Geodesic Problem in Economic General Inequality

NOTICE

• just speculation, do not trust

MOTIVATION

• Toward "unified field theory of inequality" (Krugman)

PROBLEM

• how long time should be assumed for time horizon in such a too general macroeconomic growth model?
• it could be beyond the origin/doomsday of human nature and so beyond the scope of economic theory in usual sense, but it's not a problem
• the very problem is whether time horizon could be made finite or not

on factorizing transfinites (I)

For some reasons, we should consider "factorization" of transfinite numbers.

Turing(1949) gave a interpretation of transfinite numbers as natural numbers as far as amounts of each orders are bounded

for example: 9w+35 => 9 x 100^2 + 35

we are interested in existence and uniqueness of such "factorization"

refs:
http://en.wikipedia.org/wiki/Ordinal_arithmetic
http://matwbn.icm.edu.pl/ksiazki/aa/aa4/aa432.pdf

Notes on Infinite-Dimensional Projective Algebra

I have no math/phys verifications for following speculations
I have no working knowledge so it might be apart from standard concepts

• Suppose a huge category Univ, which is category of category of sets, Set
• and its relation to projections of variations/manifolds, Proj
• axiom of regularity (foundation) and large cardinal axioms assure limit and inverse limit of Univ in Set
• they could be related to quantum field theory: regularity assures existence of vacuum (ultraviolet cutoff), large cardinals assures existence of universe (infrared cutoff)
• Proj might be also found in quantum measurement: lattice of related observations
• Proj resembles to Univ but generally lacks limits
• Algebra of integrals forms Proj
  • See Segal(1965) and its references http://www.ams.org/journals/bull/1965-71-03/S0002-9904-1965-11284-8/
  • among references, for relations to operator ring, see introductions in von Neumann(1960) http://press.princeton.edu/titles/6267.html
  • Caratheodory(1986) also explored related topics http://www.amazon.com/dp/0828401616
• Univ is a sort of compactification of Proj, but how we compactificate Proj?
• operator ring theory must have distinction between two ways of compactification, one-point and Stone-Cech
• section of infinity on circumstance, or hyperbolization of ellipse, make infinity to integrable, by counting leaves of the tree
• AdS-CFT might be thought as an example of general hyperbolization
• Can we perform hyperbolization simultaneously in both sides, at UV and IR?
• regarding any automorphic forms as "ellipses", what does it mean, existence of their (discrete) hyperbolizations?

Why Stability Matters?

Why Stability Matters: Interrelations between geom/discrete math/HEP/logic

NOTICES

• This article is almost just English version with minor refactoring of my Japanese blog post http://d.hatena.ne.jp/TuvianNavy/20141012/1413081494
• Do not trust the following

Model Theory and Geometry

• One of main motivations toward model theory is geometry
• Source of imagination: Bernhard Riemann's proof on independence of Euclid's fifth postulate (parallel postulate)
• (added) David Hilbert: the fourth problem and "tables, chairs and beer mugs" axiomatics of geometry
• Oswald Veblen originated categoricity of axiomatics
• Alfred Tarski explored decidable theories and their conservative extension
• Dana Scott: Convergent Sequences of Complete Theories (Prototype of his domain theory)

Mathematizing Kant's Transcendental Category

• A disciple of Saunders MacLane - Michael Morley: Categoricity in Power (He later joined Tarski school at UCB, under Vaught)
• From Loewenheim-Skolem, you cannot determine cardinality of infinite domain, but you can say categoricity under given cardinality
• Stability of categoricity: Saharon Shelah's efforts
• In the same period of Morley's above work, MacLane-Eilenberg theory of natural transformations turned into "category theory"
• William Lawvere, who related Quantifies and Shearves, was a disciple of Samuel Eilenberg, had "workshop" of mathematical logic at UCB Tarski school

Variational Principle and its relations to disciplines other than analysis

• PDE, Calculus of Variations in the Large, Harmonic Analysis
• giving global picture of Any potential distributions: which require infinite dimensions (non-zero energy in infrared), or, so to say, quantum dynamics
• At least two millennium problems are related: Yang-Mills and Navier-Stokes
• elliptization of parabolic PDE into infinte system of ODEs
• soliton characterization by Fermi-Pasta-Ulam and Zabusky-Kruskal
• Faddeev-Popov ghost
• In game theory, translation of n-person non-zerosum game to n+1-person zerosum game (extra player, like "natural resource", behave irrationally): In TGEB, von Neumann-Moregenstern gave a suggestion about relation to Morse theory
• In economics, explaining demands in classical general equilibrium with term structure of finance/dynamic macroeconomics

Conservation and Stability

• Stability in Model Theory
• which means degree of "power division" of time (real time)
• imaginary time, with conformal invariance, is algebraically determined, but no determinacy with real time: related to validity of path integral
• residue of energy, in order of log(t) and not absorbable to power structure (orbit), plays main role in non-linear phenomena
• In engineering, metrics in shorter time make precise control possible and which make systems more stable
• commensurability of orbits in planetary system
• ergodic theory
• basis of fourier analysis
• lattice structure of natural number >0 represented in product of primes (Dedekind)
• stability of economic equilibrium

Kakutani's contribution to understanding of stability

• I'm asking Shizuo Kakutani's significance to Princeton guys in 1940s
• origin of Kakutani's FPT and metamathematical counterpart of it

So What?

