The Philosophy of Mathematics _Auguste Comte

The Philosophy of Mathematics
Auguste Comte
Translated from Cours de Philosophie Positive by
W. M. Gillespie-Dover Publications (2005)
Format PDF 19.6 MB
Pages 284

INTRODUCTION.

GENERAL CONSIDERATIONS ON MATHEMATICAL SCIENCE 17

THE OBJECT OF MATHEMATICS 18

Measuring Magnitudes IS

Difficulties 19

General Method 20

Illustrations 21

1. Falling Bodies 21

2. Inaccessible Distances 23

3. Astronomical Facts 24

TRUE DEFINITION OF MATHEMATICS 25

A Science, not an Art 25

ITS TWO FUNDAMENTAL DIVISIONS 26

Their different Objects 27

Their different Natures 29

Concrete Mathematics 31

Geometry and Mechanics 32

Abstract Mathematics 33

The Calculus, or Analysis 33

EXTENT OF ITS FIELD 35

Its Universality 30

Its Limitations 37

BOOK I.

ANALYSIS.

CHAPTER I.

GENERAL VIEW OF MATHEMATICAL ANALYSIS . 45

THE TRUE IDEA OF AN EQUATION 46

Division of Functions into Abstract and Concrete 47

Enumeration of Abstract Functions 50

DIVISIONS OF THE CALCULUS 53

The Calculus of Values, or Arithmetic 57

Its Extent 57

Its true Nature 59

The Calculus of Functions 61

Two Modes of obtaining Equations 61

1. By the Relations between the given Quantities  61

2. By the Relations between auxiliary Quantities  64

Corresponding Divisions of the Calculus of Functions  67

CHAPTER II.

ORDINARY ANALYSIS ; OR, ALGEBRA. 69

Its Object 69

Classification of Equations 70

ALGEBRAIC EQUATIONS 71

Their Classification 71

ALGEBRAIC RESOLUTION OF EQUATIONS 72

Its Limits 72

General Solution 72

What we know in Algebra 74

NUMERICAL RESOLUTION OF EQUATIONS 75

Its limited Usefulness 76

Different Divisions of the two Systems 78

THE THEORY OF EQUATIONS 79

THE METHOD OF INDETERMINATE COEFFICIENTS 60

IMAGINARY QUANTITIES fcl

NEGATIVE QUANTITIES 81

THE PRINCIPLE or HOMOGENEITY 84

CHAPTER III.

TRANSCENDENTAL ANALYSIS : ITS DIFFERENT CONCEPTIONS 88

Preliminary Remarks 88

Its early History 89

METHOD OF LEIBNITZ 91

Infinitely small Elements

Examples, :

1 . Tangents 93

2. Rectification of an Arc 94

3. Quadrature of a Curve 95

4. Velocity in variable Motion 95

5. Distribution of Heat 96

Generality of the Formulas 97

Demonstration of the Method 98

Illustration by Tangents 102

METHOD OF NEWTON 103

Method of Limits 103

Examples :

1. Tangents 104

2. Rectifications 105

Fluxions and Fluents 106

METHOD OF LAGRANGE 108

Derived Functions*

An extension of ordinary Analysis 108

Example : Tangents 1 09

Finvlamcntal Identity of the three Methods 110

Their comparative Value 113

That of Leibnitz 113

That of Newton 115

That of Lagrange ... 117

CHAPTER IV.

THE DIFFERENTIAL AND INTEGRAL CALCULUS . 120

ITS TWO FUNDAMENTAL DIVISIONS 120

THEIR RELATIONS TO EACH OTHER 121

1. Use of the Differential Calculus as preparatory to that of the Integral 123

2. Employment of the Differential Calculus alone 125

3. Employment of the Integral Calculus alone 125

Three Classes of Questions hence resulting 126

THE DIFFERENTIAL CALCULUS 127

Two Cases : Explicit and Implicit Functions 127

Two sub-Cases : a single Variable or several  129

Two other Cases : Functions separate or combined 130

Reduction of all to the Differentiation of the ten elementary Functions 131

Transformation of derived Functions for new Variables 132

Different Orders of Differentiation 133

Analytical Applications 133

THE INTEGRAL CALCULUS 135

Its fundamental Division : Explicit and Implicit Functions 135

Subdivisions : a single Variable or several 136

Calculus of partial Differences 137

Another Subdivision : different Orders of Differentiation 138

Another equivalent Distinction 140

Quadratures 142

Integration of Transcendental Functions 143

Integration by Parts 143

Integration of Algebraic Functions 143

Singular Solutions 144

Definite Integrals 146

Prospects of the Integral Calculus 148

CHAPTER V.

