Part MMCCCXV
Future in the informatics era - Chapter 4
Chapter 4: New perception of (scientific) Informatics
CHAPTER 4: NEW PERCEPTION
of (SCIENTIFIC) INFORMATICS
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CONTENTS
Deep thoughts
Why it is very important to view informatics properly.
Old and current views of CS
Why are old views no longer acceptable and new needed.
Information processing in nature - biological and quantum
Revolutions in science and technology.
Basics of a new view of informatics
Scientific informatics - goals, features, elements, methods, paradigms.
Grand challenges of scientific informatics
Case studies: I. why we need to explore different information precessing worlds?
Case studies: II. Informatics and physics
Case studies: III. Informatics and mathematics.
Case studies: IV. Informatics and other sciences
What can be learned from the history of mathematics?
What can be learned from the history of physics?
Other big challenges of scientific informatics
Scientific informatics as the queen and servant of sciences
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PROLOGUE
PROLOGUE
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HOW CAN INFORMATICS MEET ITS CHALLENGES
In the first three chapters we tried to demonstrate that
paradigms, laws, methods and tools of information
processing and transmission are driving force of evolution
that can be seen as coming (very) soon to a Singularity
era in which a merge of biological and non-biological
intelligence and creativity can have enormous, hard to
imagine consequences.
To meet fully and fast its historical challenge, a new
(visionary) perception of informatics, much broader and
deeper, seems to be needed.
This is the subject of this and few next chapters.
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DEEP THOUGHTS - I.
Imagination is more important than knowledge.
Albert Einstein
The modern version of the grand question ”Mind or
Matter?” is ”Information or physics?
Frank Wilczek, 2004
New scientific truth does not win because its
adversaries get convinced and open their eyes. It wins
because its adversaries die and a new generation of
scientists comes who this truth already knows.
Max Planck
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DEEP THOUGHTS - Ii.
You have to know a lot to ask good questions.
You have to know a lot to state new grand challenges
before science and technology.
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ROLE of IGNORANCE in SCIENCE
For development of science it is of large
importance to know:
What we know that we do not know;
What we do not know that we do not know;
What we think that we know but don’t
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BASIC QUESTIONS
Should recent developments in science and technology
be understood as giving rise of a need for a new, much
broader and deeper, perception of Informatics?
What are current Grand Challenges of ”new”
Informatics - from theory, technology and application
points of view?
Should recent developments in Informatics and other
academic disciplines be seen as showing an emergence
of a new, Informatics driven, general methodology of
large importance for all sciences, technologies, and all
major areas of functioning of society?
Should new Informatics be seen as both a new Queen
and new Servant of sciences, technologies and so on?
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WHY IT IS VERY IMPORTANT FOR SOCIETY to PERCEIVE
PROPERLY INFORMATICS?
Wrong (too narrow and shallow) perception of a
science/technology discipline usually leads to:
Wrong emphasis on main goals, methods and value
systems within the given area of science.
Wrong/weak support of that science discipline by
society and money granting agencies.
Wrong position of such a science discipline within
the whole academic community.
Wrong orientation and low impact of the education.
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BASIC OBSERVATIONS
Old-fashioned view of any discipline usually hinders
thoughts and intellectual progress in that discipline.
Time came to clear up the philosophical and practical
confusion of contemporary computer science.
Central to the new perception of Informatics presented
here are:
A series of discoveries concerning the key role and large potential of the information
precessing in nature that are not much consistent with prevailing perception of
Computer Science.
Discoveries showing enormous potential of new, informatics based methodology.
A series of discoveries indicating that such goals as a global computer, cell-, moleculeand
brain-inspired computers seem to be realistic goals.
A series of events demonstrating that one can hardly have very significant innovations
without employment of informatics thinking and tools.
Discoveries showing that all sciences converge to Informatics and all major
technologies converge to ICT.
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TWO MEGAGOALS and THEIR NEEDS
Expected exponential growth of performance and
miniaturization of ICT, already for quite a long time,
allows science, technology and medicine to see as perhaps
their two main megagoals (what used to be seen as
science fiction till recently):
To beat human intelligence. To create
superpowerful non-biological intelligence and its merge
with biological intelligence.
To beat human death - to beat as much as possible
natural death.
These megagoals also require much broader and deeper
perception of Informatics
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EW PERCEPTION of INFORMATICS - BASIC MOTTO
When you reach for stars you may not quite get
one, but you won’t come with a handful of mud
either.
Leo Burnett
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WHY a NEW PERCEPTION? - BASIC PHILOSOPHICAL
MOTIVATION
From time to time, in any area of knowledge, usually
after a big progress, accumulated knowledge suggests
that in order to make even bigger progress a new
perception of some fundamentals of the field is needed.
To make a revolution in a field one has usually to
re-think fundamentals of the discipline.
To make a revolution in an area of science/technology,
usually an approach based on a new, deeper and
broader philosophical standpoint is needed - in short, a
visionary approach is what is much needed.
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OBSERVATIONS and LECTURE from HISTORY
Observation from 17th century:
Men who fashioned modern science, mathematicians in
their general method and concrete investigations, were
primarily speculative thinkers and visionary scientists who
expected to apprehend broad, deep, but simple, clear and
immutable mathematical principles, either through
intuition or through observations and experiments.
By John Herman Randall: Science was born of a faith in
the mathematical interpretation of nature.
New vision: New science is being born of a faith in the
information processing interpretation of nature.
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OLD/CURRENT PERCEPTIONS of INFORMATICS
OLD/CURRENT PERCEPTIONS of
INFORMATICS - COMPUTER SCIENCE
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FIRST IMPORTANT VIEW
The first widely accepted and still dominating views of computer science were presented
in a very cleverly written, and much influential, paper of Newel, Simon and Perlis,
published in Science in 1967, that well captured the perception of the field at that time.
The basic ideas presented in their paper were:
”Whenever there are phenomena there can be a science
dealing with these phenomena. Phenomena breed sciences.
Since there are computers, there is computer science. The
phenomena surrounding computers are varied, complex
and rich.”
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SECOND EARLY and INFLUENTIAL VIEWS of COMPUTER
SCIENCE
A citation from an article on ACM computer science
educational curricula (1989) - still not seen as obsolete.
Old debate continue. Is computer science a
science? An engineering discipline? Or merely a
technology, an inventor and purveyor of
computing commodities? Is it lasting or will it
fade with a generation?
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BAD EDUCATIONAL IMPACTS of the name ”COMPUTER
SCIENCE”
It leads to a too narrow, shallow, and
technology-oriented interpretation of the field.
It is reminiscent of the 19th century attempt to create
””microscope science” and the 20th century attempt to
create ”poultry science” (and to have a ”science of
robbering” as a part of the history science).
The educational impact of the name ”computer
science” could perhaps be compared with those which
the name ”drill engineering” could have on the
education of dentists.
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OTHER VIEWS of COMPUTER SCIENCE - I.
Computer science as a science is the study of scientific problems related to the
design, behaviour and utilization of computers.
Computer folklore
Computer science is a science of the artificial that studies artifacts and contributes
to the design theory and managing complexity.
Simon, 1969
Computer science as a science is, in the narrow sense, the study of symbolic
representations and manipulations of these representations, and, in a broader sense,
the study of representations and manipulations of knowledge.
