Book:Bertrand Russell/The Principles of Mathematics/Second Edition

Bertrand Russell: The Principles of Mathematics (2nd Edition)

Published $\text {1938}$

Subject Matter

• Mathematical Logic

Contents

Preface

Part $\text {I}$. The Indefinables of Mathematics

Chapter $\text {I}$. Definition of Pure Mathematics

$\S 1$. Definition of pure mathematics

$\S 2$. The principles of mathematics are no longer controversial

$\S 3$. Pure mathematics uses only a few notions, and these are logical constants

$\S 4$. All pure mathematics follows formally from twenty premisses

$\S 5$. Asserts formal implications

$\S 6$. And employs variables

$\S 7$. Which may have any value without exception

$\S 8$. Mathematics deals with types of relations

$\S 9$. Applied mathematics is defined by the occurrence of constants which are not logical.

$\S 10$. Relation of mathematics to logic.

Chapter $\text {II}$. Symbolic Logic

$\S 11$. Definition and scope of symbolic logic

$\S 12$. The indefinables of symbolic logic

$\S 13$. Symbolic logic consists of three parts

The Propositional Calculus

$\S 14$. Definition

$\S 15$. Distinction between implication and formal implication.

$\S 16$. Implication indefinable

$\S 17$. Two indefinables and ten primitive propositions in this calculus

$\S 18$. The ten primitive propositions

$\S 19$. Disjunction and negation defined

The Calculus of Classes

$\S 20$. Three new indefinables

$\S 21$. The relation of an individual to its class

$\S 22$. Propositional functions

$\S 23$. The notion of such that

$\S 24$. Two new primitive propositions

$\S 25$. Relation to propositional calculus

$\S 26$. Identity

The Calculus of Relations

$\S 27$. The logic of relations essential to mathematics

$\S 28$. New primitive propositions

$\S 29$. Relative products

$\S 30$. Relations with assigned domains

Peano's Symbolic Logic

$\S 31$. Mathematical and philosophical definitions

$\S 32$. Peano’s indefinables

$\S 33$. Elementary definitions

$\S 34$. Peano’s primitive propositions

$\S 35$. Negation and disjunction

$\S 36$. Existence and the null-class

Chapter $\text {III}$. Implication and Formal Implication

$\S 37$. Meaning of implication

$\S 38$. Asserted and unasserted propositions

$\S 39$. Inference does not require two premisses

$\S 40$. Formal implication is to be interpreted extensionally

$\S 41$. The variable in formal implication has an unrestricted field

$\S 42$. A formal implication is a single propositional function, not a relation of two

$\S 43$. Assertions

$\S 44$. Conditions that a term in an implication may be varied

$\S 45$. Formal implication involved in rules of inference

Chapter $\text {IV}$. Proper Names, Adjectives and Verbs

$\S 46$. Proper names, adjectives and verbs distinguished

$\S 47$. Terms

$\S 48$. Things and concepts

$\S 49$. Concepts as such and as terms

$\S 50$. Conceptual diversity

$\S 51$. Meaning and the subject-predicate logic

$\S 52$. Verbs and truth

$\S 53$. All verbs, except perhaps is, express relations

$\S 54$. Relations per se and relating relations

$\S 55$. Relations are not particularized by their terms

Chapter $\text {V}$. Denoting

$\S 56$. Definition of denoting

$\S 57$. Connection with subject-predicate propositions

$\S 58$. Denoting concepts obtained from predicates

$\S 59$. Extensional account of all, every, any, a and some

$\S 60$. Intensional account of the same

$\S 61$. Illustrations

$\S 62$. The difference between all, every, etc. lies in the objects denoted, not in the way of denoting them.

$\S 63$. The notion of the and definition

$\S 64$. The notion of the and identity

$\S 65$. Summary

Chapter $\text {VI}$. Classes

$\S 66$. Combination of intensional and extensional standpoints required

$\S 67$. Meaning of class

$\S 68$. Intensional and extensional genesis of classes

$\S 69$. Distinctions overlooked by Peano

$\S 70$. The class as one and as many

$\S 71$. The notion of and

$\S 72$. All men is not analyzable into all and men

$\S 73$. There are null class-concepts, but there is no null class

$\S 74$. The class as one, except when it has one term, is distinct from the class as many

$\S 75$. Every, any, a and some each denote one object, but an ambiguous one

$\S 76$. The relation of a term to its class

$\S 77$. The relation of inclusion between classes

$\S 78$. The contradiction

$\S 79$. Summary

Chapter $\text {VII}$. Propositional Functions.

