[ notation | reals ]

The Continuum

The problem

We need smooth manifolds. They require:

continuous, differentiablle

classically topology/limits; reductio :-(

measure

full Lesbesgue is dangerously powerful; too much Choice :-(

Plan

Crave less, only as much as needed; allow implementations to deliver more.

challenge-response → funtion from challenge to response

well-behaved &implies; continuous, but other things may also be continuous; badly-enough-behaved &implies; discontinuous, limiting scope on other things

measurable generated from simplices or their well-behaved images

other things may be measurable - unite, finite intersect; spheroids - subject to constraints on measures.

Details

simplices

natural to differentiable; viable seed for a topology; well-defined way to stipulate interior and voluminous, with dimension = {natural n: no n-simplex is voluminous} as free by-product.

continuous

tacitly induced from topology generated by simplices, stipulating enough that nice things work, esoteric things may, bad enough don't.

crave just enough

in each case, to reason about the property while ensuring we know the type applies to the things Physics wants to use.

The weird properties of non-exhibitable things may arise for an implementation that believes in them; but the cross-implementation API only cares about observable behaviour (including interactions with exhibitables) of whatever you claim exists to determine whether they belong in each type.

Niceness

Various categories of nice functions:

• measurable
• continuous
• [ continuously ] [ n-times ] differentiable
• smooth

Categoric rules, and rules of similar style, induce a core of each category from some seeds, including polynomials and [anti-]linear maps. Same rules guaratee required properties of the category with implementation's other nice things included. These properties tell us what we can do with at least the core, which hopefully contains all we strictly need.

Implicit Spaces

A smootth manifold is characterised by an atlas of chart, each a "smooth" bijection between patches of the manifold and some linear space. The manifold is a union of patches; the charts behave well on intersections.

The manifold's implementation is encapsulated so that we can ignore it and study the manifold's property entirely in terms of the charts in its attlas. Orthodoxy doesn't need that to describe continuity or differentiability, but I'll need such abstract spaces for manifolds, so shall not be shy of usinnng them in specifying the parts that lead to it.

Written by Eddy.