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- Herausgeber: Science & Technology Publishing
- Kategorie: Geisteswissenschaft
- Sprache: Englisch
The Philosophy of Mathematics by Auguste Comte is a foundational work that explores the logical structure, historical development, and philosophical implications of mathematics. Comte, a pioneering figure in the field of positivism, delves into the evolution of mathematical thought from its earliest origins in ancient civilizations to its sophisticated forms in the modern era. The book examines the nature of mathematical reasoning, the relationship between mathematics and the physical sciences, and the role of abstraction and generalization in mathematical progress. Comte discusses the classification of mathematical sciences, distinguishing between pure and applied mathematics, and analyzes the significance of arithmetic, geometry, algebra, and calculus. He emphasizes the cumulative and systematic nature of mathematical knowledge, arguing that mathematics serves as the foundation for all scientific inquiry. The text also addresses the psychological and educational aspects of learning mathematics, highlighting its importance in intellectual development and its influence on other disciplines. Throughout the book, Comte reflects on the philosophical questions raised by mathematics, such as the nature of mathematical objects, the meaning of mathematical truth, and the limits of mathematical knowledge. He considers the interplay between intuition and logic, the role of symbols and notation, and the impact of mathematical discoveries on society and technology. The Philosophy of Mathematics is not only a historical survey but also a profound meditation on the enduring power and mystery of mathematics, making it an essential read for anyone interested in the deeper questions surrounding this fundamental human endeavor.
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THE PHILOSOPHYOF MATHEMATICS.
THE PHILOSOPHY OF MATHEMATICS;
TRANSLATED FROM THE COURS DE PHILOSOPHIE POSITIVEOFAUGUSTE COMTE,
BY W. M. GILLESPIE,PROFESSOR OF CIVIL ENGINEERING & ADJ. PROF. OF MATHEMATICS IN UNION COLLEGE.
NEW YORK: HARPER & BROTHERS, PUBLISHERS, 82 CLIFF STREET 1851.
Entered, according to Act of Congress, in the year one thousand eight hundred and fifty-one, byHarper & Brothers. in the Clerk's Office of the District Court of the Southern District of New York.
PREFACE.
The pleasure and profit which the translator has received from the great work here presented, have induced him to lay it before his fellow-teachers and students of Mathematics in a more accessible form than that in which it has hitherto appeared. The want of a comprehensive map of the wide region of mathematical science—a bird's-eye view of its leading features, and of the true bearings and relations of all its parts—is felt by every thoughtful student. He is like the visitor to a great city, who gets no just idea of its extent and situation till he has seen it from some commanding eminence. To have a panoramic view of the whole district—presenting at one glance all the parts in due co-ordination, and the darkest nooks clearly shown—is invaluable to either traveller or student. It is this which has been most perfectly accomplished for mathematical science by the author whose work is here presented.
Clearness and depth, comprehensiveness and precision, have never, perhaps, been so remarkably united as in Auguste Comte. He views his subject from an elevation which gives to each part of the complex whole its true position and value, while his telescopic glance loses none of the needful details, and not only itself pierces to the heart of the matter, but converts its opaqueness into such transparent crystal, that other eyes are enabled to see as deeply into it as his own.
Any mathematician who peruses this volume will need no other justification of the high opinion here expressed; but others may appreciate the following endorsements of well-known authorities. Mill, in his "Logic," calls the work of M. Comte "by far the greatest yet produced on the Philosophy of the sciences;" and adds, "of this admirable work, one of the most admirable portions is that in which he may truly be said to have created the Philosophy of the higher Mathematics:" Morell, in his "Speculative Philosophy of Europe," says, "The classification given of the sciences at large, and their regular order of development, is unquestionably a master-piece of scientific thinking, as simple as it is comprehensive;" and Lewes, in his "Biographical History of Philosophy," names Comte "the Bacon of the nineteenth century," and says, "I unhesitatingly record my conviction that this is the greatest work of our age."