$\S 80$. Indefinability of such that

$\S 81$. Where a fixed relation to a fixed term is asserted, a propositional function can be analysed into a variable subject and a constant assertion

$\S 82$. But this analysis is impossible in other cases

$\S 83$. Variation of the concept in a proposition

$\S 84$. Relation of propositional functions to classes

$\S 85$. A propositional function is in general not analysable into a constant and a variable element

Chapter $\text {VIII}$. The Variable.

$\S 86$. Nature of the variable

$\S 87$. Relation of the variable to any

$\S 88$. Formal and restricted variables

$\S 89$. Formal implication presupposes any

$\S 90$. Duality of any and some

$\S 91$. The class-concept propositional function is indefinable

$\S 92$. Other classes can be defined by means of such that

$\S 93$. Analysis of the variable

Chapter $\text {IX}$. Relations

$\S 94$. Characteristics of relations

$\S 95$. Relations of terms to themselves

$\S 96$. The domain and the converse domain of a relation

$\S 97$. Logical sum, logical product and relative product of relations

$\S 98$. A relation is not a class of couples

$\S 99$. Relations of a relation to its terms

Chapter $\text {X}$. The Contradiction

$\S 100$. Consequences of the contradiction

$\S 101$. Various statements of the contradiction

$\S 102$. An analogous generalized argument

$\S 103$. Various statements of the contradiction

$\S 104$. The contradiction arises from treating as one a class which is only many

$\S 105$. Other primâ facie possible solutions appear inadequate

$\S 106$. Summary of Part $\text {I}$

Part $\text {II}$. Number

Chapter $\text {XI}$. Definition of Cardinal Numbers

$\S 107$. Plan of Part $\text {II}$

$\S 108$. Mathematical meaning of definition

$\S 109$. Definitions of numbers by abstraction

$\S 110$. Objections to this definition

$\S 111$. Nominal definition of numbers

Chapter $\text {XII}$. Addition and Multiplication

$\S 112$. Only integers to be considered at present

$\S 113$. Definition of arithmetical addition

$\S 114$. Dependence upon the logical addition of classes

$\S 115$. Definition of multiplication

$\S 116$. Connection of addition, multiplication, and exponentiation

Chapter $\text {XIII}$. Finite and Infinite

$\S 117$. Definition of finite and infinite

$\S 118$. Definition of $a_0$

$\S 119$. Definition of finite numbers by mathematical induction

Chapter $\text {XIV}$. Theory of Finite Numbers

$\S 120$. Peano's indefinables and primitive propositions

$\S 121$. Mutual independence of the latter

$\S 122$. Peano really defines progressions, not finite numbers

$\S 123$. Proof of Peano's primitive propositions

Chapter $\text {XV}$. Addition of Terms and Addition of Classes

$\S 124$. Philosophy and mathematics distinguished

$\S 125$. Is there a more fundamental sense of number than that defined above?

$\S 126$. Numbers must be classes

$\S 127$. Numbers apply to classes as many

$\S 128$ One is to be asserted, not of terms, but of unit classes

$\S 129$. Counting not fundamental in arithmetic

$\S 130$. Numerical conjunction and plurality

$\S 131$. Addition of terms generates classes primarily, not numbers

$\S 132$. A term is indefinable, but not the number $1$

Chapter $\text {XVI}$. Whole and Part

$\S 133$. Single terms may be either simple or complex

$\S 134$. Whole and part cannot be defined by logical priority

$\S 135$. Three kinds of relation of whole and part distinguished

$\S 136$. Two kinds of wholes distinguished

$\S 137$. A whole is distinct from the numerical conjunctions of its parts

$\S 138$. How far analysis is falsification

$\S 139$. A class as one is an aggregate

Chapter $\text {XVII}$. Infinite Wholes

$\S 140$. Infinite aggregates must be admitted

$\S 141$. Infinite unities, if there are any, are unknown to us

$\S 142$. Are all infinite wholes aggregates of terms?

