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