It does not form part of the task that we have undertaken further to investigate the constitution of mathematics and to determine whether there be one or several mathematical sciences; if one be fundamental and the others derived from it; if the Calculus include in itself Geometry and Mechanics, or if all three can be co-ordinated and unified in general mathematics; if Geometry and Mechanics be pure mathematics, or if they do not introduce representative and contingent elements (as seems to be without doubt the case in mathematical Physics); and so on. Suffice it that we have established the nature of mathematical science and furnished the criterion according to which it can be discerned if a given formation be mathematics or natural science, if it be pure or applied mathematics (concept or judgment of enumeration, scheme of calculation, or calculation in the act). And for this reason we shall not enter into the solution of particular questions, like those concerning the number of possible fundamental operations of arithmetic, or concerning the nature of the calculus of infinitesimals, and whether, in this, there be any place for non-mathematical concepts, that is, the philosophic, not the quantitative infinite, or, again, concerning the number of the dimensions of space. As to the use of mathematics, it concerns the mathematician who knows his business to see what arbitrary distinctions it suits him to introduce, and what arbitrary unifications to produce, in order to attain certain ends. For the philosopher, these unifications and those distinctions, if transported into philosophy, are all alike false, and all can be legitimate, if employed in mathematics. If three dimensions of space are arbitrary but convenient, four, five and _n_ dimensions will be arbitrary, and the only question that can be discussed will be whether they are convenient. Of this the philosopher knows nothing, as indeed he is sure _a priori_ is the case. [Sidenote: _Rigour of mathematics and rigour of philosophy. Loves and hates of the two forms._]