**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.**

**THE SHORT VERSION OF THE LETTER**

Athens (Greece)

April 12 Spring 2015

Dr Costas Kyritsis

Dear virtual Research Sponsor and co-author

This is Dr Constantine Kyritsis from the Technical University TEI of Epirus, in Greece.( http://ckscientific.com/ and http://www.teiep.gr/en/). I am writing this letter, so as to draw your attention to a very significant, at planetary scale, perspective, of a probable research, innovative book, and software, that through a new science of mathematics will affect all other scientific applications, education, and industries.

Actually I have already started working on it the last 15 years, since 2000, in my spare time, and the work under constructions can be found here. (http://thedigitalmathematics.blogspot.gr/2012/06/2-short-presentation.html)

By basic idea is essentially the dream of many great older mathematicians and scientists, since the time of Newton, that the history did not gave it the chance to be realized hitherto. The idea is to have as much as possible , or almost all, of the mathematics , in way that its ontology does not involve at all the infinite, and be in this way very close to the actual natural atomic physical reality, and computers digital world.

I call it

**“The New natural (or axiomatic digital) mathematics of the 2**^{nd}millennium”.
My basic heuristic rules for this work are four:

**1) We introduce new axiomatic systems where the classical Cantor’s infinite is not only is avoided but is also prohibited. This is the basic "repair of the DNA" of classical mathematics.**

2) All entities, like triangles, numbers, circles, , manifolds, fluids etc are as in the natural physical reality, in other words made from finite many invisible atoms, and have also a different visible structure.

2) All entities, like triangles, numbers, circles, , manifolds, fluids etc are as in the natural physical reality, in other words made from finite many invisible atoms, and have also a different visible structure.

3) We try to re-define and prove as many as possible areas and specializations of all the universe of classical mathematics, but with new axioms, ontology, theorems and proofs consistent with the rules 1) and 2). The practice hitherto suggests that it is possible to redefine all the essential for applications mathematics, although in a different ontology and behavior.

3) We try to re-define and prove as many as possible areas and specializations of all the universe of classical mathematics, but with new axioms, ontology, theorems and proofs consistent with the rules 1) and 2). The practice hitherto suggests that it is possible to redefine all the essential for applications mathematics, although in a different ontology and behavior.

**4) We try to get inspired also 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.**

Why do I suggest such a research, book, and software? I am convinced from the evolution of mathematics and sciences, and especially of the new science of Informatics (Computer science) and therefore I strongly believe that this is the next desirable step for mathematics too. When the basic part of such mathematics is completed, (which includes Logic, real numbers, geometry, calculus, differential equations, manifolds, statistics etc) then all the sciences like, physics, chemistry, biology, meteorology, ecology, medicine etc will have the opportunity of more than 10 times faster and more realistic applications, through software, like mathematica or matlab or other. Furthermore , the consequences will revolutionalize and will be present in the content of every book of every child, and every scientist, at least for the next decades.

The effects if this charity to humanity will start from the education of kids. The young kids are learning to reason and observe with more easiness, exactness, as if high perfection artists. The confidence to the human power of reasoning and fidelity of the senses is higher. The parallel feelings to the intellectual activity of correct reasoning are finer, elevating and refining. The automatic internal environment of the human intelligent consciousness is of higher quality. New habits of thinking, and sensing are established! As the representation of the physical reality in the human reason becomes more exact, so is its freedom of it too. The human experience is closer to the creative power of the world. Joy is the glow of it and the nocturnal rest is more relaxing.

What can be the next practical steps of cooperation?

What I “see” are quite many different options, one only or many together. You may choose which ever seem more convenient and interesting to you and the people that you like to cooperate.

1) You write at first a book with me. Just the slimmest book that “opens the door” for such a new mathematical science.

2) You write a book with me but with other coauthors too that you or I may suggest.

3) You do not participate as author of the book, but only in organizing and supporting the writing of the book. I have specific ideas of attractive details in the structure and editing of the book, but probably it is a early phase to communicate them.

4) It is not only one book but a series of books.

5) Together with the book it is created an innovative type of educational software, with new user-interactive techniques , that you supervise and sponsor

6) Besides the previous or one of the previous, it is organized a sequence of seminars across the globe (universities schools etc) to present and diffuse the new experience and reasoning-sensing habits of numbers, geometry, algebra, calculus, as natural (axiomatic digital) mathematics without the infinite and different from the classical mathematics.

7) An Institute is created to initiate, develop and support this global change in the Sciences and education.

8) We establish support and sponsoring from famous universities like Oxford, Cambridge, MIT, Harvard, or other if we think it is desirable. As the idea of the research project would come from your organizations, they would just follow, our suggestions.

Obviously all the above range from simple to complicated organization efforts, so there is plenty of room for convenient decisions

*.*
You may ask, why it is a good idea that your organizations sponsor such a research, and not the classical mathematical academic institutions?

It seems to me that there is a reason why this has not taken place inside the universities within the standard research procedures. You see, if I was a usual mathematical university researcher most probably I would refuse to initiate and integrate such a proposal as it involves many different mathematical specializations, and I would have to stick to my own specialization. Academic researchers are paid to do research in limited area of mathematics, and accepting a multi-specialization research is a violation of the standard consensus. In addition as seen from inside the academic world of mathematicians, it is not only a much great undertaking, but also one that would challenge seriously most of the fellows in the next-door office that still spend time in the mathematics with the infinite.

I must emphasize here of course that in no way, I pursuit, such a research and writing of book(s) in opposition to classical mathematics, but only as a creation of a new “island” or new specialization of mathematics.

I dare say, as it seems plausible to me, that both, Leonardo da Vinci and Isaac Newton , (if they were living today), would be highly interested , in to the opportunity that is presented to us, now , with this project, to leave a golden page and new legacy in the history of this human civilization, that its spiritual and creative glory is not atomic but collective and may last for very long time.

I think I must finish this letter somewhere here. I want to thank you again for your time to read it. It is seems to me that the proposed is a very interesting and worthy goal, a type of scientific charity but also necessary evolutional step to humanity. It resolves unhappiness and depression in life-long scale of many academic mathematical researchers too. It shall also bridge the gap between past centuries evolution of mathematics and 20

^{th}century birth of computer science. In doing that, new habits of thinking are introduced to new generations that will lead them to a happier and more perfect interaction with physical reality. This in its turn may lead to better more realistic, efficient and effective ways to succeed their goals in the arts sciences and technology It is something that always the civilization wanted and for the mathematics seems to be the only major upgrade since the ancient times of Euclid. From your position and role, you can make it happen and be part of it.
Sincerely

Prof. Costas Kyritsis

Skype DrKCostas

**DIALOGUE OF THE IMMORTALS**

**This is a fictional dialogue of the immortally famous mathematicians of the past that have significantly contribute to the mathematics of the Euclidean Geometry and comment on the new axiomatic system of the**

(

(

**Axiomatic Digital Euclidean Geometry**. The list is only indicative, not exhaustive.

**THE DIALOGUE OF THE IMMORTALS MATHEMATICIANS ON THE OCCASION OF THE NEW AXIOMS OF THE AXIOMATIC DIGITAL EUCLIDEAN GEOMETRY BY NEWCLID**

**NEWCLID after presenting the immortals the new axioms of the Axiomatic Digital Euclidean Geometry, invites them in a free discussion about it.**

**NEWCLID, is an individual representing the collective intelligence of the digital technology but also of mathematics of the 21**

^{st}century.**The participants of the discussion are the next 20.**

**1) Pythagoras**

2.

