Still another way to understand
the digital or natural mathematics is to realize that they are essential a
detailed and systematic presentation of what actually the scientists that apply
the classical mathematics (meteorologist, physicists, biologist, ecologist,
economists, etc.) are forced to do since they cannot apply neither infinite
steps, or objects of infinite information, neither pretend that they have
infinite many material atoms. Thus what they do is logically not equivalent to
what the classical mathematics are , Sometimes they call it approximation, but
essentially it deserves a totally new terminology and presentation. This adds
to the applied mathematical scientists a layer of better consciousness and
clearer awareness of what they are doing in mathematical terms. So that what
they say, what they do and what they see has congruence greater sincerity and
integrity
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 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.
THE MAIN IDEA
The
purpose of this research program is to develop through research and disseminate
by publications and books the development, of the Democritus digital calculus
without the infinite for fast and high accuracy computerized applications of
mathematics in the physical applied sciences. The Democritus calculus is by far more efficient compared to numerical
analysis, as in it, there is no concept of approximation, only exact
calculation. In the Democritus calculus the mathematical objects
exist as pure digital mathematical objects, with only finite many visible or
invisible points, digits, finite information etc. As is the case with the natural physical reality objects made from
atoms
It is also a new method of teaching, writing
developing and applying mathematics where there is subjective congruence, in
other words consistency and integrity in what we
say, write, see, think and act-apply (because it does not include the
infinite), so that the communication scientists
can trust and understand each other and the maths too.
The Democritus digital
calculus gives an application way out for the new millennium of the almost
metaphysical Newton-Leibniz calculus of infinitesimals, by far better and more
sophisticated than the classical Weierstrass calculus with limits. There is a partial
similarity of Newton-Leibniz theory and modern computerized Democritus calculus
practice captured as an abstract axiomatic system of calculi over
multi-density quantities.
As an example one of the excellent applications of
the Democritus digital calculus is the exact experimental investigation of the
non-linear partial differential equations corrections of the classical Maxwell
linear equations of electromagnetism that allow for an understanding and
measurement of the renewable energy that was discovered by N. Tesla almost one
century ago and was never really understood. The details of these corrected
equations are in the PI’s
book “Solving the global CO2 problem”
First, someone aims with a gun, and then he pulls
the trigger, not the reverse. First someone thinks in abstract way and then in
concrete way for technological inventions. In other words, one must first
invent the exact mathematical ontology of digital objects (digital numbers,
digital functions etc.) and then
implement them with digital technology. It is the goal that in the present
research we discover how we do it, and we present it by publications and boo in
detail so that more and more contemporary scientists should engage, refine and
publish further the development of the Democritus calculus without the concept
of infinity. In this way this will affect not only application in physical
sciences and social sciences but also it will become a beautiful and radical
change and improvement in teaching the mathematics at high school, and
universities that will make more students to love more the mathematical
applications.
The need for this becomes
apparent as soon as we realize that when we utilize in mathematics the ontology of the infinite then their
applications that cannot include the infinite (we cannot run infinite steps) we
become subjectively and scientifically in-congruent, in other words we thing, talk and write with infinite and we
act and apply with the finite. Thus the need of the Democritus calculus
that it will give back to us the congruence.
The mathematics as deductive discipline and
science (like the Olympic games and other cultural innovations) began in
ancient Greece, and that Democritus was famous in eliminating the superstitious
“infinite gods” from the world and replacing them with the finite material
atoms. That is why we use his name in the new finite decimal digital calculus. The Democritus calculus is essentially the
Newton-Leibniz calculus, where we substitute the literally infinitesimals of
Newton-Leibniz (smaller than any real number) with “seemingly infinitesimals
but finite differences”. It is
required though a very comprehensive and sophisticated definition when a finite
difference is to be considered “seemingly infinitesimal” when “feasible finite”
and when “seemingly infinite”. This
is what any applied scientist would be forced to do when applying the
Newton-Leibniz calculus of infinitesimals or the Weierstrass calculus of limits
and convergence. Still the Democritus digital
calculus is by far logically non-equivalent with the Newton-Leibniz calculus of
infinitesimals and logically non-equivalent with the Weierstrass calculus of
convergence and limits as it does not utilized convergence, approximations and
limits. It is a more efficient calculus in computerized applications that resembles
the above classical calculus. For example, in the Democritus calculus
there are many grades of differentiability and continuity depending on the
scale and resolution of decimal quantification, while in the classical Newton-Leibniz
and Weierstarsse calculus the differentiability and continuity is scale
independent and absolute. The similarity of the classical calculus with
the Democritus digital calculus which is essentially the similarity of
classical theory with the infinite and the practice with the finite is captured
as special instances of an axiomatic
system of abstract calculi over multi-density quantities which could be the
Newton-Leibniz with the real infinitesimals and infinite limits or the
Democritus calculus with the finite seemingly infinitesimals. A situation logically similar e.g. with the
finite dimensional and infinite dimensional vector spaces that are both
instances of the axiomatic system of linear vector space.
