25 CONCLUSION
There is no need to stress the importance that the issues examined in the course of this study present from the strictly mathematical point of view, as they contain the solution to all the problems that have been raised concerning the infinitesimal method, whether regarding its true significance or its rigor. The necessary and sufficient condition for arriving at this solution is nothing other than the strict application of true principles, but these are precisely the principles of which modern mathematicians, along with all other profane scholars, are completely ignorant. Ultimately this ignorance is the sole reason for so many of the discussions that, under these conditions, can be pursued indefinitely without ever reaching any valid conclusion, but on the contrary only further confuse the question and multiply the confusions, as the quarrel between the 'finitists' and 'infinitists' shows only too well. Nevertheless all such discussions would have been cut short quite easily had the true notion of the metaphysical Infinite and the fundamental distinction between the Infinite and the indefinite been set forth clearly and before all else. On this subject Leibnitz himself, who unlike those who have come after him at least had the merit of frankly facing certain questions, too often says things that are hardly metaphysical, and are sometimes even as clearly anti-metaphysical, as are the ordinary speculations of most modern philosophers; thus it is again this same lack of principles that prevented him from responding to his adversaries in a satisfying and as it were definitive way, and which consequently opened the door to all subsequent discussions. No doubt one can say with Carnot that, 'if Leibnitz was mistaken, it was solely in raising doubts as to the exactitude of his own analysis,
so far as he really had these doubts'; [1] but even if ultimately he did not, he was nonetheless unable to demonstrate its exactitude rigorously since his conception of continuity, which is most certainly neither metaphysical nor logical, prevented him from making the necessary distinctions and consequently from formulating a precise notion of the limit, which is as we have shown of chief importance for the foundation of the infinitesimal method.
From all of this one can see what significance the consideration of principles can have even for a specialized science considered in and of itself, and without any intention of going further in support of this science than the relative and contingent domain to which the principles are immediately applicable. Of course, this is what the moderns totally misunderstand, readily boasting as they do that with their profane conception of science they have rendered the latter independent of metaphysics, and likewise of theology, [2] while the truth of the matter is that they have thereby only deprived it of all real value as far as knowledge is concerned. In addition, once one understands the need to link science back to principles, it goes without saying that there should no longer be any reason to stop there, and one will quite naturally be led back to the traditional conception according to which a particular science, whatever it might be, is less valuable for what it is in itself than for the possibility of using it as a 'support' for elevating oneself to knowledge of a higher order. [3] Our intention here has been to present by way of a characteristic example an idea of precisely what it would be possible to do, at least in certain cases, to restore to science, mutilated and distorted by profane conceptions, its real value and scope, both from the point of view of the relative knowledge it represents directly, and from that of the higher knowledge to which it can lead through analogical transposition. In this last respect we have been able to see, notably,
what may be drawn from notions such as those of integration and 'passage to the limit'. Moreover, it should be said that, more than any other science, mathematics thus furnishes a particularly apt symbolism for the expression of metaphysical truths to the extent that the latter are expressible, as those familiar with some of our other works are aware. This is why mathematical symbolism is used so frequently, whether from the traditional point of view in general, or from the initiatic point of view in particular. [4] But it is of course understood that in order for this to be so it is above all necessary that these sciences be rid of the various errors and confusions that have been introduced by the false views of the moderns, and we should be happy if the present work is at least able to contribute in some way to this end.