5 QUESTIONS RAISED BY THE INFINITESIMAL METHOD
When Leibnitz first presented the infinitesimal method, [1] and even again in several other works that followed, [2] he particularly emphasized the uses and applications of the new calculus, in keeping with the modern tendency to attribute more importance to the practical applications of science than to science itself, as such; it would be difficult to say whether this tendency truly existed in Leibnitz, or whether this manner of presenting his method was only a sort of concession on his part. Be that as it may, in order to justify a method it certainly does not suffice to show the advantages it might have over other, previously accepted methods, or the conveniences it might furnish practically for calculation, nor even the results it might in fact have given; and the adversaries of the infinitesimal method did not fail to make use of this, and it was only their objections that persuaded Leibnitz to explain the principles, and even the origins, of his method. It is very possible, moreover, that on this last point he might never have spoken at all, but ultimately this is of little importance, for very often the occasional causes of a discovery
are in themselves only rather insignificant circumstances; at any rate, of what he wrote on the subject, [3] all that interests us is the fact that he passed from a consideration of the 'assignable' differences existing between numbers to a consideration of the 'unassignable' differences that can be conceived of between geometric magnitudes by reason of their continuity, and that he also attached great importance to this order, as being so to speak 'demanded by the nature of things'. From this it follows that for him infinitesimal quantities do not naturally appear directly to us, but only as a result of passing from a consideration of the variability of discontinuous quantity to that of continuous quantity, and from the application of the first to the measurement of the second.
What exactly is the meaning of these infinitesimal quantities Leibnitz was reproached for using without having first defined what he meant by them, and did this meaning allow him to regard his calculus as absolutely rigorous, or on the contrary merely as a method of approximation? To respond to these two questions would, by that very fact, be to resolve the most important objections raised against him; but unfortunately he himself never responded very clearly, and even his various attempts to do so do not always seem in complete accord with one another. In this connection it is worth noting that generally speaking Leibnitz was in the habit of explaining the same thing differently according to the audience he was addressing; we would certainly not hold this behavior against him, which is irritating only for systematic minds, for in principle he was only conforming to an initiatic and, more particularly, Rosicrucian precept according to which it is fitting to speak to each in his own language; only he sometimes happened to apply the precept rather poorly. Indeed, if it is obviously possible to clothe the same truth in different expressions, it is understood that this be done without ever distorting or diminishing it, being always careful to refrain from any manner of speaking that could give rise to false conceptions; in this regard Leibnitz failed in a number of instances. [4] Thus, he pushed
the idea of 'accommodation' to the point of sometimes seeming to justify those who wished to see in his calculus merely a method of approximation, for at times he presented it as being no more than a sort of abridged version of the ancients' 'method of exhaustion', useful for facilitating calculations but yielding results that have to be verified by this other method if a rigorous demonstration is desired; and it is nevertheless quite certain that this was not fundamentally what he thought, but that, in reality, he saw in it much more than a simple expedient intended to shorten calculations.
Leibnitz frequently declared that infinitesimal quantities cannot but be 'incomparable', but as to the precise meaning in which this word is to be understood, he gave an explanation that is not only rather unsatisfying, but even most regrettable, for it could not but provide ammunition to his adversaries, who, moreover, did not fail to avail themselves of it; here, again, he was certainly not expressing what he truly thought, and we can see in this another example of an excessive 'accommodation', yet more serious than the first, that would substitute erroneous views for 'adapted' expressions of the truth. Leibnitz writes:
One need not take the infinite here rigorously, but only in the manner in which one says in optics that the rays of the sun come from an infinitely distant point, and may thus be treated as parallel. And when there are several degrees of the infinite or of the infinitely small, this is like the terrestrial globe being regarded as a point with respect to the distance of the fixed stars, and a ball we might take in hand being again a point in comparison with the semi-diameter of the terrestrial globe, such that the distance of the fixed stars is like an infinite infinitude with respect to the diameter of the ball. For instead of the infinite or the infinitely small, one takes quantities as great or as small as is necessary for the error to be less than a given error, such that one differs from the style of Archimedes only in expression, which in our method is more direct, and more conformable with the art of invention. [5]
It was unfailingly pointed out to Leibnitz that however small the terrestrial globe might be with respect to the heavens, or a grain of sand in relation to the terrestrial globe, they are nonetheless fixed and determined quantities, and if one of these quantities can be regarded as practically negligible in comparison with the other, this is nevertheless only a simple approximation; his reply was that he had only wished to 'avoid the subtleties' and to 'make the reasoning evident to all, [6] which fully confirms our interpretation, and which, furthermore, is already a sort of manifestation of the 'popularizing' tendency of modern scholars. What is most extraordinary is that he was able to write afterwards: 'At any rate, there was not the slightest thing that should have caused anyone to imagine that I indeed meant a very small, but always fixed and determined, quantity,' to which he added: 'Besides, I had already written some years ago to Bernoulli of Groningen that the infinites and infinitely small might be taken for fictions, similar to imaginary roots, [7] without thereby harming our calculus, these fictions being useful and founded in reality. [8] Moreover, it seems that he never did understand exactly in what respect his comparison was flawed, for he presents it again in the same terms about ten years later; [9] but, at any rate since he expressly declared that his intention had not been to present the infinitesimal quantities as determined, we must conclude from this that, for him, the meaning of the comparison amounts to the following: a grain of sand, though not infinitely small, can, however, without appreciable disadvantage, be considered as such in relation to the earth, and thus there is no need to envisage the infinitely small 'rigorously'-they may even be regarded as mere fictions if one so desires; but however one takes them, such a consideration is nonetheless manifestly unsuitable to give any other idea of the infinitesimal calculus than that of a simple calculus of approximation, which would assuredly have been insufficient in the eyes of Leibnitz himself.