THE NEW DIGITAL THE NEW DIGITAL MATHEMATICS OF THE MATHEMATICS OF THE MILLENIUMMILLENIUMBy Dr Costas KyritsisBy Dr Costas KyritsisTEI of Epirus Dept of Finance.TEI of Epirus Dept of Finance.With the courtesy and support of the Software With the courtesy and support of the Software Laboratory, National Technical University of Athens.Laboratory, National Technical University of Athens.Spring 2011Spring 2011

The demand has been felt The demand has been felt since many yearssince many years• http://www.ted.com/talks/arthur_

benjamin_s_formula_for_changing_math_education.html

• http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers.html

E. Schroendinger1:'Nature and the Greeks' , 'Science and Humanism' 

E. Schroendinger2:'Nature and the Greeks' , 'Science and Humanism'

CAN WE ISOLATE A SIMPLE CAN WE ISOLATE A SIMPLE FACTOR TO CHANGE? FACTOR TO CHANGE?

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

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 I am better served by being accountable and holding my values consciously than being non-accountable my values consciously than being non-accountable holding my values unconsciously. I am better served by holding my values unconsciously. I am better served by examining them rather than holding them uncritically as examining them rather than holding them uncritically as not-to-be-questioned  "axioms".not-to-be-questioned  "axioms".

Nathaniel Branden Nathaniel Branden ““The six pillars od self-esteem.The six pillars od self-esteem.””

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

• Can mathematics exist without the Can mathematics exist without the infinite?infinite?

• A guiding principle: Create a A guiding principle: Create a mathematical ontology to follow the mathematical ontology to follow the wisdom of software engineering and the wisdom of software engineering and the ontology of analogue mathematical ontology of analogue mathematical entities as embedded in the operating entities as embedded in the operating system.system.

The solution: The solution: THE FINITE RESOLUTIONTHE FINITE RESOLUTION

• ““The continuum is no more a The continuum is no more a bottomless ocean: It is a sea with bottomless ocean: It is a sea with accessible and tractable bottom.”accessible and tractable bottom.”

• The Key: Two equalities : One for The Key: Two equalities : One for the visible points one for the the visible points one for the invisible pixels.invisible pixels.

The The Banach–Tarski Banach–Tarski paradoxparadox

• The axiom of choice: If S is a set that its elements are sets, there is at 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 least one set A that is made by choosing one element from each element-set of S.element-set of S.

• http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

THE TREE OF THE TREE OF MATHEMATICSMATHEMATICS

• The primary The primary foundation:foundation:

• Natural numbersNatural numbers• (similar to the CPU)(similar to the CPU)

• Meta-mathematics Meta-mathematics (Logic)(Logic)

(similar to RAM (similar to RAM memory)memory)

• Mathematics:Mathematics:• (similar to storage (similar to storage

memory)memory)• Sets, Sets, • Real numbers,Real numbers,• Statistics,Statistics,• Euclidean geometry,Euclidean geometry,• Calculus,Calculus,• Differential equationsDifferential equations• Universal AlgebraUniversal Algebra• TopologyTopology• Differential geometryDifferential geometry

NATURAL NUMBERSNATURAL NUMBERS• Natural numbers are Natural numbers are

introduced first in the introduced first in the digital mathematics with digital mathematics with the usual axioms over the the usual axioms over the initial concept of initial concept of successor (or successor (or predecessor) of a natural predecessor) of a natural numbernumber

• Is the Peano Axiom of Is the Peano Axiom of induction needed?induction needed?

• Internal( input) –external Internal( input) –external (output) natural numbers(output) natural numbers

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

LOGICLOGIC• Logic is introduced in the usual Logic is introduced in the usual

way except that all formulae of way except that all formulae of logic are finite only of a logic are finite only of a maximum natural number maximum natural number ωω0

• This finite cardinal number is This finite cardinal number is the capacity (or complexity size) the capacity (or complexity size) of the meta-mathematical of the meta-mathematical nature of Logic. The length of nature of Logic. The length of logical arguments and proofs logical arguments and proofs cannot surpass this number.cannot surpass this number.

• 11stst Order Formal Order FormalLanguage logic (similar to a Language logic (similar to a

programming language)programming language)• Admits predicates over the Admits predicates over the

terms , constant and variables terms , constant and variables but not over other predicates.but not over other predicates.

