Here is where we will need an at lest 3-levels of precision , system of real numbers, to formulate for example Riemann geometry, and differential manifolds with linear differential connections.
Here one visible point of the Riemann geometry manifold will have in it , a whole digital euclidean space, (the traditional tangent Euclidean space) which already has its two levels of precision points visible. and invisible (that are nevertheless invisible from the global Riemann geometric space).
The experience of utilizing the Google maps, in the web, gives the idea, but in addition here the scales of precision are discretized, in to at least 3.
At the end of this chapter, there is
a) Advantages-disadvantages of these new digital mathematics compared to the classical analogue, infinitary mathematics.
b) A fictional discussion in dialogue form of celebrated historic creators of the old mathematics praising or questioning the new mathematics compared to the old.
Creating 3D mental images with the imagination
The physiscists andd nobel prize winner R Feynamnn often was explaining that he was training his imagination imagining in 3D mental images the moving molecules of a gas in box, and perceiving that temperature as their avearge velocity, the ressure as the average moemntum of the walls of the box, and the density as the average number of particles per unit of volume, He was explaing that he was getting much pleasure from this spiritual excerise of craeting 3D mental images and simulating the reality.
Excatly this 3D VISUAL MENTAL IMAGES LANGUAGE parctice is UTILIZED BY ME AND is required by someone to understand and develop the digital or natural mathematics, in geometry, real numbers and analysis.
Rules for phantasy and drawing of figures.
As initially we considered a system of digital real numbers R(m,n,p,q) we consider the points of P(m), P(n) as visible in the figures while the points of P(p) as invisble pixels , and those of P(q) as invisible atoms. Therefore, even the points and seemingly infinitesimals that will be defined below, of P(n) relative to P(m) are considered as visible. This is in accordance with the habit in classical mathematics to make the points visible, although they claim that they have zero size.