$\S 143$. Grounds in favour of this view

Chapter $\text {XVIII}$. Ratios and Fractions

$\S 144$. Definition of ratio

$\S 145$. Ratios are one-one relations

$\S 146$. Fractions are concerned with relations of whole and part

$\S 147$. Fractions depend, not upon number, but upon magnitude of divisibility

$\S 148$. Summary of Part $\text {II}$

Part $\text {III}$. Quantity

Chapter $\text {XIX}$. The Meaning of Magnitude

$\S 149$. Previous views on the relation of number and quantity

$\S 150$. Quantity not fundamental in mathematics

$\S 151$. Meaning of magnitude and quantity

$\S 152$. Three possible theories of equality to be examined

$\S 153$. Equality is not identity of number of parts

$\S 154$. Equality is not an unanalyzable relation of quantities

$\S 155$. Equality is sameness of magnitude

$\S 156$. Every particular magnitude is simple

$\S 157$. The principle of abstraction

$\S 158$. Summary

Note to Chapter $\text {XIX}$.

Chapter $\text {XX}$. The Range of Quantity

$\S 159$. Divisibility does not belong to all quantities

$\S 160$. Distance

$\S 161$. Differential coefficients

$\S 162$. A magnitude is never divisible, but may be a magnitude of divisibility

$\S 163$. Every magnitude is unanalyzable

Chapter $\text {XXI}$. Numbers as Expressing Magnitudes: Measurement

$\S 164$. Definition of measurement

$\S 165$. Possible grounds for holding all magnitudes to be measurable

$\S 166$. Intrinsic measurability

$\S 167$. Of divisibilities

$\S 168$. And of distances

$\S 169$. Measure of distance and measure of stretch

$\S 170$. Distance-theories and stretch-theories of geometry

$\S 171$. Extensive and intensive magnitudes

Chapter $\text {XXII}$. Zero

$\S 172$. Difficulties as to zero

$\S 173$. Meinong's theory

$\S 174$. Zero as minimum

$\S 175$. Zero distance as identity

$\S 176$. Zero as a null segment

$\S 177$. Zero and negation

$\S 178$. Every kind of zero magnitude is in a sense indefinable

Chapter $\text {XXIII}$. Infinity, the Infinitesimal, and Continuity

$\S 179$. Problems of infinity not specially quantitative

$\S 180$. Statement of the problem in regard to quantity

$\S 181$. Three antinomies

$\S 182$. Of which the antitheses depend upon an axiom of finitude

$\S 183$. And the use of mathematical induction

$\S 184$. Which are both to be rejected

$\S 185$. Provisional sense of continuity

$\S 186$. Summary of Part $\text {III}$

Part $\text {IV}$. Order

Chapter $\text {XXIV}$. The Genesis of Series

$\S 187$. Importance of order

$\S 188$. Between and separation of couples

$\S 189$. Generation of order by one-one relations

$\S 190$. By transitive asymmetrical relations

$\S 191$. By distances

$\S 192$. By triangular relations

$\S 193$. By relations between asymmetrical relations

$\S 194$. And by separation of couples

Chapter $\text {XXV}$. The Meaning of Order

$\S 195$. What is order?

$\S 196$. Three theories of between

$\S 197$. First theory

$\S 198$. A relation is not between its terms

$\S 199$. Second theory of between

$\S 200$. There appear to be ultimate triangular relations

$\S 201$. Reasons for rejecting the second theory

$\S 202$. Third theory of between to be rejected

$\S 203$. Meaning of separation of couples

$\S 204$. Reduction to transitive asymmetrical relations

$\S 205$. This reduction is formal

$\S 206$. But is the reason why separation leads to order

$\S 207$. The second way of generating series is alone fundamental, and gives the meaning of order

Chapter $\text {XXVI}$. Asymmetrical Relations

$\S 208$. Classification of relations as regards symmetry and transitiveness

$\S 209$. Symmetrical transitive relations

$\S 210$. Reflexiveness and the principle of abstraction

$\S 211$. Relative position

$\S 212$. Are relations reducible to predications?

