The philosophy of mathematics · Auguste Comte
COMPARATIVE VALUE OF THE THREE METHODS.
If now we endeavour to estimate the comparative value of these three equivalent conceptions, we shall find in each advantages and inconveniences which are peculiar to it, and which still prevent geometers from confining themselves to any one of them, considered as final. _That of Leibnitz._ The conception of Leibnitz presents incontestably, in all its applications, a very marked superiority, by leading in a much more rapid manner, and with much less mental effort, to the formation of equations between the auxiliary magnitudes. It is to its use that we owe the high perfection which has been acquired by all the general theories of geometry and mechanics. Whatever may be the different speculative opinions of geometers with respect to the infinitesimal method, in an abstract point of view, all tacitly agree in employing it by preference, as soon as they have to treat a new question, in order not to complicate the necessary difficulty by this purely artificial obstacle proceeding from a misplaced obstinacy in adopting a less expeditious course. Lagrange himself, after having reconstructed the transcendental analysis on new foundations, has (with that noble frankness which so well suited his genius) rendered a striking and decisive homage to the characteristic properties of the conception of Leibnitz, by following it exclusively in the entire system of his _Méchanique Analytique_. Such a fact renders any comments unnecessary.
But when we consider the conception of Leibnitz in itself and in its logical relations, we cannot escape admitting, with Lagrange, that it is radically vicious in this, that, adopting its own expressions, the notion of infinitely small quantities is a _false idea_, of which it is in fact impossible to obtain a clear conception, however we may deceive ourselves in that matter. Even if we adopt the ingenious idea of the compensation of errors, as above explained, this involves the radical inconvenience of being obliged to distinguish in mathematics two classes of reasonings, those which are perfectly rigorous, and those in which we designedly commit errors which subsequently have to be compensated. A conception which leads to such strange consequences is undoubtedly very unsatisfactory in a logical point of view. To say, as do some geometers, that it is possible in every case to reduce the infinitesimal method to that of limits, the logical character of which is irreproachable, would evidently be to elude the difficulty rather than to remove it; besides, such a transformation almost entirely strips the conception of Leibnitz of its essential advantages of facility and rapidity. Finally, even disregarding the preceding important considerations, the infinitesimal method would no less evidently present by its nature the very serious defect of breaking the unity of abstract mathematics, by creating a transcendental analysis founded on principles so different from those which form the basis of the ordinary analysis. This division of analysis into two worlds almost entirely independent of each other, tends to hinder the formation of truly general analytical conceptions. To fully appreciate the consequences of this, we should have to go back to the state of the science before Lagrange had established a general and complete harmony between these two great sections.
_That of Newton._ Passing now to the conception of Newton, it is evident that by its nature it is not exposed to the fundamental logical objections which are called forth by the method of Leibnitz. The notion of _limits_ is, in fact, remarkable for its simplicity and its precision. In the transcendental analysis presented in this manner, the equations are regarded as exact from their very origin, and the general rules of reasoning are as constantly observed as in ordinary analysis. But, on the other hand, it is very far from offering such powerful resources for the solution of problems as the infinitesimal method. The obligation which it imposes, of never considering the increments of magnitudes separately and by themselves, nor even in their ratios, but only in the limits of those ratios, retards considerably the operations of the mind in the formation of auxiliary equations. We may even say that it greatly embarrasses the purely analytical transformations. Thus the transcendental analysis, considered separately from its applications, is far from presenting in this method the extent and the generality which have been imprinted upon it by the conception of Leibnitz. It is very difficult, for example, to extend the theory of Newton to functions of several independent variables. But it is especially with reference to its applications that the relative inferiority of this theory is most strongly marked.
Several Continental geometers, in adopting the method of Newton as the more logical basis of the transcendental analysis, have partially disguised this inferiority by a serious inconsistency, which consists in applying to this method the notation invented by Leibnitz for the infinitesimal method, and which is really appropriate to it alone. In designating by _dy_/_dx_ that which logically ought, in the theory of limits, to be denoted by _L_(Δ_y_/Δ_x_), and in extending to all the other analytical conceptions this displacement of signs, they intended, undoubtedly, to combine the special advantages of the two methods; but, in reality, they have only succeeded in causing a vicious confusion between them, a familiarity with which hinders the formation of clear and exact ideas of either. It would certainly be singular, considering this usage in itself, that, by the mere means of signs, it could be possible to effect a veritable combination between two theories so distinct as those under consideration. Finally, the method of limits presents also, though in a less degree, the greater inconvenience, which I have above noted in reference to the infinitesimal method, of establishing a total separation between the ordinary and the transcendental analysis; for the idea of _limits_, though clear and rigorous, is none the less in itself, as Lagrange has remarked, a foreign idea, upon which analytical theories ought not to be dependent.
_That of Lagrange._ This perfect unity of analysis, and this purely abstract character of its fundamental notions, are found in the highest degree in the conception of Lagrange, and are found there alone; it is, for this reason, the most rational and the most philosophical of all. Carefully removing every heterogeneous consideration, Lagrange has reduced the transcendental analysis to its true peculiar character, that of presenting a very extensive class of analytical transformations, which facilitate in a remarkable degree the expression of the conditions of various problems. At the same time, this analysis is thus necessarily presented as a simple extension of ordinary analysis; it is only a higher algebra. All the different parts of abstract mathematics, previously so incoherent, have from that moment admitted of being conceived as forming a single system.
Unhappily, this conception, which possesses such fundamental properties, independently of its so simple and so lucid notation, and which is undoubtedly destined to become the final theory of transcendental analysis, because of its high philosophical superiority over all the other methods proposed, presents in its present state too many difficulties in its applications, as compared with the conception of Newton, and still more with that of Leibnitz, to be as yet exclusively adopted. Lagrange himself has succeeded only with great difficulty in rediscovering, by his method, the principal results already obtained by the infinitesimal method for the solution of the general questions of geometry and mechanics; we may judge from that what obstacles would be found in treating in the same manner questions which were truly new and important. It is true that Lagrange, on several occasions, has shown that difficulties call forth, from men of genius, superior efforts, capable of leading to the greatest results. It was thus that, in trying to adapt his method to the examination of the curvature of lines, which seemed so far from admitting its application, he arrived at that beautiful theory of contacts which has so greatly perfected that important part of geometry. But, in spite of such happy exceptions, the conception of Lagrange has nevertheless remained, as a whole, essentially unsuited to applications.
The final result of the general comparison which I have too briefly sketched, is, then, as already suggested, that, in order to really understand the transcendental analysis, we should not only consider it in its principles according to the three fundamental conceptions of Leibnitz, of Newton, and of Lagrange, but should besides accustom ourselves to carry out almost indifferently, according to these three principal methods, and especially according to the first and the last, the solution of all important questions, whether of the pure calculus of indirect functions or of its applications. This is a course which I could not too strongly recommend to all those who desire to judge philosophically of this admirable creation of the human mind, as well as to those who wish to learn to make use of this powerful instrument with success and with facility. In all the other parts of mathematical science, the consideration of different methods for a single class of questions may be useful, even independently of its historical interest, but it is not indispensable; here, on the contrary, it is strictly necessary. Having determined with precision, in this chapter, the philosophical character of the calculus of indirect functions, according to the principal fundamental conceptions of which it admits, we have next to consider, in the following chapter, the logical division and the general composition of this calculus.
Text from Project Gutenberg, public domain in the United States.