The philosophy of mathematics Part 2

In the historical development of mathematical science since the time of Descartes, the advances of its abstract portion have always been determined by those of its concrete portion; but it is none the less necessary, in order to conceive the science in a manner truly logical, to consider the Calculus in all its princ.i.p.al branches before proceeding to the philosophical study of Geometry and Mechanics. Its a.n.a.lytical theories, more simple and more general than those of concrete mathematics, are in themselves essentially independent of the latter; while these, on the contrary, have, by their nature, a continual need of the former, without the aid of which they could make scarcely any progress. Although the princ.i.p.al conceptions of a.n.a.lysis retain at present some very perceptible traces of their geometrical or mechanical origin, they are now, however, mainly freed from that primitive character, which no longer manifests itself except in some secondary points; so that it is possible (especially since the labours of Lagrange) to present them in a dogmatic exposition, by a purely abstract method, in a single and continuous system. It is this which will be undertaken in the present and the five following chapters, limiting our investigations to the most general considerations upon each princ.i.p.al branch of the science of the calculus.

The definite object of our researches in concrete mathematics being the discovery of the _equations_ which express the mathematical laws of the phenomenon under consideration, and these equations const.i.tuting the true starting point of the calculus, which has for its object to obtain from them the determination of certain quant.i.ties by means of others, I think it indispensable, before proceeding any farther, to go more deeply than has been customary into that fundamental idea of _equation_, the continual subject, either as end or as beginning, of all mathematical labours. Besides the advantage of circ.u.mscribing more definitely the true field of a.n.a.lysis, there will result from it the important consequence of tracing in a more exact manner the real line of demarcation between the concrete and the abstract part of mathematics, which will complete the general exposition of the fundamental division established in the introductory chapter.

THE TRUE IDEA OF AN EQUATION.

We usually form much too vague an idea of what an _equation_ is, when we give that name to every kind of relation of equality between _any_ two functions of the magnitudes which we are considering. For, though every equation is evidently a relation of equality, it is far from being true that, reciprocally, every relation of equality is a veritable _equation_, of the kind of those to which, by their nature, the methods of a.n.a.lysis are applicable.

This want of precision in the logical consideration of an idea which is so fundamental in mathematics, brings with it the serious inconvenience of rendering it almost impossible to explain, in general terms, the great and fundamental difficulty which we find in establis.h.i.+ng the relation between the concrete and the abstract, and which stands out so prominently in each great mathematical question taken by itself. If the meaning of the word _equation_ was truly as extended as we habitually suppose it to be in our definition of it, it is not apparent what great difficulty there could really be, in general, in establis.h.i.+ng the equations of any problem whatsoever; for the whole would thus appear to consist in a simple question of form, which ought never even to exact any great intellectual efforts, seeing that we can hardly conceive of any precise relation which is not immediately a certain relation of equality, or which cannot be readily brought thereto by some very easy transformations.

Thus, when we admit every species of _functions_ into the definition of _equations_, we do not at all account for the extreme difficulty which we almost always experience in putting a problem into an equation, and which so often may be compared to the efforts required by the a.n.a.lytical elaboration of the equation when once obtained. In a word, the ordinary abstract and general idea of an _equation_ does not at all correspond to the real meaning which geometers attach to that expression in the actual development of the science. Here, then, is a logical fault, a defect of correlation, which it is very important to rectify.

_Division of Functions into Abstract and Concrete._ To succeed in doing so, I begin by distinguis.h.i.+ng two sorts of _functions_, _abstract_ or a.n.a.lytical functions, and _concrete_ functions. The first alone can enter into veritable _equations_. We may, therefore, henceforth define every _equation_, in an exact and sufficiently profound manner, as a relation of equality between two _abstract_ functions of the magnitudes under consideration. In order not to have to return again to this fundamental definition, I must add here, as an indispensable complement, without which the idea would not be sufficiently general, that these abstract functions may refer not only to the magnitudes which the problem presents of itself, but also to all the other auxiliary magnitudes which are connected with it, and which we will often be able to introduce, simply as a mathematical artifice, with the sole object of facilitating the discovery of the equations of the phenomena. I here antic.i.p.ate summarily the result of a general discussion of the highest importance, which will be found at the end of this chapter. We will now return to the essential distinction of functions as abstract and concrete.