Hopcroft-Kennedy report, 1989
Computer science is a discipline that deals with representation, implementation,
processing and communication of information.
Ullman report
Computer science is the systematic study of computing systems and computations.
The body of knowledge resulting from this discipline contains theories for
understanding computing systems and methods, design methodologies, algorithms
and tools, methods for testing concepts, for analysis and verification, and knowledge
representation and implementation.
US-Congress Report, 1992
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OTHER VIEWS of COMPUTER SCIENCE - II.
Computer science is the study of computers and what
they can do.
An influential publication of the National Research
Council in USA, in 2004, namely Computer science:
reflections on and from the field
Computer science is the study of computers, including
their designs (architecture) and their uses for
computation, data processing and systems control. Its
core disciplines are: architecture, software, operating
systems, information systems and databases, theory
(computation methods and numerical analysis, data
structures and algorithms).
Encyclopedia Britannica (2006)
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ONE SPECIAL VIEW of COMPUTER SCIENCE
Computer science is the study of computers and what they can do - the inherent
power and limitations of abstract computers, the design and characteristics of real
computers, and the innumerable applications of computers to solving problems.
Computer hardware and software have been central to computer science since its
origins.
Computer scientists seek to understand how to represent and to reason about
processes and information. They create languages for representing these phenomena
and develop also methods to analyse and create such phenomena.
They create abstractions, including abstractions that are themselves used to
compose, manipulate and represent other abstractions.
They study the symbolic representation, implementation, manipulation and
communication of information.
They create, study, experiment on, and improve real-world computation and
communication systems as well as methods to operate such artifacts.
Reflections on the Field - reflections from the field, National Research Council, US, Mary
show, 2004
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OTHER INTERESTING VIEWS of COMPUTER SCIENCE III
Dijkstra: ”Computing science is - and will always be concerned
with the interplay between mechanized and
human symbol manipulation usually referred to as
”computing”, and ”programming”, respectively. It is
located in the direction of formal mathematics and applied
logic, but ultimately, far beyond where those are now,...”
Dijkstra: Computer science is as much about computers as
astronomy is about telescopes.
Perhaps the most significant recent step in the correct
direction has been that of P.J. Denning in the ACM
Communications (July 2007), saying that science behind
computing is a natural science.
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Edgar Dijkstra
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WHAT SUPPORTS CURRENT VIEWS of COMPUTER SCIENCE
1 General views of the underlying problems.
Computer science deals with immense differences in the scale of phenomena. From
systems manipulating atoms and molecules, as well as individual bits in computers, to
trillion operations per second and highly complex software systems and
problems/solutions describing languages.
To deal with these problems computer science has to create many levels of
abstractions. It has to create intellectual tools to conceive, design, control, verify,
program and reason about the most complicated of the human creations.
All that has to be done with the unprecedent precision because the underlying
hardware is a universal computer and therefore a chaotic system.
2 Enormous power of computation and communication systems.
3 Enormous difficulties with design complex, reliable and secure software.
4 Strong feelings that computer science is deeply intertwined with engineering
concerns and considerations. That it concentrates much more on the how, than on
what, which is more the focal point of physical sciences.
5 Computer science advances are often demonstrated by dramatic demonstrations
and it is often that the (ides and concepts tested in the) dramatic demos influence
the research agenda in computer science.
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WHAT is MATHEMATICS? - OFFICIAL VIEWS
Greek views and their developments.
Mathematics as a science centers around such concepts as number and magnitude.
Mathematics as a science centers around such concepts as number, magnitude and
form.
Mathematics as a science centres around such concepts as number, magnitude, form
and change.
Encyclopedia Britannica: Mathematics is the science of structure, order and
relations that has evolved from elementary practice of counting, measuring and
describing shapes of objects - it deals with logical reasoning and quantitative
calculations, and its development has involved on an increasing degree of deduction
and abstraction of its subject matter.
Mathematics is deductive study of numbers, geometry and various abstract
constructs (the Longman Encyclopedia, 1989).
Soviet encyclopedia: Mathematics is the science to study spacial forms and
quantitative relations of the real world.
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WHAT is MATHEMATICS? - PERSONAL VIEWS
Mathematics may be defined as the subject in which we never know what we are
talking about, nor whether what we are saying is true.
B. Russel (1917)
Mathematics in general is fundamentally the science of self evident things.
F. Klein - the leading German mathematicians at the end of 19th century.
When mathematicians refer to reality, they are not certain; and as far as they are
certain they do not refer to reality
A. Einstein (19??).
Mathematics is what mathematicians carry on and mathematicians carry on a highly
sophisticated intellectual activity which is not easy to define. Mathematical folklore
Mathematics is the science that draws necessary conclusions.
B. Peirce(1809-1880)
Pure mathematics is the class of all propositions of the form p implies q, where p, q
are propositions.
F. Klein
Mathematics in general is the science of self-evident things.
F. Klein (1849-1929)
Mathematics is nothing more than a game played according to certain simple rules
with meaningless marks on paper.
D. Hilbert
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WHAT IS PHYSICS? - OFFICIAL VIEWS
Physics is the science that deals with the structure of matter and the interactions
between the fundamental constituents of the observable universe.
Encyclopedia Britannica
Physics is the study of matter, energy and the relation between them. Physics today
maybe loosely divided into classical and modern physics. Classical physics includes
branches recognized and well developed before the beginning of 20th century:
mechanics, acoustics, optics, thermodynamics and electricity and magnetism.
Modern physics is concerned with behaviour of matter and energy under extreme
conditions (small scale or rapidly moving objects).
The Longman encyclopedia
Physics involves the study of matter and its motion through space and time, along
with related concepts such as energy and force.
Wikipedia
Physics is a general analysis of nature conducted in order to understand how the
universe behaves.
Wikipedia
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WHAT IS PHYSICS? - PERSONAL VIEWS
Physics is experience arranged in economical order.
Ernst Mach
One needs to realize that not only physics determines
what computers can do, but what computers can do, in
turn, will define the ultimate nature of the laws of
physics.
R. Landauer
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WHY IS a NEW VIEW of INFORMATICS SO MUCH NEEDED? - I.
Recent results in the rapidly developing field of natural
computing, have demonstrated that radically new and
powerful information processing resources (for example
quantum non-locality, superposition and parallelism),
processes and devices exist in nature and, actually, existed
abundantly far before the modern era of electronic
computers.
Moreover, the nature made these resources and tools to
play a very important role, especially for the functioning of
living beings. Information processing is as crucial
phenomenon of life as eating and breathing.
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WHY IS a NEW VIEW of INFORMATICS SO MUCH NEEDED? -
II.
The overall development of science and technology
therefore naturally requires to explore such interesting and
important phenomena and their power, potentials, laws,
limitations and so on. To do that is also of the crucial
importance for our understanding of the universe,
evolution, life, brain, human bodies, functioning of society
....
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INFORMATION PROCESSING in NATURE
Two big discoveries led to an understanding that
natural sciences are information processing driven.
The first one was the discovery, by Francis Crick and
James Watson, in 1953, of the twin-corkscrew structure
of DNA and how genetic information is encoded into
DNA - followed by a demonstration, due to Adleman,
how DNA computing could be performed and that it
has a potential for remarkable efficiency.
