Duality in Descartes: Esprit Geometrique and Vortex Theory

NOTICE

• DO NOT TRUST THE FOLLOWING

Summary

• Rene Descartes' natural philosophy had two incompatible views of nature in it: "esprit geometrique" (as named by Blaise Pascal), or geometric methodology, and the vortex theory. Two views should have appeared in separate books if there were no political problem with exposition of Heliocenterism
• Isaac Newton took the former, although Gottfried Leibniz took the latter
• Vortex would be naturally derived from a view regarding Keplerian system as a part of a whole system of various perturbations, rather than two-body system (so Newton's success was due to his cutoff of higher degree corrections)
• Pascal, as fluid mechanician, was probably aware of the limit of efficiency of geometric, or exact, method, but his alternative "esprit finesse" (spirit of sensibility) is unclear: In [DELETED:his rarely-known book On the Spirit of Geometry and on the Art of Persuasion, he admitted the limited efficiency of Cartesian view, and suggested use of somewhat obscure art of persuasion, and] his conclusive aphorisms in Pansee, he was turned into a thorough critic to Cartesian way, and looks even almost abandoned any model building
• His awareness to the limit of Cartesian view is still hypothetical, lacking evidences that Pascal had found Cartesian view is only appliciable to isolated fluid with no vortex
• With a broadest interpretation far beyond its proper limit, we might say that Pascal's view is well consistent to such methods that simultaneously use several granularities ("finesse") of approximation, as actually done in computer simulation of multi-scale physics
• Also with too much analogy, we surmise that Descartes himself was not troubled with the apparent incompatibility between his method and the vortex theory: we still can think, as John von Neumann did, any effective model to be a projection to finite-dimensional space from infinite-dimensional nature and all we can do is, in principle, [ADDED:multi-]linear approximation

TODO

• Show evidences of Pascal's fluid-mechanical thought on non-equilibrium systems
• [ADDED:Pascal's treaties on esprit geometrique deserve thorough analysis, the other essay by him on the art of persuasion looks like a sort of logical positivism]

Utility and Measurement (part 4)

NOTICE

• May contain errors: DO NOT TRUST THE FOLLOWING
• just a survey, no new claims intended
• This is the direct sequel to part 1

For Impatiants

• Quantitative sciences, or simply any sciences, are efforts to explain changes of intensive quality and non-propotional actions with values
• Phase space as orthogonal dimensions of extensive (additive) domain, or newtonian absolute space, is the space for any quantitative effective theories of physics and other somewhat exact mathematical sciences including economics
• i.e. exact abelian category (Mac Lane 1950, Grothendieck 1957)
• Harold Hotelling once said "But we all know the world is non-linear," and John von Neumann, advocating George Dantzig against him, "If you have an application that satisfies the axioms, well use it [linear vector space]. If it does not, then don’t,” (Dantzig 1991)
• How do we approach with locally linear (zero-curvature) space/spacetime to know globally/locally curved space/spacetime with mass singularities?
• Or how do we approach with iterative approximation with higher (or, projection to lower) finite dimensional vector space to (from) infinite-dimensional nature of quantum physics?
• Or how do we measure (integrate) any [DELETED:manifolds] sets with σ-algebra?
• Or how many points (cardinality) do we need in space, sufficient for any geometry?
• All we need is: non information-losing (faithful) duality between straight space [ADDED:of unbounded dimensions] and [ADDED:locally] curved spacetime: It's [ADDED:the central drive for continuum problem and] the very matter of advanced mathematics like non-archimedean fields, cohomologies, sheaves, models (Morley 1965) and also motivated domain theory (Suppes and Scott 1958, Scott 1958)

Least Metaphysics

• Category: A jargon for Aristotelean and Kantian philosophies
• Arisototelean: Basic classification (classes) of any object, like time, position, relation, and so on
• Kantian: Mental structure underlying any logical grasp (not mere sensing, rather cognition) of external events, which are thought as a priori to any grasp

Least Keywords in History of Physics (to be written)

• Cartesian coodinate system
• Newtonian absolute space
• Closed dynamical systems, phase spaces and symmetries
• Legendre transformation and projection [ADDED:(related to monotonity in the sense of domain theory)]

