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). [Sidenote: _Decreasing utility of mathematics in the most lofty spheres of the real._]