The second one was the discovery of quantum
teleportation and of the unconditionally secure quantum
generation of shared random classical key, by Charles
Bennett et al. in 1984-1993 - followed by results of
Shor, in 1994-1996, that quantum computing could be
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These discoveries changed views on Physics and
Biology that started to be seen and explored as being,
to a significant extent, information processing driven
sciences. From that it has been only a natural and
logical step to see other natural sciences in this way, as
being to an important degree information processing
driven - and a new revolution in the study of natural
and also other sciences has emerged.
Of importance has been also an observation that there
are primitive one cell organisms, like paramecium (from
50 to 350 µm in length), that do information
processing par excellence in order to find foods, to
avoid predators, to find a mate and to have sex without
having any synapses.
All that resulted to a view that information processingprof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 33/125
WHY IS a NEW VIEW of INFORMATICS SO MUCH NEEDED? -
II.
There are also more pragmatic reasons why a new and more attractive perception of
Informatics is much needed.
With old view of the discipline, and to that adjusted education, the field stopped to
be seen as much interesting and challenging for many very bright students.
At the same time, for the overall progress of science, technology, medicine and
almost all other areas of society, is of up most importance that the field develops as
well and as fast as possible - and for that very bright students are needed.
Not appropriate and even obsolete view of a discipline leads to the wrong overall
orientation of the research and to creation of not sufficiently challenging challenges.
Not attractive enough view of the field leads usually to a weak and wrongly oriented
support of the field by society and money providing agencies.
It is getting clear that in the future in all areas of human activities one can hardly
have important developments in which ideas and tools of Informatics would not play
one of the key roles - progress in most areas of society much depends on the
progress in broadly seen Informatics.
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CASE STUDY 1 – A PERSONAL VIEW
The percentage of college freshmen planning to major
in computer science dropped by 70% between 2000 and
2005.
In an economy in which computing has become central
to innovation in nearly every sector, this decline poses a
serious threat to American competitiveness.
Indeed, it would not be an exaggeration to say that
every significant technological innovation of the 21st
century will require to use some more or less
sophisticated Informatics technology.
Bill Gates (2007)
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CASE STUDY 2 - NEEDS OF TWO ”MEGAGOALS”
Expected exponential growth of performance and
miniaturization of ICT for quite a long time allows science,
technology and medicine to see as perhaps their two main
metagoals (what used to be seen as science fiction till
recently):
To beat human intelligence - to create superpowerful
non-biological intelligence and its merge with biological
intelligence.
To fight natural death.
All that requires much broader and deeper developments
of Informatics.
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WHY IT IS NOT EASY TO SEE PROPERLY NATURE OF
INFORMATICS?
Informatics is usually presented (wrongly) as a very new science/engineering discipline
being only in the beginnings of its developments
The whole field is under enormous pressure to develop fast:
Computing and communication systems of much better performance;
Computing and communication systems of much smaller size and energy
consumption;
Faster, verifiable, reliable, secure and compatible software;
More efficient methods to design reliable, secure and maintainable software.
It is therefore a big societal pressure to see scientific base of the field as serving short
term goals of technology.
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SOURCES of NEW PERCEPTION of INFORMATICS
The recent attempts to develop scientific basis for modern computing and
information processing and communication technology and their utilisation.
Long term efforts to derive knowledge from information, to formalize knowledge and
to formalize, as well as mechanize, reasoning.
The slowly developing understanding that various science disciplines that were
motivated originally by the design and utilisation of computing technology, liberated
already themselves from seeing their main goals as serving technologies and
developed, step by step, into areas of science with very broad impacts - as have done
logic, automata and complexity theories, for example.
Old and recent attempts to understand, match and beat performance of human
intelligence and bodies.
Recent discoveries that information processing is not only inherent, but often a
driving force, in physical, biological, chemical, economical and social processes.
The subsequent understanding that the discipline is the basis of a new and very
powerful methodology and has enormous impacts in all areas of human activities,
extending old methodologies to new heights.
A realisation that the discipline could and should establish super-challenges dealing
with which requires a very broad view of the field.
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WHAT IS INFORMATICS? - NEW PERCEPTION OF
INFORMATICS
NEW PERCEPTION of INFORMATICS
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SHORT SUMMARY
Let us start with a very short summary of new perception
of Informatics.
Informatics has four very closely related components:
scientific,
engineering,
methodological,
applied.
As discussed in the following, the new perception of informatics see informatics as having
similar scientific aims as physics has and similar methodological impacts as mathematics
has.
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SCIENTIFIC INFORMATICS - briefly
The main goal of Informatics is to study laws,
limitations, paradigms and phenomena of the
information worlds.
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INFORMATICS as a FUNDAMENTAL SCIENTIFIC DISCIPLINE I
As a scientific discipline of a very broad scope and deep
nature, Informatics has many goals. Its main task is to
discover, explore and exploit in depth, the laws,
limitations, paradigms, concepts, models, theories,
phenomena, structures and processes of both natural and
virtual information processing worlds.
To achieve its tasks, scientific Informatics concentrates on
developing new, information processing based,
understanding of the universe, evolution, nature, life (both
natural and artificial), brain and mind processes,
intelligence, creativity, information storing, processing and
transmission systems and tools, complexity, security, and
other basic phenomena of information processing worlds.
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INFORMATICS as a FUNDAMENTAL SCIENTIFIC DISCIPLINE II
The development and analysis of a variety of formal,
descriptional, specification, computation, programming,
interaction, communication, security and reasoning models
and modes, languages and systems; development and
analysis of (deterministic, randomized, genetic,
evolutionary, quantum, approximation, optimization,
on-line, parallel, concurrent, distributed, continuous, ...)
algorithms, heuristics, protocols, processes and games are
some of the main tools of scientific Informatics.
Data, information, knowledge, formal languages and
systems, logics, calculi, reasoning and proof systems,
processes, resources, models and modes of information and
knowledge processing, communication and interactions are
some of the key concepts behind.
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INFORMATICS as a FUNDAMENTAL SCIENTIFIC DISCIPLINE III
Computability, efficiency, complexity (computational,
communication, interaction, descriptional,...) feasibility,
universality, emergence, security, privacy, provability,
learnability, validation and (formal) correctness, as well as
secrecy, confidentiality, privacy and anonymity are some of
the key issues.
Information and knowledge digitalization, mining, learning,
analysis, sorting, comparing, searching, compression,
representation, visualisation and transmission are some of
the primitives of information processing.
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INFORMATICS as a FUNDAMENTAL SCIENTIFIC DISCIPLINE IV
In order to meet its goals, the scientific Informatics
develops close relations with other sciences and technology
fields - currently especially with Physics, Biology and
Chemistry, on one hand, and with electronics, optics,
nano- and bio-technologies on the other hand.
The basis of the relationship between Informatics and the
natural sciences rests first of all on the fact that
information carriers are always elements of the
physical, biological or chemical worlds, and
consequently information processing is governed and
constrained by their laws and limitations. Of large
importance is also that information processes are an
inherent part of the basic aspects of the nature and life.
