Logic as the Science of the Pure Concept

by Benedetto Croce

As they are unthinkable, so are the principles of mathematics unimaginable; they have therefore been ill defined as imaginary entities, for they would in that case lose such a priori validity as they have. They are a priori, but without the character of truth—they are organized contradictions. Had mathematics (said Herbart) to die because of the contradictions of which it is composed, it would have died long ago.[1] But it does not die of them, because it does not set itself to think them, as a venomous animal does not die of its own poison, because it does not inoculate itself. Were it to pretend to think them and to give them as true, those contradictions would all become falsities.

Identification of mathematics with abstract pseudoconcepts.

Now, a function which organizes theoretic contradictions without thinking them, and so without falling into contradictions, is not a theoretic, but a practical function, and is perfectly well known to us as that particular productive form of the practical spirit which creates pseudoconcepts. But since those contradictions are a priori and not a posteriori, pure and not representative, mathematics cannot consist of those pseudoconcepts which are representative or empirical concepts. It remains, therefore, that it consists of the other form of pseudoconcepts, which are abstract concepts, which we have already defined as altogether void of truth and also void of representation, as analytic a priori and not synthetic a priori. And we have demonstrated how, in the falsification or practical reduction of the pure concept, concreteness without universality, that is to say, mere generality, belongs to empirical concepts, and universality without concreteness, that is to say, abstraction, to abstract concepts.

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.

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.

Particular questions concerning mathematics.

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.

Rigour of mathematics and rigour of philosophy. Loves and hates of the two forms.