The philosophy of mathematics Part 3

_Its Extent._ The _Calculus of Values, or Arithmetic_, would appear, at first view, to present a field as vast as that of _algebra_, since it would seem to admit as many distinct questions as we can conceive different algebraic formulas whose values are to be determined. But a very simple reflection will show the difference. Dividing functions into _simple_ and _compound_, it is evident that when we know how to determine the _value_ of simple functions, the consideration of compound functions will no longer present any difficulty. In the algebraic point of view, a compound function plays a very different part from that of the elementary functions of which it consists, and from this, indeed, proceed all the princ.i.p.al difficulties of a.n.a.lysis. But it is very different with the Arithmetical Calculus. Thus the number of truly distinct arithmetical operations is only that determined by the number of the elementary abstract functions, the very limited list of which has been given above. The determination of the values of these ten functions necessarily gives that of all the functions, infinite in number, which are considered in the whole of mathematical a.n.a.lysis, such at least as it exists at present. There can be no new arithmetical operations without the creation of really new a.n.a.lytical elements, the number of which must always be extremely small. The field of _arithmetic_ is, then, by its nature, exceedingly restricted, while that of algebra is rigorously indefinite.

It is, however, important to remark, that the domain of the _calculus of values_ is, in reality, much more extensive than it is commonly represented; for several questions truly _arithmetical_, since they consist of determinations of values, are not ordinarily cla.s.sed as such, because we are accustomed to treat them only as incidental in the midst of a body of a.n.a.lytical researches more or less elevated, the too high opinion commonly formed of the influence of signs being again the princ.i.p.al cause of this confusion of ideas. Thus not only the construction of a table of logarithms, but also the calculation of trigonometrical tables, are true arithmetical operations of a higher kind. We may also cite as being in the same cla.s.s, although in a very distinct and more elevated order, all the methods by which we determine directly the value of any function for each particular system of values attributed to the quant.i.ties on which it depends, when we cannot express in general terms the explicit form of that function. In this point of view the _numerical_ solution of questions which we cannot resolve algebraically, and even the calculation of "Definite Integrals," whose general integrals we do not know, really make a part, in spite of all appearances, of the domain of _arithmetic_, in which we must necessarily comprise all that which has for its object the _determination of the values of functions_. The considerations relative to this object are, in fact, constantly h.o.m.ogeneous, whatever the _determinations_ in question, and are always very distinct from truly _algebraic_ considerations.

To complete a just idea of the real extent of the calculus of values, we must include in it likewise that part of the general science of the calculus which now bears the name of the _Theory of Numbers_, and which is yet so little advanced. This branch, very extensive by its nature, but whose importance in the general system of science is not very great, has for its object the discovery of the properties inherent in different numbers by virtue of their values, and independent of any particular system of numeration. It forms, then, a sort of _transcendental arithmetic_; and to it would really apply the definition proposed by Newton for algebra.

The entire domain of arithmetic is, then, much more extended than is commonly supposed; but this _calculus of values_ will still never be more than a point, so to speak, in comparison with the _calculus of functions_, of which mathematical science essentially consists. This comparative estimate will be still more apparent from some considerations which I have now to indicate respecting the true nature of arithmetical questions in general, when they are more profoundly examined.

_Its true Nature._ In seeking to determine with precision in what _determinations of values_ properly consist, we easily recognize that they are nothing else but veritable _transformations_ of the functions to be valued; transformations which, in spite of their special end, are none the less essentially of the same nature as all those taught by a.n.a.lysis. In this point of view, the _calculus of values_ might be simply conceived as an appendix, and a particular application of the _calculus of functions_, so that _arithmetic_ would disappear, so to say, as a distinct section in the whole body of abstract mathematics.

In order thoroughly to comprehend this consideration, we must observe that, when we propose to determine the _value_ of an unknown number whose mode of formation is given, it is, by the mere enunciation of the arithmetical question, already defined and expressed under a certain form; and that in _determining its value_ we only put its expression under another determinate form, to which we are accustomed to refer the exact notion of each particular number by making it re-enter into the regular system of _numeration_. The determination of values consists so completely of a simple _transformation_, that when the primitive expression of the number is found to be already conformed to the regular system of numeration, there is no longer any determination of value, properly speaking, or, rather, the question is answered by the question itself. Let the question be to add the two numbers _one_ and _twenty_, we answer it by merely repeating the enunciation of the question,[6] and nevertheless we think that we have _determined the value_ of the sum.