$\S 213$. Monadistic theory of relations

$\S 214$. Reasons for rejecting the theory

$\S 215$. Monistic theory and the reasons for rejecting it

$\S 216$. Order requires that relations should be ultimate

Chapter $\text {XXVII}$. Difference of Sense and Difference of Sign

$\S 217$. Kant on difference of sense

$\S 218$. Meaning of difference of sense

$\S 219$. Difference of sign

$\S 220$. In the cases of finite numbers

$\S 221$. And of magnitudes

$\S 222$. Right and left

$\S 223$. Difference of sign arises from difference of sense among transitive asymmetrical relations

Chapter $\text {XXVIII}$. On the Difference Between Open and Closed Series

$\S 224$. What is the difference between open and closed series?

$\S 225$. Finite closed series

$\S 226$. Series generated by triangular relations

$\S 227$. Four-term relations

$\S 228$. Closed series are such as have an arbitrary first term

Chapter $\text {XXIX}$. Progressions and Ordinal Numbers

$\S 229$. Definition of progressions

$\S 230$. All finite arithmetic applies to every progression

$\S 231$. Definition of ordinal numbers

$\S 232$. Definition of nth

$\S 233$. Positive and negative ordinals

Chapter $\text {XXX}$. Dedekind's Theory of Number

$\S 234$. Dedekind's principal ideas

$\S 235$. Representation of a system

$\S 236$. The notion of a chain

$\S 237$. The chain of an element

$\S 238$. Generalized form of mathematical induction

$\S 239$. Definition of a singly infinite system

$\S 240$. Definition of cardinals

$\S 241$. Dedekind's proof of mathematical induction

$\S 242$. Objections to his definition of ordinals

$\S 243$. And of cardinals

Chapter $\text {XXXI}$. Distance

$\S 244$. Distance not essential to order

$\S 245$. Definition of distance

$\S 246$. Measurement of distances

$\S 247$. In most series, the existence of distances is doubtful

$\S 248$. Summary of Part $\text {IV}$

Part $\text {V}$. Infinity and Continuity

Chapter $\text {XXXII}$. The Correlation of Series

$\S 249$. The infinitesimal and space are no longer required in a statement of principles

$\S 250$. The supposed contradictions of infinity have been resolved

$\S 251$. Correlation of series

$\S 252$. Independent series and series by correlation

$\S 253$. Likeness of relations

$\S 254$. Functions

$\S 255$. Functions of a variable whose values form a series

$\S 256$. Functions which are defined by formulae

$\S 257$. Complete series

Chapter $\text {XXXIII}$. Real Numbers

$\S 258$. Real numbers are not limits of series of rationals

$\S 259$. Segments of rationals

$\S 260$. Properties of segments

$\S 261$. Coherent classes in a series

Chapter $\text {XXXIV}$. Limits and Irrational Numbers

$\S 262$. Definition of a limit

$\S 263$. Elementary properties of limits

$\S 264$. An arithmetical theory of irrationals is indispensable

$\S 265$. Dedekind's theory of irrationals

$\S 266$. Defects in Dedekind's axiom of continuity

$\S 267$. Objections to his theory of irrationals

$\S 268$. Weierstrass's theory

$\S 269$. Cantor's theory

$\S 270$. Real numbers are segments of rationals

Chapter $\text {XXXV}$. Cantor's First Definition of Continuity

$\S 271$. The arithmetical theory of continuity is due to Cantor

$\S 272$. Cohesion

$\S 273$. Perfection

$\S 274$. Defect in Cantor's definition of perfection

$\S 275$. The existence of limits must not be assumed without special grounds

Chapter $\text {XXXVI}$. Ordinal Continuity

$\S 276$. Continuity is a purely ordinal notion

$\S 277$. Cantor's ordinal definition of continuity

$\S 278$. Only ordinal notions occur in this definition

$\S 279$. Infinite classes of integers can be arranged in a continuous series

$\S 280$. Segments of general compact series

$\S 281$. Segments defined by fundamental series

$\S 282$. Two compact series may be combined to form a series which is not compact