Somewhat Superfluous Guessing on History of Mathematics

• First-order axioms are not meant to show "what is talked about". According to David Hilbert, you axiomatic geometer may say "tables" "chairs" and "beer glasses" to do geometry consistently, instead of "points" "lines" "planes".
• What you are allowed to do is just relabeling objects consistently, or keeping things distinct (don't relabel both "point" and "line" with a new name "table")
• In other words, keep isomorphism unchanged
• Oswald Veblen introduced categoricity of, or, in early 21st century wording, uniqueness up-to-isomorphism of model of, the theory, with suggestion by John Dewey (Veblen 1904). Dewey, as a philosopher, was familliar to Kantian terminology
• Loewenheim-Skolem theorem says that generally there's no such uniqueness: You need second-order language to lock cardinarity of models
• On the train of model-theoretic exploration of the [ADDED:cardinality-relative version of] concept of categoricity, Micheal Morley, a student of Saunders Mac Lane, wrote a paper on "categoricity in power" (Morley, ibid) under additional supervision of a Berkeley model theorist, Robert Lawson Vaught
• In this paper, he introduced a concept of "totally transcendental theory", which sounds like Kantian metaphysics. It might not a coincidence
• [STILL NEED CONFIRMATION, NEVER TRUST] Mac Lane published the first(?) paper (Mac Lane 1965) having a word "category" in its title, on the series of papers expositing his developing general theory of natural transformation beginning in 1940s, in the same year to Morley's paper. Years later, he wrote the first textbook on the subject (Mac Lane 1971, "CWM")
• Mac Lane later wrote (CWM, pp.29-30): "Now the discovery of ideas as general as there is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from philosophers: "Category" from Aristotle and Kant, "Functor" from Carnap (Logische Syntax der Sprache), and "natural transformation" from then current informal parlance."

TODO

• make the lengthy story brief
• add references

Regularity of Any Problem

NOTICE

• DO NOT TRUST THE FOLLOWING

Existence and Fixed Points

• We use Hilbert space daily and approximate it from finite dimension vector calculus
• Once our PDE problem is translated into maximization of infinite dimensional vector-valued function, logically the existence of solution is assured (even if it is not computable)
• Two fundamental problems still remain: (1) Is any thinkable PDE problem could be (with adequate topology which form $\sigma$-algebra) turned into finding fixed points in Hilbert space? (2) Can we always find, or does some transcendent principle assure adequate topology to settle arbitrary problem regularized?
• At least two Clay millennium problems are related to above problems: existence and smoothness of Yang-Mills and Navier-Stokes
• In other words, some transcendental version of fixed point theorem, which assures rigorous conservation of energy, is what we seek
• ZFC set theory might be too weak and insufficient to derive such principle

On Nodal Structure of Cofinallity of Cardinals and Dimensionality (part 1)

NOTICE

• DO NOT TRUST THE FOLLOWING

"Je le vois, mais je ne le crois pas"

• In 1877, Georg Cantor found bijection between $\mathbb{R}$ and $\mathbb{R^2}$
• He was surprised that dimensionality of space does not matter for the number of points in space, or cardinality
• In reply to him, Richard Dedekind pointed out that $\mathbb{R^2} \to \mathbb{R}$ is not a continuous mapping
• Other than cardinality, what represents dimensionality in set theory?

Singular Cardinal with Countable Cofinality

• What are $\aleph_\omega$ and $\aleph_{\omega 4}$, which appear in a Saharon Shelah's well-known result of PCF theory?
• On the way of cardinal construction, $\aleph_0, \aleph_1, .. \aleph_\omega, \aleph_{\omega + 1}, .. \aleph_{\omega + \omega} = \aleph_{\omega 2} .. \aleph_{\omega 3} .. \aleph_{\omega 4} .. \aleph_{\omega \omega} = \aleph_{\omega^2} .. \aleph_{\omega^\omega} .. \aleph_{\epsilon_0} .. \aleph_{\Gamma_0} .. \aleph_{\omega_1} .. \aleph_{\omega_1+1} .. \aleph_{\omega_1 + \omega_1} = \aleph_{\omega_1 2} ..$
• For their cofinallity, singular cardinals form "nodes", or minima in cofinality graph: $cf(\aleph_\omega) = cf(\aleph_{\omega n}) = \omega = \aleph_0$

"But we all know the world is nonlinear"

• Halord Hotelling once questioned validity of linear programming in a presentation by George Dantzig (Dantzig 1991)
• John von Neumann adovocated for Dantzig, saying "If you have an application that satisfies the axioms, well use it. If it does not, then don't"
• To believe the validity of linear spaces, we need to ask the consistency of axioms of orthogonal linear domain
• So we need to prove existence of a model of othogonal n-dimension additive space and its geometry