**Aristarchus from Samos**
3.

**Eudoxus**
4.

**Euclid**
5.

**Democritus**
6.

**Archimedes**
7.

**Apollonius**
8.

**Copernicus**
9.

**Galileo**
10.

**Newton**
11.

**Leibnitz**
12.

**Cartesius**
13.

**Cauchy**
14.

**Dedekind**
15.

**Weierstrass**
16.

**Hilbert**
17.

**Riemann**
18.

**Cantor**
19.

**von Neumann**
20.

**Newclid****NEWCLID:**

**Welcome honourable friends that you have become immortals with your fame and contribution in the creation of the science and discipline of Mathematics among the centuries on the planet earth!**

**Now that you have watched my presentation of the axiomatic system of the new Axiomatic Digital Euclidean Geometry, I would like to initiate a discussion that will involve your remarks and opinions about it. Who would like to start the conversation?**

**PYTHAGORAS:**

**Thank you Newclid for the honour in gathering us together. I must express that I like the new approach of the Axiomatic Digital Euclidean Geometry, that as you say is a resume of what already the beginning of the 21**

^{st}century in the earthly Computer Science has realized through software in the computer operating systems and computer screens and monitors.**I must say that I like the approach! In fact, I was always teaching my students that the rational numbers that are the closest direct extension of the integral natural numbers are adequate for creating a mathematical theory of the geometric space! But at that time no such detailed and elaborate axiomatic system was available in the mathematicians of the ancient Greece, Egypt or Babylon.**

**EUCLID: I am impressed Newclid for your elaborate axiomatic system. The axioms that I had gathered in my books with title “Elements” for the Euclidean geometry in my time were much less! I would like to ask you a question that puzzles me since I watched your presentation: How do we know that the more than 20 axioms of Hilbert about my Euclidean geometry, or your axioms of the Digital Euclidean Geometry are enough to prove all that we want to prove?**

**NEUCLID: This is a very good question, Euclid! Maybe our friend here Hilbert might like to answer it!**

**HILBERT: Well my friends, this is a question that I posed also to myself when writing my more than 20 axioms of the classical Euclidean Geometry! I have not read any such proof! It is by the rule of the thump as they say! I collected them , through my experience and according to the theorems of Euclidean geometry till my time but also according to the standards of proofs in my time!**

**NEWCLID: What do you mean Hilbert? That maybe in the future we might discover that we need more axioms?**

**HILBERT: Exactly! That is what the History teaches us!**

**CARTESIUS: If I may enter the discussion here, I propose that a proof that the axioms of Hilbert are enough could be proving from the Hilbert axioms, the basic numerical axioms of my Analytic Geometry with coordinates! This, in my opinion, would be a proof!**

**NEWCLID: Very good idea Cartesius! This in my opinion suggests also that my axioms of the Digital Euclidean Geometry, that involve coordinates too, most probably are enough. But I am almost sure that they are not independent and some of them can be proved from the rest. Still I cannot claim that I have any proof, more than just experience and a rule of the thump, that my axioms are adequate! Maybe in the future I may discover that I need a couple more!**

**ARISTARCHUS: May I ask Newclid if your concept of digital Euclidean space which is in the shape of a spherical ball is intended to be large enough so as to allow e.g. astronomical calculations like my calculations of the size of earth, moon , sun and their mutual distances?**

**GALILEO: I have the same question Newclid! Good that ARISTARCHUS asked it!**

**COPERNICUS: Me too Euclid!**

**NEWCLID: Certainly ARISTARCHUS! The spherical digital Euclidean space can be so large so as to include all the observable galaxies of the astronomical world as we know it! But it can be also small as a planet to accommodate for planetary calculations only too! The axioms do not specify how large or small it should be!**

**ARCHIMEDES: I like your axiomatic system and concept of space Newclid! It is as my perceptions! Actually my experimental work with solids that I was filling with sand or water to make volume comparisons is just an experimental realisation of your axioms of volumes through those of the points and finite many points!**

**DEMOCRITUS: Bravo Newclid! Exactly my ideas of atoms! Actually as in my theory of atoms, the water is made from finite many atoms, the volume experiments of Archimedes with water is rather the exact realisation of your axioms of volume through that of the invisible points! Here the atoms of the water are invisible, while the granulation of the sand may resemble your axioms of the visible points!**

**NEWCLID: Thank you, my friends! I agree!**

**EUDOXOS: Well in your digital Geometry Newclid, my definition of the ratio of two linear segments which is the base of the complete continuity of the line is not that critical in your axiomatic system, although I thing that it still holds!**

**DEDEKIND: As I reformulated the idea and definition of equality of ratio of linear segments of Eudoxus, as my concept of Dedekind cuts about the completeness of continuity of the real numbers, I must say the same thing as Eudoxus!**

**WEIESSTRASSE: The same with my definitions of convergent sequences though the epsilon-and-delta formulation! They still hold in your approach!**

**APOLLONIUS: I would like to know Newclid, if my theory of circles in mutual contact would be provable as I know it in the classical Geometry of Euclid. E.g. if tow circles are in contact externally, are they in contact in one only point, as I know it, or in more than one point in your geometry?**

**NEWCLID: I think APOLLONIUS that in my geometry what you observe in the real world is also more or less what is provable with the visible points. For sure two circles in contact even if they have only one common visible point they will have many common invisible points, all those inside the common visible point! But I am afraid that they may even have more than one common visible point , depending on their size. The reason is that it may happen that two different visible points have an error of distance from the centers of the circles which is zero in the Low precision although not zero in the High precision.**

**GALILEO: I would like to ask Newclid if your concept of invisible and visible points could be large enough and both of them visible, so as to account for the real planet earth (which is not a perfect sphere) as if a perfect sphere!**

**NEWCLID: Well GALILEO, the initial intention is the invisible points are indeed small enough to be invisible. But as you understand what is visible and invisible is not absolute and depends at least on the closeness of our eye. Theoretically one could conceive a model of my axioms where both visible and invisible points are visible and even large!**

**LEIBNITZ: I want to congratulate you Newclid for your approach! In fact my symbols of infinitesimal dx in my differential calculus suggest what I had in mind: A difference dx=x2-x1 so that it is small enough to be zero in the Low precision but still non-zero in the High precision! Certainly a finite number!**

**NEWTON: I must say here that the Leibnitz idea of infinitesimal as a finite number based on the concepts of Low and High precision is not what I had in my mind when I was writing about infinitesimals. That is why I was calling them fluxes and symbolized them differently. The theory of null sequences of numbers (converging to zero) of Cauchy and Weierstrassse is I think the correct formulation of my fluxes. So that such fluxes fit to a Geometry as Euclid and Hilbert was thinking it and not as Newclid formulated here. Still for physical applications I thing that Newclid's concept of space with finite many points only is better and closer to the physical reality! I was believing in my time that matter consist from finite many atoms , but I never dared to make a public scientific claim of it, as no easy proof would convince the scientist of my time!**

**I want to ask an important question to Newclid: Is your differential and integral calculus based on three levels of precision more difficult or simpler that the classical differential and integral calculus based on limits and infinite many real numbers?**

**NEWCLID: Well Newton thank you for the good words! Actually I have not yet developed all of a differential and integral calculus based on digital real numbers and digital Euclidean geometry, therefore the question runs ahead of our presentation. But I have thought myself about it, and I can remark the next: A differential and integral calculus based on three levels of precision is certainly less complicated than ( and also not equivalent to ) the classical calculus with infinite sequences or limits. But a differential and Integral calculus of 3 ,4 or more precision levels is by far more complicated than the classical differential and Integral calculus. Only that this further complication is a complexity that does correspond to the complexity of the physical material reality, while the complexities of the infinite differential and integral calculus ( in say Lebesgue integration theory or bounded variation functions etc) is a complexity rather irrelevant to the physical material complexity.**