HET in mathematical context: an outline

NOTICE

• DO NOT TRUST THE FOLLOWING

INTRODUCTION

• In my blog post in Nov 2010, I proposed yet another view in HET (history of economic thought) focused on mathematical context rather than contexts of ideology and social value: (in Japanese) http://d.hatena.ne.jp/TuvianNavy/20101102/1288723212
• it is partly motivated from Philip Mirowski's HET works aiming at similar target
• another source of motivation is the long-run controversy about Japan's economical slump over decades and the way out of it
• later I came to be aware of "non-separable" nature of technological knowledge and economic growth path
• parallelism between foundation of monetary theory and quantum gravity is concerned

THEMES TO BE EXAMINED

• Reality and Closed Economic Systems
• Reproduction and Iteration
• Tatonnement: Search for Self-Consistency
• Economy as Propagator, Growth as Scattering
• Utility, as Metric and Topology
• Beyond Finite Dimensions
• Homogeneity, Stability and Conservation of Wealth
• Relativity and Relaxation of Individual Values
• Logic, Knowledge and Intertemporal Substitution
• Economic Potential and Geographical Agglomeration
• Justifing Say's Law: Entropy and Demand

PARTICULAR TOPICS (very incomplete)

• Scarf's effective proof of Edgeworth conjecture
• Uzawa Equivalence
• Okishio's theorem
• Samuelson's non-substitution theorem
• Hicks' stability condition
• Arrow Impossibility
• Barro's Ricardian equivalence
• Instability of global economy / Indefinability of excess demand (SMD theorem)

CONTROVERSIES (still in Japanese)

• 重商主義に対するスミスの批判(会計的等価＝古典経済学における均衡理論)。
• 穀物法論争(リカードのグローバリズム、自由貿易に対するマルサスの批判)。
• 労働価値説(広くは生産費説)に対するオーストリア学派の批判。
• マルクスによる主観価値説への反批判。
• 社会主義計算論争。
• 失業をめぐるピグーとケインズの論争。
• ケンブリッジ資本論争。
• 変動相場制、固定相場制、金本位制
• 構造モデルと誘導モデルの乖離についてのルーカス批判。
• CAPMに対するロルの批判。

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

NOTICE

• 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?

NOTICE

• DO NOT TRUST THE FOLLOWING

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

NOTICE

• DO NOT TRUST THE FOLLOWING

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)

NOTICE

• DO NOT TRUST THE FOLLOWING

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

• DO NOT TRUST THE FOLLOWING

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

• DO NOT TRUST THE FOLLOWING

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?

John Nash and riemannian embedding (memorandum)

NOTICE

• DO NOT TRUST THE FOLLOING

HISTORICAL FACTS

• John Nash earned his degree with game theory paper
• it largely solved Francis Edgeworth's conjecture on economic competence and its equilibrium (it had been also vaguely stated by Cournot)
• von Neumann had low opinion on his result: "That's trivial, you know. That's just a fixed point theorem."
• Nash turned to differential geometry and studied riemannian embedding
• Then he went mad and could not continue his prominent career

QUESTION

• Do Nash's two principal research areas are not internally related?

SPECULATION

• Economic system could be a (near-) manifold in economic "spacetime"
• Approach to equilibrium (relaxation, or tatonnement) is performed in time, but not in same time
• Economic system as reference frame have time other than wallclock (physical time)
• above all is global view
• local view should be told with incommutable operators
• two players have strategies, some could be commutable (their payoff is independent each other),  but almost all should be incommutable
• farther they apart, less they interact
• amount of their interaction is curvature of the two-person universe
• and it approaches to zero as they get "wealthier"

Stochastic Process is Topology

NOTICE

• DO NOT TRUST THE FOLLOWING

OBSERVATIONS

• branch pattern of infinite length is stochastic process
• so stochastic process is topology
• topology have its shape, order type
• order types are ordinals
• ordinals made metamathematical (with sufficiently strong reflection principle) are forcing notions
• Lebesgue used order types to define integration
• set theorists related enumeration of integration pattern and determinacy

WHAT WE NEED TO BELIEVE IN COMPLETENESS OF PROBABILITY?

• just an assumption saying unmeasurable set would not appear
• regularity assumptions on time (transversality, no ponzi, ..)
• require smoothness of time and economic spacetime hypermanifold
• set-theoretic axioms (stronger reflection principle)
• meta set-theoretic axiom sufficient to proof Con(ZF+AD)

Continuity of Equilibrium Hypermanifold in Economic Spacetime (memorandum)

NOTICE

• DO NOT TRUST THE FOLLOWING

MOTIVATION

• Existence of Walrasian equilibrium manifold depends on continuous contraction mapping in n-space of goods with non-negativity of price and quantity
• Think in n+1-"spacetime" and see evolution along time or convergence to above manifold at a point of Ramsey's "bliss"

CONTINUITY OF TIME

• I mean dt/dx in n+1-spacetime, or in adequately blown-ups of such spacetime
• In other words, regularity of stochastic process
• In other words, completeness of expectation
• (added) dynamic constraints like transversality and no Ponzi game must be related to continuity of time

PROBLEM OF JUMP VARIABLES AND DIVERGENT PATH

• "Walrasian equilibrium" in spacetime needs continuity of time
• If no continuity of time is assured, generally divergent paths exist
• ex-ante vs ex-post (realized state of economy is always continuous evolution of the past in ex-post)
• how we treat ex-ante irregular changes as extra dimension added to continuous economic spacetime?

STRATEGIC FOUNDATION OF ECONOMIC DYNAMICS

• existence of pay-off in infinitely repeated games follows from axiom of determinacy (Mycielski-Steinhaus(1962))
• such axiomatics support is really needed? one serious problem is AD contradicts AC so it hurts Walrasian equilibrium thru Uzawa equivalence