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.
It is discussed in particular the single and double (infinitesimal) precision probabilities.
Also the paradoxes of geometric probability.
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.
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.
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 6 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 definite (but variable) finite 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
It is discussed in particular the single and double (infinitesimal) precision probabilities.
Also the paradoxes of geometric probability.
The theory of Hierarchical sampling is applied , to probabilities valued in the digital fractional real numbers or digital decimal real numbers of two levels or three levels of precision.
In particular the remarkable effects of multi-resolution sampling, where at a high resolution of the internal digital real numbers, a probability or sample statistical frequency may be say 55%, while in a low resolution or external digital real numbers the probability (of the same phenomenon) may be (in a coarser layer of the hierarchical sampling after regrouping and concluding success or failure) 85% or higher
The Theory of Probability
This very important subject changes in the same way as the Analysis. All distributions are up to a resolution. All the moments of distribution are finite many. The characteristic function of distribution is a series of finite many terms. The continuous random variables are those that the distance of two possible values may be less than the visual threshold! Again all become simpler and transparent at a single resolution. But we may have also multi-resolution probability and statistics.
Various physical, or social, or financial phenomena, have systems of stochastic equations, up-to-a resolution. Nevertheless of we change the resolution, and accept smaller and larger numbers or probabilities relative to the unit, he we may have to add some new equations to handle new areas of quantities and mutual relations. In addition the same system of equations, although with a unique solution in the classical sense , may have different solutions, at different resolutions!
In particular the paradoxes of geometric probability (Bertrand's, Buffon's needle etc) are better understood why they are met, after the specification of resolution in geometry and resolution in the quantities of probabilities. The probability sample spaces are finite, and all the paradoxes are resolved in to crystal-clear terms in a unique way, that most would find almost obvious due to new details of the geometric ontology that were not existing in the classical geometry of infinite point sets.
Although the next remark is not directed related with the changes that the ontology of finite resolution makes in maths, it is of significance to mention if we want to avoid tactics that breed intentions to almost lie with sophisticated scientific way. The standard way that statisticians or applied scientists "fit" or estimate a stochastic processes or time series over a one-element sample of paths, (a single only observed path) must be definitely be avoided! We can "fit" in this way plenty many radical different stochastic processes with high degree of classical "goodness of fit" measures and practically claim all different and opposite assertions ! Statistics requires repetition, and large samples, and this applies in the case of stochastic processes to paths and not points. So at best, a way to cut and make many elements sample of paths, is required when only only path is observed!
THE MAIN POINTS TO RE-INVENT THE PROBABILITY AND STATISTTICS AS DIGITAL PROBABILITY AND STATISTICS ARE THE NEXT
0) All the proofs are within the digital (formal) logic
1) All numberical operations and equations are in the digital real numbers thus , all equalities are of specific nature designated up-to a precision level (number of digits).
2) The axiomatic definition of probability measure are always with finite set of finite sets (fiite set theory set theory)
3) All involved functions like exponential, logarithmic etc are the corresponding digital such functions
4) All differentiations and integrations are of the digital differential and integral calculus
5) All limits like that ofthe law of large numbers are within the digotal differential and integral calculs, tsu up to aprecision level.
THE MAIN POINTS TO RE-INVENT THE PROBABILITY AND STATISTTICS AS DIGITAL PROBABILITY AND STATISTICS ARE THE NEXT
0) All the proofs are within the digital (formal) logic
1) All numberical operations and equations are in the digital real numbers thus , all equalities are of specific nature designated up-to a precision level (number of digits).
2) The axiomatic definition of probability measure are always with finite set of finite sets (fiite set theory set theory)
3) All involved functions like exponential, logarithmic etc are the corresponding digital such functions
4) All differentiations and integrations are of the digital differential and integral calculus
5) All limits like that ofthe law of large numbers are within the digotal differential and integral calculs, tsu up to aprecision level.
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.