Chapter $\text {XXXVII}$. Transfinite Cardinals

$\S 283$. Transfinite cardinals differ widely from transfinite ordinals

$\S 284$. Definition of cardinals

$\S 285$. Properties of cardinals

$\S 286$. Addition, multiplication, and exponentiation

$\S 287$. The smallest transfinite cardinal $a_0$

$\S 288$. Other transfinite cardinals

$\S 289$. Finite and transfinite cardinals form a single series by relation to greater and less

Chapter $\text {XXXVIII}$. Transfinite Ordinals

$\S 290$. Ordinals are classes of serial relations

$\S 291$. Cantor's definition of the second class of ordinals

$\S 292$. Definition of $\omega$

$\S 293$. An infinite class can be arranged in many types of series

$\S 294$. Addition and subtraction of ordinals

$\S 295$. Multiplication and division

$\S 296$. Well-ordered series

$\S 297$. Series which are not well-ordered

$\S 298$. Ordinal numbers are types of well-ordered series

$\S 299$. Relation-arithmetic

$\S 300$. Proofs of existence-theorems

$\S 301$. There is no maximum ordinal number

$\S 302$. Successive derivatives of a series

Chapter $\text {XXXIX}$. The Infinitesimal Calculus

$\S 303$. The infinitesimal has been usually supposed essential to the calculus

$\S 304$. Definition of a continuous function

$\S 305$. Definition of the derivative of a function

$\S 306$. The infinitesimal is not implied in this definition

$\S 307$. Definition of the definite integral

$\S 308$. Neither the infinite nor the infinitesimal is involved in this definition

Chapter $\text {XL}$. The Infinitesimal and the Improper Infinite

$\S 309$. A precise definition of the infinitesimal is seldom given

$\S 310$. Definition of the infinitesimal and the improper infinite

$\S 311$. Instances of the infinitesimal

$\S 312$. No infinitesimal segments in compact series

$\S 313$. Orders of infinity and infinitesimality

$\S 314$. Summary

Chapter $\text {XLI}$. Philosophical Arguments Concerning the Infinitesimal

$\S 315$. Current philosophical opinions illustrated by Cohen

$\S 316$. Who bases the calculus upon infinitesimals

$\S 317$. Space and motion are here irrelevant

$\S 318$. Cohen regards the doctrine of limits as insufficient for the calculus

$\S 319$. And supposes limits to be essentially quantitative

$\S 320$. To involve infinitesimal differences

$\S 321$. And to introduce a new meaning of equality

$\S 322$. He identifies the inextensive with the intensive

$\S 323$. Consecutive numbers are supposed to be required for continuous change

$\S 324$. Cohen's views are to be rejected

Chapter $\text {XLII}$. The Philosophy of the Continuum

$\S 325$. Philosophical sense of continuity not here in question

$\S 326$. The continuum is composed of mutually external units

$\S 327$. Zeno and Weierstrass

$\S 328$. The argument of dichotomy

$\S 329$. The objectionable and the innocent kind of endless regress

$\S 330$. Extensional and intensional definition of a whole

$\S 331$. Achilles and the tortoise

$\S 332$. The arrow

$\S 333$. Change does not involve a state of change

$\S 334$. The argument of the measure

$\S 335$. Summary of Cantor's doctrine of continuity

$\S 336$. The continuum consists of elements

Chapter $\text {XLIII}$. The Philosophy of the Infinite

$\S 337$. Historical retrospect

$\S 338$. Positive doctrine of the infinite

$\S 339$. Proof that there are infinite classes

$\S 340$. The paradox of Tristram Shandy

$\S 341$. A whole and a part may be similar

$\S 342$. Whole and part and formal implication

$\S 343$. No immediate predecessor of $\omega$ or $a_0$

$\S 344$. Difficulty as regards the number of all terms, objects, or propositions