This distinction may be established in two ways, essentially different, but complementary of each other, _a priori_ and _a posteriori_; that is to say, by characterizing in a general manner the peculiar nature of each species of functions, and then by making the actual enumeration of all the abstract functions at present known, at least so far as relates to the elements of which they are composed.

_a priori_, the functions which I call _abstract_ are those which express a manner of dependence between magnitudes, which can be conceived between numbers alone, without there being need of indicating any phenomenon whatever in which it is realized. I name, on the other hand, _concrete_ functions, those for which the mode of dependence expressed cannot be defined or conceived except by a.s.signing a determinate case of physics, geometry, mechanics, &c., in which it actually exists.

Most functions in their origin, even those which are at present the most purely _abstract_, have begun by being _concrete_; so that it is easy to make the preceding distinction understood, by citing only the successive different points of view under which, in proportion as the science has become formed, geometers have considered the most simple a.n.a.lytical functions. I will indicate powers, for example, which have in general become abstract functions only since the labours of Vieta and Descartes.

The functions _x_, _x_, which in our present a.n.a.lysis are so well conceived as simply _abstract_, were, for the geometers of antiquity, perfectly _concrete_ functions, expressing the relation of the superficies of a square, or the volume of a cube to the length of their side. These had in their eyes such a character so exclusively, that it was only by means of the geometrical definitions that they discovered the elementary algebraic properties of these functions, relating to the decomposition of the variable into two parts, properties which were at that epoch only real theorems of geometry, to which a numerical meaning was not attached until long afterward.

I shall have occasion to cite presently, for another reason, a new example, very suitable to make apparent the fundamental distinction which I have just exhibited; it is that of circular functions, both direct and inverse, which at the present time are still sometimes concrete, sometimes abstract, according to the point of view under which they are regarded.

_a posteriori_, the general character which renders a function abstract or concrete having been established, the question as to whether a certain determinate function is veritably abstract, and therefore susceptible of entering into true a.n.a.lytical equations, becomes a simple question of fact, inasmuch as we are going to enumerate all the functions of this species.

_Enumeration of Abstract Functions._ At first view this enumeration seems impossible, the distinct a.n.a.lytical functions being infinite in number. But when we divide them into _simple_ and _compound_, the difficulty disappears; for, though the number of the different functions considered in mathematical a.n.a.lysis is really infinite, they are, on the contrary, even at the present day, composed of a very small number of elementary functions, which can be easily a.s.signed, and which are evidently sufficient for deciding the abstract or concrete character of any given function; which will be of the one or the other nature, according as it shall be composed exclusively of these simple abstract functions, or as it shall include others.

We evidently have to consider, for this purpose, only the functions of a single variable, since those relative to several independent variables are constantly, by their nature, more or less _compound_.

Let _x_ be the independent variable, _y_ the correlative variable which depends upon it. The different simple modes of abstract dependence, which we can now conceive between _y_ and _x_, are expressed by the ten following elementary formulas, in which each function is coupled with its _inverse_, that is, with that which would be obtained from the direct function by referring _x_ to _y_, instead of referring _y_ to _x_.

FUNCTION. ITS NAME.

1st couple {1 _y_ = _a_ + _x_ _Sum._ {2 _y_ = _a_ - _x_ _Difference._

2d couple {1 _y_ = _ax_ _Product._ {2 _y_ = _a/x_ _Quotient._

3d couple {1 _y_ = _x^a_ _Power._ {2 _y_ = _[ath root]x_ _Root._

4th couple {1 _y_ = _a^x_ _Exponential._ {2 _y_ = _[log a]x_ _Logarithmic._

5th couple {1 _y_ = sin. _x_ _Direct Circular._ {2 _y_ = arc(sin. = _x_). _Inverse Circular._[3]

[Footnote 3: With the view of increasing as much as possible the resources and the extent (now so insufficient) of mathematical a.n.a.lysis, geometers count this last couple of functions among the a.n.a.lytical elements. Although this inscription is strictly legitimate, it is important to remark that circular functions are not exactly in the same situation as the other abstract elementary functions. There is this very essential difference, that the functions of the four first couples are at the same time simple and abstract, while the circular functions, which may manifest each character in succession, according to the point of view under which they are considered and the manner in which they are employed, never present these two properties simultaneously.

