For counting and calculating Mathematics requires formulæ, and to establish these it requires certain fundamental principles. These are called in turn definitions, axioms, and postulates. Thus arithmetic requires the number series, which beginning from unity, is obtained by always adding one unit to the preceding number. Geometry requires the conception of three dimensional spaces, with the postulates connected with it. Mechanics requires certain fundamental laws, such as the law of inertia, by which a body in motion, which is not submitted to the action of other forces, covers in equal times equal spaces. There has been much dispute as to whether these principles are _a priori_ or _a posteriori,_ pure or experimental; but the dispute must henceforth be considered settled in favour of the former alternative. Even empiricists distinguish mathematical principles from natural or empirical principles, as at least (to use their expression) _elementary experiences,_ as experiences which man completes in his own spirit, in isolation from external nature. This means, whether they like it or no, that they too distinguish them profoundly from _a posteriori_ or experimental knowledge. The _a priori_ character of mathematical principles is made manifest by every attack upon it. [Sidenote: _Contradictory nature of these a priori principles. Their unthinkability,_] But when they are recognized as being not _a posteriori_ and empirical, but _a priori,_ difficulties are not thereby at an end. The apriority of those principles possesses other most singular characteristics, which render them unlike the _a priori_ knowledge of philosophy, the consciousness of universals and of values, for instance, of logical or of moral value. For if it is impossible to think that the concepts of the true and of the good are not true, on the other hand it is _impossible to think that the principles of mathematics are trice._ Indeed, when closely considered, they prove to be all of them altogether false. The number series is obtained by starting from unity and adding always one unit; but in reality, there is no fact which can act as the beginning of a series, nor is any fact detachable from another fact, in such a way as to generate a discrete series. If mathematics abandons the discrete for the continuous, it comes out of itself, because it abandons quantity for quality, the irrational, which is its kingdom, for the rational. If it remains in the discrete, it posits something unreal and unthinkable. Space is characterized as constituted of three or more dimensions; but reality gives, not this space, thus constituted, made up of dimensions, but spatiality, that is to say, thinkability, intuitibility in general, living and organic extension, not mechanical and aggregated. Its character is not to have three dimensions, one, two, three, but to be spatiality, in which all the other dimensions are in the one, and so there are not distinguishable and enumerable dimensions.