The abstraction of the infinite seems sweet and greedy at the beginning as it reduces some complexity, in the definitions, but later on when it comes for the proofs, it turns out to be bitter, as it traps the mathematical minds in to a vast complexity irrelevant to real life applications.
At first I must say that as a student I was enjoying the infinite , and I was excited by it, while as a University professor of later I even contributed to the transfinite numbers ( I had introduced the Ordinal Real numbers see e.g. here http://www.ckscientific.com/Infinity2.htm
Also I am very well aware of its history (Logicism of Frege, Constructivism. Formalism of Hilbert etc, story of Cantor etc)., And I am aware of what you may think as benefits of it. The main benefit is that it creates a distance of your mind and the self from the physical reality.....And believe it or not in some situations this may be precious in order to be able to think fast and free. I changed my mind about the infinite year later when I was involved to more substantial contributions of mathematics to other sciences and society.
The infinite is a sweet greedy abstraction at the beginning which seems to simplify in an elegant way the definitions but it turn out to be a horrible boomerang
When it comes to proofs, and traps the mathematical mind, to complexities irrelevant to applications. The abstraction of the infinite is of less developed mentality compared to the current situation of our civilization. It is an abstraction of a mind which
1) Does not know , that physical matter, and the continuum of the physical matter is made from finite many atoms or particles (like the electrons).
2) It is not aware of the computer science, how all continuous images in the screen and digital cinema are made from finite many pixels etc In the power point at the link at the end you may find a dozen of reasons why mathematics have to be upgraded to new mathematics without the infinite.
Here are some
1) The real numbers with the infinite many digits if attempted to be represented in a computer will create a virus or worm
2) When you teach new students about the limits and derivatives they look at you with surprise, they seem to understand nothing, while we could simply teach them that the dx is a real number with finite many digits which is zero in single precision but non-zero in double precision
and the dy/dx is non-zero in single precision
3) In functional analysis you have a hell of situations with the unbounded operators, where you almost cannot prove nothing, while with digital functions, the functional linear space is of finite dimension (not infinite) thus all proofs are easier
4) You have the 3rd Hilbert problem of polyhedral that are of equal volume and still they cannot be decomposed in to an equal number of equal other polyhedra (in physical reality this is not existent) WHY ON EARTH THE MATHEMATICAL MODEL OF A PHYSICAL REALITY SITUATION HAS TO BE MORE COMPLEX THAT THE REALITY ITSELF? ONLY THE INFINITE MAKES IT SO!
5) You have crazy situations with the axiom of choice in infinite many point sets, where you can decompose a geometric sphere in to finite many pieces , re-combine them and obtain two equal spheres, and equal to the initial as if you were a magician?
6) It is supposed that there geometric constructions with ruler and compass that are impossible, like squaring he circle , while in a digital geometry where all distances are rational (all rational numbers are constructible with ruler and compass) such problems would be constructible.
7) Goedels theorem a terribly pessimistic theorem for the rational thinking is possible only because of infinite many logical propositions, while in digital logic, thinks are more optimistic and possible
etc
IN SHORT THE ABSTRACTION OF THE INFINITE IS OLD-FASHIONED ARTISTIC ABSTRACTION CREATES PROBLEMS HAS ISOLATED MATHEMATICIANS FROM OTHER SCIENTISTS AND THE 21ST CENTURY MATHEMATICS NEED UPGRADING TO MORE MATURE MATHEMATICS WITHOUT THE 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
I do not say that infinity must nor remain. It has played a useful historic role so far, and it will remain. But sciences as well as mathematics are evolving, and we need to go one in new mathematics defining an interval (0,1) that at definite resolution , has only finite many points, both visible and invisible, otherwise we are in the unrealistic fiction area as far as engineering and applications is concerned. Still the old-fashioned interval (0,1) of uncountable many points will remain for historic and museum reasons, (as we still have the old physics of fluids, without atoms, in thermodynamics when the civilization did not know what atoms are and how finite many of them make the objects of our world. If we do not realize that the absence of some decent and realistic digital mathematics are precious for a smooth evolution of physics, inventions and the civilization, we are simply too much conformists, and too much looking back instead of looking ahead. The infinite continuum of classical real numbers is one only without layers, and we need for physics a multi-layered or a (finite )sequence of different resolutions continuums, the one embedded in the other, the one denser from the other. Why? Because this is the mathematically correct way to discover, the physics of the fields (electromagnetic and gravitational) as a second density or second resolution physical material reality of finer free permanent particles and not as energy in the "vacuum" (the only free permanent particles in physics so far is the electron/proton/neutron, all other are not free or non-permanent) And discovering the 2nd field-continuum of the physical reality we may solve many practical problems like clean energy, new healing methods etc.