Some other concrete functions may be usefully introduced into the number of a.n.a.lytical elements, certain conditions being fulfilled.

It is thus, for example, that the labours of M. Legendre and of M.

Jacobi on _elliptical_ functions have truly enlarged the field of a.n.a.lysis; and the same is true of some definite integrals obtained by M. Fourier in the theory of heat.]

Such are the elements, very few in number, which directly compose all the abstract functions known at the present day. Few as they are, they are evidently sufficient to give rise to an infinite number of a.n.a.lytical combinations.

No rational consideration rigorously circ.u.mscribes, _a priori_, the preceding table, which is only the actual expression of the present state of the science. Our a.n.a.lytical elements are at the present day more numerous than they were for Descartes, and even for Newton and Leibnitz: it is only a century since the last two couples have been introduced into a.n.a.lysis by the labours of John Bernouilli and Euler.

Doubtless new ones will be hereafter admitted; but, as I shall show towards the end of this chapter, we cannot hope that they will ever be greatly multiplied, their real augmentation giving rise to very great difficulties.

We can now form a definite, and, at the same time, sufficiently extended idea of what geometers understand by a veritable _equation_. This explanation is especially suited to make us understand how difficult it must be really to establish the _equations_ of phenomena, since we have effectually succeeded in so doing only when we have been able to conceive the mathematical laws of these phenomena by the aid of functions entirely composed of only the mathematical elements which I have just enumerated. It is clear, in fact, that it is then only that the problem becomes truly abstract, and is reduced to a pure question of numbers, these functions being the only simple relations which we can conceive between numbers, considered by themselves. Up to this period of the solution, whatever the appearances may be, the question is still essentially concrete, and does not come within the domain of the _calculus_. Now the fundamental difficulty of this pa.s.sage from the _concrete_ to the _abstract_ in general consists especially in the insufficiency of this very small number of a.n.a.lytical elements which we possess, and by means of which, nevertheless, in spite of the little real variety which they offer us, we must succeed in representing all the precise relations which all the different natural phenomena can manifest to us. Considering the infinite diversity which must necessarily exist in this respect in the external world, we easily understand how far below the true difficulty our conceptions must frequently be found, especially if we add that as these elements of our a.n.a.lysis have been in the first place furnished to us by the mathematical consideration of the simplest phenomena, we have, _a priori_, no rational guarantee of their necessary suitableness to represent the mathematical law of every other cla.s.s of phenomena. I will explain presently the general artifice, so profoundly ingenious, by which the human mind has succeeded in diminis.h.i.+ng, in a remarkable degree, this fundamental difficulty which is presented by the relation of the concrete to the abstract in mathematics, without, however, its having been necessary to multiply the number of these a.n.a.lytical elements.

THE TWO PRINc.i.p.aL DIVISIONS OF THE CALCULUS.

The preceding explanations determine with precision the true object and the real field of abstract mathematics. I must now pa.s.s to the examination of its princ.i.p.al divisions, for thus far we have considered the calculus as a whole.

The first direct consideration to be presented on the composition of the science of the _calculus_ consists in dividing it, in the first place, into two princ.i.p.al branches, to which, for want of more suitable denominations, I will give the names of _Algebraic calculus_, or _Algebra_, and of _Arithmetical calculus_, or _Arithmetic_; but with the caution to take these two expressions in their most extended logical acceptation, in the place of the by far too restricted meaning which is usually attached to them.

