Showing posts from February, 2023

AxLam Example: Questions about Triangles

AxLam Example : Questions about Triangles Consider the humble, mighty triangle:  a connected shape with 3 vertices, 3 sides, and 3 angles.  We will experiment with AxLam models for describing and computing with triangles individually and in groups, at several levels of mathematical rigor.  On this road we touch briefly on some fundamental aspects of mathematics and information, while staying intuitively grounded in the triangle simplex.   In a computer-assisted modeling paradigm, it is natural to proceed constructively. An intuitive way to start modeling triangles is using the lengths of the 3 sides of the triangle. In any programming or data language, we might quickly write down an ADT, record or class containing  three numeric sides, looking something like: class  TriangleSidesRecord (sideA : Number, sideB : Number, sideC : Number) This structure is sufficient to begin computing with Triangles.  However, when we examine the semantics of this simple record structure, we quickly find m

Overview of AxLam model : Nouns, Types, Morphisms

2023-Feb-03 picking up from tweet thread of 220909  AxLam model: Noun (space), Type (paint), Morphism (arrow) AxLam is for writing models using [concepts, types, and morphisms].  Our concepts may be in the form of some data, or less specifically as things we choose to name or refer to in a model.  Over this concept space, we use types and morphisms to define an adjustable descriptive model that directs and regulates processing of the concept data (or other manifestation).  Alternatively and  more approachably we may write in terms of [nouns,  formal classes, and mappings], and this framing is mostly equivalent for informal purposes. Nouns / Concepts / Terms / Data Nouns (i.e. concepts) describe anything we wish to refer to, and need not be explicit or named.  We think of nouns as the space of things we might try to describe, name or refer to.  "Terms" in a type-theoretic sense are nouns.  Real things in the human world that someone cares about are nouns.  In practice when we