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INFORMATICS as a FUNDAMENTAL SCIENTIFIC DISCIPLINE V
Informatics as a science includes also numerous theories
much needed for its development to depth and in
broadness. Some theories are very abstract, others quite
specific, and some theories are oriented on making better
use of the outcomes of the scientific Informatics to create
a scientific basis of engineering Informatics and
Informatics-driven methodology.
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INFORMATICS as a FUNDAMENTAL SCIENTIFIC DISCIPLINE VI
To meet its scientific goals, Informatics has to develop a
whole variety of subareas. Some are deeply abstract and
appear to be, at the first sight, quite remote from the
main tasks and interest of the current engineering or
applied Informatics, yet they serve to develop deep insights
into the key problems and powerful conceptual tools.
In order to derive a deeper understanding of our real
information processing world, Informatics has to create
and explore a whole variety of other (virtual) information
processing worlds. By broadening so much its scope in this
way, Informatics can can also develop very powerful tools
to deal with information processing related problems of our
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INFORMATICS as a FUNDAMENTAL SCIENTIFIC DISCIPLINE VII
Observations, measurements, learning, analysis,
modelling, simulation and visualisation, as well as
postulation, generalization and decompositions,
constitute some of the main methodologies to
explore information processing worlds.
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INFORMATICS as the leading SCIENCE
Informatics is, without doubts, currently the leading science and technology discipline
with enormous impacts on all other sciences, technologies, industry, economics, health
and environment care, liberal art and so on - guiding them and serving them.
There has ever been a ”queen of science” with very broad impacts, also on all education.
Some examples.
Philosophy at the very beginning
Medicine in Padua and at the same time theology in Paris in 17th century
Philology at the Renaissance
Mathematics after Galileo time due to its methodological impacts and Physics in the
20th century during its impacts on industrial revolution.
By Ernest Rutherford (1912), In Science there is only Physics: all the rest is stamps
collecting.
Observation: The leading science always had crucial impacts on ALL education.
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INFORMATICS and FUTURE of SCIENCE
Three big questions:
Will science ends?
Can science still expect such great discoveries as those
of Darwin, Einstein and Watson-Crick?
Will singularity be able to provide answers to both of
previous questions?
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INFORMATICS as the FIRST and LAST SCIENCE
Informatics can be seen as the first, the oldest, are of science and technology.
Are there parts of science that can be seen as having end? Anatomy of human body,
geography, chemistry,...
There are surely areas of science that have already practically died. For example,
astrology; descriptive geometry.
If an area of science becomes without big practical contributions, as well as become
incomprehensible, it may lost support of society.
Since many areas of science can be seen as converging to informatics, it is quite
natural to explore the question whether Informatics could be seen as the last,
all-sciences-embracing, science.
Though it is true that not only information-processing phenomena are important in
all sciences it is quite safe to say that other areas of science have diminishing returns.
By far the greatest barrier to future progress in pure science is its past success.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 51/125
INFORMATICS and PHILOSOPHY
There are two major modes of human knowledge: science and philosophy. (A view
of philosophers.)
Most philosophers are deeply depressed because they cannot produce anything
worthwhile.
Each philosopher, by pushing his ideas too far, by taking them too seriously, ends up
in an absurd, self-contradicting position.
Karl Popper
In After philosophy: end of transformation, published in 1987, fourteen prominent
philosophers considered the question Has philosophy future?. The consensus was
philosophical: Maybe, maybe not.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 52/125
GRAND CHALLENGES of SCIENTIFIC INFORMATICS - I.
To explore our world as a point in a space of potential
information processing worlds.
To explore laws and limitations of information
processing that governs universe, evolution and life.
To develop theoretical foundations for specification,
design, analysis, verification, security, simulation,
modeling and visualisation of huge information
processing systems
To understand mind, consciousness, learning,
intelligence and creativity.
To make foundations for science and engineering of the
science making activities.
To understand and manage all aspects of computation,
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 53/125
GRAND CHALLENGES of MATHEMATICS
To meet 21st century, The Clay Mathematical Institute in USA formulated in 2000 the
following seven challenges solution of each is honored by one million dollar award.
P = NP
Poincar´e conjecture (solved 2003 (Perelman) , solution
verified 2006)
Hodge conjecture
Rieman hypothesis
Yang-Mills existence and mass gap
Navier-Stokes existence and smoothness
Birch and Swinnerton-Dyer conjecture
Rieman hypothesis: All nontrivial zeros of the analytical continuation of the Rieman zeta
function ∞
n=1
1
ns have a real part 1/2.
Hodge conjecture: For projective algebraic varieties, Hodge cycles are rational
combinations of algebraic cycles.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 54/125
GRAND CHALLENGES/PROBLEMS of THEORETICAL PHYSICS
Problem of the origin of universe. Can we find out the initial conditions for Big
Bang.
Problem of very basic elements of universe. What is universe made of? What are
black matter and black energy? Can we create them experimentally?
Problem of the true nature of spacetime? What is their origin and structure? How
many dimension has space? Is space an emerging phenomenon that can be derived
from something? Are time and space an illusion?
Problem of the validation of quantum theory. Is quantum theory universally valid?
Also for large complex systems and for very small distances?
Problem of the unification of particles and forces. To find out whether all particles
and forces could be described by a single theory that could explain all of them as
following from a single fundamental entity.
Problem of the constants. To explain how nature chooses values of constants in the
standard model of nature. why they have values as they do?
Challenges are from the book of Lee Smolin: The trouble with physics. The rise of string
theory, the fall of science and what comes next: 2006 and from the recent Nobel laureate
David Gross talk (2012).
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 55/125
GRAND CHALLENGES of PHYSICS in MORE DETAILS
To find out how Big Bang could occur is a new big
challenge in physics. Till quite recently this problem
has been seen as unscientific.
Concerning the structure of universe, current
understanding is that 5% is ordinary matter, 25% black
matter and 70% black energy.
David Gross considered as big challenges of physics also
questions: Can evolution be quantized and predicted?
Can we have theoretical biology? What is
consciousness? How life has been created?
In connection with such a broadening of the scope of
physics it is natural that he defines Physics is what
physicists do.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 56/125
SOME of the GRAND CHALLENGES of INFORMATICS in MORE
DETAILS - I.
To work out ways how to see our world as ”a point” in
a huge space of other related (abstract) [virtual] words
and to come up with new ways to understand and deal
with the problems of our world through the
investigation of and in this large space of worlds.
To discover and understand laws, limitations,
paradigms, processes and phenomena of the information
processing that govern the evolution of the universe
and life and play the key role in the development of
nature in general and especially of living beings.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 57/125
SOME of the GRAND CHALLENGES of INFORMATICS in MORE
DETAILS - II.
To create models, theories and (especially formal) tools
to help engineering Informatics to specify, design,
verify, analyze and maintain (especially enormously
large), concurrent and distributed (software) systems as
well as embedded systems.
To create concepts models and theories for an
understanding of intelligence and basics of intelligent
behaviour, as well as consciousness (its origin, scope
and functioning), and all that to such an extent that
”highly intelligent information processing systems,
devices and robots” become a reality.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 58/125
SOME of the GRAND CHALLENGES of INFORMATICS in MORE
DETAILS - III.
