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Title: THE NEW DIGITAL MATHEMATICS OF THE MILLENIUM


1
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

2
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

3
E. Schroendinger1'Nature and the Greeks' ,
'Science and Humanism' 
4
E. Schroendinger2'Nature and the Greeks' ,
'Science and Humanism'
5
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.

6
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.

7
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.

8
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.

9
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

10
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

11
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)

12
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.

13
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.

14
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

15
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.

16
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

17
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.

18
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.

19
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.

20
GEOMETRY B (HISTOMETRY)
21
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

22
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.

23
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

24
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.

25
CALCULUS B
26
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!

27
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

28
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.

29
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.

30
Differential geometry
  • Infinitesimal space of the surface space

31
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.

32
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.

33
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!

34
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

35
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
  •  

36
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

37
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.

38
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)
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