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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_changin g_math_education.htmlhttp://www.ted.com/talks/arthur_ benjamin_s_formula_for_changin g_math_education.htmlhttp://www.ted.com/talks/arthur_ benjamin_s_formula_for_changin g_math_education.htmlhttp://www.ted.com/talks/arthur_ benjamin_s_formula_for_changin g_math_education.html http://www.ted.com/talks/conrad _wolfram_teaching_kids_real_ma th_with_computers.htmlhttp://www.ted.com/talks/conrad _wolfram_teaching_kids_real_ma th_with_computers.htmlhttp://www.ted.com/talks/conrad _wolfram_teaching_kids_real_ma th_with_computers.htmlhttp://www.ted.com/talks/conrad _wolfram_teaching_kids_real_ma th_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.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”!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". 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?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.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 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.The Key: Two equalities : One for the visible points one for the invisible pixels.

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The Banach–Tarski 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.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/Bana ch%E2%80%93Tarski_paradoxhttp://en.wikipedia.org/wiki/Bana ch%E2%80%93Tarski_paradoxhttp://en.wikipedia.org/wiki/Bana ch%E2%80%93Tarski_paradoxhttp://en.wikipedia.org/wiki/Bana ch%E2%80%93Tarski_paradox

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THE TREE OF MATHEMATICS The primary foundation:The primary foundation: Natural numbersNatural numbers (similar to the CPU)(similar to the CPU) Meta-mathematics (Logic)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 numberNatural 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?Is the Peano Axiom of induction needed? Internal( input) – external (output) natural numbersInternal( input) – external (output) natural numbers The system of natural numbers in digital mathematics is finite, with a maximum natural number ω (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 ωLogic is introduced in the usual way except that all formulae of logic are finite only of a maximum natural number ω 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.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. 1 st Order Formal1 st Order Formal Language logic (similar to a programming language) Admits predicates over the terms, constant and variables but not over other predicates.Admits predicates over the terms, constant and variables but not over other predicates. Internal-external logicInternal-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: Goedel’s theorem revisited Storage complexity (Mathematics)and run time (RAM) complexity (Meta-mathematics).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 ω 0 > ω more sentences can be proved in a formal natural numbers theory. If ω 0 < ω less sentences can be proved.

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SETS The axiom of infinite does not exist in the digital set theoryThe 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 ω 1

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The relative size of the meta-mathematical Logic to the mathematical Set Theory: The axiom of Choice revisited if ωif ω 0 > > ω 1 Then even the axiom of choice could be a theorem not an axiom. But if ω 0 << ω 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 ω. (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 1 =2^( 0 )In classical analogue mathematics the hypothesis of the continuum is that 1 =2^( 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 betweenThat 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 ω= ω 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 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 “Ιστος”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 “Ιστος” Geometry is introduced in two ways:Geometry is introduced in two ways: 1) Directly with axioms as D. Hilbert did in his classical axiomatic definition of Euclidean geometry.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)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 3 rd 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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Hilbert’s 3 rd problem revisited Hilbert’s 3 rd problem was if two solid figures that are of equal volume are also equidecomposable.Hilbert’s 3 rd 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.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 3 rd 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 3 rd 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.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.Classical analogue mathematics: Infinitesimals, limits and finite quantities. The long historic controversy. Continuity and differentiability are different.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:The original definition of the derivative by Leibniz was though infinitesimals: dy/dxdy/dx Newton’s method of flux was reformulated later by Cauchy through infinite convergent sequences in the analogue real numbers.Newton’s 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. 0

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THE INTEGRAL In the analogue real numbers there are many types of distinct integrals:In the analogue real numbers there are many types of distinct integrals: Cauchy IntegralCauchy Integral Reimann IntegralReimann Integral Lebesque IntegralLebesque Integral Shilov IntegralShilov Integral EtcEtc 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 setsClassical 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 1 st order differential equation is laborious and complicated and involves limits of functions etcIn classical mathematics the proof of the existence and uniqueness of the solution e.g. of a 1 st 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 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. 1 st 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 2 nd equality of the real numbers) is directly exact.