To develop concepts, theories, models, methodologies
and tools that would lead to the development of very
powerful methods to search, mine, retrieve and derive
information and knowledge as well as to make
hypothesis and formulate theories in a way leading to
an engineering of the major science making activities.
To explore in depth inherent complexity of information
processing, communication and interaction phenomena
and to find ways for reasonable solutions of
computationally very intensive, or even unfeasible,
problems, in the current view of the issues, and of the
problems requiring to process enormously huge data
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 59/125
CHALLENGE 1 - TO EXPLORE THE SPACE OF POTENTIAL
INFORMATION WORLDS I.
History of science suggests that a deeper understanding of a phenomenon or of an object
comes when a way is found to see properly that phenomenon or object as only a particular
one in a very large, close and nicely smooth space of (quite) similar phenomena.
Three of the recently explored ways to see our physical world in a broader class of
potential physical worlds are;
To explore worlds with more powerful correlations than those of quantum
correlations, but that still do not contradict general relativity.
To explore worlds where various natural generalizations of quantum entanglement
are valid.
To explore worlds with the potential to beat Church-Turing barrier;
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 60/125
Case study I: HOW MUCH STUDIES of ”STRANGE” PHYSICAL
WORLDS HELP PHYSICS?
or, more technically,
HOW MANY GEOMETRIES
OF THE PHYSICAL WORLDS ARE USEFUL?
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 61/125
DEVELOPMENTS in GEOMETRY - I.
The first important step was the development of
Euclidean geometry that was actually a useful
abstraction of our basic vision of our 3-dimensional
physical world.
Euclidean geometry can be seen as the most basic and
the first perfect physics theory.
Arithmetization of Euclidean geometry by Descarets
(1596-1650) open the door to the study of much larger
class of geometrical objects of the two and
three-dimensional worlds and also paved the way for
study of more (even infinite) dimensional worlds.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 62/125
DEVELOPMENTS in GEOMETRY - II.
The development of non-Euclidean geometries
(1826-29) by Lobachevsky (and others showed that
apparently very strange virtual worlds can represent
features of very real world..
Another big revolution in geometry started Rieman
(1826-1866) by considering geometries as n-dimensional
manifolds with a general quadratic differential form as
its metric. This allowed to generalize much the concept
of curvature, to show that there are different kinds of
space of three-dimensions, what could soon be used by
Einstein in his theory of relativity.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 63/125
DEVELOPMENTS in GEOMETRY - III.
Another big step in the area of virtualization and
abstraction in geometry was initiated by F. Klein
(1849-1929) who started to see the task of geometry as
the study of geometrical objects as invariants of some
virtual worlds (manifolds) with respect to special
groups of transformations. The field developed as
driven by the principles of the mathematical art and by
an understanding that peculiar geometries may reflect
features of some physical worlds.
Klein’s approach allowed to see dimensionality of a
virtual world as depending of chosen class of elementary
objects. As a consequence our physical world can be
seen as three, four and even five dimensional.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 64/125
SOME PROBLEMS SCIENTIFIC INFORMATICS DEALS WITH - I.
What are the laws, limits, elements, structures, processes and methods of
information world and what are their properties?
What is feasible?
What is feasibly constructable, solvable, computable, decidable?
What has (and is) a feasible evidence, a proof?
What is feasibly learnable, expressible, communicable?
To study space of information processing problems, its structure, hierarchies,
reductions, equivalences, hardest problems,...
To study the space of information processing resources. What are their properties
and power? Which relations and trade-offs are among them? How powerful are such
resources as time, space, randomness, interactions, parallelism, concurrency,
nondeterminism, alteration , reversibility, entanglement, non-local correlations?
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 65/125
SOME PROBLEMS SCIENTIFIC INFORMATICS DEALS WITH - II.
To study space of information processing machines,
their power, mutual simulations. Models of machines
that fit existing technologies, physical theories,....
To study the space of descriptional systems: logics,
languages, rewriting systems, automata,....
To study inherent complexity of objects, processes
(computational, communication, descriptional,...
To study information, knowledge, reasoning, intellectual
properties,...
To develop learning methods and systems and of their
properties (uncontrolled learning, deep learning,...)
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 66/125
SOME PROBLEMS SCIENTIFIC INFORMATICS DEALS WITH -
III.
To develop methods of design and analysis of
deterministic, randomized, quantum and genetic
algorithms.
To develop theory and tools for management, analysis
and processing huge (extremely large) data sets and
streams.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 67/125
COMMENTS I - INFORMATICS versus PHYSICS
The main goal of Physics can be seen as to study laws,
limitations and phenomena of the physical worlds.
The main goal of Informatics can be seen as to study laws,
limitations and phenomena of the information worlds.
Physics and Informatics can therefore be seen as
representing two windows through which we try to
perceive and understand the world around us.
In a similar way we can see life-sciences and Informatics as
providing two windows and tools with which we try to
understand, imitate and outperform the biological world
and its highlights - human brain, mind, consciousness, and
cognitive capabilities.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 68/125
WHEELER’s VIEW I
I think of my lifetime in physics as divided into
three periods
In the first period ...I was in the grip of the idea
that
EVERYTHING IS PARTICLE
I call my second period
EVERYTHING IS FIELDS
Now I am in the grip of a new vision, that
EVERYTHING IS INFORMATION
John Archibald Wheeler
Gems, Black Holes and Quantum Foam
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WHEELER’s VIEW II
”I have been led to think of analogies between
the way a computer works and the way the
universe works. The computer is built on yes-no
logic. So, perhaps is the universe ... The universe
and all that it contains (”it”) may arose from the
myriad yes-no choices of measurements (the
”bits”);
By Wheeler, information has some connection to
existence, a view he advertised with the slogan
”It from bit” - or, in other words, that
”Everything is information”.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 70/125
WHEELER’s ”IT FROM BIT”
IT FROM BIT symbolizes the idea that every item
of the physical world has at the bottom - at the
very bottom, in most instances - an immaterial
source and explanation.
Namely, that which we call reality arises from
posing of yes-no questions, and registering of
equipment-invoked responses.
In short, that things physical are information
theoretic in origin.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 71/125
WHO WAS JOHN ARCHIBALD WHEELER (1912-2008)?
A man who named black holes.
A man who helped to explain nuclear fusion.
A leading expert in quantum physics, relativity theory.
Wheeler is considered among physicists as one of the
greatest teachers of the last century.
Richard Feynman was one of his students.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 72/125
OTHER VIEWS on NATURE and SCIENCE
Democritos; Nothing exists except atoms and empty
space; everything else is opinion.
Pythagoreans: Everything is number
Philolaus: All things that can be known have number;
for it is not possible that without number anything can
be either conceived or known.
Plato created a beautiful unified theory of matter
according to which everything is constructed of the
ideal right triangles.
Cezanne: Everything in nature can be modeled from a
sphere, a cone and a cylinder.
Galileo: The book of universe was written in
mathematical language and its alphabet consists of
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 73/125
OTHER VIEWS
I think that modern physics has definitely decided in
favour of Plato. In fact the smallest units of matter are
not physical objects in the ordinary sense: they are
forms, ideas which can be expressed unambiguously
only in mathematical language.
W, Heisenberg
Assumption that our world is a finite system does not
just hint that information aspects of physics are
important, it insists that information aspects are all
there is to physics at the most microscopic level.