The complete work of M. Comte—his "Cours de Philosophie Positive"—fills six large octavo volumes, of six or seven hundred pages each, two thirds of the first volume comprising the purely mathematical portion. The great bulk of the "Course" is the probable cause of the fewness of those to whom even this section of it is known. Its presentation in its present form is therefore felt by the translator to be a most useful contribution to mathematical progress in this country. The comprehensiveness of the style of the author—grasping all possible forms of an idea in one Briarean sentence, armed at all points against leaving any opening for mistake or forgetfulness—occasionally verges upon cumbersomeness and formality. The translator has, therefore, sometimes taken the liberty of breaking up or condensing a long sentence, and omitting a few passages not absolutely necessary, or referring to the peculiar "Positive philosophy" of the author; but he has generally aimed at a conscientious fidelity to the original. It has often been difficult to retain its fine shades and subtile distinctions of meaning, and, at the same time, replace the peculiarly appropriate French idioms by corresponding English ones. The attempt, however, has always been made, though, when the best course has been at all doubtful, the language of the original has been followed as closely as possible, and, when necessary, smoothness and grace have been unhesitatingly sacrificed to the higher attributes of clearness and precision.
Some forms of expression may strike the reader as unusual, but they have been retained because they were characteristic, not of the mere language of the original, but of its spirit. When a great thinker has clothed his conceptions in phrases which are singular even in his own tongue, he who professes to translate him is bound faithfully to preserve such forms of speech, as far as is practicable; and this has been here done with respect to such peculiarities of expression as belong to the author, not as a foreigner, but as an individual—not because he writes in French, but because he is Auguste Comte.
The young student of Mathematics should not attempt to read the whole of this volume at once, but should peruse each portion of it in connexion with the temporary subject of his special study: the first chapter of the first book, for example, while he is studying Algebra; the first chapter of the second book, when he has made some progress in Geometry; and so with the rest. Passages which are obscure at the first reading will brighten up at the second; and as his own studies cover a larger portion of the field of Mathematics, he will see more and more clearly their relations to one another, and to those which he is next to take up. For this end he is urgently recommended to obtain a perfect familiarity with the "Analytical Table of Contents," which maps out the whole subject, the grand divisions of which are also indicated in the Tabular View facing the title-page. Corresponding heads will be found in the body of the work, the principal divisions being in small capitals, and the subdivisions in Italics. For these details the translator alone is responsible.
ANALYTICAL TABLE OF CONTENTS.
INTRODUCTION.
Page
GENERAL CONSIDERATIONS ON MATHEMATICAL SCIENCE
17
The Object of Mathematics
18
Measuring Magnitudes
18
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
36
Its Limitations
37
BOOK I. ANALYSIS.
CHAPTER I.
Page
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
80
Imaginary Quantities
81
Negative Quantities
81
The Principle of Homogeneity
84
CHAPTER III.
TRANSCENDENTAL ANALYSIS:
its different conceptions
88
Preliminary Remarks
88
Its early History
89
Method of Leibnitz
91
Infinitely small Elements
91
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
108
An extension of ordinary Analysis
108
Example
: Tangents
109
Fundamental 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
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 Field
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
195
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
1⁰. Different Questions with respect to the same Figure
204
2⁰. Similar Questions with respect to different Figures
204
Geometry of the Ancients
204
Geometry of the Moderns
206
Superiority of the Modern
207
The Ancient the base of the Modern
209
CHAPTER II.
ANCIENT OR SYNTHETIC GEOMETRY
212
Its proper Extent
212
Lines; Polygons; Polyhedrons
212
Not to be farther restricted
213
Improper Application of Analysis
214
Attempted Demonstrations of Axioms
216
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
232
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
THEPHILOSOPHY OF MATHEMATICS.
INTRODUCTION.
GENERAL CONSIDERATIONS.
Although Mathematical Science is the most ancient and the most perfect of all, yet the general idea which we ought to form of it has not yet been clearly determined. Its definition and its principal divisions have remained till now vague and uncertain. Indeed the plural name—"The Mathematics"—by which we commonly designate it, would alone suffice to indicate the want of unity in the common conception of it.
In truth, it was not till the commencement of the last century that the different fundamental conceptions which constitute this great science were each of them sufficiently developed to permit the true spirit of the whole to manifest itself with clearness. Since that epoch the attention of geometers has been too exclusively absorbed by the special perfecting of the different branches, and by the application which they have made of them to the most important laws of the universe, to allow them to give due attention to the general system of the science.
