Topological Ontology and Logic of Qualitative quantity

Hegel's great work on Logic, the “Science of Logic” (Wissenschaft der Logik), appeared in
three volumes in 1812, 1813 and 1816, contains long discussions of mathematics. In
particular in “Science of Logic” Hegel develops philosophies of the natural numbers, the
negative numbers and also of the rational numbers. In Hegel’s logic we will find an
extensive and erudite account (with the real numbers presupposed for the sake of the
argument) of how this philosophical understanding of mathematics ``makes sense'' of the
(informal) infinitesmimal manipulations of the differential and integral calculus.
Alan Paterson argues that Hegel has much to say to modern mathematical philosophy,
although the Hegelian perspective needs to be substantially developed to incorporate within
it the extensive advances in post-Hegelian mathematics and its logic. According to Paterson,
the Hegelian approach provides a framework for answering the philosophical problems,
discussed by Kurt Gödel in his paper on Bertrand Russell, which arise out of the existence in
mathematics of self-referential, non-constructive concepts (such as class).
In his paper “Does Hegel have anything to say to modern mathematical philosophy”,
Paterson insists that “it is time to grapple with what Hegel was trying to do in his treatment
of mathematics. Of course a lot of the details of Hegel's view of mathematics belong to his
time and sound quaint(!) (but no more so than Newton's discussions of fluxions), and Hegel
did not have the benefit of coming after Weierstrass, Dedekind, Cantor and Frege. But
Hegel's method of inquiry in terms of a conceptual development is the correct method for
investigating the human activity that is called mathematics simply because it is intrinsic and
lets the subject develop itself (as a genuine infinite, a circle closed on itself) rather than
assuming an external ``objective'' pose in which it is put in front of the investigator and
inspected ``from the outside'', only for it to creep in from the back in the very act of
investigation.” 115
For the purpose of the question above and in particular for the issue of human infinity as
conceptual infinity, I will focus on the discussion given by Alan Paterson in his paper
“Towards a Hegelian philosophy of mathematics”.
According to Alan Paterson, as he establishes in “Towards a Hegelian philosophy of
mathematics” 116 46 - “the well-known difficulties in modern mathematical logic require a
Hegelian grounding for their resolution and that, in the terminology of Hegelian logic”
(Hegel's “Science of Logic”). In his paper Paterson describes how “the discussion of the
problems with the formal approach to mathematics generates a Hegelian approach to
mathematics, one in which the great results of mathematical logic are aufgehoben, sublated
their integrity preserved, while at the same time, the rigid separation of mechanical logic
and mathematical intuition - that is, the obsessive abstractive operation of Verstand, the
Understanding – is overcome.”
Paterson establishes that “the Hegelian Concept, however, does not have just two moments.
It has three, those of Universality, Particularity and Singularity (Allgemeinheit,
Besonderheit, Einzelheit).” The same statement is valid in thinking of the categories as
“quality” and “quantity”.
Akademie Verlag, http://www1.fee.uva.nl/pp/bin/311fulltext.pdf; Qualitative and
Quantitative Analysis in Semantic Dialectics: Marx vs. Hegel and Arthur vs. Smith (2010);
On the Dialectical Foundations of Mathematics.(2006);
115
Alan Paterson, Does Hegel have anything to say to modern mathematical philosophy,
Idealistic Studies 32:2, 2002, 143-158.
116
Alan Paterson, “Towards a Hegelian philosophy of mathematics”, Idealistic Studies,
27(1997)
49