E. Fredkin
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 74/125
RELATIONS of INFORMATICS to OTHER SCIENCES
There are several basic reasons why relations between Informatics and other sciences are
getting stronger and stronger, deeper and deeper.
It starts to be understood that other sciences are in their fundamentals much
information processing driven and they converge to the state that studying their
information processing aspects is becoming one of the main way to get a deeper
insight into them. (Both physics and biology are perhaps main ones of that type.)
For example, it starts to be understood that information processing is for life as
important as eating and breathing.
Because of strong impacts of informatics, especially of its methodology, on all
sciences, one can talk about their convergence to and increasingly strong relation
with Informatics.
Informatics has been designing systems that are so huge and ever evolving in a not
much coordinated distributive way, as internet, silicon compilers and so on, that they
need to be studied by observational and experimental methods of natural and social
sciences. As a consequence, that brings Informatics closer to these disciplines. All
that also creates a space for influencing Informatics development on the basis of
laws, limitations and findings of natural and social sciences.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 75/125
WHAT CAN BE LEARNED from HISTORY of OTHER SCIENCES?
WHAT CAN BE LEARNED
from the HISTORY of OTHER SCIENCES?
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 76/125
A GENERAL OBSERVATION
A success of a science depends much on how large and at
the same time intellectually simple, smooth and dense
space it is able to create and investigate in comparison to
that observable by human senses and common sense.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 77/125
LESSONS LEARNED from the HISTORY Of PHYSICS
Physics was first named as natural philosophy.
For a long time physics was seen as dealing with stuff
we can handle between fingers.
Quantum physics and relativity theory opened a new
dimension to physics with big impacts on the stuff we
can handle between fingers.
Discoveries in cosmology shifted scientific interest to
the study of black holes, Planck scale space and time,
black matter and energy,....
Moral: Physics got a new dimension, regarding knowledge
and usefulness, by extending VERY MUCH its scope and
research space.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 78/125
LESSONS LEARNED FROM HISTORY of PHYSICS II
In the 17th century physics solved the long standing
problem of the understanding of the direct motion.
In the process of doing this physics was able to replace
a mysterious concept of impetus by the modern
concept of inertia and to come to the modern theory of
motion and by doing this to drive out spirits from the
scientific thoughts and open the door to see the
universe as running as a piece of clockwork.
Physics did that by geometrizing the problem. That is
by solving the problem in a virtual (geometric)
frictionless, directionless, gravitationless and empty
space.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 79/125
LESSONS LEARNED FROM HISTORY of PHYSICS III
Physics as the science has had so far practical applications
exceeding that of any science before.
In spite of that, the need to deal with immediate problems
of the practice can hardly be seen as the main, or as the
only, driving force of its development.
It has been rather curiosity and the need to extend our
knowledge and understanding of the physical world that
was behind its main discoveries and contributions.
Physics has made big progress even by exploring properties
of particles that are not sure to exist (in some reasonable
sense) or to develop theories we see no way to verify
experimentally.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 80/125
OTHER FUNDAMENTAL PROBLEMS of PHYSICS
What is space? Has space a beginning, development, end, borders? What are and
from where came space development driving forces?
What is time? Has time a beginning, development, end? What are and from where
come time development driving forces?
What is (our) universe? What is in it? What we can say about its beginning,
development, end, laws and limitations?
Is something more fundamental than time and space? Are there some other
universes?
Why is our universe as it is? What can we know about our universe?
Why we are in our universe?
Does there exist a universal theory of our universe, of its structure, development,
laws and limitations? Of all universes?
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 81/125
WHAT CAN WE LEARN from HISTORY of MATHEMATICS
WHAT CAN BE LEARNED
from the HISTORY of MATHEMATICS
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 82/125
APPROACHES to the PERCEPTION of MATHEMATICS
There are several positions from which one can characterize mathematics.
1 By specifying subjects of mathematics
2 By specifying the essence of mathematics
3 By specifying methods of mathematics
4 By specifying goals of mathematics
5 Folklore views of mathematics
6 Peculiar views of mathematics
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 83/125
HOW TO PERCEIVE MATHEMATICS? I. Greek times
developments
Mathematics as a science centers around such
concepts as number and magnitude.
Mathematics as a science centers around such
concepts as number, magnitude and form.
Mathematics as a science centres around such
concepts as number, magnitude, form and
change.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 84/125
HOW TO PERCEIVE MATHEMATICS? II
Greek period: Mathematics used to be seen as the science centred around such
concepts as quantity, structure, form and change.
Encyclopedia Britannica: Mathematics is the science of structure, order and
relations that has evolved from elementary practice of counting, measuring and
describing shapes of objects - it deals with logical reasoning and quantitative
calculations, and its development has involved on an increasing degree of deduction
and abstraction of its subject matter.
Soviet encyclopedia: Mathematics is the science to study spacial forms and
quantitative relations of the real world
Folklore: Mathematics is what mathematicians carry on and mathematicians carry
on a highly sophisticated intellectual activity which is not easy to define
Academia Europaea, 2010: Mathematics is centred around such concepts as
quantity, structure, form and change.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 85/125
MORE PERSONAL VIEWS of MATHEMATICS
As far as laws of mathematics refer to reality,
they are not certain; and as far as they are
certain they do not refer to reality (Einstein).
Mathematics may be defined as the subject in
which we never know what we are talking
about, nor whether what we are saying is true
(B. Russel - 1917).
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 86/125
DEFINITION of MATHEMATICS by its SUBJECTS
Mathematics is science of quantities. (Otto Encyclopedia - 1930 - Czech).
Mathematics is the body of knowledge centered on such concepts as quantity,
structure, space and change, and also an academic discipline that studies them
(Wikipedia, 2008).
Major disciplines within mathematics fist arose out of the need to do calculations in
commerce, to understand relationships between numbers, to measure land and to
predict astronomical events.
Mathematics is deductive study of numbers, geometry and various abstract
constructs (the Longman Encyclopedia, 1989).
Die moderne Mathematik sieht ihre Aaufgabe v.a. in der Untersuchung sogenante
Strukturen, die durch die in einervorgegebenen Menge beliebiger Objekte definierten
Relationen und Verkn¨upfungen bestimt sind. Nach traditioneller Einteilung gliedert
sich die Mathematik in die Arithmetik, die Geometrie, die Algebra und die Analysis.
(Meyers Grosses Taschen-Lexicon).
Science qui ´etudie par le moyendu raisonnement d´eductif les propri´et´es d’ˆetres
abstraits (nombres, figures ge´eom´etriques, fonctions,spaces, etc.) ainsi que les
relations qui s’´etablissent entre eux (Le petit Larousse).
Mathematics is science to study spacial forms and quantitative relations of the real
world. (Soviet encyclopedia).
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 87/125
MATHEMATICS by its ESSENCE
Whereas mathematics began merely as a calculation tool for computation and
tabulation of quantities, it has blossomed into an extremely rich and diverse set of
tools, terminologies, and approaches which range from the purely abstract to the
utilitarian.