• Internal-external logicInternal-external logic

• The quantification The quantification (for all, for every, (for all, for every, there is…)there is…)

• is a symbolic is a symbolic shortcut to avoid the shortcut to avoid the complexity of complexity of “scans” of the same “scans” of the same size as the objects of size as the objects of study.study.

The relative size of the meta-mathematical Logic The relative size of the meta-mathematical Logic to the mathematical Natural Numbers: Goedel’s to the mathematical Natural Numbers: Goedel’s theorem revisitedtheorem revisited

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

• An optimistic solution:An optimistic solution:• No Goedel type absolute No Goedel type absolute

impossibility to prove impossibility to prove theorems in digital theorems in digital mathematics. Only relative mathematics. Only relative to “resources” impossibility to “resources” impossibility or possibilityor possibility

• E.g. if E.g. if ωω0 > ω more sentences can be proved in a formal natural numbers theory. If ωω0 < ω

less sentences can be less sentences can be proved.proved.

SETSSETS• The axiom of The axiom of

infinite does not infinite does not exist in the digital exist in the digital set theoryset theory

• All axioms of digital All axioms of digital set theory are set theory are referring to finite referring to finite sets .sets .

• Internal-external setsInternal-external sets• All sets are of All sets are of

cardinality less than cardinality less than a maximum finite a maximum finite cardinal number cardinal number ωω11

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

• ifif ωω0 > > ω1 Then even the axiom of choice could be a theorem not an axiom.

• But if ωω0 << ω1 we prefer to put it as axiom.

REAL NUMBERSREAL 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 There are no irrational numbers. All numbers are numbers. All numbers are essentially rational.essentially rational.

• The maximum integer within The maximum integer within the real numbers is the real numbers is symbolized by symbolized by ω.ω. (the same (the same with that of the axiomatic with that of the axiomatic system of the natural system of the natural numbers. This guarantees numbers. This guarantees the Archimedean axiom)the Archimedean axiom). . Internal-external real Internal-external real numbersnumbers

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

• In classical analogue In classical analogue mathematics the hypothesis mathematics the hypothesis of the continuum is that of the continuum is that 11=2^(=2^(00))

• That is that the cardinality That is that the cardinality of the power set of the of the power set of the natural numbers is the next natural numbers is the next cardinal number after the cardinal number after the cardinality of the natural cardinality of the natural numbers. Nothing in numbers. Nothing in betweenbetween

• In Digital mathematics the In Digital mathematics the corresponding axiom is that corresponding axiom is that the maximum natural the maximum natural number within the real number within the real numbers is equal numbers is equal

• ωω= = ωω11 (or of the order size) (or of the order size) of the maximum cardinal of the maximum cardinal number of the digital set number of the digital set theory.theory.

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

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

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

GEOMETRY (HISTOMETRY)GEOMETRY (HISTOMETRY)• The ancient word for geometry The ancient word for geometry

(e.g. at the time of Pythagoras) (e.g. at the time of Pythagoras) was History, (because of figures was History, (because of figures and the lines that look liked the and the lines that look liked the mast of a ship, and the Greek mast of a ship, and the Greek word for mast was the word word for mast was the word ““ΙστοςΙστος””

• Geometry is introduced in two Geometry is introduced in two ways:ways:

• 1) Directly with axioms as D. 1) Directly with axioms as D. Hilbert did in his classical Hilbert did in his classical axiomatic definition of Euclidean axiomatic definition of Euclidean geometry. geometry.

• 2) As 3 dimensional vector 2) As 3 dimensional vector space over the real numbers space over the real numbers (analytic Cartesian geometry)(analytic Cartesian geometry)

• The digital geometry can be The digital geometry can be defined again in both ways as defined again in both ways as above.above.

• If defined directly by axioms over If defined directly by axioms over (visible) points linear segments (visible) points linear segments and planes, the axioms of and planes, the axioms of betweeness are changed. betweeness are changed. Between two (visible) points Between two (visible) points does not exist always a 3does not exist always a 3rdrd visible point.visible point.

• Defined as 3 dimensional vector Defined as 3 dimensional vector space over the digital real space over the digital real numbers (analytic Cartesian numbers (analytic Cartesian geometry) is easier and the geometry) is easier and the distinction of smallest visible distinction of smallest visible points and invisible pixels is points and invisible pixels is inherited here too.inherited here too.

• Internal-external space.Internal-external space.