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Differential geometry Infinitesimal space of the surface spaceInfinitesimal 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.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 17 th century. Physical Nano-worlds, micro-worlds and macro-worlds for the 1 st 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 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.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.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 Rozsa Peter “Playing with Infinity” Dover Publications 1961 R. L. Wilder “Evolution of mathematical Concepts” Transworld Publishers LTD 1968 R. L. Wilder “Evolution of mathematical Concepts” Transworld Publishers LTD 1968 Howard Eves “An Introduction to the History of Mathematics”,4 th edition 1953 Holt Rinehart and Winston publications Howard Eves “An Introduction to the History of Mathematics”,4 th edition 1953 Holt Rinehart and Winston publications Howard Eves “Great Moments in Mathematics” The Mathematical Association of America 1980 Howard Eves “Great Moments in Mathematics” The Mathematical Association of America 1980 Hans Rademacher-Otto Toeplitz “The Enjoyment of Mathematics”Princeton University Press 1957. Hans Rademacher-Otto Toeplitz “The Enjoyment of Mathematics”Princeton University Press 1957. R. Courant and Herbert Robbins “What is Mathematics” Oxford 1969 R. Courant and Herbert Robbins “What is Mathematics” Oxford 1969 A.D. Aleksndrov, A.N. Kolmogorov, M.A. Lavrentev editos A.D. Aleksndrov, A.N. Kolmogorov, M.A. Lavrentev editos “Mathematics, its content, methods, and meaning” Vol 1,2,3 MIT press 1963 “Mathematics, its content, methods, and meaning” Vol 1,2,3 MIT press 1963 Felix Kaufmann “The Infinite in Mathematics” D. Reidel Publishing Company 1978 Felix Kaufmann “The Infinite in Mathematics” D. Reidel Publishing Company 1978 Edna E. Kramer “The Nature and Growth of Modern Mathematics” Edna E. Kramer “The Nature and Growth of Modern Mathematics” Princeton University Press 1981 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)A New Kind of ScienceStephen Wolfram Wolfram Media. www.wolframscience.comwww.wolframscience.com

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References 2 1) G. H. Hardy A course in Pure Mathematics Cambridge 10 th edition 1975 1) G. H. Hardy A course in Pure Mathematics Cambridge 10 th edition 1975 2) T. Jech Set theory Academic Press 1978 2) T. Jech Set theory Academic Press 1978 3) Robert R. Stoll Sets, Logic and Axiomatic Theories Freeman 1961 3) Robert R. Stoll Sets, Logic and Axiomatic Theories Freeman 1961 4) M. Carvallo “Logique a trois valeurs loguique a seuil” Gauthier-Villars 1968 4) M. Carvallo “Logique a trois valeurs loguique a seuil” Gauthier-Villars 1968 5) D. Hilbert- W. Ackermann “Principles of Mathematical Logic” 5) D. Hilbert- W. Ackermann “Principles of Mathematical Logic” Chelsea publishing Company N.Y. 1950 Chelsea publishing Company N.Y. 1950 6) H. A. Thurston “The number system” Dover 1956 6) H. A. Thurston “The number system” Dover 1956 7) J.H. Conway “On Numbers and Games” Academic Press 1976 7) J.H. Conway “On Numbers and Games” Academic Press 1976 8) D. Hilbert “Grundlangen der Geometrie” Taubner Studienbucher 1977 8) D. Hilbert “Grundlangen der Geometrie” Taubner Studienbucher 1977 9) V. Boltianskii “Hilbert’s 3 rd problem” J. Wesley & Sons 1978 9) V. Boltianskii “Hilbert’s 3 rd problem” J. Wesley & Sons 1978 10)E. E. Moise “Elementary geometry from an advanced standpoint” Addison –Wesley 1963 10)E. E. Moise “Elementary geometry from an advanced standpoint” Addison –Wesley 1963 11) Euclid The 13 books of the Elements Dover 1956 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 MathematicsAnalogue Mathematics Physical Reality Irrelevant complexity. The model of reality may be more complex than reality itself.Physical Reality Irrelevant complexity. The model of reality may be more complex than reality itself. The infinite “feels” goodThe infinite “feels” good Hard to understand mechanism of limits,approximations and not easy to teachHard to understand mechanism of limits,approximations and not easy to teach Pessimistic theorems (Goedel, paradoxes, many axioms etc)Pessimistic theorems (Goedel, paradoxes, many axioms etc) Difficult proofsDifficult proofs Many never proved conjecturesMany never proved conjectures Simple algebra of closure of operationsSimple algebra of closure of operations Absence of the effects of multi- resolution continuumAbsence 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 MeteorologyMeteorology PhysicsPhysics BiologyBiology EngineeringEngineering EcologyEcology SociologySociology EconomicsEconomics 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)^2=2(M/N)^2=2 M,N with no common prime divisorM,N with no common prime divisor M*M=2*N*NM*M=2*N*N 2/M2/M 2/N2/N ContradictionContradiction This proof would no hold in digital real numbers: Two reasons A) although (M/N)^2 may be a digital real number, M^2 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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