In overall we realize that this research project
is not only ingenious , but it will
also have great impact to young scientists (faster , and more clear methods of
applications) , but also it is of
low risk and cost because after
giving the definitions without the infinite in the Democritus calculus the new
proofs in are derived by a straightforward transcription from the corresponding
proofs in classical analogue Newton-Leibniz and Weierstrass calculus, In
spite of this we should stress here that : The Democritus digital calculus created as in the current project, although
similar to classical mathematics, is logically and in the details different.
They are not logically equivalent to classical mathematics. The Democritus calculus
has a new complexity which is though directly relevant to the material physical
reality which also consists from finite many atoms, while the classical
calculus has a complexity irrelevant to the material reality and relevant to
the infinite.
Almost everyone understands the difference between
analogue images and audio recordings from digital images and digital audio
recordings. As is known, almost all the mathematical objects of classical
(otherwise analogue) mathematics, such as numbers (e.g. numbers pi, e,
irrational numbers etc.), geometric triangles, circles, functions, differential
manifolds etc., are just like analogue images and sounds because they are
defined with infinite many elements, and recent times, the applications with
modern digital technology require an APROXIMATE ONLY computer digital
representation. Special
mathematical areas and disciplines have been developed for this, such as
Numerical Analysis, Approximation
Solution by the Computer of the Differential Equations, etc. But this is very
complicated and difficult, it costs time, learning, risky mathematical
improvisation on the part of software developers etc. In addition, it requires
scientists to learn mathematics twice. One as analogue objects that include
infinity and then as digital approximations in the computer applications (or
physical reality applications). Thus the gap between theory and practice is sadly
perpetuated. The Democritus digital mathematical calculus solves this problem fundamentally.
By going back and deeply in to the
very foundations and definitions of mathematics, to formal logic, to integers
and real numbers to the axioms of geometry, to the definition of functions,
etc., and banning the use of infinity from their “DNA”, we re-define all
ordinary mathematical objects as pure digital mathematical objects, with only
finite many visible or invisible points, digits, finite information etc. As is
the case with the natural atomic physical reality objects or as is the case
with the bits and pixels of software objects in the computers.
So when it comes the time for computer-based
applications, they DON'T NEED A DIGITAL APPROXIMATION since they are already
digital, but only their EXACT UPLOAD in to the computer (such as is the
emailing transfer of text-files or transfer of digital images-files, digital
recorded sound-files etc.).
The gain in the reduction of the
conceptual complexity, the gain in physical realism, the gain in less effort,
the gain in time, and the gain in efficiency on the part of scientists or
students is large and obvious. Moreover, when one understands what this is
about, and how it is done systematically (i.e. applying mathematical analysis through the Democritus digital calculus
) it is after all an easier practice compared to learning to approximate by
digitization the classical analogue mathematical objects. The next diagram
shows the difference between the current innovative method beyond the present
state of the art in digital applications that the current research creates and
the current state of the art and practice in digital applications.
The vision of discovering and founding the Democritus mathematical calculus as digital mathematics without the infinite is an old vision of most mathematicians of history. But in the past, as the digital technology had not developed, it was not easy to conceive how this could be done. It was an old dream of mathematics, starting with the Pythagoreans in ancient Greece to create mathematical models that would contain only finite entities, and would not use the infinite (a rather phenomenological mental abstraction) which has proven to create unsurpassed difficulties in the development of mathematical knowledge of the physical and social reality.
Many were the mathematicians and other scientists who have suggested in the past time that some mathematical objects can be re-invented as digital. An example was the mathematician and philosopher Bertrand Russel. However, it usually concerns standardized digital approaches to the corresponding analogue mathematical objects. Also the term digital mathematics in literature is usually used mainly only for the approximate digital representation of analogue mathematical objects.
