Godina izdanja: 2010
ISBN: 9780415487412
Jezik: Engleski
Autor: Strani

BERTRAND RUSSELL

PRINCIPLES OF MATHEMATICS

Izdavač - Routledge, London

Godina - 2010

552 strana

21 cm

Edicija - Routledge Classics

ISBN - 9780415487412

Povez - Broširan

Stanje - Kao na slici, tekst bez podvlačenja



SADRŽAJ:
Introduction to the Edition
Introduction to the Second Edition
Preface

PART I THE INDEFINABLES OF MATHEMATICS
Definition of Pure Mathematics
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

Symbolic Logic
Definition and scope of symbolic logic
The indefinables of symbolic logic
Symbolic logic consists of three parts

A The Propositional Calculus
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
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
The logic of relations essential to mathematics
New primitive propositions
Relative products
Relations with assigned domains

D Peano`s Symbolic Logic
Mathematical and philosophical definitions
Peano`s indefinables
Elementary definitions
Peano`s primitive propositions
Negation and disjunction
Existence and the null-class

Implication and Formal Implication
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

Proper Names, Adjectives and Verbs
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

Denoting
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, etclies in the objects denoted, not in the way of denoting them
The notion of the and definition
The notion of the and identity
Summary

Classes
Combination of intensional and extensional standpoints required
Meaning of class
Intensional and extensional genesis of class
Distinctions overlooked by Peano
The class as one and as many
The notion of and
All men is not analysable 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
The contradiction
Summary

Propositional Functions
Indefinability of such that
Where a fixed relation to a fixed term is asserted, a propositional function can be analysed 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 analysable into a constant and a variable element

The Variable
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

Relations
Characteristics of relations
Relations of terms to themselves
The domain and the converse domain of a relation
Logical sum, logical product and relativeproduct of relations
A relation is not a class of couples
Relations of a relation to its terms

The Contradiction
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
Definition of Cardinal Numbers
Plan of Part II
Mathematical meaning of definition
Definition of numbers by abstraction
Objections to this definition
Nominal definition of numbers

Addition and Multiplication
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

Finite and Infinite
Definition of finite and infinite
Definition of a
Definition of finite numbers by mathematical induction

Theory of Finite Numbers
Peano`s indefinables and primitive propositions
Mutual independence of the latter
Peano really defines progressions, not finite numbers
Proof of Peano`s primitive propositions

Addition of Terms and Addition of Classes
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

Whole and Part
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

Infinite Wholes
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

Ratios and Fractions
Definition of ratio
Ratios are one-one relations
Fractions are concerned with relations of whole and part
Fractions depend, not upon number, but upon magnitude of divisibility
Summary of Part II

PART III QUANTITY
The Meaning of Magnitude
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 unanalysable relation of quantities
Equality is sameness of magnitude
Every particular magnitude is simple
The principle of abstraction
Summary

The Range of Quantity
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 unanalysable

Numbers as Expressing Magnitudes: Measurement
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

Zero
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

Infinity, the Infinitesimal and Continuity
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
And the use of mathematical induction
Which are both to be rejected
Provisional sense of continuity
Summary of Part III

PART IV ORDER
The Genesis of Series
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

The Meaning of Order
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

Asymmetrical Relations
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

Difference of Sense and Difference of Sign
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

On the Difference Between Open and Closed Series
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

Progressions and Ordinal Numbers
Definition of progressions
All finite arithmetic applies to every progression
Definition of ordinal numbers
Definition of `nth`
Positive and negative ordinals

Dedekind`s Theory of Number
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

Distance
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
The Correlation of Series
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

Real Numbers
Real numbers are not limits of series of rationals
Segments of rationals
Properties of segments
Coherent classes in a series

Limits and Irrational Numbers
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

Cantor`s First Definition of Continuity
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

Ordinal Continuity
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

Transfinite Cardinals
Transfinite cardinals differ widely from transfinite ordinals
Definition of cardinals
Properties of cardinals.
Addition, multiplication and exponentiation
The smallest transfinite cardinal
Other transfinite cardinals
Finite and transfinite cardinals form a single series by relation to greater and less


Transfinite Ordinals
Ordinals are classes of serial relations
Cantor`s definition of the second class of ordinals
Definition of w
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

The Infinitesimal Calculus
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
The Infinitesimal and the Improper Infinite
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

Philosophical Arguments Concerning the Infinitesimal
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
The Philosophy of the Continuum
Philosophical sense of continuity not here in question
The continuum is composed of mutually external units
Zeno and Weierstrass
The argument of dichotomy nisb
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

The Philosophy of the Infinite
Historical retrospect
Positive doctrine of the infinite
Proof that there are infinite classes
The paradox of Tristram Shandy
A whole and a part may be similar
Whole and part and formal implication
No immediate predecessor of or a
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
Dimensions and Complex Numbers
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

Projective Geometry
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

Descriptive Geometry
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
Ideal lines
Ideal planes
The removal of a suitable selection of points renders a projective space descriptive

Metrical Geometry
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 temu
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

Relation of Metrical to Projective and Descriptive Geometry
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

Definitions of Various Spaces
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

The Continuity of Space
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

Logical Arguments Against Points
Absolute and relative position
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

Kant`s Theory of Space
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
Matter
Dynamics is here considered as a branch of load 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

Motion
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

Causality
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

Definition of a Dynamical World
Kinematical motions
Kinetic motions

Newton`s Laws of Motion
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

Absolute and Relative Motion
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

Hertz`s Dynamics
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

APPENDICES
List of Abbreviations

APPENDIX A
The Logical and Arithmetical Doctrines of Frege
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 & 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
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


`The Principles of Mathematics (PoM) is a 1903 book by Bertrand Russell, in which the author presented his famous paradox and argued his thesis that mathematics and logic are identical.

The book presents a view of the foundations of mathematics and Meinongianism and has become a classic reference. It reported on developments by Giuseppe Peano, Mario Pieri, Richard Dedekind, Georg Cantor, and others.

In 1905 Louis Couturat published a partial French translation that expanded the book`s readership. In 1937 Russell prepared a new introduction saying, `Such interest as the book now possesses is historical, and consists in the fact that it represents a certain stage in the development of its subject.` Further editions were published in 1938, 1951, 1996, and 2009.`



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