mathematics & dialectics

Juan Inigo jinigo at
Thu Mar 2 23:47:37 MST 1995

When we face the magnitude that the necessity of our action has, to cognize
it as such, we have to deal with a real form in which the general
self-affirming by means of self negation has developed into its specific
form of self-affirming by means of the negation of self-negation. In other
words, we have to deal with a real form in which the general necessity to
become has acquired the specific form of the difference determined as
indifference. Hence, we have relations to discover, but we cannot discover
them by following their development with our thought. We can only discover
the quantitative relations in a completely external way with respect to the
unfolding of their own necessity as the abstract real forms they are.

Lacking a real necessity whose development it can ideally follow, our
cognition process has to provide itself the necessity of its course; that
is, a strictly constructive necessity. Inasmuch as we unfold the ideal
reproduction of reality as a process of measuring, as mathematical
cognition, it necessarily takes the form of an ideal representation of
reality. Nevertheless, the nature itself the object of this representation
has, where the content appears in its form by means of the negation of its
own negation, overcomes any restriction to the validity of the logically
represented relations.

Freed from the inversion due to its use as a representation of qualitative
determination in general, logic directly reflects the specific quality of
its object, quantitative determination. The logical relations do not appear
now as representing attributes inherent to abstractly mental forms, to the
truth or falsity of our logical thoughts. These relations immediately show
themselves as what they are: representations of the concrete forms of the
real quantitative determination.

The need to empty logic of the specificity of its object, forces the
representation of the number by the set in the mathematics produced by
scientific theory. Nevertheless, the set is the unity of continuity and
discontinuity represented under its concrete form of the term of the unity
between intensiveness and extensiveness. The development of the
representation of the concrete forms that mediate between this unity and
the true concrete form that is represented by the number, i.e., the
identity of the unit and the multiplicity where the exteriority of the
quantitative determination appears completely unfolded, is consequently
inverted. It does not appear under its proper shape of a simple logical
development, but as a representation of the magnitude of an abstract form
by the relations of measure of its concrete forms, as mathematical

Thus proceeds the universally dominant mathematics today. It starts with
the representation of the number as a set (G. Frege), follows with the
relations between transfinite magnitudes (G. Cantor), and ends up by
representing the realization of the relation of the unit with itself in the
degree of multiplicity as the limit of the change of the ratio between
variables (K. Weierstrass). Hence, the development of dialectical cognition
carries the necessity to revolutionize scientific cognition even in the
part in which this cognition keeps representation as its necessary form.

Juan Inigo
jinigo at

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