$\S 345$. Cantor's first proof that there is no greatest number

$\S 346$. His second proof

$\S 347$. Every class has more sub-classes than terms

$\S 348$. But this is impossible in certain cases

$\S 349$. Resulting contradictions

$\S 350$. Summary of Part $\text {V}$

Part $\text {VI}$. Space

Chapter $\text {XLIV}$. Dimensions and Complex Numbers

$\S 351$. Retrospect

$\S 352$. Geometry is the science of series of two or more dimensions

$\S 353$. Non-Euclidean geometry

$\S 354$. Definition of dimensions

$\S 355$. Remarks on the definition

$\S 356$. The definition of dimensions is purely logical

$\S 357$. Complex numbers and universal algebra

$\S 358$. Algebraical generalization of number

$\S 359$. Definition of complex numbers

$\S 360$. Remarks on the definition

Chapter $\text {XLV}$. Projective Geometry

$\S 361$. Recent threefold scrutiny of geometrical principles

$\S 362$. Projective, descriptive, and metrical geometry

$\S 363$. Projective points and straight lines

$\S 364$. Definition of the plane

$\S 365$. Harmonic ranges

$\S 366$. Involutions

$\S 367$. Projective generation of order

$\S 368$. Möbius nets

$\S 369$. Projective order presupposed in assigning irrational coordinates

$\S 370$. Anharmonic ratio

$\S 371$. Assignment of coordinates to any point in space

$\S 372$. Comparison of projective and Euclidean geometry

$\S 373$. The principle of duality

Chapter $\text {XLVI}$. Descriptive Geometry

$\S 374$. Distinction between projective and descriptive geometry

$\S 375$. Method of Pasch and Peano

$\S 376$. Method employing serial relations

$\S 377$. Mutual independence of axioms

$\S 378$. Logical definition of the class of descriptive spaces

$\S 379$. Parts of straight lines

$\S 380$. Definition of the plane

$\S 381$. Solid geometry

$\S 382$. Descriptive geometry applies to Euclidean and hyperbolic, but not elliptic space

$\S 383$. Ideal elements

$\S 384$. Ideal points

$\S 385$. Ideal lines

$\S 386$. Ideal planes

$\S 387$. The removal of a suitable selection of points renders a projective space descriptive

Chapter $\text {XLVII}$. Metrical Geometry

$\S 388$. Metrical geometry presupposes projective or descriptive geometry

$\S 389$. Errors in Euclid

$\S 390$. Superposition is not a valid method

$\S 391$. Errors in Euclid (continued)

$\S 392$. Axioms of distance

$\S 393$. Stretches

$\S 394$. Order as resulting from distance alone

$\S 395$. Geometries which derive the straight line from distance

$\S 396$. In most spaces, magnitude of divisibility can be used instead of distance

$\S 397$. Meaning of magnitude of divisibility

$\S 398$. Difficulty of making distance independent of stretch

$\S 399$. Theoretical meaning of measurement

$\S 400$. Definition of angle

$\S 401$. Axioms concerning angles

$\S 402$. An angle is a stretch of rays, not a class of points

$\S 403$. Areas and volumes

$\S 404$. Right and left

Chapter $\text {XLVIII}$. Relation of Metrical to Projective and Descriptive Geometry

$\S 405$. Non-quantitative geometry has no metrical presuppositions

$\S 406$. Historical development of non-quantitative geometry

$\S 407$. Non-quantitative theory of distance

$\S 408$. In descriptive geometry

$\S 409$. And in projective geometry

$\S 410$. Geometrical theory of imaginary point-pairs

$\S 411$. New projective theory of distance

Chapter $\text {XLIX}$. Definitions of Various Spaces

$\S 412$. All kinds of spaces are definable in purely logical terms

$\S 413$. Definition of projective spaces of three dimensions

$\S 414$. Definition of Euclidean spaces of three dimensions

$\S 415$. Definition of Clifford's spaces of two dimensions

Chapter $\text {L}$. The Continuity of Space

$\S 416$. The continuity of a projective space

$\S 417$. The continuity of metrical space

$\S 418$. An axiom of continuity enables us to dispense with the postulate of the circle

$\S 419$. Is space prior to points?