This signifies that in this case the first expression of the function had no need of being transformed, while it would not be thus in adding twenty-three and fourteen, for then the sum would not be immediately expressed in a manner conformed to the rank which it occupies in the fixed and general scale of numeration.

[Footnote 6: This is less strictly true in the English system of numeration than in the French, since "twenty-one" is our more usual mode of expressing this number.]

To sum up as comprehensively as possible the preceding views, we may say, that to determine the _value_ of a number is nothing else than putting its primitive expression under the form

_a_ + _bz_ + _cz_ + _dz_ + _ez4_ ... . . + _pz^m_,

_z_ being generally equal to 10, and the coefficients _a_, _b_, _c_, _d_, &c., being subjected to the conditions of being whole numbers less than _z_; capable of becoming equal to zero; but never negative. Every arithmetical question may thus be stated as consisting in putting under such a form any abstract function whatever of different quant.i.ties, which are supposed to have themselves a similar form already. We might then see in the different operations of arithmetic only simple particular cases of certain algebraic transformations, excepting the special difficulties belonging to conditions relating to the nature of the coefficients.

It clearly follows that abstract mathematics is essentially composed of the _Calculus of Functions_, which had been already seen to be its most important, most extended, and most difficult part. It will henceforth be the exclusive subject of our a.n.a.lytical investigations. I will therefore no longer delay on the _Calculus of Values_, but pa.s.s immediately to the examination of the fundamental division of the _Calculus of Functions_.

THE CALCULUS OF FUNCTIONS, OR ALGEBRA.

_Principle of its Fundamental Division._ We have determined, at the beginning of this chapter, wherein properly consists the difficulty which we experience in putting mathematical questions into _equations_.

It is essentially because of the insufficiency of the very small number of a.n.a.lytical elements which we possess, that the relation of the concrete to the abstract is usually so difficult to establish. Let us endeavour now to appreciate in a philosophical manner the general process by which the human mind has succeeded, in so great a number of important cases, in overcoming this fundamental obstacle to _The establishment of Equations_.

1. _By the Creation of new Functions._ In looking at this important question from the most general point of view, we are led at once to the conception of one means of facilitating the establishment of the equations of phenomena. Since the princ.i.p.al obstacle in this matter comes from the too small number of our a.n.a.lytical elements, the whole question would seem to be reduced to creating new ones. But this means, though natural, is really illusory; and though it might be useful, it is certainly insufficient.

In fact, the creation of an elementary abstract function, which shall be veritably new, presents in itself the greatest difficulties. There is even something contradictory in such an idea; for a new a.n.a.lytical element would evidently not fulfil its essential and appropriate conditions, if we could not immediately _determine its value_. Now, on the other hand, how are we to _determine the value_ of a new function which is truly _simple_, that is, which is not formed by a combination of those already known? That appears almost impossible. The introduction into a.n.a.lysis of another elementary abstract function, or rather of another couple of functions (for each would be always accompanied by its _inverse_), supposes then, of necessity, the simultaneous creation of a new arithmetical operation, which is certainly very difficult.

If we endeavour to obtain an idea of the means which the human mind employs for inventing new a.n.a.lytical elements, by the examination of the procedures by the aid of which it has actually conceived those which we already possess, our observations leave us in that respect in an entire uncertainty, for the artifices which it has already made use of for that purpose are evidently exhausted. To convince ourselves of it, let us consider the last couple of simple functions which has been introduced into a.n.a.lysis, and at the formation of which we have been present, so to speak, namely, the fourth couple; for, as I have explained, the fifth couple does not strictly give veritable new a.n.a.lytical elements. The function _a^x_, and, consequently, its inverse, have been formed by conceiving, under a new point of view, a function which had been a long time known, namely, powers--when the idea of them had become sufficiently generalized. The consideration of a power relatively to the variation of its exponent, instead of to the variation of its base, was sufficient to give rise to a truly novel simple function, the variation following then an entirely different route. But this artifice, as simple as ingenious, can furnish nothing more; for, in turning over in the same manner all our present a.n.a.lytical elements, we end in only making them return into one another.

We have, then, no idea as to how we could proceed to the creation of new elementary abstract functions which would properly satisfy all the necessary conditions. This is not to say, however, that we have at present attained the effectual limit established in that respect by the bounds of our intelligence. It is even certain that the last special improvements in mathematical a.n.a.lysis have contributed to extend our resources in that respect, by introducing within the domain of the calculus certain definite integrals, which in some respects supply the place of new simple functions, although they are far from fulfilling all the necessary conditions, which has prevented me from inserting them in the table of true a.n.a.lytical elements. But, on the whole, I think it unquestionable that the number of these elements cannot increase except with extreme slowness. It is therefore not from these sources that the human mind has drawn its most powerful means of facilitating, as much as is possible, the establishment of equations.

