Sources on Hegelian Dialectics and Logical Completeness

NOTICE

• this post is meant to be an index of modern mathematical thoughts related to Hegelian dialectics
• items might be added and corrected without notice

BACKGROUND

• for logical completeness or algebraic closure, we often require "contradict", "limit" or "ideal" mathematical elements needed in some form of compactification
• one can extend "set of xx" to "complete or perfect set including any possible xx and elements which could never be xx but limits of them"
• such ideal elements break finiteness in some sense of xx, so do not meet criteria of being xx
• such as: infs/sups, point at infinity, infinitesimals, non-standard infinities of natural/real number, non-measurable sets, inaccessible ordinals, inaccessible cardinals, etc.
• we also require closed and curved compact space to be open and straight additive space: Riemann had shown that such spaces (in finite dimensions) are mutually embeddable
• contradiction in Hegelian sense should be distinguished from logical unsatisfiability
• Hegelian contradiction should not also be confused with simply limit
• what is unsatisfiable in old definition is always made consistent with new definition
• new definition make it possible to explore toward new limit or "contradiction"
• we can list examples of (nearly) contradict concepts and summarize there treatment in mathematics

QUOTES

"If one has really technically penetrated a subject, things that previously seemed in complete contrast, might be purely mathematical transformations of each other."
(attributed to John von Neumann, [Bródy 1970])
"When the main contradictions of a thing have been found, the scientific procedure is to summarize them in slogans which one then constantly uses as an ideological weapon for the further development and transformation of the thing."
(William Lawvere, [Lawvere 1970])

References

• [Bródy 1970] Proportion, prices and planning: A mathematical restatement of the labor theory of value, North-Holland, 1970
• [Dantzig 1991] Linear Programming: The Story about How It Began, http://users.eecs.northwestern.edu/~morales/PSfiles/dantzig.pdf
• [Gratian] Concordia discordantium canonum (Gratian's Decretum)
• [Kunen 1971] Elementary embeddings and infinitary combinatorics, JSL 36(3), pp. 407-413, http://www.jstor.org/stable/2269948
• [Nicholas 1440] De Docta ignorantia (On Learned Ignorance)
• [Lawvere 1970] Quantifiers and Sheaves, Actes, Congrès intern, math., 1970. Tome 1, pp. 329-334.
• [Marx 1881] Mathematical Manuscripts, New Park, 1983, http://www.marxists.org/archive/marx/works/download/Marx_Mathematical_Manuscripts_1881.pdf
• [Riemann 1868] über die Hypothesen, welche der Geometrie zu Grunde liegen, http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=35634
• [Rodin 2013] Categorical Logic and Hegelian Dialectics, 2013, http://philomatica.org/wp-content/uploads/2013/01/dialectics.pdf
• [Shelah 1996] Existence of almost free abelian groups and reflection of stationary set, http://arxiv.org/abs/math/9606229
• [Souza 2015] On limit behavior in space-time, Boletim da Sociedade Paranaense de Matemática, 33(1), 2015, http://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/23011