196 B. RUSSELL:
The Law of Continuity is also discussed in chapter iv. The single concept, we are told, in order to be understood in its origin, must no longer be regarded as a rigid and immovable logical entity: its being is only determined in connexion with a logical system, and the system of concepts must assimilate the notion of logical development. The postulate of continuity is not intelligible if a given material is to be described, but only because it is one of the fundamental acts by which consciousness conditions the object. In more special forms, the law of continuity asserts that extreme cases, from some points of view excluded, may yet be included in general theorems, e.g., propositions concerning the ellipse will hold for the parabola. The general statement is: Datis ordinatis etiam quæsita sunt ordinata. M. Couturat points out (p. 233), what Dr. Cassirer appears not to have observed, that this principle is regarded by Leibniz as a consequence of the principle of reason; the deduction, however, unlike most of the others, is invalid.[1] Moreover the principle is false in fact, unless it means, what would be perfectly trivial, that the consequents are ordered by the mere correlation with the data. Take, for example, the series of rational fractions in order of magnitude, each in its lowest terms. The numerators of these fractions are one-valued functions of the fractions, but have no order except that resulting from the correlation itself. Again, in the case of the ellipse and the parabola, the latter has some but not all of the properties of the former, and the mathematician's desire to treat such different cases together, though praised by Dr. Cassirer (p. 221), has been a source of constant and most pernicious fallacies. The principle of continuity, therefore, must be regarded as one of the most unfortunate parts of Leibniz's philosophy. Mathematically, it is false; and the philosophical meaning suggested by our author seems to amount to the assertion that everything is really something else — a principle whose merit is, that it excuses us from the necessity of understanding anything because it isn't really the thing we don't understand.
Part ii., on Mechanics, opens with a chapter on Space and Time. Time, it says, is the independent variable in regard to all related magnitudes (p. 257). This assertion is often made, without, I believe, any knowledge of its exact meaning. The only exact meaning of which it is capable is, that any relation relating all the moments of time respectively to various magnitudes of a given kind may be many-one, but cannot be one-one or one-many.[2] This is of course more or less true of important relations; but if there is any material particle which is never twice in the same position in space, then, as far as that particle is concerned, the