$\S 420$. Empirical premisses and induction

$\S 421$. There is no reason to desire our premisses to be self-evident

$\S 422$. Space is an aggregate of points, not a unity

Chapter $\text {LI}$. Logical Arguments Against Points

$\S 423$. Absolute and relative position

$\S 424$. Lotze's arguments against absolute position

$\S 425$. Lotze's theory of relations

$\S 426$. The subject-predicate theory of propositions

$\S 427$. Lotze's three kinds of Being

$\S 428$. Argument from the identity of indiscernibles

$\S 429$. Points are not active

$\S 430$. Argument from the necessary truths of geometry

$\S 431$. Points do not imply one another

Chapter $\text {LII}$. Kant's Theory of Space

$\S 432$. The present work is diametrically opposed to Kant

$\S 433$. Summary of Kant's theory

$\S 434$. Mathematical reasoning requires no extra-logical element

$\S 435$. Kant's mathematical antinomies

$\S 436$. Summary of Part $\text {VI}$

Part $\text {VII}$. Matter and Motion

Chapter $\text {LIII}$. Matter

$\S 437$. Dynamics is here considered as a branch of pure mathematics.

$\S 438$. Matter is not implied by space

$\S 439$. Matter as substance

$\S 440$. Relations of matter to space and time

$\S 441$. Definition of matter in terms of logical constants

Chapter $\text {LIV}$. Motion

$\S 442$. Definition of change

$\S 443$. There is no such thing as a state of change

$\S 444$. Change involves existence

$\S 445$. Occupation of a place at a time

$\S 446$. Definition of motion

$\S 447$. There is no state of motion

Chapter $\text {LV}$. Causality

$\S 448$. The descriptive theory of dynamics

$\S 449$. Causation of particulars by particulars

$\S 450$. Cause and effect are not temporally contiguous

$\S 451$. Is there any causation of particulars by particulars?

$\S 452$. Generalized form of causality

Chapter $\text {LVI}$. Definition of a Dynamic World

$\S 453$. Kinematical motions

$\S 454$. Kinetic motions

Chapter $\text {LVII}$. Newton's Laws of Motion

$\S 455$. Force and acceleration are fictions

$\S 456$. The law of inertia

$\S 457$. The second law of motion

$\S 458$. The third law

$\S 459$. Summary of Newtonian principles

$\S 460$. Causality in dynamics

$\S 461$. Accelerations as caused by particulars

$\S 462$. No part of the laws of motion is an à priori truth

Chapter $\text {LVIII}$. Absolute and Relative Motion

$\S 463$. Newton and his critics

$\S 464$. Grounds for absolute motion

$\S 465$. Neumann's theory

$\S 466$. Streintz's theory

$\S 467$. Mr Macaulay's theory

$\S 468$. Absolute rotation is still a change of relation

$\S 469$. Mach's reply to Newton

Chapter $\text {LIX}$. Hertz's Dynamics

$\S 470$. Summary of Hertz's system

$\S 471$. Hertz's innovations are not fundamental from the point of view of pure mathematics

$\S 472$. Principles common to Hertz and Newton

$\S 473$. Principle of the equality of cause and effect

$\S 474$. Summary of the work

Part Appendices.

Appendix $\text {A}$. The Logical and Arithmetical Doctrines of Frege

$\S 475$. Principal points in Frege's doctrines

$\S 476$. Meaning and indication

$\S 477$. Truth-values and judgment

$\S 478$. Criticism

$\S 479$. Are assumptions proper names for the true or the false?

$\S 480$. Functions

$\S 481$. Begriff and Gegenstand

$\S 482$. Recapitulation of theory of propositional functions

$\S 483$. Can concepts be made logical subjects?

$\S 484$. Ranges

$\S 485$. Definition of $\in$ and of relation

$\S 486$. Reasons for an extensional view of classes

$\S 487$. A class which has only one member is distinct from its only member

$\S 488$. Possible theories to account for this fact

$\S 489$. Recapitulation of theories already discussed

$\S 490$. The subject of a proposition may be plural

$\S 491$. Classes having only one member

$\S 492$. Theory of types

$\S 493$. Implication and symbolic logic

$\S 494$. Definition of cardinal numbers

$\S 495$. Frege's theory of series

$\S 496$. Kerry's criticisms of Frege

Appendix $\text {B}$. The Doctrine of Types

$\S 497$. Statement of the doctrine

$\S 498$. Numbers and propositions as types

$\S 499$. Are propositional concepts individuals?

$\S 500$. Contradiction arising from the question whether there are more classes of propositions than propositions

Further Editions

• 1903: Bertrand Russell: The Principles of Mathematics