2. _By the Conception of Equations between certain auxiliary Quant.i.ties._ This first method being set aside, there remains evidently but one other: it is, seeing the impossibility of finding directly the equations between the quant.i.ties under consideration, to seek for corresponding ones between other auxiliary quant.i.ties, connected with the first according to a certain determinate law, and from the relation between which we may return to that between the primitive magnitudes.

Such is, in substance, the eminently fruitful conception, which the human mind has succeeded in establishing, and which const.i.tutes its most admirable instrument for the mathematical explanation of natural phenomena; the _a.n.a.lysis_, called _transcendental_.

As a general philosophical principle, the auxiliary quant.i.ties, which are introduced in the place of the primitive magnitudes, or concurrently with them, in order to facilitate the establishment of equations, might be derived according to any law whatever from the immediate elements of the question. This conception has thus a much more extensive reach than has been commonly attributed to it by even the most profound geometers.

It is extremely important for us to view it in its whole logical extent, for it will perhaps be by establishing a general mode of _derivation_ different from that to which we have thus far confined ourselves (although it is evidently very far from being the only possible one) that we shall one day succeed in essentially perfecting mathematical a.n.a.lysis as a whole, and consequently in establishing more powerful means of investigating the laws of nature than our present processes, which are unquestionably susceptible of becoming exhausted.

But, regarding merely the present const.i.tution of the science, the only auxiliary quant.i.ties habitually introduced in the place of the primitive quant.i.ties in the _Transcendental a.n.a.lysis_ are what are called, 1, _infinitely small_ elements, the _differentials_ (of different orders) of those quant.i.ties, if we regard this a.n.a.lysis in the manner of LEIBNITZ; or, 2, the _fluxions_, the limits of the ratios of the simultaneous increments of the primitive quant.i.ties compared with one another, or, more briefly, the _prime and ultimate ratios_ of these increments, if we adopt the conception of NEWTON; or, 3, the _derivatives_, properly so called, of those quant.i.ties, that is, the coefficients of the different terms of their respective increments, according to the conception of LAGRANGE.

These three princ.i.p.al methods of viewing our present transcendental a.n.a.lysis, and all the other less distinctly characterized ones which have been successively proposed, are, by their nature, necessarily identical, whether in the calculation or in the application, as will be explained in a general manner in the third chapter. As to their relative value, we shall there see that the conception of Leibnitz has thus far, in practice, an incontestable superiority, but that its logical character is exceedingly vicious; while that the conception of Lagrange, admirable by its simplicity, by its logical perfection, by the philosophical unity which it has established in mathematical a.n.a.lysis (till then separated into two almost entirely independent worlds), presents, as yet, serious inconveniences in the applications, by r.e.t.a.r.ding the progress of the mind. The conception of Newton occupies nearly middle ground in these various relations, being less rapid, but more rational than that of Leibnitz; less philosophical, but more applicable than that of Lagrange.

This is not the place to explain the advantages of the introduction of this kind of auxiliary quant.i.ties in the place of the primitive magnitudes. The third chapter is devoted to this subject. At present I limit myself to consider this conception in the most general manner, in order to deduce therefrom the fundamental division of the _calculus of functions_ into two systems essentially distinct, whose dependence, for the complete solution of any one mathematical question, is invariably determinate.

In this connexion, and in the logical order of ideas, the transcendental a.n.a.lysis presents itself as being necessarily the first, since its general object is to facilitate the establishment of equations, an operation which must evidently precede the _resolution_ of those equations, which is the object of the ordinary a.n.a.lysis. But though it is exceedingly important to conceive in this way the true relations of these two systems of a.n.a.lysis, it is none the less proper, in conformity with the regular usage, to study the transcendental a.n.a.lysis after ordinary a.n.a.lysis; for though the former is, at bottom, by itself logically independent of the latter, or, at least, may be essentially disengaged from it, yet it is clear that, since its employment in the solution of questions has always more or less need of being completed by the use of the ordinary a.n.a.lysis, we would be constrained to leave the questions in suspense if this latter had not been previously studied.