GEOMETRY B GEOMETRY B (HISTOMETRY)(HISTOMETRY)

GEOMETRY C GEOMETRY C (HISTOMETRY)(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

Hilbert’s 3Hilbert’s 3rdrd problem revisited problem revisited• Hilbert’s 3Hilbert’s 3rdrd problem was if two problem was if two

solid figures that are of equal solid figures that are of equal volume are also volume are also equidecomposable.equidecomposable.

• Two figures F, H are said to be Two figures F, H are said to be equidecomposable if the figure F equidecomposable if the figure F can be suitably decomposed can be suitably decomposed into a finite number of pieces into a finite number of pieces which can be reassembled to which can be reassembled to give the figure H.give the figure H.

• The 3The 3rdrd Hilbert problem was Hilbert problem was proven by Dehn in 1900 in the proven by Dehn in 1900 in the negative: There are figures of negative: There are figures of equal volume that are not equal volume that are not equidecomposable. E.g. A Cube equidecomposable. E.g. A Cube and a regular tetrahedron of and a regular tetrahedron of equal volume are not equal volume are not equidecomposable.equidecomposable.

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

STATISTICSSTATISTICS• Statistics and probability are Statistics and probability are

defined in the usual way in defined in the usual way in the digital mathematics too.the digital mathematics too.

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

CALCULUSCALCULUS• Classical analogue Classical analogue

mathematics: Infinitesimals, mathematics: Infinitesimals, limits and finite quantities. limits and finite quantities. The long historic The long historic controversy.controversy.

• Continuity and Continuity and differentiability are different. differentiability are different.

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

• Continuity and one sided Continuity and one sided differentiability are identical in differentiability are identical in the digital calculus.the digital calculus.

CALCULUS BCALCULUS B

THE DERIVATIVETHE DERIVATIVE• The original definition of the The original definition of the

derivative by Leibniz was derivative by Leibniz was though infinitesimals:though infinitesimals:

• dy/dxdy/dx• Newton’s method of flux Newton’s method of flux

was reformulated later by was reformulated later by Cauchy through infinite Cauchy through infinite convergent sequences in the convergent sequences in the analogue real numbers.analogue real numbers.

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

THE INTEGRALTHE INTEGRAL• In the analogue real In the analogue real

numbers there are many numbers there are many types of distinct integrals: types of distinct integrals:

• Cauchy IntegralCauchy Integral• Reimann IntegralReimann Integral• Lebesque IntegralLebesque Integral• Shilov IntegralShilov Integral• EtcEtc

• In the calculus of digital In the calculus of digital real numbers of a single real numbers of a single resolution, there is only resolution, there is only one type of integral:one type of integral:

• The digital (single The digital (single resolution) Integralresolution) Integral

TOPOLOGYTOPOLOGY• Classical mathematics: The Classical mathematics: The

topology is defined though topology is defined though Axioms of Open setsAxioms of Open sets

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

DIFFERENTIAL EQUATIONSDIFFERENTIAL EQUATIONS• In classical mathematics the In classical mathematics the

proof of the existence and proof of the existence and uniqueness of the solution e.g. uniqueness of the solution e.g. of a 1of a 1stst order differential order differential equation is laborious and equation is laborious and complicated and involves limits complicated and involves limits of functions etcof functions etc

• In the nalogue mathematics we In the nalogue mathematics we must introduce a separate must introduce a separate course that of numerical course that of numerical analysis to compute the analysis to compute the solutions.solutions.

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

Differential geometryDifferential geometry• Infinitesimal space of the surface Infinitesimal space of the surface

spacespace

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

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

The applications of multiresolution The applications of multiresolution mathematics in the physical mathematics in the physical sciences are entirely beyond sciences are entirely beyond classical analogue mathematics, classical analogue mathematics, and may bring such a and may bring such a revolution in physics as the revolution in physics as the Newtonian and Leibnizian Newtonian and Leibnizian calculus did in the 17calculus did in the 17thth century. century. Physical Nano-worlds, micro-Physical Nano-worlds, micro-worlds and macro-worlds for the worlds and macro-worlds for the 11stst time can be treated time can be treated quantitatively in a mutual quantitatively in a mutual consistent and integrated way.consistent and integrated way.

STOCHASTIC DIFFERETIAL EQUATIONS: A STOCHASTIC DIFFERETIAL EQUATIONS: A radical simplification radical simplification

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

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

UNIVERSAL ALGEBRAUNIVERSAL ALGEBRA• Except of the fact the closure of Except of the fact the closure of

algebraic operations is almost algebraic operations is almost always conditional in the digital always conditional in the digital mathematics, universal algebra mathematics, universal algebra is the subject that has few only is the subject that has few only alterations besides the standard alterations besides the standard alterations of the underlying alterations of the underlying digital set theory.digital set theory.