**CARTESIUS: I want to congratulate you Newclid for your practical , finite but axiomatic too approach for the physical space, and the introduction of my idea of rectangular coordinates right from the beginning of the axioms! I have a question though! You correspond points to coordinates, but they also have volume. If we think of a cubic lattice with its points and coordinates, which of the 8 cubes that surround the point you assume as voluminous point in your geometry?**

**NEWCLID: If I understand your question well CARTESIUS, it is the cube that its left upper corner is the point. Thanks for your praise!**

**CAUCHY: I wish I had thought of such an axiomatic system of space with finite many only points, and the concept of infinitesimals as Leibnitz mentioned with your Local, Low and High precision levels! But there is a reason for this! Your axioms are much more elaborate and complicated that the Hilbert axioms of Euclidean Geometry!**

**NEWCLID: Indeed CAUCHY! But later the proofs of many other theorems, on areas, volumes and even derivatives, will become much simpler!**

**HILBERT: I like your brave and perfect approach Newclid! No infinite in your axioms so as to have easy physical applications, as nothing in the physical material reality is infinite. Congratulations! I am glad that my axioms of the classical Euclidean Geometry were of a good use to your work.**

**Von NEUMANN: I like tooyou axiomatic system Newclid! I believe that I could easily make it myself, except at that time I was busy in designing a whole generation of computers! I believe your works is a direct descendant of my work on computers. As you said your ideas came from software developers in the operating system of a computer!**

**NEWCLID: Indeed von Neumann! Thank you!**

**CANTOR: Pretty interesting your axiomatic system Newclid! But what is wrong with the infinite? Why you do not allow it in your axiomatic system? I believe that the infinite is a legitimate creation of the human mind! Your Digital Euclidean Geometry lacks the magic of the infinite!**

**PYTHAGORAS: Let me, Newclid, answer this question of CANTOR! Indeed CANTOR the human mind may formulate with a consistent axiomatic way what it wants! E.g. an axiomatic theory of the sets where infinite sets exist! And no doubt that the infinite is a valuable and sweet experience of the human consciousness! But as in the physical material reality there is nowhere infinite many atoms, mathematical models that in their ontology do not involve the infinite, will be more successful for physical applications! In addition there will not be any irrelevant to the physical reality complexity as in the mathematical models of e.g. of physical fluids that use infinite many points with zero dimensions in the place of the finite many only physical atoms with finite dimensions. The infinite may have its magic, but the axioms of the Digital Euclidean Geometry have their own and different magic!**

**RIEMANN: Very impressive Newclid your logical approach to the Euclidean space! But what about my Riemannian geometric spaces? Could they be formulated also with Local, Low and High precision levels and finite many visible and invisible points?**

**NEWCLID: Thank you Riemann! Well my friend any axiomatic system of your Riemannian Geometric spaces, with finite many points would require at least 4 precision levels! The reason is that at any A point of a Riemannian Space, the tangent or infinitesimal space at A is Euclidean! And here the interior of the point A will be a whole spherical Euclidean space which already requires two precision levels and both the visible and invisible points of the tangent Euclidean space will have to be invisible, while the point A visible point! But let us have patience! When I will be able to develop fully the digital differential calculus on a digital Euclidean Geometry we will reach and answer your question with clarity!**

**NEWCLID: If there no more questions or remarks, let us end here our discussion, and let us take a nice and energizing walk under the trees in the park close to our building.**

**AT THIS POINT THE DISCUSSION ENDS.**

**THE LONG VERSION OF THE LETTER**

Odense, University of Southern Denmark and Athens (Greece) Spring 2008

Dr Costas Kyritsis

"The real wealth is not really money but it is more the creative abilities of people to choose and evolve their self and destiny"

- Dear Mr. Research Sponsor and probable co-author

I am honored that you spend the time to read these pages of my letter and I thank you for thinking about what I am writing. I shall try to make it worth reading this letter.

The contents of the letter are:

**§ 1 Introduction and preview**

**§ 2 The golden- fair new concept of natural (axiomatic digital) mathematics that eliminates the infinite**

**§ 3 Consequences in the arts and sciences**

**§ 4 Consequences inside classical mathematics**

**§ 5 Practical Steps.**

**§ 6 Objections and risks**

**§ 7 Review and summary**

**§ 8**Epilogue

**and close of the letter**

**§ 1 Introduction and preview**

I dare say, as it seems plausible to me, that both, Leonardo da Vinci and Isaac Newton , (if they were living today), would be highly interested , in to the opportunity that is presented to us, now , to leave a golden page in the history of this human civilization, that its spiritual glory may last for thousands of years, and which I shall try to present to you in this letter.

It seems to me that, what I shall try to communicate, may seem quite familiar to you, as you have spent many years for the business and for the high standards of excellence, simplicity, and effectiveness, in holding essentially the major part of the gate that permits the execution, in this planet, of any software in the computer hardware (in other words of the MS operating system).

What I communicate to you through this letter, seems to me that is not exactly a business deal , neither exactly a philanthropic funding, but can be also considered a proposal for a new hobby and spiritual adventure, as it may involve your creative participation too. To save your time, I shall outline in the first three pages of the letter the main ideas, while in the remaining pages, when you will find time, I shall enlarge on the key technical details and possible organizational and business procedures.

I hope, though, that it is of high interest to you, in your powerful role. A good reason is that if we accomplish what I shall propose, it shall bring a great liberation and perfection, to the current collective scientific thinking and it will bring an educational revolution from the level of early high school.

The consequences will be present in the content of every book of every child, and every scientist, not only for the next decades, but I dare say for the next centuries. Of course I must mention that as it seems to me, the birth and growth of the software, and operating system of computers, as basis of the digital world of information, in which you have had a protagonist role, already has left a long lasting mark in the civilization, and in the future books of Schools, and Universities. What I suggest is to use the success of this effect, further in a level closer to the roots of the network of various sciences.

It will be also an advancement that will accelerate and will integrate the use of software and the computers in this civilization, with the accumulated creative work of the past history in the arts and sciences. This may make the evolution of the civilization smoother.

*This integration will resolve a serious gap between, from the one side, the 20*

^{th}century developments of the computer and software industry, and from the other side, the present intellectual and conceptual state of the sciences as accumulated through the past centuries**.**
Well you may ask, what exactly are you talking about? How such a grandiose development is related to what we can do? What is the serious gap between the software and computer industry from the one side and the intellect of the scientific world from the other side?

So as to understand me better, let me make metaphors, relevant to software and computers. You know better than me, the power of a core of simple ideas, like the Basic programming language or the DOS that are inflated to global developments in the whole industry.

If we compare the physical reality with the hardware, and the scientific theories with the software, then the gate that links them, as an operating system is nothing else than the discipline of mathematics.

