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

How entropy incarnates

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• DO NOT TRUST THE FOLLOWING

CREDIBILITY

• what is money? what is credit?
• credible information must be verifiable
• some evidences may contradict each other
• some evidences may disappear
• history must be carved on stone
• bank account balance must be carved on magnetic media and papers
• information with economic value must have firm physical support, such as dumb-looking mainframe and its huge disks
• information theoretic entropy is not always physical entropy, but information with economic value has physical entropy
• why: money have its exchange rate to various sources of energy

THE SUREST THING

• the universe is
• its probability of existence must be larger than anything inside it
• its mass must be largest too

Why things which are unlikely to happen would never happen?

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MOTIVATION

• what is "not likely" to happen?
• for example:
• sufficiently big wave would destroy reactor plant: likely to happen
• huge big wave may happen in certain period: likely to happen
• reactor plants used to go normally and seemed to be apart from catastrophe
• in other words, it seemed unlikely to happen in ex-ante
• another example:
• thermodynamically, one would never find ice in operating boiler: unlikely to happen
• one cannot imagine extremely non-equilibrium situation (intentionally or naturally) which enable ice exist in operating boiler
• in future days, technology might make it possible to keeping a piece of ice midst of burning fire
• another example:
• someday one might find an ever-unseen decay mode of well-known particle
• it might be a red alert to theorists
• it might be thought as a ever-hypothetical high energy (so rare) action and evidence of such excitation
• another example:
• someday we would form a new kind of business models which could be well understood as fitting better to new technological and social environment

KNOWN PITFALL OF PROBABILITY CALCULUS

• some unlikelihood turns out to be quite a likelihood from an wider viewpoint
• narrower, or ex-ante, view contrasts with wider, or ex-post, view

WHAT IS TECHNOLOGY?

• finer measurement and control
• higher energy time quanta

WHAT IS POLITICAL?

• political complication make things unpredictable
• just like as in turbulence, or in viscose vacua: too many causes contribute
• highly frictional universe is non-Newtonian: physicist of fishes should not be alike to Isaac Newton
• if galaxies think, they might think like fishes, unlike human

recurrence, measurability, determinacy and decidability

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OBSERVATIONS

• numerical integration: some computation would never halt
• someone guesses approximate fixed point of above computation
• what is the "probability" whether the guesser is right
• as far as the system is integrable, this would make sense
• automata: cyclic group could be represented as circular state machine
• apply homology to nodes of an automaton: nodes are loops
• John von Neumann: he worked on cellular automata after his (mathematical) effort on ergodic theory and operator ring theory
• John Nash: existence of core of game / Riemannian embedding
• William Thurston: geometrization conjecture and following studies of automata

SPECULATION

• there must be an inclusive viewpoint for understanding them all as one:
• ergodic theory
• probability theory (especially, stochastic process)
• metamathematics, set-theoretic and lattice-theoretic
• model theory
• (non-relativistic) quantum field theory
• general relativity
• automata theory
• thermodynamics
• dynamic macroeconomics
• financial and monetary economics

TURBULENCE/RANDOMNESS/NON-INTEGRABILITY

• what is "mean coordinates" of chaotic orbit?

measuring value of broken tools (memorandum)

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MOTIVATION

• a la Heidegger, we could say tools notice us when they do not work properly
• think direct sum decomposition of tool: tool = broken tool + its functionality
• other than its functionality, value of broken tool might be zero
• repairer might price broken tool as >0

(added) KNOWLEDGE AND NON-COMMUTATIVITY

• determining (valuation) of broken tool is dependent on repairer's knowledge and available repair process (both are continuously changing)
• some repair depends on too specific supply of spare parts recycled from another broken tool

Statistical Dynamics of Radioactive Pollution (memorandum)

NOTICE

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Motivation

• Any event occurs in this universe minimize the statistical Lagrangian of the universe
• What is minimized, or how is it minimized, in spacial scattering of unstable isotopes as consequence in huge diastrophism (displacement of crustal plates)?
• scattering of magma (containing heavy elements like thorium and uranium) and nuclear fuel works in same way
• solar neutrinos and beta-decay isotopes might affect local Higgs field
• and they all, more or less, modify state of so-called causality (time asymmetry)

Recent Scientific Findings

• Existence of Higgs boson was confirmed (Mar 2013)
• A report that say stem cells are reduced from differentiated cells easily and more reversibly (Jan 2014)

Potential background effect of more symmetric vacuum physics

• crazy idea, I agree

Harmless to time arrow?

• even if it is so, who cares? or who could care?
• If our universe is really infinite dimensional, you always find another force and blame it for still remaining errors, so the time still flows
• If our universe is finite dimensional and you find the very last force, the universe might stop: no one would feel nor think any more

Cardinal Geometry (memorandum)

NOTICE

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Cardinal Geometry

• Desirable geometric view/interpretation for set-theoretic elementary embedding of universe into universe, not new math
• Infinite dimensional analogue of Riemannian embedding
• Ignoring finite sets, set-theoretic universe consists of ordinal "lines"
• Measurable cardinals or $\aleph_\omega$ (first singular cardinal) might work as "vanishing lines" or "horizon"
• But why ZFC? Categorical topology actually works and needs no additional concern about cardinality. Continuous geometry was invented for vaguely same objective, but disappeared abandoned

So what?

• Just thinking

Update: curse of dimensionality, integrability and chaos

• the very first result in dimension theory (Cantor(1877)): $card(\mathbb{R}^2) = \mathfrak{c}$
• existence of discontinuous mapping between 1-d real line and 2-d real area
• generally speaking, elementary embedding of universe into universe does not conserve information of dimensionality
• as change of topology, it changes continuity into singularity (making neighbors apart), it might also change singularity into continuity (making new neighborhood with aliens from distant galaxy)
• if you change order of enumeration, measure is not conserved (see summation of infinite series)
• curse of dimensionality: with requirement of conserving dimensional order information (for integrability) elementary embedding require further huge universe (if GCH holds it must be the power of current universe)
• continuum problem in numerical context: can you find measure-conserving chaotic (turbulent) circumstance characterized as having fractional dimensionality?