Empiricism has certainly sought to cure its own insufficiencies, of which it was more or less conscious, and _evolutionist positivism_ must be numbered among these attempts. This form proposed to correct the anti-historical character of positivism by providing a _history_ of reality. But this history was always based upon empirical presuppositions, and was therefore a history of classifications, not of concrete reality; an extravagant caricature of the philosophy of becoming, from whose breast comes History rightly and truly so-called. Another attempt was that of _rationalist positivism,_ which sought to check the degeneration of positivism toward dualism, sentimentalism and superstition, by appealing to the absolute rights of reason. But this reason is nevertheless always empirical reason, limited to certain series of facts, extrinsic, classificatory, unintelligent. Absolute authority can well be attributed to it in words, but such an attribution does not confer the power of exercising it. This kind of positivism, therefore, meets in our day with favour in freemasonry (at least of the Franco-Italian sort). This is a sect, which is annoying, chiefly because, heedless of facts, it preserves and defends the habit of making use of empty formulas and phrases, and because when it has insulted some priestly vestment, it believes that it has successfully destroyed superstition and obscurantism in man, or when it has declaimed about liberty, it imagines that by this slight effort, liberty has been won and established. True _reason_ abhors _rationalism,_ if it be rationalism of that sort. [Sidenote: _Mathematicism_] _Mathematicism_ is much rarer than empiricism, because the confusion between thinking and calculating is less easy than that between thinking and classifying. Owing to its rarity and paradoxical character, mathematicism has something aristocratic about it, resembling in this the other extreme error, of æstheticism; whereas the intermediate error, empiricism, just because of its mediocrity, is popular and indeed vulgar. [Sidenote: _Symbolical mathematics._] We cannot properly consider as mathematicism that form of philosophy which appeared in antiquity as _Pythagoreanism_ and _Neopythagoreanism_ and has reappeared in our days as a doctrine of the mathematical relations of the universe and the harmony of the world. In this conception, numbers are not numbers, but symbols; the numerical relations are not arithmetical, but æsthetic. The pretended mathematical philosophers of this type are neither philosophers nor mathematicians, nor are they arbitrary combiners of these two methods. They would be better described as poets or semi-poets. [Sidenote: _Mathematics as demonstrative form of philosophy._]