The philosophy of mathematics · Auguste Comte
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 analysis, geometers count this last couple of functions among the analytical 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 analytical 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 analysis; 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 analytical combinations. No rational consideration rigorously circumscribes, _à priori_, the preceding table, which is only the actual expression of the present state of the science. Our analytical 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 analysis 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 passage from the _concrete_ to the _abstract_ in general consists especially in the insufficiency of this very small number of analytical 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 analysis have been in the first place furnished to us by the mathematical consideration of the simplest phenomena, we have, _à priori_, no rational guarantee of their necessary suitableness to represent the mathematical law of every other class of phenomena. I will explain presently the general artifice, so profoundly ingenious, by which the human mind has succeeded in diminishing, 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 analytical elements.
Text from Project Gutenberg, public domain in the United States.