Descartes, Spinoza, Method

Spinoza's advocacy of the 'geometric' method of proof implies a rejection of Descartes' method of doubt. The latter is, according to Spinoza, procedurally inadequate, for two main reasons. First, it employs as primitive, i. e. 'certainty' and 'doubt', notions that, at minimum, require definition, if not derivation, and, likewise, 'Mind' and 'Body' are soon introduced without preparation. Second, since the geometric method advances from certain proposition to certain proposition, 'I doubt X' cannot appear in a sound proof. Also unsound, according to Spinoza, is any attempt to prove the existence of anything without the existence of God as its initial premise. Conversely, any proof in which the existence of God is a conclusion is likewise unsound. Furthermore, Spinoza shows that any thought of the thought of X is no more certain than the thought of X, which means that Descartes cannot be both certain that he thinks that he thinks that his body exists and doubt that his body exists. Finally, Spinoza argues that every representation of X is an affirmation of the existence X, which means, first, that Descartes' affirmation of his conclusion that he exists is superfluous, and, second, that Descartes cannot simultaneously think X and doubt the existence of X, which his method seems to entail.