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.
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 same, yet under the form of quantity, and, as
will be shown, of multiplicity. An infinity resides in both; in the
Real an endlessness of individual forms; in the Ideal but one endless
form; in the latter case an eternity, in the former an infinity. The
quantity and multiplicity of the whole of mathematics is contained in
the same manner in the 0, that the quantity and multiplicity of the
triangles are in the ideal or primary triangle. Mathematics is a system
of nullities or nothings, and this admits of being easily proved.
37. Zero is indeed the universality of mathematics, this, however,
is not real, but only ideal. Every number issues out of zero, like
the multiplicity of the real triangles out of the primary triangle.
This progression of numbers out of zero takes place through a process
of becoming determinate and limited; just as the real triangles are
only definitions of the absolute triangle. The process of becoming
determined is identical with becoming a Finite; becoming real is called
becoming finite. Mathematical singulars or numbers can, therefore, be
nothing else than zero disintegrated, or rendered real by determination.
Public-domain text, read in full here on John Shaqi.
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