THE NEW DIGITAL MATHEMATICS OF THE MILLENIUM

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THE NEW DIGITAL MATHEMATICS OF THE MILLENIUM

• By Dr Costas Kyritsis
• TEI of Epirus Dept of Finance.
• With the courtesy and support of the Software
  Laboratory, National Technical University of
  Athens.
• Spring 2011

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The demand has been felt since many years

• http//www.ted.com/talks/arthur_benjamin_s_formula
  _for_changing_math_education.html
• http//www.ted.com/talks/conrad_wolfram_teaching_k
  ids_real_math_with_computers.html

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E. Schroendinger1'Nature and the Greeks' ,
'Science and Humanism' 
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E. Schroendinger2'Nature and the Greeks' ,
'Science and Humanism'
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CAN WE ISOLATE A SIMPLE FACTOR TO CHANGE?

• Global changes in a whole science would be
  chaotic if a single key factor was isolated and
  assessed to eliminate or change through out.

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The factor The infinite

• The Odysseus's Lotus of the infinity!
• I am better served by being accountable and
  holding my values consciously than being
  non-accountable holding my values unconsciously.
  I am better served by examining them rather than
  holding them uncritically as not-to-be-questioned
   "axioms".
• Nathaniel Branden The six pillars od
  self-esteem.

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CAN WE DEVISE A SIMPLE KEY SOLUTION AND CHANGE
APPROPRIATELY THE DNA OF MATHEMATICS?

• Can mathematics exist without the infinite?
• A guiding principle Create a mathematical
  ontology to follow the wisdom of software
  engineering and the ontology of analogue
  mathematical entities as embedded in the
  operating system.

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The solution THE FINITE RESOLUTION

• The continuum is no more a bottomless ocean It
  is a sea with accessible and tractable bottom.
• The Key Two equalities One for the visible
  points one for the invisible pixels.

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The BanachTarski paradox

• The axiom of choice If S is a set that its
  elements are sets, there is at least one set A
  that is made by choosing one element from each
  element-set of S.
• http//en.wikipedia.org/wiki/BanachE28093Tarski
  _paradox

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THE TREE OF MATHEMATICS

• The primary foundation
• Natural numbers
• (similar to the CPU)
• Meta-mathematics (Logic)
• (similar to RAM memory)

• Mathematics
• (similar to storage memory)
• Sets,
• Real numbers,
• Statistics,
• Euclidean geometry,
• Calculus,
• Differential equations
• Universal Algebra
• Topology
• Differential geometry

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NATURAL NUMBERS

• Natural numbers are introduced first in the
  digital mathematics with the usual axioms over
  the initial concept of successor (or predecessor)
  of a natural number
• Is the Peano Axiom of induction needed?
• Internal( input) external (output) natural
  numbers

• The system of natural numbers in digital
  mathematics is finite, with a maximum natural
  number ? (of unknown or variable size but fixed
  finite number)

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LOGIC

• Logic is introduced in the usual way except that
  all formulae of logic are finite only of a
  maximum natural number ?0
• This finite cardinal number is the capacity (or
  complexity size) of the meta-mathematical nature
  of Logic. The length of logical arguments and
  proofs cannot surpass this number.
• 1st Order Formal
• Language logic (similar to a programming
  language)
• Admits predicates over the terms , constant and
  variables but not over other predicates.
• Internal-external logic

• The quantification (for all, for every, there
  is)
• is a symbolic shortcut to avoid the complexity of
  scans of the same size as the objects of study.

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The relative size of the meta-mathematical Logic
to the mathematical Natural Numbers Goedels
theorem revisited

• Storage complexity (Mathematics)and run time
  (RAM) complexity (Meta-mathematics).

• An optimistic solution
• No Goedel type absolute impossibility to prove
  theorems in digital mathematics. Only relative to
  resources impossibility or possibility
• E.g. if ?0 gt ? more sentences can be proved in a
  formal natural numbers theory. If ?0 lt ?
• less sentences can be proved.

