6 MATHESIS. 



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 essen- 

 tially one and the same, and yet are different, the diver- 

 sity 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. 



The Real and Ideal are one and the same, only under 

 two kinds of form. The latter is the same under an 

 indefinite, eternal, single form ; but the Real is also the 