As far as we know, the above innovative approach of the Democritus digital calculus has not been fully introduced (starting from the axioms of mathematical theories as a re-foundation of mathematics) and systematically and integrally to all basic mathematical analysis, by other researchers internationally except of the Professor, Kyritsis Constantine (see references [1], [2]).
This program develops fully the new Democritus digital mathematical calculus and shows to contemporary scientists how to reinvent the mathematical analysis as digital and greate a brifge for direct material applications , with all due respect to historical classical mathematics, since it requires very good knowledge of both classical mathematics and digital computer technology. After all as spiritual existences that we are we can measure our spirituality by the ability of our consciousness (e.g. mathematical science) to control the material reality (through the Democritus digital calculus for the applied material sciences)
The basic philosophical, logical and conceptual principles and methodology of the Democritus calculus 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 Democritus digital mathematical calculus thus created, although similar to classical mathematics, is logically and in the details different. It is 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 and relevant to the infinite, the digital mathematics always has complexities related to the complexities of the physical reality and complexities of the scientific quantitative practice
These principles allow for an alternative and more realistic universe of mathematics, directly usable in artificial intelligence of computers. The development of Democritus digital mathematical calculus is somehow more elaborate in the definitions but radically easier in the proofs compared to classical mathematics. It is certainly not logically equivalent to the classical mathematics. But here we give emphasis only to the theorems that are very similar and very familiar with corresponding to classical mathematics.
THE IMPACT IN THE MATHEMATICAL, PHYSICAL
AND SOCIAL SIENCES.
The web portal
that will be created will enable reference and asynchronous access to the core
philosophy and techniques of the basic mathematical areas, for future research,
synergies, lectures and development. The same with the book that will be published. The title of it could be
THE NEWTON-LEIBNIZ CALCULUS OF INFINITESIMALS AND THE NEW APPLIED
DEMOCRITUS DIGITAL CALCULUS OF THE 21ST CENTURY A partial similarity of theory
and practice captured as instances of the axiomatic system of abstract calculi
over multi-density quantities.
Although the proposed program is relatively low-cost
and of a spermatic nature, its quality
is highlighted. If in the future this current Democritus calculus way of
teaching and applying mathematics (as pure digital mathematics) with digital
technology in the applied sciences is widely disseminated, the mission of this
research project would have been accomplished.
WE SUMMARIZE SOME OF THE IMPORTANT
POINTS OF THIS RESEARCH PROJECT
1.Ingenious
project that covers a long standing gap in the evolution of mathematics and
their applications.
This research project is
an initiative and spermatic project and has no exact previous in the current
state of the art of the mathematics relevant to the digital world. It is not a
research that continuous the research of other researchers. Except of course
that it does continuous and re-founds the mathematics after the dissemination
of the digital technology, as pure digital mathematics. Because it is radical
and goes utterly deeply at the very axioms of the basic mathematics and
furthermore it digitalizes the meta-mathematics as well through the digital
formal logic, it should be understood that it is a very great and ambitious
project. The previous such great initiation was 2,000 years ago with the
initiation of mathematics by the mathematicians that contributed to Euclid’s
geometry, the mathematics of Pythagoras etc. For this reason, the current small
project is only spermatic and initiative.
2.Virtual
impact: Ingenious and desirable.
The germ-like importance
of this research project will be realized my the many scientists only after the
gradual dissemination of its ideas and techniques. But this dissemination will
utilize the increasing momentum of the digital technology which persuades more
and more people that it is valuable and more realistic. The current project of
Democritus digital calculus will create an acceleration of the scientific
applications that will be implemented faster and conceptually with a more
transparent and logically consistent mathematical context.
3.The
project as low risk research project.
Once a mature
mathematician understands the basic 7-principles A1-A7ofthe Democritus calculus,
the development of the basic pure Democritus digital mathematics (numbers,
geometry, analysis, differential equations etc.) is quite straightforward with
rather small surprises. It only takes time to ponder about the new ontology,
love in doing it and patience to write all the details.
All the basic results and
theorems of the classical calculus are already written in standard books and he
only needs to re-write the proofs rather slightly differently with the new
ontology and definitions and discover new theorems that do not hold in the
classical calculus. But as we mentioned as the new pure Democritus digital
calculus is not logically equivalent to the classical analogue mathematics,
some great deviations will occur. We mention two of them.