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SETS

• The axiom of infinite does not exist in the
  digital set theory

• All axioms of digital set theory are referring to
  finite sets .
• Internal-external sets
• All sets are of cardinality less than a maximum
  finite cardinal number ?1

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The relative size of the meta-mathematical Logic
to the mathematical Set Theory The axiom of
Choice revisited

• if ?0 gt gt ?1 Then even the axiom of choice
  could be a theorem not an axiom.

• But if ?0 ltlt ?1 we prefer to put it as axiom.

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REAL NUMBERS

• The digital Real numbers are defined as a finite
  system of decimal numbers based on the concept of
  finite resolution.
• There are two equivalence relations That of the
  (smallest) visible points (of the real line) and
  that of the invisible pixels.
• There is no unconditional closure of the usual
  operations within the real numbers.

• There are no irrational numbers. All numbers are
  essentially rational.
• The maximum integer within the real numbers is
  symbolized by ?. (the same with that of the
  axiomatic system of the natural numbers. This
  guarantees the Archimedean axiom).
  Internal-external real numbers

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The relative size meta-mathematical logic to the
resolution size of the real numbers The
continuum hypothesis revisited

• In classical analogue mathematics the hypothesis
  of the continuum is that ?12(?0)
• That is that the cardinality of the power set of
  the natural numbers is the next cardinal number
  after the cardinality of the natural numbers.
  Nothing in between

• In Digital mathematics the corresponding axiom is
  that the maximum natural number within the real
  numbers is equal
• ? ?1 (or of the order size) of the maximum
  cardinal number of the digital set theory.

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A new type of proof Mathematical Induction on
the pixels of the resolution.

• In classical mathematics the real numbers have
  uncountable cardinality and the points of the
  real line are not well ordered. So no finite or
  transfinite inductions is readily applicable to
  the real lile points.

• In the digital real numbers the (invisible)
  pixels are finite linearly ordered and well
  ordered, so mathematical induction on them does
  apply. The latter is a powerful tool for proofs
  that can prove many propositions hard to prove in
  the analogue real numbers.

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GEOMETRY (HISTOMETRY)

• The ancient word for geometry (e.g. at the time
  of Pythagoras) was History, (because of figures
  and the lines that look liked the mast of a ship,
  and the Greek word for mast was the word ?st??
• Geometry is introduced in two ways
• 1) Directly with axioms as D. Hilbert did in his
  classical axiomatic definition of Euclidean
  geometry.
• 2) As 3 dimensional vector space over the real
  numbers (analytic Cartesian geometry)

• The digital geometry can be defined again in both
  ways as above.
• If defined directly by axioms over (visible)
  points linear segments and planes, the axioms of
  betweeness are changed. Between two (visible)
  points does not exist always a 3rd visible point.
• Defined as 3 dimensional vector space over the
  digital real numbers (analytic Cartesian
  geometry) is easier and the distinction of
  smallest visible points and invisible pixels is
  inherited here too.
• Internal-external space.

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GEOMETRY B (HISTOMETRY)
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GEOMETRY C (HISTOMETRY)

• Analogue geometry Antiquity insoluble problems
  with ruler and compass
• A) Squaring the circle
• B) Trisection of an angle

• Digital mathematics
• All rational numbers are constructible with ruler
  an compass
• A) Squaring the circle is constructible with
  ruler and compass
• B) Trisection of an angle is constructible with
  ruler and compass

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Hilberts 3rd problem revisited

• Hilberts 3rd problem was if two solid figures
  that are of equal volume are also
  equidecomposable.
• Two figures F, H are said to be equidecomposable
  if the figure F can be suitably decomposed into a
  finite number of pieces which can be reassembled
  to give the figure H.
• The 3rd Hilbert problem was proven by Dehn in
  1900 in the negative There are figures of equal
  volume that are not equidecomposable. E.g. A Cube
  and a regular tetrahedron of equal volume are not
  equidecomposable.

• The situation is not the same in the digital
  Euclidean geometry. Two figure of equal volume
  are also equidecomposiable! The reason is that
  according to the resolution of the Euclidean
  geometry rational numbers only up to a decimal
  can be volumes of figures. E.g. visible points do
  have volume, and two figures of equal volume are
  also of equal number of visible points. So the
  visible points make the equidecomposability.

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STATISTICS

• Statistics and probability are defined in the
  usual way in the digital mathematics too.