And inside the mathematics there is a core of simple concepts and axioms (in Number theory, Logic and Set theory) that derive all other branches, which in their turn serve many different sciences. Thus, the developments that I dream in all the sciences and for many centuries, require to

*“coin a golden initial mathematical concept”*at the critical axiomatic systems of mathematics, in other words at this very simple core of mathematics.*This is the definite cure from the need of the infinite*. It can be though also as a*re-creation*of all mathematics (from its simple core) which is a repetition, (in a deeper intellectual level and as open source on paper) of the creation and development of the operating system in the industry of computers. We may compare such a new concept, with the introduction in mathematics and algebra of the use of the “**unknown x**”. An ingenious simple technique that introduced a new reasoning experience and habit, not applied before, with great advantages in the applications.
As you may have probably already guessed, the gap between the present state of the scientific intellect and the world of computers (as well as of world of physical reality) that I mentioned before is created from the formal use of the concept of the infinite at the core of the mathematics. The infinite was not a conceptual child neither of ancient Archimedes, nor of Euclid, neither of Leonardo da Vinci, neither of Isaac Newton, neither of many other great scientists. It was an imperfection of the creative work in mathematics of the 19Greece that during her life in the second part of the 20

^{th}century, which became legitimate through the Set theory of G. Cantor, and became formal in the beginning of the 20^{th}century by Zermelo and Frankel. Although this core of mathematics was called the “Cantor’s Paradise” it is easily realizable by any modern computer programmer, that literally the definition of the infinite is much like a “virus” or “worm” that blocks all direct application of the classical mathematics in the operating system of any computer. This has vast paralyzing consequences in the applications of all other sciences, and results to much despair to many mathematicians. I personally knew of a woman and good mathematician, in^{th}century he was so much depressed from the inability of mathematics to handle infinitesimal and the calculus without the infinite, so as to go to a psychiatrist. Modern mathematicians face more and more difficult situations of harder to prove or non-provable statements and an increasing complexity all because of the infinite that consumes the time of their life and their intellectual efforts without any correspondence in to physical reality and applications and consequently to other scientists and groups in the societies. Each time an applied scientist must put in the computers quantitative and numerical calculations, he has to “jump” from the world of the mathematics with the infinite, to a different world of an operating system without infinite. And this “jump” is a serious gap in the collective scientific intellect, and also an evolutional gap of the civilization.
How therefore can we

*re-create*, and*re-define*the basic core of all mathematics, without using of the concept of infinite? What is this “*golden new initial concept*” that we have to coin so as to “cure” them from the infinite? How can we create it in such a way that is possible to represent it in an exact (not approximate) way within a computer operating system and run it?
How in this way can we make the mathematics happier and more perfect world not only for high school pupils but also for NASA scientists, and clinical doctors? A new development that would make the spirit of Euclid , Archimedes, da Vinci, Cartesius, Newton , Leibniz, etc rejoice?

Will the above accelerate the use of software and computers in the civilization in all educational levels?

How in addition, can new people, understand the very logical power and key ontology of all software and mathematics, without even having a computer, but reasoning in an exact way even on paper or board?

How can we transform the reasoning and sensual awareness habits, of traditional mathematical thinkers (and this includes the kids that learn calculus in high school) to one that is aligned, with the experience of software and computers in the modern world? An experience that may turn after all to be independent from the particular hardware?

It seems to me that I have the answer to all most all of the above questions! And this is the content of my letter and proposal. It was for me a journey in discoveries and new reasoning awareness from 1995 to 2005, in which the developments of the industry of software, and the Microsoft Windows operating system, contributed a lot.

*From your powerful role, you can make it happen and be part of it.*

*Let us leave one more long lasting honored mark in the civilization at liberating mathematician and scientists from the gloomy castle of the present mathematics with the infinite, towards the new millennium and fast growing digital paradise. This will ease in an incredible way the computation of scientific mathematical models of Physics, Chemistry, Medicine, Meteorology, economics etc. in an operating system. And we may do it with, Propriety, Prudence and Benevolence. It may be considered*

**a type of charity to humanity**, a charity, that has nevertheless with less funding, potentially greater benefits, as it is worked out gradually in time, by worthy minds, and in worthy intelligent, areas of the societies. This is accelerating a smoother evolution in the civilization. An evolution where the works of human hands, go parallel with the awareness of the works of the human mind.
I hope also that with your vast practical experience in business and your outstanding mind you may suggest managerial practices and procedures to accomplish the above with the least possible frictions, in a smart, elegant, beautiful and efficient way, which will make almost all parts happy.

My motivation to accomplish the above is also relevant to the feelings described in famous quote of the remarkable American multi-scientist Buckminster Fuller :

“The Things to do are: the things that need doing, that you see need to be done, and that no one else seems to see need to be done. Then you will conceive your own way of doing that which needs to be done — that no one else has told you to do or how to do it. This will bring out the real you that often get buried inside a character that has acquired a superficial array of behaviors induced or imposed by others on the individual.”

**§ 2 The golden- fair new concept of natural (axiomatic digital) mathematics that eliminates the infinite.**

In order to build the new “

**invisible city**” of all mathematics without the infinite, in other words , the**digital mathematics**, that are directly fully loadable, in a computer operating system, but also existing in the collective conceptual intelligence, of the present and future generations of the next centuries, we must devise new fair and practical concepts and definitions. As I see it, we must**coin the new golden concepts**that resolve the old “infinite” (world of analogue mathematics) in to the finite (world of digital mathematics) ! It is a concept respecting the**measure of the human mind**, avoiding, according to the opinion of some people, the “hybris” or “eschatology” of the infinite. The mathematics can be founded again not on the “**sand**” as the neo-Pythagoreans at the beginning of the 20^{th}century had accused G. Cantor’s infinite, but on the solid “**rock**” of the finite.
I shall try in the next pages of this letter to discuss, and speculate what can be done, what can be the role of Microsoft and why, how you personally can be possibly creatively involved as author and benefit in new honor and respect in all the scientific world, while entertain yourself by rediscovering all of mathematics as a different kind of open “operating system” for all the sciences. And also what are the exact key technical mathematical details (what new concept must be coined) that I have figured out, define the intended metamorphosis or rather conceptual morphing of the source of mathematics, to a great liberation, through higher perfection . I shall try also to speculate why it seems a good idea to do it now, and why it has not be done till now.

But before we enter in to the technical details, lets have a large scale overview of the consequences in the net of arts and sciences in the civilization.

**§ 3 Consequences in the arts and sciences**

Effect in THE NETWORK OF DIFFERENT ARTS AND SCIENCES

The effect of such changes in the present mathematics and given the consequent development of new mathematics that will follow shall affect dramatically some sciences. It seems to me that the disciplines that apply differential equations and stochastic processes shall have the strongest boost. In these disciplines are included

a) Dynamic Systems with applications in meteorology, weather forecasting and applications in star systems dynamics. Given the future of extreme weather phenomena, this seems to be very valuable.

b) Fluid dynamics with applications in ship and airplane design and manufacturing. The benefits for faster and safer transportation will be great. The academic sciences will come closer to discovering extraction of energy from the potential energy of the planet’s universal attraction, with marvelous consequences while the use of oil is gradually reducing due to its increased scarcity. The effects in the macroeconomics will also be stabilizing with fair distribution of energy resources. In addition new understanding of the electromagnetic field and field-propulsion may allow as even travel in an elegant and silent way outside the planet without the use of rockets.

c) Signal theory , filtering and new animation techniques with many applications in communications and entertainment.

d) Engineering design and manufacturing of devices including micro chips.

But it is not only the standard engineering that will greatly benefit! Even classical social sciences and disciplines shall have a revolution. The area of functional analysis, distributions, stochastic processes , stochastic differential equations etc in mathematics seems to be yet entirely unrealistic, ambiguous and very complex. Mostly because of the complications of the infinite. Consequently the other sciences that apply them like

anti-seismic design,

social simulation,

Econometry,

Ecology,

Biology.

Medicine.

Will benefit a lot.

The main reason is the all of the above sciences sooner or later have to formulate and utilize quantitative mathematical models.

