Title: THE NEW DIGITAL MATHEMATICS OF THE MILLENIUM
1THE 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
2The 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
3E. Schroendinger1'Nature and the Greeks' ,
'Science and Humanism'
4E. Schroendinger2'Nature and the Greeks' ,
'Science and Humanism'
5CAN 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.
6The 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.
7CAN 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.
8The 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.
9The 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
10THE 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
11NATURAL 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)
12LOGIC
- 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.
13The 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.
14SETS
- 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
15The 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.
16REAL 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
17The 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.
18A 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.
19GEOMETRY (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.
20GEOMETRY B (HISTOMETRY)
21GEOMETRY 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
22Hilberts 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.
23STATISTICS
- 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
24CALCULUS
- 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.
25CALCULUS B
26THE 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!
27THE 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
28TOPOLOGY
- 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.
29DIFFERENTIAL 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.
30Differential geometry
- Infinitesimal space of the surface space
31Are 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.
32STOCHASTIC 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.
33UNIVERSAL 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!
34References 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
35References 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
-
36Comparisons 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
37The 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.
38Reexamining 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)