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