http://www.youtube.com/watch?v=7Xu2cSEko6M
Comparisons of the classical analogue and the new
digital mathematics
•Analogue Mathematics
•Physical Reality Irrelevant
complexity. The model of reality may be more complex than reality itself.
•The infinite “feels” good
•Hard to understand mechanism
of limits,approximations and not easy to teach
•Pessimistic theorems (Goedel,
paradoxes, many axioms etc)
•Difficult proofs
•Many never proved conjectures
•Simple algebra of closure of
operations
•Absence of the effects of
multi-resolution continuum
•Digital
Mathematics
•Physical Reality Relevant complexity
•No infinite only finite invisible resolution.
Realistic
•No limits, or approximations,only pixels and
points, easy to teach
•Optimistic facts (realistic balance of “having”
resources and “being able to derive” in results. Less axioms)
•Easier proofs (a new method induction on the
pixels)
•Famous conjectures are easier to prove or
disprove
•Not simple algebra of closure of operations.
Internal-external entities.
•A new enhanced reality of multi-resolution
continuum
The effect in applied
sciences of the Digital mathematics
•Meteorology
•Physics
•Biology
•Engineering
•Ecology
•Sociology
•Economics
•More
realistic
•Ontology closer to that as represented in an
computer operating system
•Faster to run computations in computers
•Easier cooperation among physical
scientists-mathematicians-software engineers
•Easier learning of the digital mathematics
•Any additional complexity is a reality relevant
complexity not reality irrelevant complexity.
References
1) Rozsa Peter “Playing with Infinity” Dover Publications 1961
2) R. L. Wilder “Evolution of mathematical Concepts” Transworld Publishers LTD 1968
3) Howard Eves “An Introduction to the History of Mathematics”,4th edition 1953 Holt Rinehart and Winston publications
4) Howard Eves “Great Moments in Mathematics” The Mathematical Association of America 1980
5) Hans Rademacher-Otto Toeplitz “The Enjoyment of Mathematics”Princeton University Press 1957.
6) R. Courant and Herbert Robbins “What is Mathematics” Oxford 1969
7) A.D. Aleksndrov, A.N. Kolmogorov, M.A. Lavrentev editos
“Mathematics, its content, methods, and meaning” Vol 1,2,3 MIT press 1963
8) Felix Kaufmann “The Infinite in Mathematics” D. Reidel Publishing Company 1978
9) Edna E. Kramer “The Nature and Growth of Modern Mathematics”
Princeton University Press 1981
10) G. Polya “Mathematics and plausible reasoning” Vol 1, 2 1954 Princeton University press
11) Maurice Kraitchik “Mathematique des Jeux” 1953 Gauthier-Villars
12) Heinrich Dorrie “100 Great Problems of Elementary Mathematics”
Dover 1965
13) Imre Lakatos “Proofs and Refutations” Cambridge University Press 1976
“Dtv-Atlas zur Mtahematik” Band 1,2,1974
14) Struik D. J. A Concise History of Mathematics Dover 1987
15) S. Bochner The Role of Mathematics in the Rise of Science Princeton 1981
16) D.E. Littlewood “Le Passé-Partout Mathematique” Masson et c, Editeurs Paris 1964
17) A concise history of mathematics , by D. J. Struik, Dover 1987
At the end of this chapter, there is
a) Advantages-disadvantages of these new digital mathematics compared to the classical analogue, infinitary mathematics.
b) A fictional discussion in dialogue form of celebrated historic creators of the old mathematics praising or questioning the new mathematics compared to the old.