It seems to me, that modern econometry for example is in the age of the chaotic Ptolemaic system. They are still waiting for a Newton to put simple universal laws that will bring simplicity and order in the description of the phenomena. The understanding of the social phenomena, macroeconomic phenomena, and ecological phenomena may benefit tremendously, and universally acceptable consensus may start to reign in the scientific journals, books and the web.

The new digital mathematics without the infinite , I estimate that shall make some areas obsolete and shall transform other like numerical analysis, to elegant, simpler and very effective methods directly executable in computers from the very definition of their mathematical ontology. A lot of essentially meaningless complexities that consume the life time of many academic researchers and also the money of the states, shall disappear, leaving time for more realistic transparent , happy and valid research work.

As Andrew Carnegie was explaining to Napoleon Hill (in the book “The wisdom of Andrew Carnegie” see references) the mind is like a very powerful explosive. This is the way to approach it, and small changes in details may have large effects. The above are some of the long-lasting and large scale consequences that seem plausible to me.

**§ 4 Consequences inside classical mathematics**

How the KEY CHANGES affect the various specialization areas in mathematics (ontogenesis and phylogenesis: The ontogenesis may be a summary of the phylogenesis)

If you like biology here is a biological metaphor that helps to understand the relation of the history of a science, with its logical ontology while learning it.

Biologist believe and prove somehow, that the phases of the development of the embryo of a species in the womb of its mother (in other words ontogenesis) is a summary of the phases of the historic development, of the species as a whole (in other words phylo-genesis). When during my lecturing in the University of the Aegean, I had to design a program of courses for all graduate studies of mathematics, the previous principle was a guide to me, and the students were somehow taught the mathematics twice. The first two years as it was the historic development of the science of mathematics (phylogenesis) and the final two years as modern mathematics are logically re-created and developed from various axiomatic systems. So in the mind of the students an ontogenesis of mathematics occurred summarizing the historic phylogenesis of them during the past centuries. When the science of computers was born, and especially when through personal PC’s as public, we could easily watch the development of the operating systems, programming languages, windows, multimedia etc, I realized that I was watching an ontogenesis of a new science, that somehow was summarizing the long history of the mathematical though of the civilization. In the page below I give in the balloons the correspondence between this ontogenesis of software, with the corresponding phylogenesis of mathematics.

Within this correspondence and metaphor we may trace the effect of the Cantor’s infinite in the ontology of mathematics. Before Cantor, and since ancient Pythagoras, mathematics was based mainly on the numbers (corresponding to bits or bytes in the universe of software). After Cantor, mathematics were based on set theory, which corresponds to Data Bases in the universe of software. The keys of how through only logic and the numbers ,all set theory and thus mathematics too, could be created were lost. But the ontogenesis of the universe of the software, revealed exactly how it could be done. Anyone experienced in software, knows how a data bases, with its records in tables (or in a lower layer as files) is created from bytes. Still if set theory was of finite many finite sets, the change would not be essential. Set theory nevertheless, in modern mathematics is were all ranks of the infinite start.

In this paragraph I shall outline in a sketchy way what are the key-changes in the core of basic axiomatic theories in mathematics that eliminate infinite, while rendering mathematics, simpler more realistic, and still enhanced, in addition more sophisticated and perfect and closer to software.

The golden

*initial concept*that has to be coined so as to eliminate the infinite is nothing else (as you may have already guessed) than*This is the wisdom and technology in the software, and this is what I think is to be done in mathematics too. We may safely make the bold equivalence that the infinite is (or permits) essentially the introduction of a continuum that we are incapable of resolving in to (finite) pixels. The word infinite in Greek (άπειρον) is identical with the word ‘inexperienced’. We may shift from the inexperienced stage of the infinite phenomenology of the continuum (e.g. of the lines of an image that we think have infinite many points…) to the creators experienced stage of re-creating the ontology and phenomenology of a continuous line, through the invisible pixels of a resolution, using only the finite. In addition we save ourselves from the extravagant ultraism of the speaking modes of the infinite, to the tempered , within human measure, excellence of exactness of an invisible finite resolution.*__the concept of (finite) resolution.__
Before we go to the consequences and changes inside the various specialisations of the mathematics, let me write the praise or you may say the obituary of the infinite. I have an image that helps me understand the role of the infinite in the history of mathematics. Let us assume that we want to design and manufacture a car. We shall obviously start with an artistic sketch of it. This seems to me the mathematics of the infinite. It is the artistic sketch before the detailed quantitative designs for the engineers that correspond the coming digital mathematics without the infinite. The twist here seems to be that the engineers first manufactured the car, before they devised any detailed design on paper. (The software industry comes prior to the yet unborn digital mathematics). There reasons why the by now hell of Cantor’s mathematics was called Cantor’s Paradise:

1) The mathematics based on the infinite were phenomenological and seemed simple, although unfamiliar!

2) They were “feeling good” when developed.

3) They were free from any “hardware” practice of engineers or accountants of that time,(1

^{st}part of the 20^{th}century) that would “make the waters unclear” during the creativity and discovery process. The fancy world of computer hardware was not in sight at that time.
4) The infinite had an aurora that is mystifying and metaphysical with an attractive charm for strange discoveries, even if this creation was restricted to a minority of scientific creators isolated from the other sciences. It was often related to the glory of a “holy graal” for the mathematicians. Although the concept of infinite was somehow metaphysically totalitarian, it still was feeling good.

5) It was faster than the creation of the industry and practice of the computers , so it was anticipating somehow the coming innovation of the computers and the coming democracy of the digital world, half or one century earlier. From this point of view it was smart.

Here is a sketchy reference to the key changes in the core axiomatic systems of mathematics, that derive all other mathematics. (You can find more in the address http://www.ckscientific.com/23MathInfinity.htm)

#
1) NUMBERS: The axioms of the natural numbers remain essentially the same. They are stated only up to a maximum (but variable and denoted by a variable symbol) number. In this way the axiom of induction is a theorem.

#
2) LOGIC: The axioms of logic do not change except all symbols, predicates, and element of the language of logic are finite with a variable unknown maximum, and with careful comparison with the maximum (unknown) number of natural numbers that may be objects of logic , as the relative comparison created ability or inability to prove propositions. (Geodel’s impossibilities resolve!)

3) SET THEORY: All Cantorian set theory is restated in finite terms. No axiom of infinite.

4) REAL NUMBERS: The real numbers are defined with the usual operations and order, but there are two equalities (or rather equivalence relations) that of the visible points in a representing axis, or essentialy of order of precision in decimals, and the invisible resolution equality of numbers ( a finer final precision). The universal predicates of the operations are interpreted relative to these tow equivalence relations or resolutions, and differ from classical closure of the operations. Obviously all numbers are rational numbers. This is so for square root of 2 too.

6) EUCLIDEAN GEOMETRY: Following Cartesius, or rather modern computer screen programmers, Euclidean geometry is redefined with two different equivalence relations of points, that of visible points (and congruence of lines triangles etc), and that of invisible pixels of fixed (but may be variable and unknown) resolution. Again lines are essential linear segments as in ancient Euclid. Axioms like that ‘between any two points there exists a third’ , do not hold, without subtle changes in the statements. The number pi as quotient of the length of circumference of a circle to its diameter is rational number, with digits depending on the, in advance fixed, resolution of the geometry

**.**So for the first time after Euclid (2,000 ago!) we will be in a position to have a major “upgrade” of the geometry. The reason that I mention such long past or future horizons for the digital mathematics , is that by the very nature of physical and spiritual reality, there are not really too many convenient Logically different ways to create the very core of the digital mathematics. This guarantees its endurance in time, in the same way that Euclidean geometry not mainly up to Logic , but mainly as intended applications , can be essentially one for the past 2000 years and the next millenniums too.
7) CALCULUS AND DIFFERENTIAL EQUATIONS: Functions become subtler and of enhanced behavior. There is no need of limits and the infinite to define continuity and differentiability. The infinitesimal dx, dy of Newton and Leibniz are ordinary rational numbers, non-zero in the equality of invisible resolution of the real numbers, but zero in the equality of visible numbers up to the significant number of digits (E.g. if they were dollars or euros they would be with non-zero digits beyond cents, and up to the maximum allowed length.)

