity Three antinomies Of which the antitheses depend upon an axiom of finitude

10 xviii Bertrand Russell The Principles of Mathematics xix 183. And the use of mathematical induction Which are both to be rejected Provisional sense of continuity Summary of Part III PART IV ORDER CHAPTER XXIV THE GENESIS OF SERIES 187. Importance of order Between and separation of couples Generation of order by one-one relations By transitive asymmetrical relations By distances By triangular relations By relations between asymmetrical relations And by separation of couples CHAPTER XXV THE MEANING OF ORDER 195. What is order? Three theories of between First theory A relation is not between its terms Second theory of between There appear to be ultimate triangular relations Reasons for rejecting the second theory Third theory of between to be rejected Meaning of separation of couples Reduction to transitive asymmetrical relations This reduction is formal But is the reason why separation leads to order The second way of generating series is alone fundamental, and gives the meaning of order CHAPTER XXVI ASYMMETRICAL RELATIONS 208. Classification of relations as regards symmetry and transitiveness Symmetrical transitive relations Reflexiveness and the principle of abstraction Relative position Are relations reducible to predications? Monadistic theory of relations Reasons for rejecting this theory Monistic theory and the reasons for rejecting it Order requires that relations should be ultimate CHAPTER XXVII DIFFERENCE OF SENSE AND DIFFERENCE OF SIGN 217. Kant on difference of sense Meaning of difference of sense Difference of sign In the cases of finite numbers And of magnitudes Right and left Difference of sign arises from difference of sense among transitive asymmetrical relations

11 xx Bertrand Russell The Principles of Mathematics xxi CHAPTER XXVIII ON THE DIFFERENCE BETWEEN OPEN AND CLOSED SERIES 224. What is the difference between open and closed series? Finite closed series Series generated by triangular relations Four-term relations Closed series are such as have an arbitrary first term 238 CHAPTER XXIX PROGRESSIONS AND ORDINAL NUMBERS 229. Definition of progressions All finite arithmetic applies to every progression Definition of ordinal numbers Definition of nth Positive and negative ordinals CHAPTER XXX DEDEKIND S THEORY OF NUMBER 234. Dedekind s principal ideas Representation of a system The notion of a chain The chain of an element Generalized form of mathematical induction Definition of a singly infinite system Definition of cardinals Dedekind s proof of mathematical induction Objections to his definition of ordinals And of cardinals CHAPTER XXXI DISTANCE 244. Distance not essential to order Definition of distance Measurement of distances In most series, the existence of distances is doubtful Summary of Part IV PART V INFINITY AND CONTINUITY CHAPTER XXXII THE CORRELATION OF SERIES 249. The infinitesimal and space are no longer required in a statement of principles The supposed contradictions of infinity have been resolved Correlation of series Independent series and series by correlation Likeness of relations Functions Functions of a variable whose values form a series Functions which are defined by formulae Complete series CHAPTER XXXIII REAL NUMBERS 258. Real numbers are not limits of series of rationals Segments of rationals Properties of segments Coherent classes in a series Note

12 xxii Bertrand Russell The Principles of Mathematics xxiii CHAPTER XXXIV LIMITS AND IRRATIONAL NUMBERS 262. Definition of a limit Elementary properties of limits An arithmetical theory of irrationals is indispensable Dedekind s theory of irrationals Defects in Dedekind s axiom of continuity Objections to his theory of irrationals Weierstrass s theory Cantor s theory Real numbers are segments of rationals CHAPTER XXXV CANTOR S FIRST DEFINITION OF CONTINUITY 271. The arithmetical theory of continuity is due to Cantor Cohesion Perfection Defect in Cantor s definition of perfection The existence of limits must not be assumed without special grounds CHAPTER XXXVI ORDINAL CONTINUITY 276. Continuity is a purely ordinal notion Cantor s ordinal definition of continuity Only ordinal notions occur in this definition Infinite classes of integers can be arranged in a continuous series Segments of general compact series Segments defined by fundamental series Two compact series may be combined to form a series which is not compact CHAPTER XXXVII TRANSFINITE CARDINALS 283. Transfinite cardinals differ widely from transfinite ordinals Definition of cardinals Properties of cardinals Addition, multiplication and exponentiation The smallest transfinite cardinal α Other transfinite cardinals Finite and transfinite cardinals form a single series by relation to greater and less CHAPTER XXXVIII TRANSFINITE ORDINALS 290. Ordinals are classes of serial relations Cantor s definition of the second class of ordinals Definition of ω An infinite class can be arranged in many types of series Addition and subtraction of ordinals Multiplication and division Well-ordered series Series which are not well-ordered Ordinal numbers are types of well-ordered series Relation-arithmetic Proofs of existence-theorems There is no maximum ordinal number Successive derivatives of a series

