The philosophy of mathematics · Auguste Comte
THE CALCULUS OF VALUES, OR ARITHMETIC.
_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 principal difficulties of analysis. 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 analysis, such at least as it exists at present. There can be no new arithmetical operations without the creation of really new analytical 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 classed as such, because we are accustomed to treat them only as incidental in the midst of a body of analytical researches more or less elevated, the too high opinion commonly formed of the influence of signs being again the principal 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 class, 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 quantities 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 homogeneous, 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 analysis. 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.
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_ + _cz²_ + _dz³_ + _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 quantities, 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 analytical investigations. I will therefore no longer delay on the _Calculus of Values_, but pass immediately to the examination of the fundamental division of the _Calculus of Functions_.
Ideas discussed in this chapter
Text from Project Gutenberg, public domain in the United States.