1)
The inability to “square the circle” does not exist in
the Democritus digital geometry as all lengths are rational numbers and all
rational numbers are constructible by ruler and compass.
2) The 3rd Hilbert
problem that has been solved in the direction of the existence of solids of
equal volume but non-equidecomposable, is solvable in Democritus digital
geometry in the opposite direction. (All
equal volume solids are equidecomposable)
If we are careful to avoid
such critical areas, and concentrate in re-producing the basic theorems of
mathematics in the pure Democritus digital mathematics as well, it is obvious
that the research is of low risk. The
only risk is not finding the time or the financial funding for this.
4.
Appropriateness and feasibility of the methodology.
The previous principles require a rewriting of
the classical axiomatic systems and areas of basic mathematics to those of
1) the Democritus
natural digital numbers,
2) the Democritus digital 1st and 2nd
order formal logic,
3) the Democritus digital set theory,
4) the Democritus digital real numbers,
5) The Democritus digital analytic Cartesian
geometry
6) The axiomatic
system of abstract calculi over multi-density quantities.
7) the Democritus digital ordinary and basic
partial differential equations
Etc.
The above sequence of the basic areas of
mathematics is known to be, since many decades of research and by almost all
Universities and department of mathematics the methodology of best and most
appropriate logical order to develop the basic mathematics for applications in
the sciences.
The title of the resulting book could be
THE NEWTON-LEIBNIZ CALCULUS OF INFINITESIMALS AND THE NEW APPLIED
DEMOCRITUS DIGITAL CALCULUS OF THE 21ST CENTURY A partial similarity of theory
and practice captured as instances of the axiomatic system of abstract calculi
over multi-density quantities.
5.Scientific
Achievements relevant to the research project and maturity and ability to
conduct this research
It should be guessed that
for someone to discover and develop the Democritus calculus as pure mathematics, the next abilities are
required
1)
MATHEMATICS: A thorough and mature knowledge of the
areas of the basic mathematics.
2)
PHYSICS: A good perception of the atomic structure of
physical matter and how nature creates material continuous shapes from finite
numbers of atoms
3)
COMPUTER SCIENCE: A good perception of how the
software developers, and software engineers have created the operating system
in the computer so that in the computer screen are displayed continuous
mathematical objects, made though from finite number of bits and pixels.
The initiator of this
research professor Constantinos Kyritsis, happens to have all the above
abilities as he is an interdisciplinary
researcher. He has publications in mathematics, and physics, and he has
been proved to be a skillful computer programmer. Furthermore he has solved and published 2 of the 7 Clay
millenniums problems (the 3rd and 4th Clay millennium
problems https://www.claymath.org/millennium-problems )
about the complexity of the computer algorithms and about the Navier-Stokes
equations. (see references [12], [13], [14] ). Few of the researchers may have
all the above competencies and this was one of the reasons that such an initiative and ingenious research projects was not dared
so far.
REFERENCES
[1]
AN AXIOMATIC SYSTEM FOR A PHYSICAL OR DIGITAL BUT CONTINUOUS
3-DIMENSIONAL EUCLIDEAN GEOMETRY, WITHOUT INFINITE MANY POINTS.
By K. Kyritsis World Journal of Research and Review (WJRR) ISSN:2455-3956, Volume-5, Issue-4, October 2017 Pages 31-43
[2]
OUTLINE OF THE INTRODUCTION TO THE DIGITAL DIFFERENTIAL AND INTEGRAL CALCULUS.
By K. Kyritsis. Proceedings of the 3RD INTERNATIONAL CONFERENCE ON
QUANTITATIVE, SOCIAL, BIOMEDICAL AND ECONOMIC ISSUES 2019,pp
123-141 http://books.google.com/books/about?id=WrGlDwAAQBAJ
Banach-Tarski
paradox:
[3] Banach,
Stefan; Tarski, Alfred (1924). "Sur la décomposition des
ensembles de points en parties respectivement
congruentes" (PDF). Fundamenta
Mathematicae (in French). 6: 244–277.
