“Philosophy of science is about as useful to scientists as ornithology is to birds,”--Richard Feynman. Maybe, maybe not. At least he wasn't talking about the philosophy of mathematics.
Isidore Auguste Marie François Xavier Comte (19 January 1798 – 5 September 1857), better known as Auguste Comte, may well be described as being among the first modern philosophers of science, as well as being one of the founders of the field of sociology and the philosophical/logical doctrine of positivism. His book on the philosophy of mathematics is the first of its kind in the modern sense of the study.
The Philosophy of Mathematics, translated from the Cours de Philosophie Positive...translated by W.M. Gillespie...New York, Harper & Brothers, 1851. xvi, 17-260pp, 6+6pp ads at end. This is evidently the U.S. copyright deposit copy, or at least so it seems from the notation on the title page, making it a rather unique copy of a somewhat significant book in the history of the philosophy of mathematics. The copy is rebound in simple black cloth, with new endpapers. Very good copy. $200
Comte defines math as a science and not an art early on in the work:
"TRUE DEFINITION OF MATHEMATICS We are now able to define mathematical science with precision by assigning to it as its object the indirect measurement of magnitudes and by saying it constantly proposes to determine certain magnitudes from others by means of the precise relations existing between them."
"This enunciation instead of giving the idea of only an art as do all the ordinary definitions characterizes immediately a true science and shows it at once to be composed of an immense chain of intellectual operations which may evidently become very complicated because of the series of intermediate links which it will be necessary to establish between the unknown quantities and those which admit of a direct measurement of the number of variables ccexistent in the proposed question and of the nature of the relations between all these different magnitudes furnished by the phenomena under consideration According to such a definition the spirit of mathematics consists in always regarding all the quantities which any phenomenon can present as connected and interwoven with one another with the view of deducing them from one another..."
Contents and almost-full text below:
GENERAL CONSIDERATIONS ON MATHEMATICAL SCI-
True Definition Of Mathematics 25
Its Two Fundamental Divisions 26
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
The Calculus of Values, or Arithmetic 57
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
ORDINARY ANALYSIS; OR, ALGEBRA. 69
Classification of Equations 70
Algebraic Resolution Of Equations 72
Numerical Resolution Of Equations 75
Different Divisions of the two Systems 78
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
2. Employment of the Differential Calculus alone. . 125
3. Employment of the Integral Calculus alone .... 125
Three Classes of Questions hence resulting .. . 126
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 ele-
Transformation of derived Functions foi new Variables 132
Different Orders of Differentiation 133
Its fundamental Division: Explicit and Implicit Func-
Subdivisions: a single Variable or several 136
Calculus of partial Differences 137
Another Subdivision: different Orders of Differentiation 138
Another equivalent Distinction 140
Integration of Transcendental Functions 143
THE CALCULUS OF VARIATIONS 151
Problems Giving Rise To It 151
Ordinary Questions of Maxima, and Minima 151
Solid of least Resistance; Brachystochrone; Isope-
Analytical Nature Of These Questions 154
Methods Of The Older Geometers 155
1. Absolute Maxima and Minima 157
A more general Consideration 159
2. Relative Maxima and Minima 160
THE CALCULUS OF FINITE DIFFERENCES 167
Its Identity with this Calculus 172
Periodic Or Discontinuous Functions 173
Applications Of This Calculus 173
Approximate Rectification, &c 174
A GENERAL VIEW OF GEOMETRY 179
The true Nature of Geometry 179
2. Different kinds of Extension 182
The Final Object Of Geometry 184
Nature of Geometrical Measurement 185
The Infinite Extent Of Its Field 190
Analytical Invention of Curves, &c 193
Expansion Of Original Definition 193
Properties of Lines and Surfaces 195
1. To find the most suitable Property 195
2. To pass from the Concrete to the Abstract 197
The Two General Methods Of Geometry 202
Their fundamental Difference 203
1°. Different Questions with respect to the same
2°. Similar Questions with respect to different Figures 204
The Ancient the base of the Modern 209
Lines; Folypons; Polyhedrons 212
Not to be farther restricted 213
Improper Application of Analysis -214
Attempted Demonstrations of Axioms 216
Geometry op The Right Line 217
Two Methods of introducing Angles 226
2. By trigonometrical Lines 226
Its Division of trigonometrical Questions 227
1. Relations between Angles and trigonometrical
2. Relations between trigonometrical Lines and
The Analytical Representation Of Figures 232
Reduction of Figure to Position 233
Determination of the position of a Point 234
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
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
Determination of a Point in Space 251
Expression of Surfaces by Equations 253
Expression of Equations by Surfaces 253
Imperfections of Analytical Geometry 258
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