8. The Digital Analytic Democritus-Cartesian or Vector Space Euclidean Geometry.

THIS PROJECT IS UNDER THE NEXT PHILOSOPHICAL PRINCIPLES

1) CONSCIOUSNESS IS INFINITE. CONVERSELY  THE INFINITE IS A FUNCTION AND PROPERTY OF THE CONSCIOUSNESSES.

2) BUT THE PHYSICAL MATERIAL  WORLD IS FINITE.

3) THEREFORE MATHEMATICAL MODELS IN THEIR ONTOLOGY SHOULD CONTAIN ONLY FINITE ENTITIES AND SHOULD NOT INVOLVE THE INFINITE. 

THIS PROJECT THEREFORE IS CREATING AGAIN THE BASIC OF MATHEMATICS AND ITS ONTOLOGY WITH NEW AXIOMS THAT DO NOT INVOLVE THE INFINITE AT ALL.

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.

The basic philosophical, logical and conceptual principles and methodology that reinvent and re-found the basic mathematics as digital, starting from their foundations and axioms include the following 7 principles

 

A1) Each finite quantity of physical reality material is made up of a finite number of atoms, so the (digital) mathematical models of the physical reality must in turn only be based on finite sets of entities, such as finite number points, digits, etc.

 

A2) The Infinite is an important subjective experience for the scientist who studies the natural world, but it must remain in the realm of consciousness and not exist in the field of objective ontology of the (digital) mathematical models.

 

A3) The quantification by measurements in the (digital) mathematics of the physical reality has many parallel simultaneous levels of precision levels or resolutions (number of digits in the measurements), but always stops up to a finite maximum level of accuracy. For the basic digital mathematics, 2-4 precision levels or resolutions are sufficient. The exact number of digits is left variable but finite.

 

A4) Any equality in the quantification and measurements of (digital) mathematics must determine the finite level of accuracy otherwise it is undefined. We do not allow accuracy of infinite many digits.

 

A5) The infinite is not allowed not only in (digital) mathematical objects to be studied, but also in the (digital) mathematical formal logic used by (digital) mathematics. All symbols and formal propositions of a mathematical theory is a carefully defined) finite (but variable) number.

 

A6) Although all digital mathematics ontology is finite, and the infinite is not allowed, still useful concepts are introduced such as "seemingly infinite number", "seemingly infinitesimal number", "seemingly irrational number" etc. The way these concepts are introduced is through the coexistence of different finite precision levels or resolutions of vast differences in the number of digits.

 

A7). The digital mathematics thus created, although similar to classical mathematics, are logically and in the details different. They are not logically equivalent to classical mathematics. It may be either more difficult or easier than classical mathematics, but while classical mathematics has complexities irrelevant to the physical reality, the digital mathematics always has complexities related to physical reality complexities and complexities of the scientific quantitative practice

 

Α8) When stating a classical familiar  theorem within the digital mathematics, we start with the corresponding in classical mathematics and then we are considering how it is applied in real human practice with numbers of finite many digits, and so we find the closest and most meaningful and elegant formulation of it within the elaborate ontology of digital mathematics

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.

A guide to re-write the basic theorems of analytic geometry to a digital analytic geometry is the next

 

1) We read carefully the corresponding theorem of the classical Cartesian analytic geometry and we think how applied scientists like engineers , etc would apply it, with real numbers of finite many digits, up to a precision level and so we try to re-state it within the context of the digital Cartesian analytic geometry

2) We make sure to put assumptions that make the figures exist fully inside the digital Cartesian space

3) We distribute the various magnitudes and numbers of the theorem and relevant figure at the 4 precision levels, so that it makes sense for the applications of the engineer and applied scientist.

4) We specify the occurring equality and order (or inclusion) relations to the 4 precision levels again so that it makes sense for the applications of the engineer and applied scientist.

5) If there are more than one choices for the previous we choose the simplest formulation which occurs most often in practical applications.

6) We review our statement so that we make sure that it is valid and easily provable, along similar lines of the classical proof.

7) We make sure that the balance of utilized symbols and terminology is not too complicated compared to the intended interpretation and application actions.