[4]
Boltianskii V. (1978)“Hilbert’s 3rd problem” J. Wesley & Sons
1978
[5]
Euclid The 13 books of the Elements Dover 1956
[6]
Hilbert D. (1977)“Grundlangen der Geometrie” Taubner Studienbucher
1977
Hausdorf
paradox:
[7]
Hausdorff Felix (1914). "Bemerkung über den Inhalt von
Punktmengen". Mathematische Annalen. 75:
428–434. doi:10.1007/bf01563735
[8]
Moise E. E. (1963) “Elementary geometry from an advanced standpoint” Addison
–Wesley 1963
[9]
Von Neumann paradox:
On
p. 85 of: von Neumann, J. (1929), "Zur allgemeinen Theorie
des Masses" (PDF), Fundamenta Mathematicae, 13: 73–116
[10]
Pawlikowski, Janusz (1991). "The Hahn–Banach theorem implies the
Banach–Tarski paradox" (PDF). Fundamenta Mathematicae. 138:
21–22.
[11]
Wilson, Trevor M. (September 2005). "A continuous movement version of the
Banach–Tarski paradox: A solution to De Groot's problem". Journal of
Symbolic Logic. 70 (3): 946–952. JSTOR 27588401. doi:10.2178/jsl/1122038921.
[12]
The solution to the 3RD Clay Millennium Problem. A short proof that P not equal
to NP and NP = Exptime in the context of Zermelo-Frankel set theory.
International Journal of Pure and Applied Mathematics Volume 120 No. 3 2018,
497-510 ISSN: 1311-8080.
[13] A Short and
Simple Solution of the Millennium Problem about the Navier-Stokes Equations and
Similarly for the Euler Equations Journal of Applied Mathematics
and Physics 2022 10(8):2538-2560 DOI: 10.4236/jamp.2022.108172
[14] The solutions of the 3rd and 4th Clay Millennium problems.
https://arxiv.org/abs/1902.07265
The Solutions
of the 3rd and 4th Millennium Mathematical Problems
The
solutions of the Millennium Problem P vs NP in computational complexity and the
Millennium problem in fluid dynamics
LAP Lambert Academic
Publishing ( 2021-11-25 )
|
ISBN: |
978-620-4-72562-8 |
[15] Solving
the CO2 Problem Solar Energy Stored in the Gravitational Field
The unification of electromagnetism, gravitation and inertia
LAP Lambert Academic Publishing (
2021-12-27 )
|
ISBN: |
978-620-4-21004-9 |
This post is some early philosophical and educational outline of the idea of axiomatic digital mathematics. But by now there have been formulated the new axioms of the digital mathematics (Digital natural numbers, Digital Logic, Digital Set Theory, Digital Euclidean Geometry, Digital Real numbers etc) , that can be found in the posts 3,4,5,7,7 respectively.
See also the post 21
http://thedigitalmathematics.blogspot.gr/2015/04/21-letter-to-research-sponsor-and.html
We can view the presentation as ppt file here
https://dl.dropboxusercontent.com/u/107295772/Digital_maths/Digital_mathematics_lectureNTUA.ppt
Our perception and experience of the reality, depends on the system of beliefs that we have. In mathematics, the system of spiritual beliefs is nothing else than the axioms of the axiomatic systems that we accept. The rest is the work of reasoning and acting.
There is nothing wrong in thinking infinite many mathematical entities, as our subjective mind is essentially free. The Infinite is mainly a phenomenological concept and by including it in the ontology of a mathematical model is as we simply include the confession that there are aspect of the model that we cannot measure. But when the infinite is included in the ontology of the mathematical entities and objects, problems start. When the infinite is a mathematical object we give also the opportunity to practical operations to manipulate our minds instead of leaving them free to think.
Whenever classical mathematics, models are applied to physical reality, they include the infinite. And if we look carefully we will discover that this infinite models a situation of a very large finite number in the physical reality. But we do know also from mathematics that the infinite and the finite have completely different ontology. Therefore, it is unavoidable that our classical mathematics model which involves the infinite will not fit well to the physical reality. On the contrary models of mathematics that do not involve the infinite at all, in an axiomatic way, have the ability to fit well to the physical reality.
The Infinite is mainly a phenomenological concept and by including it in the ontology of a mathematical model is as we simply include the confession that there are aspect of the model that we cannot measure.
When perceiving the world with the digital mathematics without the infinite we do not perceive a smaller world. It is the same world hat the traditional mathematics of the infinite describe.
I do not want to eliminate or disregard the classical mathematics with the infinite. but I want to point out that in too many cases of applications in the physical world, we need a more realistic system of mathematics that does not use the infinite.