• Algebra is not mainly based on Algebra is not mainly based on the continuum that is why the the continuum that is why the digital mathematics do not digital mathematics do not affect it much.affect it much.

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

References 1References 1• Rozsa Peter “Playing with Infinity” Dover Rozsa Peter “Playing with Infinity” Dover

Publications 1961 Publications 1961 • R. L. Wilder “Evolution of mathematical R. L. Wilder “Evolution of mathematical

Concepts” Transworld Publishers LTD 1968Concepts” Transworld Publishers LTD 1968• Howard Eves “An Introduction to the History Howard Eves “An Introduction to the History

of Mathematics”,4of Mathematics”,4thth edition 1953 Holt edition 1953 Holt Rinehart and Winston publicationsRinehart and Winston publications

• Howard Eves “Great Moments in Howard Eves “Great Moments in Mathematics” The Mathematical Association Mathematics” The Mathematical Association of America 1980of America 1980

• Hans Rademacher-Otto Toeplitz “The Hans Rademacher-Otto Toeplitz “The Enjoyment of Mathematics”Princeton Enjoyment of Mathematics”Princeton University Press 1957.University Press 1957.

• R. Courant and Herbert Robbins “What is R. Courant and Herbert Robbins “What is Mathematics” Oxford 1969Mathematics” Oxford 1969

• A.D. Aleksndrov, A.N. Kolmogorov, M.A. A.D. Aleksndrov, A.N. Kolmogorov, M.A. Lavrentev editosLavrentev editos

• ““Mathematics, its content, methods, and Mathematics, its content, methods, and meaning” Vol 1,2,3 MIT press 1963meaning” Vol 1,2,3 MIT press 1963

• Felix Kaufmann “The Infinite in Mathematics” Felix Kaufmann “The Infinite in Mathematics” D. Reidel Publishing Company 1978D. Reidel Publishing Company 1978

• Edna E. Kramer “The Nature and Growth of Edna E. Kramer “The Nature and Growth of Modern Mathematics”Modern Mathematics”

• Princeton University Press 1981Princeton University Press 1981

• G. Polya “Mathematics and plausible G. Polya “Mathematics and plausible reasoning” Vol 1, 2 1954 Princeton reasoning” Vol 1, 2 1954 Princeton University press University press

• Maurice Kraitchik “Mathematique des Jeux” Maurice Kraitchik “Mathematique des Jeux” 1953 Gauthier-Villars 1953 Gauthier-Villars

• Heinrich Dorrie “100 Great Problems of Heinrich Dorrie “100 Great Problems of Elementary Mathematics”Elementary Mathematics”

• Dover 1965Dover 1965• Imre Lakatos “Proofs and Refutations” Imre Lakatos “Proofs and Refutations”

Cambridge University Press 1976Cambridge University Press 1976• ““Dtv-Atlas zur Mtahematik” Band 1,2,1974Dtv-Atlas zur Mtahematik” Band 1,2,1974• Struik D. J. A Concise History of Mathematics Struik D. J. A Concise History of Mathematics

Dover 1987Dover 1987•   • S. Bochner The Role of Mathematics in the S. Bochner The Role of Mathematics in the

Rise of Science Princeton 1981Rise of Science Princeton 1981• D.E. Littlewood “D.E. Littlewood “Le Passé-Partout Le Passé-Partout

Mathematique” Masson et c, Editeurs Paris Mathematique” Masson et c, Editeurs Paris 19641964

•   • A New Kind of Science by  by Stephen Wolfram (2002) (2002)• Wolfram Media. Wolfram Media. www.wolframscience.com

References 2References 2• 1)1)          G. H. Hardy A course in Pure G. H. Hardy A course in Pure

Mathematics Cambridge 10Mathematics Cambridge 10thth edition 1975 edition 1975• 2)2)          T. Jech Set theory Academic Press 1978 T. Jech Set theory Academic Press 1978• 3) Robert R. Stoll Sets, Logic and Axiomatic 3) Robert R. Stoll Sets, Logic and Axiomatic

Theories Freeman 1961Theories Freeman 1961•   • 4) M. Carvallo “Logique a trois valeurs 4) M. Carvallo “Logique a trois valeurs

loguique a seuil” Gauthier-Villars 1968loguique a seuil” Gauthier-Villars 1968• 5) D. Hilbert- W. Ackermann “Principles of 5) D. Hilbert- W. Ackermann “Principles of