13 xxiv Bertrand Russell The Principles of Mathematics xxv CHAPTER XXXIX THE INFINITESIMAL CALCULUS 303. The infinitesimal has been usually supposed essential to the calculus Definition of a continuous function Definition of the derivative of a function The infinitesimal is not implied in this definition Definition of the definite integral Neither the infinite nor the infinitesimal is involved in this definition CHAPTER XL THE INFINITESIMAL AND THE IMPROPER INFINITE 309. A precise definition of the infinitesimal is seldom given Definition of the infinitesimal and the improper infinite Instances of the infinitesimal No infinitesimal segments in compact series Orders of infinity and infinitesimality Summary CHAPTER XLI PHILOSOPHICAL ARGUMENTS CONCERNING THE INFINITESIMAL 315. Current philosophical opinions illustrated by Cohen Who bases the calculus upon infinitesimals Space and motion are here irrelevant Cohen regards the doctrine of limits as insufficient for the calculus And supposes limits to be essentially quantitative To involve infinitesimal differences And to introduce a new meaning of equality He identifies the inextensive with the intensive Consecutive numbers are supposed to be required for continuous change Cohen s views are to be rejected CHAPTER XLII THE PHILOSOPHY OF THE CONTINUUM 325. Philosophical sense of continuity not here in question The continuum is composed of mutually external units Zeno and Weierstrass The argument of dichotomy The objectionable and the innocent kind of endless regress Extensional and intensional definition of a whole Achilles and the tortoise The arrow Change does not involve a state of change The argument of the measure Summary of Cantor s doctrine of continuity The continuum consists of elements CHAPTER XLIII THE PHILOSOPHY OF THE INFINITE 337. Historical retrospect Positive doctrine of the infinite Proof that there are infinite classes The paradox of Tristram Shandy

14 xxvi Bertrand Russell The Principles of Mathematics xxvii 341. A whole and a part may be similar Whole and part and formal implication No immediate predecessor of ω or α Difficulty as regards the number of all terms, objects, or propositions Cantor s first proof that there is no greatest number His second proof Every class has more sub-classes than terms But this is impossible in certain cases Resulting contradictions Summary of Part V PART VI SPACE CHAPTER XLIV DIMENSIONS AND COMPLEX NUMBERS 351. Retrospect Geometry is the science of series of two or more dimensions Non-Euclidean geometry Definition of dimensions Remarks on the definition The definition of dimensions is purely logical Complex numbers and universal algebra Algebraical generalization of number Definition of complex numbers Remarks on the definition CHAPTER XLV PROJECTIVE GEOMETRY 361. Recent threefold scrutiny of geometrical principles Projective, descriptive and metrical geometry Projective points and straight lines Definition of the plane Harmonic ranges Involutions Projective generation of order Möbius nets Projective order presupposed in assigning irrational coordinates Anharmonic ratio Assignment of coordinates to any point in space Comparison of projective and Euclidean geometry The principle of duality CHAPTER XLVI DESCRIPTIVE GEOMETRY 374. Distinction between projective and descriptive geometry Method of Pasch and Peano Method employing serial relations Mutual independence of axioms Logical definition of the class of descriptive spaces Parts of straight lines Definition of the plane Solid geometry Descriptive geometry applies to Euclidean and hyperbolic, but not elliptic space Ideal elements Ideal points

