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.
The individual objects or figures of mathematics thus attain existence, so far only as the idea comprising them emerges out of itself and assumes an individual character.
It is clear that all individual triangles taken together closely resemble the ideal triangle, or, to express the same in more general terms, that the Real is equivalent to the Ideal, that the former is but the latter which has become dissevered and finite, and that the aggregate of every Finite is equivalent to the Ideal. This will probably be rendered still more distinct by the example of ice and water. The crystals of ice are nothing else than water bounded by definite lines. So, also, are the Real and Ideal no more different from each other than ice and water; both of these, as is well known, are essentially one and the same, and yet are different, the diversity consisting only in the form. It will be shown in the sequel that everything which appears to be essentially different from another, is so only in the form.