The mathematical sciences Plato has expressly excluded from philosophy. He considers them as helps to philosophical thinking (_Rep._ VII. 526), as necessary steps of knowledge, without which no one can come to philosophy (_Ib._ VI. 510); but mathematics with him is not philosophy, for it assumes its principles or axioms, without at all accounting for them, as though they were manifest to all, a procedure which is not permitted to pure science; it also serves itself for its demonstrations, with illustrative figures, although it does not treat of these, but of that which they represent to the understanding (_Ib._). Plato thus places mathematics midway between a correct opinion and science, clearer than the one, but more obscure than the other. (_Ib._ VII. 533.) IV. THE PLATONIC DIALECTICS. 1. CONCEPTION OF DIALECTICS.—The conception of dialectics or of logic, is used by the ancients for the most part in a very wide sense, while Plato employs it in repeated instances interchangeably with philosophy, though on the other hand he treats it also as a separate branch of philosophy. He divides it from physics as the science of the eternal and unchangeable from the science of the changeable, which never is, but is only ever becoming; he distinguishes also between it and ethics, so far as the latter treats of the good not absolutely, but in its concrete exhibition in morals and in the state; so that dialectics may be termed philosophy in a higher sense, while physics and ethics follow it as two less exact sciences, or as a not yet perfected philosophy. Plato himself defines dialectics, according to the ordinary signification of the word, as the art of developing knowledge by way of dialogue in questions and answers. (_Rep._ VII. 534). But since the art of communicating correctly in dialogue is according to Plato, at the same time the art of thinking correctly, and as thus thinking and speaking could not be separated by the ancients, but every process of thought was a living dialogue, so Plato would more accurately define dialectics as the science which brings speech to a correct issue, and which combines or separates the species, _i. e._ the conceptions of things correctly with one another. (_Soph._ p. 253. _Phædr._ p. 266). Dialectics with him has two divisions, to know what can and what cannot be connected, and to know how division or combination can be. But as with Plato these conceptions of species or ideas are the only actual and true existence, so have we, in entire conformity with this, a third definition of dialectics (_Philebus_ p. 57), as the science of being, the science of that which is true and unchangeable, the science of all other sciences. We may therefore briefly characterize it as the science of absolute being or of ideas.