The leading interpretation principles, so as to derive results in natural (axiomatic digital) mathematics without the infinite from results in classical mathematics are
A) The countable infinite is replaced with a too large finite number, that we cannot count with our resources of formal logic, and we do have information of how large it is (that is why we may denote it by ω)
B) For larger than countable , cardinal or ordinal e.g. α,β , with α<β and ω<=α again we replace them with too large finite numbers, that we cannot count with our resources of formal logic, or have information of how large they are, but we do have information that α<β and ω<=α.
Classical mathematics with the infinite have more complicated logical setting and less degrees of freedom of actions in applications. Natural (Axiomatic Digital) mathematics have simpler logical setting and more degrees of freedom in actions of applications .
4) We try to get inspired as much as possible, by the practical ways (like invisible pixels, resolutions etc) that usual software programmers have used to represent classical mathematical entities (like numbers, triangles, circles, manifolds, fluids etc) in the computer operating systems and on the computer screen, which necessarily does not involve the ontology of infinite.
If you simply , create finite digital-models within the mathematics of infinite sets, for practical applications, is not a clean solution because you mix again the infinite in the arguments and definitions (even so as to control the finite) which makes a slippery way again. The clean solution is to re-create , re--found with new axioms the basic University mathematics, as upgrade to the old mathematics, without the infinite. It is an effort with a little more laborious definitions, but the effort is worth it, as the proofs become natural and easier in the end. In fact my initial term for such mathematics without the infinite was NATURAL MATHEMATICS . But I do not want to be one-sided only and only against the mathematics with the infinite. I believe that BOTH are necessary in the evolution of the civilization. At first the mathematics with the infinite, with their great advantage: They keep the mind and self far away from the material reality (And believe me this is precious for fast and free thinking.) But once we have a whole new world of the technology of digital-everything, I think it is time to upgrade the mathematics to the new, greater, both more complex and simpler, the mathematics without the infinite. And this I believe is real need, that is why we have created this blog
In a discussion in a forum of Linkdin for mathematicians a fellow mathematician expressed the argument that being human means also that there are situation that we cannot count and measure with numbers, therefore a situation of infinite And this inability to count is true and, very probable in our human condition. Nevertheless this does not make us not be able to assume that if we could count it, it would be a finite number. Furthermore, assuming that it is infinite is something very concrete and specific according to the axioms of the Cantor theory, which may not be reasonable at all to assume in a situation of a vast number of entities that we are not in a position to count. In summary, a vast number of entities that we are not in a human position to count, and which very often we call "infinite", is not really an infinite according to the axioms of Cantor's theory but simply a vast , not possible to estimate finite number ,that we can very well symbolize and denote with a variable or unknown x!
Many people believe that when the formulate and prove theorems over infinite sets, they get validity over larger collections of entities. In other words that the mathematics of the infinite are larger and of larger extension than mathematics of the finite. BUT THIS IS SIMPLY AN ILLUSION! Under the view that the initial pre-Cantorian intention was that by infinite it was intended to mean either a very very very large finite , or a statement about every finite collection, (independent of its size), we understand that statements about countable infinite can also be formulated within the new Natural (axiomatic digital) mathematics with appropriate logic. On the other hand pinning down this very very very large finite number x, as the Cantorian infinite is a kind of mind-trap (because now the infinite is ontologically different than the finite) and requires larger infinites (uncountable, inaccessible cardinal etc) to talk about very very very large finite numbers that are larger than the previous very very large finites. Furthermore this trap gets worse as the larger the infinites the less, we can prove about t hem. While in the Natural mathematics without the Cantorian infinite, all types of very large numbers are tractable in the proofs as are always finite.
In conclusion, we must understand that the applications of the natural Mathematics that do not have anything infinite, not only are not smaller extensions in real life applications compared to classical mathematics, but in addition the proofs are not as impossible as in the infinitary mathematics. In short, the Natural Finite mathematics are no-smaller, or more restricted, as one might initially think, than the classical mathematics that contain the infinite. On the contrary , the infinites might be perceived as ghost part of mathematics , that in unfair way, has absorbed the part of mathematics that were destined to be for very very large finite sets, and unfortunately this in the Cantorian way that blocks most of what we would need to prove. Although my perspective right now is negative for the classical Cantorian mathematics of the infinite, it is only now (after 2000) that the civilization has much changed. Historically the introduction of the infinite might have been the best solution at that time (end of 19th beginning of 20th century). But I believe not anymore and as time passes the more we would do better as scientists and mathematicians in a universe of mathematics that does not have any Cantorian infinite.
