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)

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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)

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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?

John Nash and riemannian embedding (memorandum)

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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

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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)

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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

So many quantizations (memorandum)

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• glücklichneujahr!!!

QUANTIZATION

• mathematical form which represent infinite dimensional space
• must be separable (and summable) under certain restriction

INFINITELY MANY WAYS OF QUANTIZATION

• wave functions
• linear operators (hamiltonian, probability density, observations, ..)
• paths (in path integral)
• homotopies (might be used in generalization of path integral)
• wavelets
• triangulations of relativistic universe (simplicial decomposition)
• chain complexes (in homology algebra, as generalization of triangulation and observation, and these might be used in future categorical formulations)
• quantum sets (a la Takeuti and Ozawa, as parametrized boolean algebra and generalization of von Neumann's quantum logic)
• formulae valid in various systems of non-standard analysis allowing infinitely many infinitesimals
• (anything that forms topology and can be completed when bounded, given adequate interpretation)

What would be the "22nd century maths"?

• it looks just a matter of encoding (only after when throughly carried out, it should likely be realized that also old maths can be fully employed to formulate and solve)
• similarly in definition of Borel sets, it might include somewhat disguised (and likely finitized in properties of prime numbers) form of transfinite ordinals (order types) to order topologies
• it means: we might need meta-topology, which is just a topology itself, over topologies

Why?

• Since it must be a first-order approximation of "framework subsuming any possible mathematically-definable compatible and incompatible views as reflecting the only reality"
• "views" might be considered as topological spaces
• And it must eliminate infinities (poles) with finitization and interpretation