The complete solution of every question of the _calculus_, from the most elementary up to the most transcendental, is necessarily composed of two successive parts, whose nature is essentially distinct. In the first, the object is to transform the proposed equations, so as to make apparent the manner in which the unknown quant.i.ties are formed by the known ones: it is this which const.i.tutes the _algebraic_ question. In the second, our object is to _find the values_ of the formulas thus obtained; that is, to determine directly the values of the numbers sought, which are already represented by certain explicit functions of given numbers: this is the _arithmetical_ question.[4] It is apparent that, in every solution which is truly rational, it necessarily follows the algebraical question, of which it forms the indispensable complement, since it is evidently necessary to know the mode of generation of the numbers sought for before determining their actual values for each particular case. Thus the stopping-place of the algebraic part of the solution becomes the starting point of the arithmetical part.

[Footnote 4: Suppose, for example, that a question gives the following equation between an unknown magnitude x, and two known magnitudes, _a_ and _b_,

_x_ + 3_ax_ = 2_b_,

as is the case in the problem of the trisection of an angle. We see at once that the dependence between _x_ on the one side, and _ab_ on the other, is completely determined; but, so long as the equation preserves its primitive form, we do not at all perceive in what manner the unknown quant.i.ty is derived from the data. This must be discovered, however, before we can think of determining its value.

Such is the object of the algebraic part of the solution. When, by a series of transformations which have successively rendered that derivation more and more apparent, we have arrived at presenting the proposed equation under the form

_x_ = ?(_b_ + v(_b_ + _a_)) + ?(_b_ - v(_b_ + _a_)),

the work of _algebra_ is finished; and even if we could not perform the arithmetical operations indicated by that formula, we would nevertheless have obtained a knowledge very real, and often very important. The work of _arithmetic_ will now consist in taking that formula for its starting point, and finding the number _x_ when the values of the numbers _a_ and _b_ are given.]

We thus see that the _algebraic_ calculus and the _arithmetical_ calculus differ essentially in their object. They differ no less in the point of view under which they regard quant.i.ties; which are considered in the first as to their _relations_, and in the second as to their _values_. The true spirit of the calculus, in general, requires this distinction to be maintained with the most severe exact.i.tude, and the line of demarcation between the two periods of the solution to be rendered as clear and distinct as the proposed question permits. The attentive observation of this precept, which is too much neglected, may be of much a.s.sistance, in each particular question, in directing the efforts of our mind, at any moment of the solution, towards the real corresponding difficulty. In truth, the imperfection of the science of the calculus obliges us very often (as will be explained in the next chapter) to intermingle algebraic and arithmetical considerations in the solution of the same question. But, however impossible it may be to separate clearly the two parts of the labour, yet the preceding indications will always enable us to avoid confounding them.

In endeavouring to sum up as succinctly as possible the distinction just established, we see that ALGEBRA may be defined, in general, as having for its object the _resolution of equations_; taking this expression in its full logical meaning, which signifies the transformation of _implicit_ functions into equivalent _explicit_ ones. In the same way, ARITHMETIC may be defined as destined to _the determination of the values of functions_. Henceforth, therefore, we will briefly say that ALGEBRA is the _Calculus of Functions_, and ARITHMETIC the _Calculus of Values_.

We can now perceive how insufficient and even erroneous are the ordinary definitions. Most generally, the exaggerated importance attributed to Signs has led to the distinguis.h.i.+ng the two fundamental branches of the science of the Calculus by the manner of designating in each the subjects of discussion, an idea which is evidently absurd in principle and false in fact. Even the celebrated definition given by Newton, characterizing _Algebra_ as _Universal Arithmetic_, gives certainly a very false idea of the nature of algebra and of that of arithmetic.[5]

[Footnote 5: I have thought that I ought to specially notice this definition, because it serves as the basis of the opinion which many intelligent persons, unacquainted with mathematical science, form of its abstract part, without considering that at the time of this definition mathematical a.n.a.lysis was not sufficiently developed to enable the general character of each of its princ.i.p.al parts to be properly apprehended, which explains why Newton could at that time propose a definition which at the present day he would certainly reject.]

Having thus established the fundamental division of the calculus into two princ.i.p.al branches, I have now to compare in general terms the extent, the importance, and the difficulty of these two sorts of calculus, so as to have hereafter to consider only the _Calculus of Functions_, which is to be the princ.i.p.al subject of our study.

THE CALCULUS OF VALUES, OR ARITHMETIC.