15 xxviii Bertrand Russell The Principles of Mathematics xxix 385. Ideal lines Ideal planes The removal of a suitable selection of points renders a projective space descriptive CHAPTER XLVII METRICAL GEOMETRY 388. Metrical geometry presupposes projective or descriptive geometry Errors in Euclid Superposition is not a valid method Errors in Euclid (continued) Axioms of distance Stretches Order as resulting from distance alone Geometries which derive the straight line from distance In most spaces, magnitude of divisibility can be used instead of distance Meaning of magnitude of divisibility Difficulty of making distance independent of stretch Theoretical meaning of measurement Definition of angle Axioms concerning angles An angle is a stretch of rays, not a class of points Areas and volumes Right and left CHAPTER XLVIII RELATION OF METRICAL TO PROJECTIVE AND DESCRIPTIVE GEOMETRY 405. Non-quantitative geometry has no metrical presuppositions Historical development of non-quantitative geometry Non-quantitative theory of distance In descriptive geometry And in projective geometry Geometrical theory of imaginary point-pairs New projective theory of distance CHAPTER XLIX DEFINITIONS OF VARIOUS SPACES 412. All kinds of spaces are definable in purely logical terms Definition of projective spaces of three dimensions Definition of Euclidean spaces of three dimensions Definition of Clifford s spaces of two dimensions CHAPTER L THE CONTINUITY OF SPACE 416. The continuity of a projective space The continuity of a metrical space An axiom of continuity enables us to dispense with the postulate of the circle Is space prior to points? Empirical premisses and induction There is no reason to desire our premisses to be self-evident Space is an aggregate of points, not a unity CHAPTER LI LOGICAL ARGUMENTS AGAINST POINTS 423. Absolute and relative position

16 xxx Bertrand Russell The Principles of Mathematics xxxi 424. Lotze s arguments against absolute position Lotze s theory of relations The subject-predicate theory of propositions Lotze s three kinds of Being Argument from the identity of indiscernibles Points are not active Argument from the necessary truths of geometry Points do not imply one another CHAPTER LII KANT S THEORY OF SPACE 432. The present work is diametrically opposed to Kant Summary of Kant s theory Mathematical reasoning requires no extra-logical element Kant s mathematical antinomies Summary of Part VI PART VII MATTER AND MOTION CHAPTER LIII MATTER 437. Dynamics is here considered as a branch of pure mathematics Matter is not implied by space Matter as substance Relations of matter to space and time Definition of matter in terms of logical constants CHAPTER LIV MOTION 442. Definition of change There is no such thing as a state of change Change involves existence Occupation of a place at a time Definition of motion There is no state of motion CHAPTER LV CAUSALITY 448. The descriptive theory of dynamics Causation of particulars by particulars Cause and effect are not temporally contiguous Is there any causation of particulars by particulars? Generalized form of causality CHAPTER LVI DEFINITION OF A DYNAMICAL WORLD 453. Kinematical motions Kinetic motions CHAPTER LVII NEWTON S LAWS OF MOTION 455. Force and acceleration are fictions The law of inertia The second law of motion The third law Summary of Newtonian principles Causality in dynamics Accelerations as caused by particulars No part of the laws of motion is an à priori truth

17 xxxii Bertrand Russell The Principles of Mathematics xxxiii CHAPTER LVIII ABSOLUTE AND RELATIVE MOTION 463. Newton and his critics Grounds for absolute motion Neumann s theory Streintz s theory Mr Macaulay s theory Absolute rotation is still a change of relation Mach s reply to Newton CHAPTER LIX HERTZ S DYNAMICS 470. Summary of Hertz s system Hertz s innovations are not fundamental from the point of view of pure mathematics Principles common to Hertz and Newton Principle of the equality of cause and effect Summary of the work APPENDIX A THE LOGICAL AND ARITHMETICAL DOCTRINES OF FREGE 475. Principal points in Frege s doctrines Meaning and indication Truth-values and judgment Criticism Are assumptions proper names for the true or the false? Functions Begriff and Gegenstand Recapitulation of theory of propositional functions Can concepts be made logical subjects? Ranges Definition of ϵ and of relation Reasons for an extensional view of classes A class which has only one member is distinct from its only member Possible theories to account for this fact Recapitulation of theories already discussed The subject of a proposition may be plural Classes having only one member Theory of types Implication and symbolic logic Definition of cardinal numbers Frege s theory of series Kerry s criticisms of Frege APPENDIX B THE DOCTRINE OF TYPES 497. Statement of the doctrine Numbers and propositions as types Are propositional concepts individuals? Contradiction arising from the question whether there are more classes of propositions than propositions Index