Functions can be differentiable in a coarser resolution, but even discontinuous in a finer resolution, and vice versa.

8) STATISTICS AND PROBABILITY: It will be revealing when we discover that the probabilistic and statistical behavior of a phenomenon social or physical depends more than we imagine on the resolution were we measure it.

I cannot escape the temptation to make a remark of the similarity of statistics of physical and social phenomena: You may think that if you take a drop of water from your glass on the table and throw it on the computer screen, the large number of small drops on the screen that will be created, would follow a normal (Gaussian) distribution with respect to their size. But it is not so! If you really measure their size you will discover that they follow rather a Pareto or Power distribution. The same distribution that follows the size of websites, or the size of companies, or of income of individuals, or of size of cities, and of size of the stars! An interesting question is of course: Why?

9)

**THE OPERATING SYSTEM AND THE DIGITAL MATHEMATICS**.
More could be written about more specializations of mathematics, but I guess you get the idea already.

I must mention that as I understand it, the easiness with which any geometric or functional mathematical model would be realizable in the computer operating system, is after the user specifies, in what scale, the basic underlying mathematical resolution (a key concept of the new digital mathematics) will be realized by a software resolution in the computer. In this way, no limits or approximations are required.

**In this way the mathematical ontology and software ontology in the operating system correspond one-to-one in a exact way.**This creates the easy and direct loading of all mathematics in the operating system.**An appropriate software that will show how all mathematical ontology from the most abstract modern creations , to the classical elementary mathematics, can be loaded in an exact non-approximate way in the operating system, is certainly a significant innovation that Microsoft can get exclusive rights.**
10. Finally let us consider the new math’s also as new values of more convenient and refined experience, in the senses, in speaking, in practicing, and in reasoning. There is a famous aphorism by H. Poicare’s which sais that ‘Geometry is the art of correct reasoning on incorrect figures”. Well it seems that the time has come that we can say that “Geometry is the art of correct reasoning on correct figures”. That the way we reason and feel based on the sensual data is very-very important is also the doctrine of the modern discipline for human excellence under the name NLP or Neuro-linguistic Programming. Logical reasoning corresponds exactly to the observed by vision reality, while after the operational intervention of zoom-in , with previously invisible reality too! The otology but also the subjective human phenomenology are both represented in the new Logical-conceptual scheme. The old philosophical war of experiential phenomenology and transcendental ontologism ends in to a peace of perfection. From well know mythology I can recall the tales of Guiliver’s Lilliputians , and H.C. Andersen’s Thumpelina: They give a hint of the world existing at least in two different scales. In the present case of digital mathematics an ontological micro-scale of normally invisible bits and pixels, and a scale of normally visible elements. For both, exact logically correct reasoning with theorems and corollaries, is to be created. I put here a poem made for this paragraph that is inspired from the experience of the two levels or scales in the digital mathematics.

**§ 5 Practical Steps.**

In this paragraph I shall try to suggest a list of alternative scenario, of what I think can be done as a project, to realize the ideas presented in the letter.

*What I “see” are quite many different options, one only or many together. You may choose which ever seem more convenient and interesting to you and the people that you like to cooperate.*

*1)*

*You write at first a book with me.:*

*Just the slimmest book that “opens the door”.*

*2) You write a book with me but with other coauthors too that you or I may suggest.*

*3) You do not participate as author of the book, but only in organizing and supporting the writing of the book.*

*I have specific ideas of attractive details in the structure and editing of the book, but probably it is a early phase to communicate them.*

*4) It is not only one book but a series of books.*

*5) Together with the book it is created an innovative type of educational software, with new user-interactive techniques , that you supervise and sponsor in your software company. Would the software be in the form of a game?*

*Would it be attached to existing software for mathematics? These are decisions to be considered in later phase. I may suggest ideas, but I am open in your expertise too.*

*6)*

*You agree deals with companies that have software with reputation in the educational and scientific world, like Matlab, Mathematica, etc to create modules or separate software based in the new definitions and mathematical concepts of digital mathematics, without the infinite.*

*7) Besides the previous or one of the previous, it is organized a sequence of seminars across the globe (universities schools etc) to present and diffuse the new experience and reasoning-sensing habits of numbers, geometry, algebra, calculus, as digital mathematics without the infinite.*

*8)*

*An Institute is created to initiate, develop and support this global change in the Sciences and education.*

*9)*

*We establish support and sponsoring from famous universities like Oxford, Cambridge, MIT, Harvard, or other if we think it is desirable. As the idea of the research project would come from Microsoft, they would just follow, our suggestions.*

*Obviously all the above range from simple to complicated organization efforts, so there is plenty of room for convenient decisions.*
Most probably in achieving such a goal, the benefits in humanity are deep and great, as it enhances the benefits of the software, to deeper conceptual, scientific, and educational layers. So it may be considered

**a type of charity to humanity**, a charity, that has nevertheless with less funding, potentially greater benefits, as it is worked out gradually in time, by worthy minds, and in worthy intelligent, areas of the societies. This type of “charity” as Andrew Carnegie mentions in his little book “The gospel of wealth”, (see references) is helping the evolution of humanity, compared to the dangers of blind charity that may by accident, empower the unworthiness of the people in need, thus slowing down the evolution.
The effects if this charity to humanity will start from the education of kids. The young kids are learning to reason and observe with more easiness, exactness, as if high perfection artists. The confidence to the human power of reasoning and fidelity of the senses is higher. The parallel feelings to the intellectual activity of correct reasoning are finer, elevating and refining. The automatic internal environment of the human intelligent consciousness is of higher quality. New habits of thinking, and sensing are established! As the representation of the physical reality in the human reason becomes more exact, so is its freedom of it too. The human experience is closer to the creative power of the world. Joy is the glow of it and the nocturnal rest is more relaxing.

**§ 6 Objections.**

In this paragraph I shall try to analyze possible objections that may come to the mind while reading this letter. E.g.

1) Why not to try to fund and run such a research in the standard research funds of the Department of mathematics in the Universities?

2) Why a software company is the right organization for this?

3) I am not in mathematics, it is not my competence circle.

4) I have not time, why me?

5) Why to trust your talents ?

6) It is too great, why to do it now?

7) It is may not bring profits

8) There are Risks like:

1) It may fail as an initiation of new millennium mathematics

2) There may not be appropriate co-authors

3) It may be considered as a war to classical mathematics

4) Do we really deserve this role?

