| Does anyone know of recent, up-to-date scholarly discussion of Hegel's
| treatment of the calculus in the Logic? I know he got bashed for it by
| Russell and Whitehead, although Whitehead later learned his own philosophy
| resembled Hegel's in some ways.
I am not quite sure whether I can cast any beam of light on the questions
you raise. Anyhow... I could start, and others better informed than me may
share more on these issues.
*Certain* philosophical investigations on the mathematical logic of both
Georg Wilhelm Friedrich Hegel (1770-1831) and Friedrich Wilhelm Joseph von
Schelling (1775-1854) were addressed by Bernhard Taureck in his thesis at
the University of Tuebingen, 1971 [1].
Another work of interest was written by Antonio Moretto, who refers to
Hegel and the _mathematics of the infinite_ [2]. The book was reviewed
for ZfM by Aldo Ursini (Dipartimento di Matematica, Universita' di Siena)
as follows:
The theme of this book is the importance in Hegel's thought
of his philosophy of mathematics. The book's core is an
outline of the development of Hegel's reflections on the basic
concepts of what we call now mathematical analysis, from the
first papers of 1800-1802 to the Wissenschaft der Logik. The
author's conclusions are that two main Hegelian notions are
involved into the matter as a red thread: contradiction
(connected to infinitesimals) and relation (which Hegel sees
from the beginning as the true realization of infinity in
science). In the last chapters, the author compares in detail
some of Hegel's contemporary mathematical approaches to the
notions of function and of infinitesimal with his conception
of relation, which is explored most thoroughly in the Logik.
This book is mainly intended for philosophers, as shown by the
many explanations and illustrations of certain mathematical
items, which often reproduce the original "classical" arguments
of the late 18th century mathematics, and hence may be of some
interest to the non historian reader.
The present reviewer: (1) is not expert in judging about the
properness of some philological interpretations proposed by the
author; (2) would have liked a section comparing Hegel's view
to his contemporary philosophers or mathematicians on those
matters and (3) would remark that since methodological issues
are central in the philosophy of mathematics (as of knowledge in
general), the very neat critiques by Hegel of some mathematical
methods, especially being presented in the Phenomenology of
Spirit, should have been taken into account in some way.
Anyway, may be via a suitable curtailment of some mathematical
explanations, historical reconstructions and philological
exaggerations in footnotes, and may be via a stylistic improvement
in the exposition, the bulk of this book should deserve a wide
diffusion among thinkful mathematicians.
The author points out some very interesting ideas of Hegel, for
instance: (i) considering "rational infinity" to be realized in
the mathematical concepts of ratio (in measure theory) and law
(subsequently subsumed under the notion of relation); (ii) the
importance tribute to the ability of mathematics in being a
conceptual science, in some of its branches; (iii) Hegel's
critique of the "Bad Infinity" in the theory of series, and in
general in the mathematical notion of "going to a limit"; (iv)
Hegel's reflections on the "part-whole" issue as a foundation of
his notion of relation.
These hints should not only be of interest to philosophers or
historians of science, but also to any "working mathematician".
If only for the sake of contrasting those views, one is leaded
to think about. The invitation to "hardcore" thinking is a
permanent Hegelian heritage in our culture. Our usual contempt
and sense of superiority in front of the "mathematical nonsense",
which allegedly is spread in some Hegel's works, might be damned
to an inglorious burial: what, for instance, if some of his ideas
will show well alive, and will be taken over by Artificial
Intelligence people, before mathematicians will become aware of
their value...?
A scholarly discussion of Hegel's treatment of mathematical understanding
and interpretation of mathematical considerations is Kleinert's "Hegel
ueber die Mathematik". The paper is mainly about philosophy of geometry,
if I recall correctly [3].
Paul Ziche (Philosophische Fakultaet, Universitaet Jena) has researched
on the mathematical and scientific models in the philosophy of Schelling
and Hegel [4]. In his paper, Ziche addresses the following issues:
(a) The relation of Schelling and Hegel with mathematics and natural
sciences, and their acknowledgement to Christoph Friedrich von
Pfleiderer (1736-1821), a mathematician and bibliophile. (Our
friend Walter Felscher may provide further notice of him, I'm
sure. There are surely many books and papers of Pfleiderer at
the University Library of Tuebingen.)
(b) The relation of finiteness and infinity in the argument between Fichte,
Schelling, and Hegel.
(c) Schelling & Hegel and the role of Kepler's laws.
(d) Schelling & Hegel and _Drehmoment und Hebelgesetz_.
(e) The performance of models for the _Verstaendnis des Absoluten_ in
Schelling and Hegel.
References quoted (and more!):
[1] Taureck, Bernhard:
"Mathematische und transzendentale Identitaet. Philosophische
Untersuchungen ueber den Identitaetsbegriff der mathematischen Logik
sowie bei Schelling und Hegel", Ueberlieferung und Aufgabe 11,
Wien/Muenchen: R. Oldenbourg Verlag, 182 pages, 1973.
[2] Moretto, Antonio:
"Hegel e la _matematica dell'infinito_", Pubblicazioni di Verifiche 8,
Trento: Verifiche, 327 pages, 1984.
[3] Kleinert, Ernst:
"Hegel ueber die Mathematik", _Math. Semesterber_ 38 (1991), no 2, 151-174.
[4] Ziche, Paul:
"Mathematische und naturwissenschaftliche Modelle in der Philosophie
Schellings und Hegels", Spekulation und Erfahrung. Abteilung II:
Untersuchungen. 39. Stuttgart: Friedrich Frommann Verlag Guenther Holzboog
GmbH & Co., 1996 [ISBN 3-7728-1769-6/hbk].
[5] Hansen, Frank-Peter:
"G.W.F. Hegel: Wissenschaft der Logik. Ein Kommentar", Wuerzburg:
Koenigshausen & Neumann, 192 pages, 1996 [ISBN: 3-8260-1252-6/pbk].
[6] Rebuffo, Franco:
"Hegel e il pensiero matematico della sua epoca", Firenze: La Nuova
Italia, xiv, 156 pages, 1989. Series: Pubblicazioni della Facolta\ di
lettere e filosofia dell'Universita\ di Milano 130; Sezione a cura del
Dipartimento di filosofia 16. [ISBN: 882210708X]
With best regards,
Julio Gonzalez Cabillon
NB Please, avoid diacritical marks (accents, tildes, cedillas, ...) at
ALL events -- unfortunately, they do NOT travel nicely by email. Thanks.