_Corresponding Divisions of the Calculus of Functions._ It follows from the preceding considerations that the _Calculus of Functions_, or _Algebra_ (taking this word in its most extended meaning), is composed of two distinct fundamental branches, one of which has for its immediate object the _resolution_ of equations, when they are directly established between the magnitudes themselves which are under consideration; and the other, starting from equations (generally much easier to form) between quant.i.ties indirectly connected with those of the problem, has for its peculiar and constant destination the deduction, by invariable a.n.a.lytical methods, of the corresponding equations between the direct magnitudes which we are considering; which brings the question within the domain of the preceding calculus.

The former calculus bears most frequently the name of _Ordinary a.n.a.lysis_, or of _Algebra_, properly so called. The second const.i.tutes what is called the _Transcendental a.n.a.lysis_, which has been designated by the different denominations of _Infinitesimal Calculus_, _Calculus of Fluxions and of Fluents_, _Calculus of Vanishing Quant.i.ties_, the _Differential and Integral Calculus_, &c., according to the point of view in which it has been conceived.

In order to remove every foreign consideration, I will propose to name it CALCULUS OF INDIRECT FUNCTIONS, giving to ordinary a.n.a.lysis the t.i.tle of CALCULUS OF DIRECT FUNCTIONS. These expressions, which I form essentially by generalizing and epitomizing the ideas of Lagrange, are simply intended to indicate with precision the true general character belonging to each of these two forms of a.n.a.lysis.

Having now established the fundamental division of mathematical a.n.a.lysis, I have next to consider separately each of its two parts, commencing with the _Calculus of Direct Functions_, and reserving more extended developments for the different branches of the _Calculus of Indirect Functions_.

CHAPTER II.

ORDINARY a.n.a.lYSIS, OR ALGEBRA.

The _Calculus of direct Functions_, or _Algebra_, is (as was shown at the end of the preceding chapter) entirely sufficient for the solution of mathematical questions, when they are so simple that we can form directly the equations between the magnitudes themselves which we are considering, without its being necessary to introduce in their place, or conjointly with them, any system of auxiliary quant.i.ties _derived_ from the first. It is true that in the greatest number of important cases its use requires to be preceded and prepared by that of the _Calculus of indirect Functions_, which is intended to facilitate the establishment of equations. But, although algebra has then only a secondary office to perform, it has none the less a necessary part in the complete solution of the question, so that the _Calculus of direct Functions_ must continue to be, by its nature, the fundamental base of all mathematical a.n.a.lysis. We must therefore, before going any further, consider in a general manner the logical composition of this calculus, and the degree of development to which it has at the present day arrived.

_Its Object._ The final object of this calculus being the _resolution_ (properly so called) of _equations_, that is, the discovery of the manner in which the unknown quant.i.ties are formed from the known quant.i.ties, in accordance with the _equations_ which exist between them, it naturally presents as many different departments as we can conceive truly distinct cla.s.ses of equations. Its appropriate extent is consequently rigorously indefinite, the number of a.n.a.lytical functions susceptible of entering into equations being in itself quite unlimited, although they are composed of only a very small number of primitive elements.

_Cla.s.sification of Equations._ The rational cla.s.sification of equations must evidently be determined by the nature of the a.n.a.lytical elements of which their numbers are composed; every other cla.s.sification would be essentially arbitrary. Accordingly, a.n.a.lysts begin by dividing equations with one or more variables into two princ.i.p.al cla.s.ses, according as they contain functions of only the first three couples (see the table in chapter i., page 51), or as they include also exponential or circular functions. The names of _Algebraic_ functions and _Transcendental_ functions, commonly given to these two princ.i.p.al groups of a.n.a.lytical elements, are undoubtedly very inappropriate. But the universally established division between the corresponding equations is none the less very real in this sense, that the resolution of equations containing the functions called _transcendental_ necessarily presents more difficulties than those of the equations called _algebraic_. Hence the study of the former is as yet exceedingly imperfect, so that frequently the resolution of the most simple of them is still unknown to us,[7] and our a.n.a.lytical methods have almost exclusive reference to the elaboration of the latter.

[Footnote 7: Simple as may seem, for example, the equation

_a^x_ + _b^x_ = _c^x_,

we do not yet know how to resolve it, which may give some idea of the extreme imperfection of this part of algebra.]

ALGEBRAIC EQUATIONS.

Considering now only these _Algebraic_ equations, we must observe, in the first place, that although they may often contain _irrational_ functions of the unknown quant.i.ties as well as _rational_ functions, we can always, by more or less easy transformations, make the first case come under the second, so that it is with this last that a.n.a.lysts have had to occupy themselves exclusively in order to resolve all sorts of _algebraic_ equations.