But at the present time the progress of the special departments is no longer so rapid as to forbid the contemplation of the whole. The science of mathematics is now sufficiently developed, both in itself and as to its most essential application, to have arrived at that state of consistency in which we ought to strive to arrange its different parts in a single system, in order to prepare for new advances. We may even observe that the last important improvements of the science have directly paved the way for this important philosophical operation, by impressing on its principal parts a character of unity which did not previously exist.
To form a just idea of the object of mathematical science, we may start from the indefinite and meaningless definition of it usually given, in calling it "The science of magnitudes," or, which is more definite, "The science which has for its object the measurement of magnitudes." Let us see how we can rise from this rough sketch (which is singularly deficient in precision and depth, though, at bottom, just) to a veritable definition, worthy of the importance, the extent, and the difficulty of the science.
THE OBJECT OF MATHEMATICS.
Measuring Magnitudes. The question of measuring a magnitude in itself presents to the mind no other idea than that of the simple direct comparison of this magnitude with another similar magnitude, supposed to be known, which it takes for the unit of comparison among all others of the same kind. According to this definition, then, the science of mathematics—vast and profound as it is with reason reputed to be—instead of being an immense concatenation of prolonged mental labours, which offer inexhaustible occupation to our intellectual activity, would seem to consist of a simple series of mechanical processes for obtaining directly the ratios of the quantities to be measured to those by which we wish to measure them, by the aid of operations of similar character to the superposition of lines, as practiced by the carpenter with his rule.
The error of this definition consists in presenting as direct an object which is almost always, on the contrary, very indirect. The direct measurement of a magnitude, by superposition or any similar process, is most frequently an operation quite impossible for us to perform; so that if we had no other means for determining magnitudes than direct comparisons, we should be obliged to renounce the knowledge of most of those which interest us.
Difficulties. The force of this general observation will be understood if we limit ourselves to consider specially the particular case which evidently offers the most facility—that of the measurement of one straight line by another. This comparison, which is certainly the most simple which we can conceive, can nevertheless scarcely ever be effected directly. In reflecting on the whole of the conditions necessary to render a line susceptible of a direct measurement, we see that most frequently they cannot be all fulfilled at the same time. The first and the most palpable of these conditions—that of being able to pass over the line from one end of it to the other, in order to apply the unit of measurement to its whole length—evidently excludes at once by far the greater part of the distances which interest us the most; in the first place, all the distances between the celestial bodies, or from any one of them to the earth; and then, too, even the greater number of terrestrial distances, which are so frequently inaccessible. But even if this first condition be found to be fulfilled, it is still farther necessary that the length be neither too great nor too small, which would render a direct measurement equally impossible. The line must also be suitably situated; for let it be one which we could measure with the greatest facility, if it were horizontal, but conceive it to be turned up vertically, and it becomes impossible to measure it.
The difficulties which we have indicated in reference to measuring lines, exist in a very much greater degree in the measurement of surfaces, volumes, velocities, times, forces, &c. It is this fact which makes necessary the formation of mathematical science, as we are going to see; for the human mind has been compelled to renounce, in almost all cases, the direct measurement of magnitudes, and to seek to determine them indirectly, and it is thus that it has been led to the creation of mathematics.
General Method. The general method which is constantly employed, and evidently the only one conceivable, to ascertain magnitudes which do not admit of a direct measurement, consists in connecting them with others which are susceptible of being determined immediately, and by means of which we succeed in discovering the first through the relations which subsist between the two. Such is the precise object of mathematical science viewed as a whole. In order to form a sufficiently extended idea of it, we must consider that this indirect determination of magnitudes may be indirect in very different degrees. In a great number of cases, which are often the most important, the magnitudes, by means of which the principal magnitudes sought are to be determined, cannot themselves be measured directly, and must therefore, in their turn, become the subject of a similar question, and so on; so that on many occasions the human mind is obliged to establish a long series of intermediates between the system of unknown magnitudes which are the final objects of its researches, and the system of magnitudes susceptible of direct measurement, by whose means we finally determine the first, with which at first they appear to have no connexion.
Illustrations. Some examples will make clear any thing which may seem too abstract in the preceding generalities.
1. Falling Bodies. Let us consider, in the first place, a natural phenomenon, very simple, indeed, but which may nevertheless give rise to a mathematical question, really existing, and susceptible of actual applications—the phenomenon of the vertical fall of heavy bodies.