THE CALCULUS OF VARIATIONS 151

PROBLEMS GIVING RISE TO IT 151

Ordinary Questions of Maxima and Minima 151

A new Class of Questions 152

Solid of least Resistance ; Brachystochrone ; Isoperimeters 153

ANALYTICAL NATURE OF THESE QUESTIONS 154

METHODS OF THE OLDER GEOMETERS 155

METHOD OF LAGRANGE 156

Two Classes of Questions 157

1. Absolute Maxima and Minima 157

Equations of Limits 159

A more general Consideration 159

2. Relative Maxima and Minima 160

Other Applications of the Method of Variations 162

ITS RELATIONS TO THE ORDINARY CALCULUS 163

CHAPTER VI.

THE CALCULUS OF FINITE DIFFERENCES 167

Its general Character 167

Its true Nature 168

GENERAL THEORY OF SERIES 170

Its Identity with this Calculus 172

PERIODIC OR DISCONTINUOUS FUNCTIONS 173

APPLICATIONS OF THIS CALCULUS 173

Series 173

Interpolation 173

Approximate Rectification, &c 174

ANALYTICAL TABLE OF CONTENTS.

BOOK II.

GEOMETRY

CHAPTER I.

A GENERAL VIEW OF GEOMETRY  179

The true Nature of Geometry  179

Two fundamental Ideas   181

1. The Idea of Space   181

2. Different kinds of Extension  182

THE FINAL OBJECT OF GEOMETRY 184

Nature of Geometrical Measurement 185

Of Surfaces and Volumes   185

Of curve Lines   187

Of right Lines   189

THE INFINITE EXTENT OF ITS FlELD  190

Infinity of Lines  190

Infinity of Surfaces  191

Infinity of Volumes 192

Analytical Invention of Curves, &c  193

EXPANSION OF ORIGINAL DEFINITION  193

Properties of Lines and Surfaces  1 95

Necessity of their Study  195

1. To find the most suitable Property  195

2. To pass from the Concrete to the Abstract  197

Illustrations :

Orbits of the Planets  198

Figure of the Earth 199

THE TWO GENERAL METHODS OF GEOMETRY  202

Their fundamental Difference 203

1. Different Questions with respect to the same Figure  204

2. Similar Questions with respect to different Figures 204

Geometry of the Ancients 204

Geometry of the Moderns  205

Superiority of the Modern  207

The Ancient the base of the Modern 209

CHAPTER II.

ANCIENT OR SYNTHETIC GEOMETRY

ITS PROPER EXTENT 212

Lines ; Polygons ; Polyhedrons 212

Not to be farther restricted 213

Improper Application of Analysis 214

Attempted Demonstrations of Axioms 210

GEOMETRY OF THE RIGHT LINE 217

GRAPHICAL SOLUTIONS 218

Descriptive Geometry 220

ALGEBRAICAL SOLUTIONS 224

Trigonometry 225

Two Methods of introducing Angles 226

1. By Arcs 226

2. By trigonometrical Lines 226

Advantages of the latter 226

Its Division of trigonometrical Questions 227

1. Relations between Angles and trigonometrical Lines 228

2. Relations between trigonometrical Lines and Sides 228

Increase of trigonometrical Lines 228

Study of the Relations between them 230

CHAPTER III.

MODERN OR ANALYTICAL GEOMETRY

THE ANALYTICAL REPRESENTATION OF FIGURES 232

Reduction of Figure to Position 233

Determination of the position of a Point 234

PLANE CURVES 237

Expression of Lines by Equations 237

Expression of Equations by Lines 238

Any change in the Line changes the Equation 240

Every  “Definition” ; of a Line is an Equation 241

Choice of Co-ordinates 245

Two different points of View 245

1. Representation of Lines by Equations 246

2. Representation of Equations by Lines 246

Superiority of the rectilinear System 248

Advantages of perpendicular Axes 249

SURFACES 251

Determination of a Point in Space 251

Expression of Surfaces by Equations 253

Expression of Equations by Surfaces 253

CURVES IN SPACE 255

Imperfections of Analytical Geometry 258

Relatively to Geometry 258

Relatively to Analysis 258