Such indeed are the fictions of mathematics;--they have universality without concreteness, and therefore feigned universality. Inversely to the natural sciences, which give the value of the concept to representations of the singular, although they succeed in doing so only by convention, mathematics gives the value of the single to concepts, also succeeding in this only by convention. Thus it divides spatiality into dimensions, individuality into numbers, movement into motion and rest, and so on. It also creates fictitious beings, which are neither representations nor concepts, but rather concepts treated as representations. It is a devastation, a mutilation, a scourge, penetrating into the theoretical world, in which it has no part, being altogether innocuous, because it affirms nothing of reality and acts as a simple practical artifice. The general purpose of that artifice is known; it is to aid memory. And the particular mnemonic purpose of this is at once evident; it is to aid the recall to memory of series of representations, previously collected in empirical concepts and thus rendered homogeneous. That is to say, they serve to supply the abstract concepts, which make possible the judgment of enumeration; to construct instruments for counting and calculating and for composing that sort of false _a priori_ synthesis, which is the enumeration of single objects. [Sidenote: _The ultimate end of mathematics: to enumerate and consequently to aid the determination of the single. Its place._] Applying thus to mathematics what has been said of the judgment of enumeration, it is now clear that it facilitates the manipulation of knowledge as to individual reality. Calculation indeed presupposes: (i) perceptions (individual judgments); (2) classifications (judgments of classification); and only by means of these latter does it attain to the first. But it must attain to the first, because were there no single things to recall to the mind, calculation would be vain. Quantification would be sterile fencing, if it did not eventually arrive at qualification. Mathematics is sometimes conceived as the special instrument of the natural sciences, _appendix magna_ to the natural sciences, as Bacon called it; but from what has been said, we must not forget that both taken together, because co-operating, constitute an _appendix magna_ or an _index locupletissimus_ to history, which is full knowledge of the real. It is further altogether erroneous to present mathematics as a prologue to all knowledge of the real, to philosophy and to the sciences, for this confuses head with tail, _appendix_ and _index,_ with text and preface. [Sidenote: _Particular questions concerning mathematics._]