Practical convenience suggests the postulates to mathematics; but the purity of the elements that it manipulates gives to them the rigour of demonstrations, the force of truth. It is a curious force, that has a weakness for point of support,—the non-truth of the postulate, and reduces itself to a perpetual tautology, by which it is recorded that what has been granted has been granted. But the rigour of the demonstrations and the arbitrariness of the foundations explain how philosophers have been in turn attracted and repelled by mathematics. Mathematics operating with pure concepts is a true simia philosophiae (as it was said of the devil that he was simia Dei), and philosophers have sometimes seen in it the absoluteness of thought and have saluted it as sister or as the first-born of philosophy. Other philosophers have recognized the devil in that divine form, and have addressed to it the far from pleasant words that saints and ascetics used to employ on similar occasions. Hence mathematics has been accused of not being able to justify its own principles, notwithstanding its rigorous procedure; and of constructing empty formulæ and of leaving the mind vacant. It has been accused of promoting superstition, since the whole of concrete reality lies outside its conventions, an unattainable mystery; and of being too difficult for lofty spirits, just because it is too easy.[2] Gianbattista Vico confessed that having applied himself to the study of Geometry, he did not go beyond the fifth proposition of Euclid, since "that study, proper to minute intellects, is not suitable to minds already made universal by metaphysic."[3] But these accusations are not accusations, and simply confirm the peculiar nature of those spiritual formations, eternal as the nature of the spirit is eternal.
Impossibility of reducing the empirical sciences to mathematics, and empirical limits of the mathematical science of nature.
The nature of mathematics being explained, we can now resume the thread of the narrative, left hanging loose, and discover how inadmissible is the claim for a mathematical science of nature, which should be the true end and the inner soul of the empirical and natural sciences. It is said that this mathematical science presides, as an ideal, over all the particular natural sciences, but it should be added, as an unrealized and unrealizable ideal, and therefore rather an illusion and a mirage than an ideal. It is urged that this ideal has been partially realized, and that therefore nothing prevents its being altogether realized. But, indeed, whoever looks closely will see that it has not been even partially realized, because mathematical formulæ of natural facts are always affected by the empirical and approximate character of the naturalistic concepts which they use, and by the intuitive element upon which these are based. When it is sought to establish in all its rigour the ideal of the mathematical science of nature, it becomes necessary to assume as a point of departure elements that are distinct, but perfectly identical and therefore unthinkable; quantity without quality, which are nothing but those mathematical fictions of which we have spoken. The idea of a mathematical science is thus resolved into the idea simply of mathematics, and the much-vaunted universality of that science is the universal applicability of mathematics, wherever there are things and facts to number, to calculate and to measure. The natural sciences will never lose their inevitable intuitive and historical foundation, whatever progress may be made in the calculus and in the application of the calculus. They will remain, as has been said, descriptive sciences (and this time it has been well said, as it prevents the failure to recognize the intuitive elements, of which they are composed).
Decreasing utility of mathematics in the most lofty spheres of the real.
We have already illustrated the slight perceptibility of differences (or the slight interest that we take in individual differences), as we gradually descend into what is called nature or inferior reality. On this is founded the illusion that nature is invariable and without history. And it also explains why mathematics has seemed more applicable to the globus naturalis than to the globus intellectualis, and in the globus naturalis, to mineralogy more than to zoology, to physics more than to biology. Still, mathematics is equally applicable to the globus intellectualis, as, for instance, in Economics and Statistics. And, on the other hand, it is inapplicable to both spheres, when they are considered in their effective truth and unity as the history of nature or the history of reality, in which nothing is repeated and therefore nothing is equal and identical. Beneath that difference of applicability there is nothing but a consideration of utility. If the grains of sand on which we tread can be considered (although they are not) equal to one another, it happens less frequently that we regard those with whom we associate and act in the same light. Hence the decreasing utility of naturalistic constructions (and of mathematical calculation), as we gradually approach human life and the historical situation in which we find ourselves. Decreasing but never non-existent, for otherwise, neither empirical sciences (grammars, books on moral conduct, psychological types, etc.) nor calculations (statistics, economic calculations, etc.,) would continue in use. A constructor of machines needs little intuition, but much physics and mechanics. A leader of men needs very little mathematics, little empirical science, but much intuitive and perceptive faculty for the vices and value of the human individuals with whom he has to do. But both little and much are empirical determinations; the Spirit, which is the whole spirit in every particular man and at every particular instant of life, is never composed of measurable elements.
[1] Introduction to Philosophy, Italian tr., Vidossich, p. 272.
[2] There is a curious collection of judgments adverse to mathematics in Hamilton, Fragments philosophiques, tr. Plisse, Paris, 1840, pp. 283-370.
[3] Autobiography in Works, Ferrari, 2nd edition, iv. p. 336.