The mind the most unused to mathematical conceptions, in observing this phenomenon, perceives at once that the two quantities which it presents—namely, the height from which a body has fallen, and the time of its fall—are necessarily connected with each other, since they vary together, and simultaneously remain fixed; or, in the language of geometers, that they are "functions" of each other. The phenomenon, considered under this point of view, gives rise then to a mathematical question, which consists in substituting for the direct measurement of one of these two magnitudes, when it is impossible, the measurement of the other. It is thus, for example, that we may determine indirectly the depth of a precipice, by merely measuring the time that a heavy body would occupy in falling to its bottom, and by suitable procedures this inaccessible depth will be known with as much precision as if it was a horizontal line placed in the most favourable circumstances for easy and exact measurement. On other occasions it is the height from which a body has fallen which it will be easy to ascertain, while the time of the fall could not be observed directly; then the same phenomenon would give rise to the inverse question, namely, to determine the time from the height; as, for example, if we wished to ascertain what would be the duration of the vertical fall of a body falling from the moon to the earth.
In this example the mathematical question is very simple, at least when we do not pay attention to the variation in the intensity of gravity, or the resistance of the fluid which the body passes through in its fall. But, to extend the question, we have only to consider the same phenomenon in its greatest generality, in supposing the fall oblique, and in taking into the account all the principal circumstances. Then, instead of offering simply two variable quantities connected with each other by a relation easy to follow, the phenomenon will present a much greater number; namely, the space traversed, whether in a vertical or horizontal direction; the time employed in traversing it; the velocity of the body at each point of its course; even the intensity and the direction of its primitive impulse, which may also be viewed as variables; and finally, in certain cases (to take every thing into the account), the resistance of the medium and the intensity of gravity. All these different quantities will be connected with one another, in such a way that each in its turn may be indirectly determined by means of the others; and this will present as many distinct mathematical questions as there may be co-existing magnitudes in the phenomenon under consideration. Such a very slight change in the physical conditions of a problem may cause (as in the above example) a mathematical research, at first very elementary, to be placed at once in the rank of the most difficult questions, whose complete and rigorous solution surpasses as yet the utmost power of the human intellect.
2. Inaccessible Distances. Let us take a second example from geometrical phenomena. Let it be proposed to determine a distance which is not susceptible of direct measurement; it will be generally conceived as making part of a figure, or certain system of lines, chosen in such a way that all its other parts may be observed directly; thus, in the case which is most simple, and to which all the others may be finally reduced, the proposed distance will be considered as belonging to a triangle, in which we can determine directly either another side and two angles, or two sides and one angle. Thence-forward, the knowledge of the desired distance, instead of being obtained directly, will be the result of a mathematical calculation, which will consist in deducing it from the observed elements by means of the relation which connects it with them. This calculation will become successively more and more complicated, if the parts which we have supposed to be known cannot themselves be determined (as is most frequently the case) except in an indirect manner, by the aid of new auxiliary systems, the number of which, in great operations of this kind, finally becomes very considerable. The distance being once determined, the knowledge of it will frequently be sufficient for obtaining new quantities, which will become the subject of new mathematical questions. Thus, when we know at what distance any object is situated, the simple observation of its apparent diameter will evidently permit us to determine indirectly its real dimensions, however inaccessible it may be, and, by a series of analogous investigations, its surface, its volume, even its weight, and a number of other properties, a knowledge of which seemed forbidden to us.
3. Astronomical Facts. It is by such calculations that man has been able to ascertain, not only the distances from the planets to the earth, and, consequently, from each other, but their actual magnitude, their true figure, even to the inequalities of their surface; and, what seemed still more completely hidden from us, their respective masses, their mean densities, the principal circumstances of the fall of heavy bodies on the surface of each of them, &c.
By the power of mathematical theories, all these different results, and many others relative to the different classes of mathematical phenomena, have required no other direct measurements than those of a very small number of straight lines, suitably chosen, and of a greater number of angles. We may even say, with perfect truth, so as to indicate in a word the general range of the science, that if we did not fear to multiply calculations unnecessarily, and if we had not, in consequence, to reserve them for the determination of the quantities which could not be measured directly, the determination of all the magnitudes susceptible of precise estimation, which the various orders of phenomena can offer us, could be finally reduced to the direct measurement of a single straight line and of a suitable number of angles.
