Page:Mind (New Series) Volume 8.djvu/349

 THE PHILOSOPHY OF SPINOZA AND LEIBNIZ. 335 continuous. Thus while Leibniz is at one with Spinoza in seeking not mere speculative probability but " demonstra- tion " in philosophy, he is not to be regarded as thinking of demonstration in exactly the same way as Spinoza did. 1 The form of Spinoza's Ethics makes it evident that he regarded demonstration in philosophy as a process analogous to the synthetic method in geometry, which endeavours to apply a canon of pure self-consistency to a variety of given geometrical figures. The aim of the inquiry is to ascertain the properties or qualities of the figures, and a property is shown to belong to a figure when it is proved to be consistent with the definition of that figure. Each kind of figure is treated as a distinct and separate species and their inter- relations are considered in a purely external way. The demonstrations are supposed to be pure, direct deductions from given premisses. But in reality there is a continual reference to experience, to the system of space, certain of the relations of which are expressed by the figures. The proof of each proposition requires a " construction " of some kind to be made, such as the producing of lines or the superposition of figures, and this construction is simply a reference to the unity of the system of space, in which the particular figure is an element (or combination of elements) related to others, and by which all the kinds of figures are ultimately deter- mined. For instance, if you produce two sides of a triangle in order to prove something about its angles, you implicitly recognise that the triangle is not a self-complete system, the properties of which may be directly deduced from its defini- tion, but that it is an element in a surface and that its internal properties are logically dependent on its external relations, or, at least, are in the most intimate connexion with them. Thus the synthetic method in geometry pre- supposes the system of space in its definitions and postulates, without showing how the figures described in the definitions or the right to demand these postulates follow from the nature of space itself. Now the mathematical form of Spinoza's Ethics is modelled upon that of Euclid's Geometry. There are numerous definitions of more or less independent things or ideas. Certain axioms are also assumed as self- evident, and from a combination of the axioms with the definitions the whole philosophy is regarded as necessarily following. The definitions are the substantial part of the 1 Spinoza's demonstrations have, for the most part, the character of reductio ad absurdum. Leibniz writes of them : Ce Spinosa est plein de reveries bien embarassees et ses pretcndues demonstrations de Deo n'en ont pas seuleinent le semblant " (Gerhardt, ii., 133).