Mathematics is a broad-ranging field of study in which the properties and
interactions of idealized objects are examined (CRS Concise (3942 pages)
Encyclopedia of Mathematics, 2003)
This, therefore, is the mathematics: she remains you of the invisible form of soul;
she gives life to her own discoveries; she awakens the mind and purifies the intellect;
she brings light to our intrinsic ideas; she abolishes oblivion and ignorance which are
our by birth. Proclus (410-485 A.D.)
Mathematics may be defined as the subject in which we never know what we are
talking about, nor whether what we are saying is true. B. Russel (1917)
Mathematics in general is fundamentally the science of self evident things.
F. Klein - the leading German mathematicians at the end of 19th century.
Mathematics is often seen as the science of numbers, quantities and measurements.
These things are indeed an important part of materials to which mathematics has
been applied. But they are no more mathematics than are the paints in an artist’s
tubes the masterpiece he is painting.
As far as laws of mathematics refer to reality, they are not certain; and as far as they
are certain they do not refer to reality A. Einstein (19??).
Mathematics is Queen of the Sciences and Arithmetic the Queen of mathematics.prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 88/125
FOLKLORE VIEWS of MATHEMATICS
Mathematics is what mathematicians carry on and mathematicians carry on a highly
sophisticated intellectual activity which is not easy to define
Creative mathematicians are inspired primarily by the art of mathematics, rather
than by any prospect of immediate utility in sciences and technologies - they are
guided only by their feeling for symmetry, simplicity, generality and an indefinable
sense of fitness of things and mathematical beauty Bell 1951.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 89/125
PECULIAR DEFINITIONS of MATHEMATICS
Mathematics is the science that draws necessary conclusions. B. Peirce(1809-1880)
Pure mathematics is the class of all propositions of the form p implies q, where p, q
are propositions.
Mathematics in general is the science of self-evident things F. Klein (1849-1929)
These and scores of other definition of mathematics illustrate the hopelessness of
trying to paint brilliant sunrise in one color. Bell, 1953
The attempt to compress the free spirit of modern mathematics into an inch in a
dictionary is as futile as trying to squeeze an ever-expanding thundercloud into a
pint bottle.
Mathematics is nothing more than a game played according to certain simple rules
with meaningless marks on paper D. Hilbert
In the pure mathematics we contemplate absolute truths, which existed in the Divine
Mind before the morning stars sang together, and which will continue to exist there,
when the last of their radiant host shall have fallen from heaven.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 90/125
Is the end of mathematics visible
It appears to me that the mine of mathematics is
already very deep and unless one discovers new veins it
will be necessary, sooner or later to abandon it.
Lagrange, 1781
I dare say that in less than century we shall not have
three great mathematicians left in Europe,..
Diderot, 1754
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 91/125
WHAT IS MATHEMATICS ABOUT?
Up to 1550 the concepts of mathematics were
immediate idealizations of or abstractions from
experience.
After 1550 mathematics started to be dominated by
intellectual constructs derived from the recesses of
human mind. (One can say for the genesis of its ideas
mathematics gradually turned from the sensory to
intellectual faculties.)
As a result mathematicians started to realise that
mathematics is not a body of truth about nature.
Around 1900 mathematics broke away not only from
philosophy, but also from nature and science.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 92/125
LESSONS LEARNED from the DEVELOPMENT Of
MATHEMATICS
Mathematics has made an enormous progress especially by
extremely enlarging and completing its research space.
M moved from dealing with integers to rationals,
irrationals, complex numbers,..
M moved to the study of arbitrarily large cardinality
M moved from the study of 2D and 3D spaces to
spaces of arbitrary dimensions and metrics.
M moved from the study of 2d and 3D figures to the
study of curves and bodies in any space.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 93/125
LESSONS LEARNED from the DEVELOPMENT Of
MATHEMATICS - I.
It took centuries for M to move from considering a
number as a fundamental object to seeing continuum
as the fundamental object and set theory and logic as
its building blocks.
In all such cases extensions of the scope brought also
new techniques to solve old problems, and that was
actually the main underlying goal, inspired mainly by a
centuries old experience with the power of
generalisation and abstraction. Similarly we could
analyse developments in other sciences.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 94/125
LESSON LEARNED FROM GOLDEN ERA of GREEK
MATHEMATICS
LESSON LEARNED FROM
GOLDEN ERA of GREEK MATHEMATICS
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 95/125
GREEK MATHEMATICS - CLASSICAL PERIOD (Pythagoreans,
600-300 B.C.)
Greek mathematics was based on and helped to develop a new doctrine of nature namely
that nature is orderly and develops according to a plan. Old doctrine, but in
Greek society actually dominating at that time, was that gods manipulate nature
and men according their whims.
Its protagonists were Thales and Pythagorean and it was highlighted by works of
Eudoxus, Euclid, Plato and Aristotle.
Greeks created, for the first time, mathematics as an organized, independent and
reasoned discipline.
Greeks made mathematics abstract - to see mathematical entities, numbers and
geometrical objects as abstractions, sharply distinguished from physical objects.
Greek made mathematics deductive, deriving truth in theorems by deduction from
axioms.
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CONTINUATION
Greeks came with the idea to prove existence by construction.
Their mathematics was mainly (well founded) geometry, (actually motivated by
astronomy).
Main goal of Mathematics was seen as to understand functioning of universe - they
believed that mathematics is the key to comprehension of universe, for mathematical
laws are the essence of its design.
Greeks made mathematics to be a liberal art closely related (and a preparation) to
philosophy.
For Greeks arithmetic, geometry and astronomy were considered as the art of the
mind and soul. It is believed that it was the aesthetic appeal of the subject that
caused Greek mathematicians to carry the exploration of particular topics beyond
their use in the understanding of the physical wo r1d.
Greeks made enormous contributions to the philosophy of science.
Their mathematics was much inspired by the fact that phenomena that are much
diverse from qualitative point of view exhibit identical mathematical properties.
Their position was based on a belief that mind is capable to recognize truth,
observation of physical world is not needed. As a consequences their outcomes were
a combination of ingenious ideas, bold speculations and shrewd guesses.
Greek mathematicians mixed deep and serious thoughts with what we could consider
as fanciful, useless, and unscientific doctrines. sic for the soul.
Greek mathematicians ”reduced” astronomy and music to numbers and therefore
astronomy and music was considered to be a part of mathematics.prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 97/125
PLATO’s IDEAS
He was the most influential propagator of the doctrine that the reality and
intelligibility of the physical world can be comprehended only through mathematics.
He was convinced that the world was mathematically designed, for God eternally
geometrizes
He believed that the world perceived by the senses is confused and deceptive and in
any case imperfect and impermanent.
He believed that physical knowledge is unimportant, because material objects
change and decay; therefore direct study of nature and purely physical investigations
are worthless.
Plato believed that physical world is but an imperfect copy of the ideal world, the
one mathematicians and philosophers should study.
He believed that mathematical laws, eternal and unchanging, are the essence of
reality.
Plato encouraged devotion to a theoretical astronomy, whose problems please the
mind not eye, and whose objects are apprehended by the mind, not visually.
Plato not only tried to understand nature through mathematics, he actually tried to
substitute mathematics for nature itself.
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CONTINUATION
Plato believed that there is an objective, universally valid reality consisting of the
forms, or ideas.
For Plato mathematics was also a methodology that helped to train the mind to see
eternal ideas.