• An advantage of the digital probability theory
  over say the digital Euclidean geometry is that
  the classical paradoxes of geometric probability
  of analogue mathematics have nice and rational
  explanation in the digital mathematics

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CALCULUS

• Classical analogue mathematics Infinitesimals,
  limits and finite quantities. The long historic
  controversy.
• Continuity and differentiability are different.

• Digital mathematics Invisible Pixels and visible
  points 2 equivalence relations. Both finite
  many. The calculus in the digital mathematics is
  neat easy to understand in the screen of a
  computer and easy to teach. No limits and
  infinite sequences convergence. Differentiability
  is left and right.
• Continuity and one sided differentiability are
  identical in the digital calculus.

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CALCULUS B
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THE DERIVATIVE

• The original definition of the derivative by
  Leibniz was though infinitesimals
• dy/dx
• Newtons method of flux was reformulated later by
  Cauchy through infinite convergent sequences in
  the analogue real numbers.

• In the digital real numbers infinitesimal dx are
  the pixel real numbers (e.g. double precision
  numbers) , that are less than any visible point
  real number x (single precision number) still
  greater than zero. 0ltdxltx And a quotient of such
  double precision umbers, can very well be a
  single precision number. The derivative in the
  digital real numbers is not defined though limits!

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THE INTEGRAL

• In the analogue real numbers there are many types
  of distinct integrals
• Cauchy Integral
• Reimann Integral
• Lebesque Integral
• Shilov Integral
• Etc

• In the calculus of digital real numbers of a
  single resolution, there is only one type of
  integral
• The digital (single resolution) Integral

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TOPOLOGY

• Classical mathematics The topology is defined
  though Axioms of Open sets

• Digital mathematic The topology is defined by
  Axioms for a (visible) point being in contact
  with a set of (visible) points.

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DIFFERENTIAL EQUATIONS

• In classical mathematics the proof of the
  existence and uniqueness of the solution e.g. of
  a 1st order differential equation is laborious
  and complicated and involves limits of functions
  etc
• In the nalogue mathematics we must introduce a
  separate course that of numerical analysis to
  compute the solutions.

• In digital mathematics the solution of a (e.g.
  1st order) differential equation is directly
  constructed in a recursive method as difference
  equations over the pixels of the real numbers. It
  is simple and the existence , uniqueness, and at
  the same time direct calculation of the solution
  by computer are all simultaneous. Numerical
  analysis here is not different that differential
  equations analysis, there are no limits
  approximations etc. The solutions (up to the 2nd
  equality of the real numbers) is directly exact.

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Differential geometry

• Infinitesimal space of the surface space

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Are there multi-resolution real numbers? The
difference that makes the difference in the
physical applications of the digital mathematics.

• In digital mathematics Differential manifolds and
  geometry already may require double resolution
  real numbers. (a curvilinear line on the manifold
  representing real numbers has to be of higher
  resolution to the real numbers represented on the
  straight line of the infinitesimal or tangent
  space.

• The applications of multiresolution mathematics
  in the physical sciences are entirely beyond
  classical analogue mathematics, and may bring
  such a revolution in physics as the Newtonian and
  Leibnizian calculus did in the 17th century.
  Physical Nano-worlds, micro-worlds and
  macro-worlds for the 1st time can be treated
  quantitatively in a mutual consistent and
  integrated way.

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STOCHASTIC DIFFERETIAL EQUATIONS A radical
simplification

• In classical analogue mathematics, the ITO
  calculus defines and solves the stochastic
  differential equations though a highly
  complicated system of elaborate probabilistic
  limits and convergence.

• In the digital mathematics, the stochastic
  differential equations become time-series over
  the pixels, and the relevant statistics and
  probability very significantly simpler. I have
  programmed much of such real time monitoring or
  simulated examples, of which the realism and
  value in direct financial applications is great.

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UNIVERSAL ALGEBRA

• Except of the fact the closure of algebraic
  operations is almost always conditional in the
  digital mathematics, universal algebra is the
  subject that has few only alterations besides the
  standard alterations of the underlying digital
  set theory.
• Algebra is not mainly based on the continuum that
  is why the digital mathematics do not affect it
  much.