Let us discuss them one by one

*1)**Why such a research and creative work cannot be funded easily in standard academic research institutes or University Departments of pure mathematics:*It seems to me that there is a reason why this has not taken place inside the universities within the standard research procedures. You see, if I was a standard university researcher I would refuse to initiate and integrate such a proposal as it involves many different mathematical specializations, and I would have to stick to my own specialization. Academic researchers are paid to do research in limited area of mathematics, and accepting a multi-specialization research is a violation of the standard consensus. In addition as seen from inside the academic world of mathematicians, it is not only a too great undertaking, but also one that would challenge seriously most of the fellows in the next-door office that still spend time in the mathematics with the infinite. It is not of course entirely impossible to develop such a “flower” of innovation inside a university mathematical institute, but it seems, to my experience, that it is a lot easier to plant it and grow it outside the classical environment of a University Department of Mathematics. So it seems to me that it requires creative minds that have grown outside the academic environment, that would give them a bold courage for concentrated great changes. I am half such a case. I started my carrier from a University Department of Mathematics, but soon I shifted to software companies and financial services. One further reason is that creators to participate and accept to contribute, would love to secure an IP right, together with financial benefits. Something, not always meaningful in University Research of pure Mathematics. As the ancient mythological tale of Jason and the heroes in the ship of Argo, there always has to be a “Golden-Fleece” to motivate for the expedition. And in the software industry such “Golden-Fleece” is more likely to be found as financial prosperity and publicity, rather than in Academic Institutes. It is important to make clear, that such an innovation , like the proposed digital mathematics, is not at all a war against classical mathematics. One good reason is that they can immediately be introduced as only a special area of classical mathematics, under the possible title “Digital Mathematics” or “Computer Mathematics” etc.

2)

*Why a Software company?*###
We may notice that the science of physics has spread in the world the idea that physical reality consists of atoms since 1900. And also that the art and technology of the cinema has also spread around the world the idea that the continuum of the motion is created by the discrete of finite many picture frames also since the early part of the 20^{th} century. But it seems to me that even more than the previous two influence factors, the computer operating systems have made a common public experience that all the visual, audio, etc continuum as experienced on the computer screen is created in appropriate way from the finite of bits or pixels. I find it therefore most plausible that it should be a software company with its powerful presence, uniqueness and authority, that should also sponsor the initiation of a more profound and subtler change at the very conceptual foundations of mathematics as applied in many different sciences. And of course I chose to address this letter to you, as I believe that you are a very smart person, an outstanding mind , with many of the talents of Archimedes and Leonardo da Vinci, an excellent businessman, and most probably with a love in mathematics and sciences but also with love to genuine honor and respect from the best parts of humanity.

3)

*I am not in mathematics, it is not my competence circle of me and my group.*
You may say that you are not in mathematics, and that might be true. But it seems to me that you and your group have a deep knowledge of the digital world, as it was born in the software, and operating systems, since the decade of 90’s. And that is already more than adequate to outline or supervise the creation of the new digital mathematics. The rest is simple ideas of numbers, geometry and calculus that exist not only in high school books, but also in an indirect way in manuals of programming languages, like VBA, software like MS-excel and its charting, or software for data bases like Access. And remember that although VBA, Excel, and Access, might not be around in a recognizable form after 20 or 50 years, a book with the appropriate definitions of bits, and pixels in numbers, Euclidean geometry and calculus, seems to me that will remain (practically without “upgrades”) for centuries. As the whole endeavor is a lot simpler than designing and creating, VBA, MS-Excel, and Access, I think it must be considered a smart creative choice as far as honor and respect is concerned.

*9)*

*I have not time.*

This might be certainly true. But If I am not wrong, there was never time for all the issues appearing in our life. While there is always sufficient time for the significant and entertaining issues, as such issues are always few, by definition.

In addition you may just decide favorably to sponsor it, and supervise it, while not participate creatively. There are always sufficient many writers that can accomplish the volume of the work, once I give you the basic keys of how the definitions (of old known concepts, like real number, geometric line, continuous or differentiable function in calculus) are to be stated in the new digital mathematics.

*10)*

*Why to trust your talents?*

I believe that I have the necessary experience, talent, courage, and persistence to accomplish the creation and publishing of a first adequate core of the new digital mathematics without the infinite. The rest will be inflation by many other future authors, that many of them will be better than me. And I believe that I can convince you, that given your expertise in the digital world, as created in the software like MS-Office and Operating Systems like MS-Windows , such a goal is definitely within your reach, and somehow smart and easy.

I was not considering myself a smart person, but my studies in the “forest of mathematics and sciences” taught me to create always an invisible smart nucleus for all the scanned complexity.

Here are a few words about me:

I was born in 1959 in the island of Crete in Greece. I studied mathematics in the University of Athens, from which I received also my PhD in Mathematics. I Lectured since 1989, in many different Greek and foreign Universities (like University of Portsmouth, Coventry, New York state University, etc). I lectured in undergraduate and graduate studies initially on Mathematics, then on Computer Science and finally on Business administration and Finance. You can find about me more in my pages http://www.softlab.ntua.gr/~kyritsis or http://preveza.teiep.gr/kyritsis or http://www.ckscientific.com .

The idea that I propose in this letter came to me in the decade of the 90’s while the windows 3.x were used, and I first published in my pages a sketch of how it could be done and consequences in all of mathematics during 2002-2005. You can find more in my pages e.g. at the address http://www.ckscientific.com/23MathInfinity.htm

Since 2005 I lecture Financial Mathematics and other courses in a small university college of west Greece (TEI of Epirus, School of Management and Economics, Department of Finance and Auditing , at Preveza). I stay mainly in Athens and lecture in Preveza. During my visits in Denmark I was acquainted with

*Centre fpr Playware*,at the University of Southern Denmark www.sdu.dk/palyware and discussed the possibility of cooperation in a creative project as this that I propose in this letter.
But what I consider more important is that years ago, since the decade of the 90’s, I realized that the academic carrier was not what I really wanted, and my indented fulfillment. In addition I realized that free non-academic researchers, especially in the new world of software industry, and the internet, could contribute more in the sciences than a classical academic professor. Motives of profit, prosperity and innovation, instead of prohibiting genuine significant research may very well accelerate and permit research of highest quality standards and excellence. And as Thomas Edison wrote, one of the hidden values of technology and its products, is the indirect education they provide to the masses while utilizing them.

In trying to answer the question why it is you only that you come to make such a suggestion for such a goal I must admit that, it is probably more difficult to write the right thing compared to the rest of the questions in this letter. It seems to me that I have some common points with the ancient Odyssey that had to cross many wavy seas before he returns to his home island Ithaca. In my life also I crossed many wavy times trying to be faithful to my creative ideas. I shifted from mathematics to computer science and then to business administration, but I never gave up or forgot the power of the mathematical ideas. As I started Lecturing at the University of the Aegean in the island of Samos (birth island of the ancient Pythagoras) I realized how much from ancient times the power of mathematics was honored in Greece. I realized also that what was missing to the creative environment of mathematics to go on, was the happy, light, colorful and rock mood that was often found in the experience of creating a software, and also the joy of financial benefits secured by protected intellectual property rights. As my adventure gave me the above experiences it may be that I am one of the many other who are appropriate to make it at least in a pilot way. Nevertheless probably I am the only one suggesting it to you. And how can I know? Maybe there are other reasons that I cannot rationalize. Maybe subconsciously I had the ambition to do something significant. I hope though that you shall help me.

