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

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