• The algebra of the fields of geometrically
  constructible real numbers, or algebraic real
  numbers is of course much changed in the digital
  mathematics. All real numbers (being rational
  numbers) are geometrically constructible. The
  Squaring of the circle is a fact in the digital
  Euclidean geometry!

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References 1

• Rozsa Peter Playing with Infinity Dover
  Publications 1961
• R. L. Wilder Evolution of mathematical Concepts
  Transworld Publishers LTD 1968
• Howard Eves An Introduction to the History of
  Mathematics,4th edition 1953 Holt Rinehart and
  Winston publications
• Howard Eves Great Moments in Mathematics The
  Mathematical Association of America 1980
• Hans Rademacher-Otto Toeplitz The Enjoyment of
  MathematicsPrinceton University Press 1957.
• R. Courant and Herbert Robbins What is
  Mathematics Oxford 1969
• A.D. Aleksndrov, A.N. Kolmogorov, M.A. Lavrentev
  editos
• Mathematics, its content, methods, and meaning
  Vol 1,2,3 MIT press 1963
• Felix Kaufmann The Infinite in Mathematics D.
  Reidel Publishing Company 1978
• Edna E. Kramer The Nature and Growth of Modern
  Mathematics
• Princeton University Press 1981

• G. Polya Mathematics and plausible reasoning
  Vol 1, 2 1954 Princeton University press
• Maurice Kraitchik Mathematique des Jeux 1953
  Gauthier-Villars
• Heinrich Dorrie 100 Great Problems of Elementary
  Mathematics
• Dover 1965
• Imre Lakatos Proofs and Refutations Cambridge
  University Press 1976
• Dtv-Atlas zur Mtahematik Band 1,2,1974
• Struik D. J. A Concise History of Mathematics
  Dover 1987
•  
• S. Bochner The Role of Mathematics in the Rise of
  Science Princeton 1981
• D.E. Littlewood Le Passé-Partout Mathematique
  Masson et c, Editeurs Paris 1964
•  
• A New Kind of Science by Stephen Wolfram (2002)
• Wolfram Media. www.wolframscience.com

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References 2

• 1)     G. H. Hardy A course in Pure Mathematics
  Cambridge 10th edition 1975
• 2)     T. Jech Set theory Academic Press 1978
• 3) Robert R. Stoll Sets, Logic and Axiomatic
  Theories Freeman 1961
•  
• 4) M. Carvallo Logique a trois valeurs loguique
  a seuil Gauthier-Villars 1968
• 5) D. Hilbert- W. Ackermann Principles of
  Mathematical Logic
• Chelsea publishing Company N.Y. 1950
• 6) H. A. Thurston The number system Dover 1956
• 7) J.H. Conway On Numbers and Games Academic
  Press 1976
• 8) D. Hilbert Grundlangen der Geometrie Taubner
  Studienbucher 1977
•  
• 9) V. Boltianskii Hilberts 3rd problem J.
  Wesley Sons 1978
• 10)E. E. Moise Elementary geometry from an
  advanced standpoint Addison Wesley 1963
• 11) Euclid The 13 books of the Elements Dover
  1956

• 1) Michael Spivak Calculus Benjamin 1967
• 2) Ivan N. Pesin Classical and modern Integration
  theories Academic Press 1970
• 3) T. Apostol Mathematical Analysis Addison
  Wesley 1974
• 4) G.E. Shilov-BL. Gurevich Integral Measure
  Derivative a unified approach Dover 1977
• 5) M Spivak Calculus on Manifolds Benjamin 1965
• 6) W. Hurevicz
• Lectures on ordinary Differential Equations
•  

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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

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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.

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Reexamining the classical proof that the square
root of 2 is not a rational number within digital
mathematics

• (M/N)22
• M,N with no common prime divisor
• MM2NN
• 2/M
• 2/N
• Contradiction

• This proof would no hold in digital real numbers
  Two reasons
• A) although (M/N)2 may be a digital real number,
  M2 may be outside the digital real numbers
• B) There are two equalities in he digital real
  numbers that of single precision quantities
  (visible points) that double precision (pixels)