Mathematical Logic”Mathematical Logic”• Chelsea publishing Company N.Y. 1950Chelsea publishing Company N.Y. 1950• 6) 6) H. A. Thurston “The number system” H. A. Thurston “The number system”

Dover 1956Dover 1956• 7) J.H. Conway “On Numbers and Games” 7) J.H. Conway “On Numbers and Games”

Academic Press 1976Academic Press 1976• 8) D. Hilbert “Grundlangen der Geometrie” 8) D. Hilbert “Grundlangen der Geometrie”

Taubner Studienbucher 1977Taubner Studienbucher 1977•   • 9) V. Boltianskii “Hilbert’s 39) V. Boltianskii “Hilbert’s 3rdrd problem” J. problem” J.

Wesley & Sons 1978Wesley & Sons 1978• 10)E. E. Moise “Elementary geometry from 10)E. E. Moise “Elementary geometry from

an advanced standpoint” Addison –Wesley an advanced standpoint” Addison –Wesley 19631963

• 11) Euclid The 13 books of the Elements 11) Euclid The 13 books of the Elements Dover 1956Dover 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•   

Comparisons of the classical analogue Comparisons of the classical analogue and the new digital mathematicsand the new digital mathematics

• Analogue Analogue MathematicsMathematics

• Physical Reality Irrelevant complexity. The Physical Reality Irrelevant complexity. The model of reality may be more complex model of reality may be more complex than reality itself.than reality itself.

• The infinite “feels” goodThe infinite “feels” good• Hard to understand mechanism of Hard to understand mechanism of

limits,approximations and not easy to limits,approximations and not easy to teachteach

• Pessimistic theorems (Goedel, paradoxes, Pessimistic theorems (Goedel, paradoxes, many axioms etc)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 Absence of the effects of multi-resolution

continuumcontinuum

• Digital MathematicsDigital Mathematics• Physical Reality Relevant complexityPhysical Reality Relevant complexity• No infinite only finite invisible resolution. No infinite only finite invisible resolution.

RealisticRealistic• No limits, or approximations,only pixels No limits, or approximations,only pixels

and points, easy to teachand points, easy to teach• Optimistic facts (realistic balance of Optimistic facts (realistic balance of

“having” resources and “being able to “having” resources and “being able to derive” in results. Less axioms)derive” in results. Less axioms)

• Easier proofs (a new method induction on Easier proofs (a new method induction on the pixels)the pixels)

• Famous conjectures are easier to prove or Famous conjectures are easier to prove or disprovedisprove

• Not simple algebra of closure of Not simple algebra of closure of operations. Internal-external entities.operations. Internal-external entities.

• A new enhanced reality of multi-resolution A new enhanced reality of multi-resolution continuumcontinuum

The effect in applied sciences of the The effect in applied sciences of the Digital mathematicsDigital mathematics

• MeteorologyMeteorology• PhysicsPhysics• BiologyBiology• EngineeringEngineering• EcologyEcology• SociologySociology• EconomicsEconomics

• More realisticMore realistic• Ontology closer to that as Ontology closer to that as

represented in an computer represented in an computer operating systemoperating system

• Faster to run computations in Faster to run computations in computerscomputers

• Easier cooperation among Easier cooperation among physical scientists-physical scientists-mathematicians-software mathematicians-software engineersengineers

• Easier learning of the digital Easier learning of the digital mathematicsmathematics

• Any additional complexity is a Any additional complexity is a reality relevant complexity not reality relevant complexity not reality irrelevant complexity.reality irrelevant complexity.

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

• (M/N)^2=2(M/N)^2=2• M,N with no M,N with no

common prime common prime divisordivisor

• M*M=2*N*NM*M=2*N*N• 2/M2/M• 2/N2/N• ContradictionContradiction

• This proof would no hold in This proof would no hold in digital real numbers: Two digital real numbers: Two reasonsreasons

• A) although (M/N)^2 may be A) although (M/N)^2 may be a digital real number, M^2 a digital real number, M^2 may be outside the digital may be outside the digital real numbersreal numbers

• B) There are two equalities B) There are two equalities in he digital real numbers in he digital real numbers that of single precision that of single precision quantities (visible points) quantities (visible points) that double precision that double precision (pixels)(pixels)