Page:Mind (New Series) Volume 12.djvu/206

 192 B. EUSSELL : After a long Introduction on Descartes' critique of mathematical .and scientific knowledge, the body of the work is divided into four parts, dealing respectively with Mathematics, Mechanics, Meta- physics, and the growth of Leibniz's system. All knowledge, the Introduction asserts, is for Descartes really mathematics, and magnitude is the fundamental concept of mathematics. More- over, magnitude is essentially connected with space, and is by Descartes almost identified with extension. By attempting to reduce everything to space, he failed to give due weight to time, and so failed to found Dynamics : his notion of force is only valid for Statics. In his notion of substance he failed to hold fast its deepest meaning, which is (p. 60) "to postulate as a condition of the object the thorough-going unity of knowledge ". In Part I. the first chapter deals with the relation of mathe- matics and logic. Leibniz assigned to Aristotle the merit of having first written mathematically outside mathematics. All certain knowledge, Leibniz says, incorporates logical forms (of which, however, some are not Aristotelian). Dr. Cassirer, in a true Kantian spirit, remarks that this view is problematical, if Algebra and Geometry contain an independent contribution to method : to reduce mathematics to logic is to loosen its connexion with the sciences of experience andnature (pp. 107-108). To this we must reply that it is now known, with all the certainty of the multiplica- tion-table, that Leibniz is in the right and Kant in the wrong on this point : Algebra and Geometry do not contain an independent contribution to method ; and as for the connexion of mathematics with the sciences of experience, this is precisely the same as that of logic with the said sciences, i.e., they cannot violate mathe- matics, which is concerned wholly and solely with logical implica- tions, but also they all of them, including the geometry of actual space, require premisses which mathematics cannot supply. This conclusion, originally suggested by non-Euclidean geometry, has now, by the labours of Weierstrass, Cantor and Peano, been wholly removed from the region of dubitable hypothesis. The author proceeds to discuss the relative importance of defini- tions and identical principles in Leibniz's proofs of axioms. He decides (p. 109) that the true principles are definitions, while the identical propositions are mere auxiliaries. I do not know whether this view is more tenable than the opposite : Leibniz's opinions could not be clear, as either alternative was absurd, for an identical proposition, if there were any such thing, would be perfectly trivial, while a definition is merely a statement of a symbolic abbreviation, giving information as to symbols, not as to what is symbolised. But here Leibniz's doctrine as to the possibility of ideas becomes relevant his theory that all (complex) ideas involve a judgment. Dr. Cassirer speaks as though, in this notion, there were for Leibniz no difficulties : the mutual compatibility of all simple ideas is not mentioned. This is an instance (of which others might be given) 4>f failure to apprehend the reasons why Leibniz's system cannot