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 coexistent 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. Now there is evidently no phenomenon which cannot give rise to considerations of this kind; whence results the naturally indefinite extent and even the rigorous logical universality of mathematical science. We shall seek farther on to circumscribe as exactly as possible its real extension.
The preceding explanations establish clearly the propriety of the name employed to designate the science which we are considering. This denomination, which has taken to-day so definite a meaning by itself signifies simply science in general. Such a designation, rigorously exact for the Greeks, who had no other real science, could be retained by the moderns only to indicate the mathematics as the science, beyond all others—the science of sciences.
Indeed, every true science has for its object the determination of certain phenomena by means of others, in accordance with the relations which exist between them. Every science consists in the co-ordination of facts; if the different observations were entirely isolated, there would be no science. We may even say, in general terms, that science is essentially destined to dispense, so far as the different phenomena permit it, with all direct observation, by enabling us to deduce from the smallest possible number of immediate data the greatest possible number of results. Is not this the real use, whether in speculation or in action, of the laws which we succeed in discovering among natural phenomena? Mathematical science, in this point of view, merely pushes to the highest possible degree the same kind of researches which are pursued, in degrees more or less inferior, by every real science in its respective sphere.
ITS TWO FUNDAMENTAL DIVISIONS.
We have thus far viewed mathematical science only as a whole, without paying any regard to its divisions. We must now, in order to complete this general view, and to form a just idea of the philosophical character of the science, consider its fundamental division. The secondary divisions will be examined in the following chapters.
This principal division, which we are about to investigate, can be truly rational, and derived from the real nature of the subject, only so far as it spontaneously presents itself to us, in making the exact analysis of a complete mathematical question. We will, therefore, having determined above what is the general object of mathematical labours, now characterize with precision the principal different orders of inquiries, of which they are constantly composed.
Their different Objects. The complete solution of every mathematical question divides itself necessarily into two parts, of natures essentially distinct, and with relations invariably determinate. We have seen that every mathematical inquiry has for its object to determine unknown magnitudes, according to the relations between them and known magnitudes. Now for this object, it is evidently necessary, in the first place, to ascertain with precision the relations which exist between the quantities which we are considering. This first branch of inquiries constitutes that which I call the concrete part of the solution. When it is finished, the question changes; it is now reduced to a pure question of numbers, consisting simply in determining unknown numbers, when we know what precise relations connect them with known numbers. This second branch of inquiries is what I call the abstract part of the solution. Hence follows the fundamental division of general mathematical science into two great sciences—ABSTRACT MATHEMATICS, and CONCRETE MATHEMATICS.
This analysis may be observed in every complete mathematical question, however simple or complicated it may be. A single example will suffice to make it intelligible.
Taking up again the phenomenon of the vertical fall of a heavy body, and considering the simplest case, we see that in order to succeed in determining, by means of one another, the height whence the body has fallen, and the duration of its fall, we must commence by discovering the exact relation of these two quantities, or, to use the language of geometers, the equation which exists between them. Before this first research is completed, every attempt to determine numerically the value of one of these two magnitudes from the other would evidently be premature, for it would have no basis. It is not enough to know vaguely that they depend on one another—which every one at once perceives—but it is necessary to determine in what this dependence consists. This inquiry may be very difficult, and in fact, in the present case, constitutes incomparably the greater part of the problem. The true scientific spirit is so modern, that no one, perhaps, before Galileo, had ever remarked the increase of velocity which a body experiences in its fall: a circumstance which excludes the hypothesis, towards which our mind (always involuntarily inclined to suppose in every phenomenon the most simple functions, without any other motive than its greater facility in conceiving them) would be naturally led, that the height was proportional to the time. In a word, this first inquiry terminated in the discovery of the law of Galileo.
When this concrete part is completed, the inquiry becomes one of quite another nature. Knowing that the spaces passed through by the body in each successive second of its fall increase as the series of odd numbers, we have then a problem purely numerical and abstract; to deduce the height from the time, or the time from the height; and this consists in finding that the first of these two quantities, according to the law which has been established, is a known multiple of the second power of the other; from which, finally, we have to calculate the value of the one when that of the other is given.
