24 THE TRUE CONCEPTION OF 'PASSAGE TO THE LIMIT'

The consideration of 'passage to the limit', we said above, is necessary, if not to the practical applications of the infinitesimal method, then at least to its theoretical justification, and this justification is precisely the only thing that concerns us here, for simple practical rules of calculation that succeed in an as it were 'empirical' manner and without our knowing exactly why, are obviously of no interest from our point of view. Undoubtedly, in order to perform the calculations, and even to follow them through to the end, there is in fact no need to raise the question as to whether the variable reaches its limit, or how it can do so; nevertheless, if it does not reach its limit, such a calculus will only have value as a simple calculus of approximation. It is true that here we are dealing with an indefinite approximation, since the very nature of infinitesimal quantities allows the error to be rendered as small as one might wish, without it being possible to eliminate it entirely, since despite the indefinite decrease, these same infinitesimal quantities never become nothing. Perhaps one might say that, practically speaking, this is the equivalent of a perfectly rigorous calculation; but, besides the fact that this is not what matters to us, such is not in question, can the indefinite approximation itself retain meaning if, with respect to the desired results, one no longer envisages variables, but rather fixed and determined quantities? Under these conditions, one cannot escape the following alternative as far as the results are concerned: either the limit is not reached, in which case the infinitesimal calculus is then only the least crude of various methods of approximation; or the limit is reached, in which case one is dealing with a method that is truly rigorous. But we have seen that limits, by their very definition, can never exactly be reached by variables; how, then, do we have the right to say that they are nonetheless reached? This can be precisely accomplished, not in the course of the calculation, but in the results, since only fixed and determined quantities, like the limit itself, must figure therein, while variables no longer do so; consequently the distinction between variable and fixed quantities, which is a strictly qualitative distinction, moreover, is the only true justification for the rigor of the infinitesimal calculus, as we have already said. Thus, let us repeat it again, a limit cannot be reached within a variation, and as a term of the latter; it is not the final value the variable takes on, and the idea of a continuous variation arriving at any 'final value', or 'final state', would be as incomprehensible and contradictory as that of an indefinite sequence arriving at a 'final term', or of the division of a continuum arriving at 'final elements'. Therefore a limit does not belong to the sequence of successive values of the variable, but it falls outside of this series, and that is why we said that 'passage to the limit' essentially implies a discontinuity. Were it otherwise, we would be faced with an indefinitude that could be exhausted analytically, and this can never happen. Here the distinction we previously established in this regard takes on its full significance, for we find ourselves in one of those cases in which it is a question of reaching the limits of a given indefinite quantity, according to an expression we have already used; it is therefore not without reason that the same word 'limit' comes up again, but with another, more specialized meaning, in the particular case we shall now consider. The limit of a variable must truly limit, in the general sense of the word, the indefinitude of the states or possible modifications comprised within the definition of this variable; and it is precisely for this reason that it must necessarily be located outside of that which it limits. There can be no question of exhausting this indefinitude through the very course of the variation by which it is constituted; in reality, it is a question of passing beyond the domain of this variation, in which the limit is not contained, and this is the result that is obtained, not analytically and by degrees, but synthetically and in a single stroke, in a manner that is as it were 'sudden' and corresponds to the discontinuity produced in passing from variable to fixed quantities. ^[1] Limits pertain essentially to the domain of fixed quantities; this is why 'passage to the limit' logically demands the simultaneous consideration of two different and as it were superimposed modalities existing within quantity; it is nothing other than passage to the higher modality, in which what exists only as the state of a simple tendency in the lower modality, is fully realized; to use the Aristotelian terminology, it is a passage from potentiality to actuality, which assuredly has nothing in common with the simple 'compensation of errors' that Carnot had in mind. The mathematical notion of the limit implies by its very definition a character of stability and equilibrium, which applies to permanent and definite things, and which obviously cannot be realized by quantities insofar as one considers them in the lower of the two modalities, as essentially variable; the limit can therefore never be reached gradually, but only immediately by the passage from one modality to the other, which alone allows the omission of all intermediate stages, since it includes and embraces synthetically all of their indefinitude; in this way, what was and could only be but a tendency within the variable, is affirmed and fixed in a real and definite result. Otherwise, 'passage to the limit' would always be an illogicality pure and simple, for it is obvious that, insofar as one keeps to the domain of variables, one cannot obtain the fixity appropriate to limits, since the quantity previously considered to be variable would precisely have to lose its transitory and contingent character. The state of variable quantities is indeed an eminently transitory and as it were imperfect state, since it is only the expression of a 'becoming', as we have likewise found to be the case with the idea at the root of indefinitude itself, which, moreover, is closely linked to the state of variation. The calculation will thus only be perfect, or truly completed, when it arrives at results in which there is no longer anything variable or indefinite, but only fixed and determined quantities; and we have already seen how this can be applied through analogical transposition beyond the quantitative order-which latter will then have no more than a symbolic value-and will extend even to that which directly concerns the metaphysical 'realization' of being.