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

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

Published $\text {1938}$

### 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