In this example the concrete question is more difficult than the abstract one. The reverse would be the case if we considered the same phenomenon in its greatest generality, as I have done above for another object. According to the circumstances, sometimes the first, sometimes the second, of these two parts will constitute the principal difficulty of the whole question; for the mathematical law of the phenomenon may be very simple, but very difficult to obtain, or it may be easy to discover, but very complicated; so that the two great sections of mathematical science, when we compare them as wholes, must be regarded as exactly equivalent in extent and in difficulty, as well as in importance, as we shall show farther on, in considering each of them separately.
Their different Natures. These two parts, essentially distinct in their object, as we have just seen, are no less so with regard to the nature of the inquiries of which they are composed.
The first should be called concrete, since it evidently depends on the character of the phenomena considered, and must necessarily vary when we examine new phenomena; while the second is completely independent of the nature of the objects examined, and is concerned with only the numerical relations which they present, for which reason it should be called abstract. The same relations may exist in a great number of different phenomena, which, in spite of their extreme diversity, will be viewed by the geometer as offering an analytical question susceptible, when studied by itself, of being resolved once for all. Thus, for instance, the same law which exists between the space and the time of the vertical fall of a body in a vacuum, is found again in many other phenomena which offer no analogy with the first nor with each other; for it expresses the relation between the surface of a spherical body and the length of its diameter; it determines, in like manner, the decrease of the intensity of light or of heat in relation to the distance of the objects lighted or heated, &c. The abstract part, common to these different mathematical questions, having been treated in reference to one of these, will thus have been treated for all; while the concrete part will have necessarily to be again taken up for each question separately, without the solution of any one of them being able to give any direct aid, in that connexion, for the solution of the rest.
The abstract part of mathematics is, then, general in its nature; the concrete part, special.
To present this comparison under a new point of view, we may say concrete mathematics has a philosophical character, which is essentially experimental, physical, phenomenal; while that of abstract mathematics is purely logical, rational. The concrete part of every mathematical question is necessarily founded on the consideration of the external world, and could never be resolved by a simple series of intellectual combinations. The abstract part, on the contrary, when it has been very completely separated, can consist only of a series of logical deductions, more or less prolonged; for if we have once found the equations of a phenomenon, the determination of the quantities therein considered, by means of one another, is a matter for reasoning only, whatever the difficulties may be. It belongs to the understanding alone to deduce from these equations results which are evidently contained in them, although perhaps in a very involved manner, without there being occasion to consult anew the external world; the consideration of which, having become thenceforth foreign to the subject, ought even to be carefully set aside in order to reduce the labour to its true peculiar difficulty. The abstract part of mathematics is then purely instrumental, and is only an immense and admirable extension of natural logic to a certain class of deductions. On the other hand, geometry and mechanics, which, as we shall see presently, constitute the concrete part, must be viewed as real natural sciences, founded on observation, like all the rest, although the extreme simplicity of their phenomena permits an infinitely greater degree of systematization, which has sometimes caused a misconception of the experimental character of their first principles.
We see, by this brief general comparison, how natural and profound is our fundamental division of mathematical science.
We have now to circumscribe, as exactly as we can in this first sketch, each of these two great sections.
CONCRETE MATHEMATICS.
Concrete Mathematics having for its object the discovery of the equations of phenomena, it would seem at first that it must be composed of as many distinct sciences as we find really distinct categories among natural phenomena. But we are yet very far from having discovered mathematical laws in all kinds of phenomena; we shall even see, presently, that the greater part will very probably always hide themselves from our investigations. In reality, in the present condition of the human mind, there are directly but two great general classes of phenomena, whose equations we constantly know; these are, firstly, geometrical, and, secondly, mechanical phenomena. Thus, then, the concrete part of mathematics is composed of Geometry and Rational Mechanics.
This is sufficient, it is true, to give to it a complete character of logical universality, when we consider all phenomena from the most elevated point of view of natural philosophy. In fact, if all the parts of the universe were conceived as immovable, we should evidently have only geometrical phenomena to observe, since all would be reduced to relations of form, magnitude, and position; then, having regard to the motions which take place in it, we would have also to consider mechanical phenomena. Hence the universe, in the statical point of view, presents only geometrical phenomena; and, considered dynamically, only mechanical phenomena. Thus geometry and mechanics constitute the two fundamental natural sciences, in this sense, that all natural effects may be conceived as simple necessary results, either of the laws of extension or of the laws of motion.
