We want to begin creating formal sequential definitions for constructing the fundamental objects of this system. Do we want to create all these objects from something absolutely basic like a bit -- which creates an alphabet of terminal characters -- from which are created all higher-level composite objects? Or start with a more traditional basis like a set or an ordered class? Maybe the answer to this question in involves two levels 1) the actual construction of the object or definition in terms of "data structures" -- in terms of bits, terminal characters 2) the symbolic abstract meaning of some structure in a sequence that builds all these definitions Something like 1) the absolute physical construction of the object -- the technical/physical construction of a word -- its letters selected from a terminal alphabet, the definition of the font that constructs those letters, the bit/pixel pattern that constructs the word 1a) this borders on issues in semantics about "sounds" -- phonemes, that create words 2) the mathematical properties of this object What is an axiom? Abstract object

What is an axiom? Placeholder | Back The Wikipedia reference makes a couple of important distinctions Logical axioms Thu, Apr 8, 2021 Reference An axiom, postulate or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíma, 'that which is thought worthy or fit' or 'that which commends itself as evident.' The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. As used in modern logic, an axiom is a premise or starting point for reasoning. As used in mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, "axiom", "postulate", and "assumption" may be used interchangeably. In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., parallel postulate in Euclidean geometry). To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms), and there may be multiple ways to axiomatize a given mathematical domain. Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the philosophy of mathematics. URL https://en.wikipedia.org/wiki/Axiom Formal systematic definitions Axiomatic systems Equality

Abstract object Placeholder | Back Generally we mean something represented by a symbol in a medium -- a "map" A "concrete object" is the territory that the map (symbol) is pointing to (representing) The dimensional construction of abstract objects -- a subject we want to explore as foundational to this method Thu, Apr 8, 2021 URL https://en.wikipedia.org/wiki/Abstract_and_concrete Core vocabulary Formal systematic definitions