Now then comes the question, what is the first principle of Mathematics?
31. The highest mathematical idea, or the fundamental principle of all mathematics is the zero = 0.
The whole science of mathematics depends upon zero. Zero alone determines the value in mathematics.
32. Zero is in itself nothing. Mathematics is based upon nothing, and, consequently, arises out of nothing.
33. Out of nothing, therefore, it is possible for something to arise, for mathematics, consisting of propositions, is a something in relation to 0. Mathematics itself were nothing if it had none other than its highest principle zero. In order, therefore, that mathematics may become a real science, it must, in addition to its highest principle, subdivide into a number of details, namely, first of all into numbers, and, finally, into propositions. What is tenable in regard to mathematics must be equally so of all the sciences; they must all resemble mathematics.
34. The first act towards realization or the becoming something, is an origination of Many. All reality can, accordingly, manifest itself only in multiplicity.
That which belongs to the Many is a Definite; this again is a Limited; the Limited is a Finite. The Finite only is real.
The question now arises, how it happens that mathematics becomes a multiplicity, or, what is the same thing, a reality, a something.
35. The reality of mathematics consists in the universality of its quantities; viz. numbers or figures. Every number, and every thing which belongs to mathematics, can be derived from no other source than zero.
Mathematical multiplicity, or its reality, must have proceeded, therefore, out of zero.
36. Zero, however, contains no number and no figure really in itself; it contains, forsooth, neither 1 nor 2, neither a point nor a line within itself. The Singulars or details cannot, therefore, reside in a real, but only an ideal manner in zero; or, in other words, not _actually_, but only _potentially_. The conditions here are the same as with all mathematical ideas. We may conceive, e. g., an idea or definition of a triangle in so general a sense that it shall comprehend all triangles, without, however, a definite triangle being actually intended, or without even a triangle actually existing. In order that the idea of the triangle be realized, it must become a definite, in other words, an obtuse or an acute triangle. In short, the idea of the triangle must multiply itself, be self-evolved, or else it is as naught in reference to mathematics, or only a geometrical zero.