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Adam I. Gerard
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NIU
CS MS TBD

Half Dimensions #4

Expanding on some nascent comments I made previously.

Some Conventions

I shall let:

  1. PA stand for Peano Arithmetic.
  2. RA stand for Robinson Arithmetic.

Also:

  1. I will use the term Float to refer to Numeric Type suffifient for representing Signed/Unsigned Float Arithmetic and varying-degrees of Decimal Point Precision. I choose this term over Reals which (although typically innocuous) imports some distracting historical baggage W.R.T. to the historic debates around the actual existence of Integrals, Fluxions, and Infinitesimals (or the manner of their specification/definition) but they should be understood as equivalent to the Reals here (for all intents and purposes).
  2. I will use the term Integral or Infinitesimal Calculus to mean the general manner in which increasingly smaller Numeric quantities are calculated, determined, or evaluated.

Some Historic Predecessors

  1. Leibniz himself argued that his general techniques around Integral and Infinitesimals were universally applicable anywhere continutity and terimination come into play. "Proposito quocunque transitu continuo in aliquem terminum desinente, liceat raciocinationem communem instituere, qua ultimus terminus comprehendatur." pp. 1551 ("In any supposed continuous transition, ending in any terminus, it is permissible to institute a general reasoning, in which the final terminus may also be included.")
  2. RA is a weak Number Theory and cannot provide a Model for standard Arithmetic. Basic properties of Multiplication and Addition fail within it. As such, I'll restrict the topic below to PA.
  3. I consider the Integral or Infinitesimal Calculus to be properly defined within (or against) the background universe of the Reals (Floats, per the above) rather than in some anciliary kind of Theory (popular in Leibniz and Newton's time).

Float Attachment

I shall here take to calling the Satisfaction of a Theory TH by a Model Mₙ when Mₘ ⊨ TH and Mₘ ⊂ Mₙ Attachement to convey the idea that:

  1. Mₘ ⊨ TH entails that Mₙ ⊨ TH.
  2. In other words, if Mₘ Satisfies TH, Mₙ does - e.g. that Mₙ is also "attached" to TH in the same semantically-relevant way.
  3. Expressed more formally, succinctly: (Mₘ ⊂ Mₙ ⋀ Mₘ ⊨ TH) → Mₙ ⊨ TH
  4. It leaves certain specifics about the exact nature of the "attachment" unspecified (e.g. - the exact nature about the relationship between Infinitesimals and the Reals) but shows that such a linkage substantially exists.
  5. This is a fairly trivial kind of Model-Theoretic relationship but one that's useful to drive home a particular point/relationship.

Call Float Attachment whenever a Model Mₘ Satisfying Theory TH will entail Mₙ ⊨ TH per the above.

  1. If a particular Theory is Satisfied by a suitable Float Model, it implies that certain Float-extensions will also hold;
  2. Namely, the Integral or Infinitesimal Calculus (being grounded in the Reals or not).
  3. This appears to be in the spirit of Leibniz's views (above) since a Theory being Satisfied by a suitable Model exhibiting continutity and terimination makes it a legitimate playground for the Integral or Calculus.

The aim here is to identify and define a formal property that can be used to reason about various Infinitesimal Rule Failures.

Cartesian, Leibnizian Failure and Fuzzy Sets

Fuzzy Set Theory essentially introduces a Fuzzy Elemental Inclusion Operator that maps A ∈բ Ω (the Elemental Inclusion Relation) to the Inclusive Interval [0,1].

These seem to be a natural goto. The path forward (to making sense of say Riemannian Manifolds, Minkowski Spaces with 3.14 Dimensions) is to use the "obviously" relevant conceptual unit: Fuzzy Sets (they're Sets and they're Fuzzy at that)! Indeed, Fuzzy and Partial Sets have been longed explored (and are now being extended into Topology).

I think there are some deeper issues that linger and start to rear up the moment we look more closely at the conjunctive triad:

  1. Fuzzy Set Theory
  2. Half-Dimensions (Non-Integered Quantities of Space Time Dimensions)
  3. PA Float Attachment

Dimensional Indeterminancy and Cartesian Multiplication:

  1. Consider standard-fare Cartesian Multiplication: R ✕ … ✕ R. Fuzzy Set Theory generally "normalizes" or "generalizes" the result depending on the specifics of the Fuzzy Theory at hand.
  2. min is typically selected - e.g. given two Sets A, B such that A ✕ B and a ∈բ A = .5, b ∈բ B = .1 then [a,b] = .1<.
  3. Another, perhaps more popular approach maps out the full gambit of possibilities (looking at some combination of both min and max values of all combinations within the Product).

We now arrive at the first concern:

  1. Per theTotality of Presence, the quantity of Dimensions an Object has is determined by the number of Coordinates it has.
  2. But this means some Coordinates within the definition of an Object can exhibit varying degrees of Elemental Inclusion depending on the method or full range of possibilities.
  3. So, either this reveals a limitation of current Fuzzy Semantics and extensions to our present considerations (possibly explaining why no one has thus far asserted a connection between them) or, more radically, implies that Objects have Indeterminate quantities of Dimensions after Fuzzy Cartesian Multiplication.
  4. This also applies to any Point in the Space that is Fuzzy Cartesian Product of any Non-Integered Dimension (so represented, with Fuzzy Sets).
  5. Moreover, the resolved Dimesion of some Coordinate would be influenced by other Coordinates based on the degree of coercion introduced by the choice of min and max.
  6. That is to say the Dimesions of an Object or Point in the Space would either be Indeterminate or Dependent on other Objects in the Space.

Pretty heady stuff. Even with standard-fare Fuzzy Semantics used (and considered unproblematic), there's another line of questioning that stems from how typical Integral and Derivative Chain, Product, and other Rules can be derived - how traditional Float Concepts can be respected in such a scenario (using Fuzzy Semantics):

  1. The argument here begins with the observations about Float Attachment. If we allow a Float-quantitied number of Dimensions, then so too should Leibnizian-considerations around Integrals and Infinitesimals.
  2. But then, multiplication of Non-Integered Dimensions defined by Limits, Derivatives, and Integrals should obey the Chain, Product, and other Rules.
  3. Fuzzy Set Theory has no such in-built facility (although one could potentially equip Fuzzy Multiplication Rules to respect standardfare Integral or Infinitesimal Calculus) and I know of no such system that's addressed these specific concerns.
  4. In tandem with the above concern, the computation of the resultant values would also be indeterminate or give rise to a local range (among other oddities).

Interesting Reads

  1. pp. 2 provides a concise formulation of so-called Infinitesimal Rules of Neglect (whereby Newton would treat some calculations as being 0-ed out).
  2. Also detailed here - a ways down in a tidy table.
  3. https://www.ams.org/notices/201211/rtx121101550p.pdf

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