And if the three or more dimensions as attributes of space prove to be unthinkable, and also the point without extension, the line without superficies, and the superficies without solidity--so too in consequence are all the concepts derived from them, such as those of geometrical figures, none of which has, or can have, reality. No triangle has, or can have, the sum of its angles equal to two right angles, because no triangle has existence. Hence those geometrical concepts are not completely expressed in any real fact, since they are in none, thereby differing from the philosophic concepts, which are all in every instant and are not completely expressed in any instant. Similar results follow in the case of the principles of Mechanics. No body can be withdrawn from the action of external forces, because every body is connected with all the others in the universe; hence the law of inertia is unthinkable. [Sidenote: _and not intuitible._] 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. [Sidenote: _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.