*11)*

*It is too great, why to do it now?*

From the macroscopic time scale point of view the “now” is both late and fast and early. Late as it should be done earlier almost at the time that the idea of atomic structure of the physical matter was a common knowledge or as the technology of the picture-frames of the cinema was becoming popular experiences, to avoid the crystallization of the “infinite” in to formal axioms at the beginning of the 20

^{th}century. It is also fast and early as it is almost a little more than one decade after the windows are used around the globe and thus the experience of the continuum in the computer screen as generated by bits and pixels is again a popular experience. I want to be honest with you about me so I must say that from the personal time scale point of view the “now” is a good time because any financial support to me as participating in it is of higher value now that another time. I might also speculate that the “now” might be a good time for you too, as it may be that you have accomplished long ago and in great measure your business success plans, thus leaving room for let us say more subtle ambitions relevant e.g. to honor in the sciences, honor through beneficence etc.*12)*

*It is may not bring good profits.*

It will certainly not bring the profits that MS-Excel has brought. And from the complexity point of view it is a goal a lot easier to accomplish. In addition as one of its main deliverables will be published book(s), there are no good enough IP contents, like business secrete, or other licensable IP, except of course the copyright IP, to exploit in the book(s). But as I mentioned it is a good idea to design and produce relevant software too, that may bring profits. I know that you know more about me on this. After all you have plenty of money, it might not be the first priority now. The most attractive part in it, I believe, is the long lasting honor and respect that it will bring from highly esteemed parts of the intelligent humanity.

*13)*

*There are Risks like:*

*1)*

*It may fail as an initiation of new millennium mathematics.*

I believe that it shall not fail, as initiation of the new digital mathematics of the new millennium. There are simple reasons for this:

1.1) At first, the need to escape from the gloomy castle of Cantorian mathematics with the infinite, was felt as early as the decade of 1960, when the non-standard mathematics of A. Robinson were created. Still the attempt was half the way, and it remained a strange branch of mathematics reflecting exactly their title. It was early at that time as the experience of using computers was limited in the public, and the software industry was not sufficiently developed.

1.2) It is a common experience during the 20

^{th}century in mathematics and other sciences like physics, chemistry, meteorology etc that utilize mathematics, that the calculus with the dx, of Leibniz or ox of Newton was not really absorbed with the Cantorian infinite and the concept of Limits of infinite sequences. Something else and simpler is required. In my proposals, this is nothing else than e.g. the Single precision and double precision of quantities in VBA, so that dx is zero in the single precision but not zero in the double precision. In short the need for the digital mathematics was already felt during the second part of the 20^{th}century
1.3) What is a more important argument in favor of the success of the digital mathematics is the great success of the digital world of electronics and software since 1980. Somehow this digital world of electronics and software is nothing else than the application of the not-yet born digital mathematics. We have here one more case where the practical creativity goes faster than the intellectual conceptual creativity. Not something strange in this peculiar civilization we live. The new conceptual and reasoning techniques that I propose to introduce in mathematics, have already been a successful software technology, in modern computer science. From this point of view the proposed is almost an inspired manual for all of the mathematics in the digital world of computers and the internet. It introduces new definitions of older concepts and solves many traditionally unsolved problems of the mathematics with the infinite. It is a happier universe of mathematics that gives better more realistic efficient and effective ways for the people to succeed their goals in the arts sciences and technology Something that always the civilization wanted and is the only major upgrade of mathematics since the time of Euclid.

Therefore imitating one of the main secrets of success of

**Shakespeare**, we shall**simply “re-state in our own smart and enlightening words and concepts, some, by now already old, successful and popular stories”.****1.4)**A final argument is that once it is realized by the scientific world, how simpler easier and more direct is to formulate models , equations etc in the digital mathematics and passing them for computation in an operating system, compared to the same models in the classical mathematics, more and more creators will start contributing in the new digital mathematics. More and more software it will be invented, and more and more people will not only prefer it, but will considered it the only natural, and realistic mathematics, in the new society of information. Of course the coin has an other side too. The new digital mathematics, are also subtler than the classical mathematics of the infinite, subtler in a different more realistic way (as subtle is the software, chemistry and biology) But this difficulty is only the reflection that reality (“the common country” of all scientists as H.Poincare was putting it) is indeed subtle , deep and difficult.

To guarantee success , while keeping some software an exclusive privilege for Microsoft, we will target as much free conformance as possible through the book(s) with all relevant other Educational or Scientific organizations, institutes or companies.

*2)*

*There may not be appropriate co-authors.*

Well I guarantee that even if none from your circle accepts to co-author I can write a slim small book that will contain the basic keys for the new digital mathematics. But it seems to me that by instructing good researchers in the appropriate way, or even asking famous authors just to review and make the proof-reading, will be adequate.

*3)*

*It may be considered as a war to classical mathematics*

Not at all! It shall start as a specialization of modern mathematics, that most of the classical mathematicians will sympathize with.

*4)*

*Do we really deserve this role?*

I am sure there shall be more authors that will create and recreate all these again in the near or distant future. Has history reserved to other spirits this role? I cannot answer it! But it seems to me , if I am nor wrong, that what we may do, does not exclude futures creators to reestablish it, probably in a more elaborate way. After all what we shall do ? Simply write a book and probably create a software! This does not seem great by itself. If it shall follow now or later an inflation of similar ideas and further development, that will be greatness in manifestation, and it will be the civilization doing it. If not, then still in our hearts will know that it is great, but probably not exactly timed or populated. In both cases we will have taken a greater or smaller share of the glory! It seems to me that it is worth doing it, and try to initiate such a great change, at this abstract level, within the duration of our mortal lives. I should refer again here to the quote by

**F. Scott Fitzgerald***“…his dream must have seemed so close that he could hardly fail to grasp it..”*

**§ 7 Review and summary**

In the previous paragraphs :

1) I introduced to you the intellectual environment and basic personal benefits of my idea:

**§ 1,****§ 2**
2) I described the consequences in the various sciences, and the increased integration with the utilization of computer software:

**§ 3**
3) I described some of the technical details of the idea, within the specializations of the mathematics:

**§ 4**
4) I suggested optional and alternative practical steps to realize the idea with books, software, and seminars:

**§ 5**
5) I analyzed the risks, possible objections, and why such a project is easier than it seems, has greater effects than it seems, and is more likely to have accelerating success:

**§ 6,****§ 8****§ 8**

**Epilogue and close of the letter**

I think I must finish this letter somewhere here. I want to thank you again for your time to read it. It is seems to me that the proposed is a very interesting and worthy goal a type of scientific charity to humanity. It resolves unhappiness and depression in life-long scale of many academic researchers too. It shall also bridge the gap between past centuries evolution of mathematics and 20

^{th}century birth of computer science. In doing that, new habits of thinking are introduced to new generations that will lead them to a happier and more perfect interaction with physical reality. This in its turn may lead to better more realistic, efficient and effective ways to succeed their goals in the arts sciences and technology It is something that always the civilization wanted and for the mathematics seems to be the only major upgrade since the ancient times of Euclid. From your powerful role, you can make it happen and be part of it. Let us leave one more long lasting honored mark in the civilization at liberating mathematician and scientists from the gloomy castle of present mathematics with the infinite, towards the new millennium growing digital paradise. And let’s do it with, Propriety, Prudence and Benevolence.
After the increasing complexity (and the success too) of the software and the digital world, a simple epitome in the context of mathematical concepts, open and free for all, is as it seems to me, a smart creative choice, as far as honor and respect is concerned, and hopefully from the most spiritual minority of creators in the civilization. But aside the honor and respect, and the benefits in mathematics, I believe that there are great benefits for all the sciences that utilize mathematical models, and require computing in an operating system. E.g. Physics, Chemistry, Medicine, Meteorology, Seismology, Environmental sciences, Economics etc.

Sincerely

Dr Costas Kyritsis

**References**

**[2] Andrew Carnegie “The Gospel of Wealth”**

**Applewood 1998**

**[3] Napoleon Hill “The wisdom of Andrew Carnegie” 1981 (N. Hill Foundation)**

**[4] R. Buckminster Fuller “Critical Path” St. Martin’s Griffin N.Y. 1981**