Miser Project: ‹ob› Primitive Functions

2019-11-05 Update: Complete migration from the WordPress post
2019-11-02 Update: Tiny edits and migration to Blogger

Preliminaries

• Hark! Is That a Theory I See Before Me?
  The first-order logic with equality notation for Ot, the mathematical theory for ‹ob›

Characterizing the mathematical structure ‹ob› = 〈Ob,Of,Ot〉involves first-order-logic with equality as the project notation for expressing  an applied logic, Ot, in which Ob is the entire domain of discourse.   Predicates and functions (in Of) are also introduced by characterization in Ot.

Four distinct functions, along with equality, are sufficient to distinguish among all of the Ob obs.  It will be the case that characterization of obs, including as transformations of other obs, will reduce to compositions of these functions and cases of equality/inequality.

The notions of individual, enclosure, singleton, and pair have their inspiration in the nested array work of Trenchard More and  the foundation structure of LISP by John McCarthy.  It is intentional that these structural characteristics are limited and simple.

An essential characteristic is that the primitive notions have the structure be closed and the primitives be total.  One cannot in any way “fall out” or “fall through” a primitive.  And if the operands are defined, so are the results, viewing the primitives as operations.  This tidiness will be valuable in demonstrating and reasoning about practical computational interpretations of the mathematical entities of ‹ob›.

The elaborate treatment of just this much is also indicative of how much one must assert to narrow the objects of discourse to ones that are suitable for underlying a model of computation.  And we are not quite there yet.

At this point, consider that mathematical treatment is rather different than construction of computer programs that embody a mathematical theory in some sense.   It is one purpose of The Miser Project to sharpen and clarify that condition.
  

Continuation

1. Narrowing  ‹ob› for Computational Interpretation
   Summarizing the primitive notions and positioning for computational interpretation with additional restraints
2. Representing Data as Obs
   Expanding on the difference between a logical mathematical theory and computational interpretation, noticing that obs themselves can be interpretations of data, that interpretation being carried over to a computational interpretation.  SML is used to demonstrate one operational interpretation in a programming language.
3. Representing Functions in ‹ob›’s Abstract World
   Reasoning about functions on obs and then about interpretation of other structures by representation of functions in interpretations.
4. Interpretation, Representation, Computation, and Manifestation
   Reprise and account for specialized concepts in Miser Project, setting up for representation of the computation model