At older historic times, that the frequency (relevant to the spin of electrons protons etc) of the human bodies was lower, the correlation with the material reality was often prohibiting clear thinking and the abstraction of the infinite that was obviously non-realistic for physical reality, was keeping the thinking mind in to safe distance from the physical reality. But now the frequency of the human civilization is higher, and thinking mathematically in a more realistic way, for the physical reality, only with finite avoiding the infinite seem possible and desirable.
I advise the readers, when he read the word finite in this book, not to have spontaneous unconscious mental images, of a small number of solid objects (e.g. of their room) but rather a mental image of the water molecules of the oceans, or of the oxygen molecules of the the atmosphere, or even better of the total number electrons of the planet, or of the electrons and photons of the galaxy, etc.
I do not want to eliminate or disregard the classical mathematics with the infinite. It would be unwise, as it contains the work of hundreds and thousands of fine mathematicians. But I want to point out that in too many cases of applications in the physical world, we need a more realistic system of mathematics that does not use the infinite.
The leading interpretation principles, so as to derive and re-produce results in digital mathematics from results in classical mathematics are
A) The countable infinite is replaced with a too large finite number, that we cannot count with our resources of formal logic, and we do not have information of how large it is (that is why we may denote it by ω)
B) For larger than countable , cardinal or ordinal e.g. α,β , with α<β and ω<=α again we replace them with too large finite numbers, that we cannot count with our resources of formal logic, or have information of how large they are, but we do have information that α<β and ω<=α.
An interesting question is what is the relation of the proposed natural (or axiomatic digital) mathematics with the philosophy and mathematics of Intuitionism (Brouwder, Heyting etc) and neo-Pythagoreans. It seems to me that many although probably not all of the objections of Intuitionism to methods of classical mathematics (like proofs by induction etc) are explained why, one might have an objection, and certainly the natural (or axiomatic digital) mathematics, are closer to Intuitionism, that to classical Cantorian mathematics.
I do not claim here that the axioms of the digital natural numbers, digital real numbers, digital set theory, digital Euclidean geometry , below, are independent within each system of them, in other words none of them can be proved from the others. As the elements are finite, there may be such a case. But I am strongly interested
a) at first that are non-contradictory, and
b) second that are adequate many, so as to describe the intended structure.
Later simplified and improved in elegance versions of the axioms may be given. These systems of digital mathematics should not be considered formally as parts of the traditional mathematics that have the infinite. That is why it is taken the trouble, to re- introduce the set theory, logic and natural numbers in their digital version. So as to have an almost complete system of basic mathematics, outside the traditional axioms,of the traditional mathematics. The Digital mathematics must be consider an independent system of axioms, as non-euclidean (e.g. hyperbolic) geometry is considered an independent axiomatic system compared to the axiomatic system of the traditional Euclidean geometry.
Therefore here the Term Digital Mathematics does not mean traditional mathematics lectured with digital computer tools but something ontological radically different.
Axioms of the digital natural numbers (small changes only to the known axioms)
https://dl.dropboxusercontent.com/u/107295772/Digital_maths/Axioms_of_digital_natural_numbers.pdf
Axioms of the digital real numbers (sophisticated changes here of the axioms, initial entities and initial relations).
https://dl.dropboxusercontent.com/u/107295772/Digital_maths/Axioms_of_digital_real_numbers.pdf
Axioms of the digital set theory (same axioms as the Zermelo-Frankel axioms, except there is no axiom of the infinite. And another difference is that while in the trraditional Zermelo-Frankel axiomatic system, the formal logic that is utilized does use the infinite, in the present approach, the digital set theory uses the digital formal logic , that has no infinite sequences of propositions etc, but all is finite)
https://dl.dropboxusercontent.com/u/107295772/Digital_maths/Axioms_of_digital_set_theory.pdf
Finally an initial outline of the project during a sabbatical in 2011, in ppt file
https://dl.dropboxusercontent.com/u/107295772/Digital_maths/Digital_mathematics_lectureNTUA.ppt
What is missing is the axioms of the digital formal logic, which is of course very imporatnt especially in the digital set theory, where most of the sets are defined through logic and logical formulas. But I have not worked out in publishable form, all its details yet.
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