For Plato nature was mathematically designed and this design was harmonious,
aesthetically pleasing, and the inner truth.
Platonic dialogue Philebus first gave expression to the thought that each science is a
science only so far it contains mathematics.
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Aristotle’s IDEAS
He believed in material things as the primary substance and source of reality.
He believed that science must study the physical world to obtain truth.
He believed that genuine knowledge is obtained from sense experience by intuition
and abstraction.
He believed that matter alone is not significant. It becomes significant when it is
organized into various forms. Form and the changes in matter that give rise to new
forms are the interesting features of reality and a real concern of science.
He viewed matter as consisting of earth, water, air and fire.
He believed that science has to study causes of changes.
He believed that mathematics alone can never provide an adequate definition of
substance.
He distinguished sharply between physics and mathematics and assigned a minor
role to mathematics.
He gave 6 non-mathematical arguments why space is finite.
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LIMITATIONS OF CLASSICAL GREEK MATHEMATICS
They reduced mathematics to geometry dealing with simple curves, areas and bodies
only.
By insisting on a unity, completeness and simplicity, and by separating speculative
thoughts from utility, they narrowed people’s vision and closed their minds to new
thoughts and methods.
Their insistence on exact concepts and proofs was also a defect so far as creative
mathematics is concerned.
They were not able to accept irrational numbers in arithmetic.
Their concept of proof was too restrictive concerning creative mathematics, and so
was their concept of constructability.
They were not able to accept infinity. Neither infinity of large not of small objects
and not infinite processes.
They could not accept continuity because of their emphasis on atomism.
They believed that mathematical facts are not created by men, that they exists and
can only be discovered.
They, including Aristotle, rejected practical utility as the purpose of inquiry and
believed that philosophical investigations carry their own reward.
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CONTINUATION
All that happened in spite of considerable Greek achievement in engineering. For
example, Archimedes designed various war-making machines - however, he thought
so little of this as a subject of ”philosophical discourse” that he included not a word
about mechanics or engineering in the vast body of theoretical writing he left behind.
Exception were Hippocratic physicians (5th century B.C.). Greek medicine of that
time had inquire into a wide range of human anatomy and physiology to develop the
medical and surgical means of dealing with wounds, fractures and diseases.
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GREEK MATHEMATICS - ALEXANDRIAN PERIOD (300 B.C. -
600 A.C.)
Mathematics produced was quite different in character from that of the classical
period, there is an absence of deductive structure.
Main protagonists were Archimedes, Diophantine and Ptolemy.
From time of Nichomachus, who wrote a textbook on arithmetic used for 1000
years, arithmetic became the main subject of study.
Geometers of that period concentrated on quantitative geometry, mainly on the
computation of the lengths, areas and volumes.
Philosophical differences in two periods of Greek mathematics - an example: Euclid
was happy to prove that area of circle is proportional to the square of diameter,
Archimedes concentrated on approximation of the constant π.
Mathematics made many contributions to mechanics, optics, geodesy, canonic
(science of musical harmony), ...
Archimedes used ideas from mechanics to derive mathematical theorems.
Big achievements - spherical trigonometry
Main achievement - first truly quantitative astronomical theory (using trigonometry).
Mathematicians of that period severe their relations with philosophy and allied
themselves with engineering.
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WHY WAS GREAT GREEK MATHEMATICS IGNORED FOR 2000
One of the most puzzling things in history of science is
why ingenious Greek mathematics of its classical period
was then practically ignored for about 2000 years.
It was actually already ignored in the Alexandrian
period (300 B.C. - 600 A.C.)
One reason is nicely put together by Cicero: The Greeks
held the geometers in the highest honour; accordingly,
nothing made more brilliant progress among them than
mathematics. But we have established as the limit of
this art its usefulness in measuring and counting.
Other reasons: it ignored computational needs of
society; it was based on a wrong view of importance of
observations and so it could hardly help other sciences;
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CONTINUATION
Christianity decreased interest in physical world,
preparation of the soul for after-life in the heaven was
the main concern.
Christianity brought a new belief concerning ways one
seek for truth.
Theology was seen as embracing all knowledge.
New revival of the Classical Greek period appeared
after a new doctrine was developed that saw God as
the one creating mathematical nature and as seeing
search for mathematical laws of nature as religious
quest. A discovery of a mathematical law was seen as a
further discovery of the greatness of the God - and
therefore God was to be praised after each discovery of
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SUBAREAS OF GREEK and MEDIEVAL AGE MATHEMATICS
Classical period ; arithmetic, geometry, astronomy, music
Alexandrian period : arithmetic (number theory),
geometry, mechanics, astronomy, optics,
geodesy, canonic and logistic (applied
arithmetic).
Medieval age: Astrology was considered as a subarea of
mathematics
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ASTROLOGY as a COMPANION of MATHEMATICS
Astrology received new dimension in Alexandrian Greek period.
Ptolemy wrote five books on astrology.
Astrology nourished astronomy as much as alchemy nourished chemistry.
Alexandrian astrology was personal. It predicted future and fate of individual on the
basis of the computed position of the sun, moon and five planets in the zodiac at
the moment of birth.
Errors in astrological predictions used to be ascribed to errors in astronomic
calculations, not in unreliability of astrological doctrines.
The basic astrological doctrine was that heavenly bodies influence and controlled
human bodies and fortunes.
Galileo lectured to medical students on astronomy, but for the sake of astrology.
Physicians of medieval time had to be more astrologers and mathematicians than
experts in human bodies.
Newton was also much involved in astrology.
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SOME OTHER COMMENTS on MEDIEVAL SCIENCE and
MATHEMATICS
Medieval science was for long time concerned with understanding of the role of God
in creation and running the world
For example, Galileo expresses a widely spread view when he interpreted the
difference between human and divine understanding: we proceed in step-by-step
discussion from inference to inference, whereas He conceives through mere intuition.
Some of important concepts of medieval science disappeared from modern scientific
considerations. For example: ether, substance, Some other concepts disappeared as
crucial. For example: left-to-right,
In medieval time mathematics was much intertwined with physics.
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SECOND GOLDEN AGE of MATHEMATICS
This was during the scientific revolution in the 16th and
17th century. At that time:
A very important starting steps was development of the
analytical geometry by Descartes - as one of the
greatest mathematical achievements of all times.
As one of the outcomes metaphysics of space could be
ignored. It is now never necessary to require what
”space” is or whether it ”exists”. We need only to
know how to manipulate equations whose variables
denote outcomes of measurements or observations, As
a consequence metaphysics of space is a luxury only few
creative physicists can safely afford to bother about.
The introduction of postulation/axiomatic method for
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CONTINUATION
Mathematical concepts started to be more and more
purely intellectual constructs that were created by ”the
recesses” of human minds and they were no longer only,
or mainly, an idealization of some immediate experience
(irrational and complex numbers, integral,...).
That was then accompanied by a complete change of
the relation between geometry and algebra and the last
one, and as its branch at first and very powerful child
later, analysis, started to play a dominating role.
Analysis then dominated for centuries and strated to
play more and more the role of a servant of physics, to
produce (computational concepts and tools) to deal
with more and more involved problems of physics.
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THIRD GOLDEN AGE of MATHEMATICS
The third most prolific period in the history of mathematics started by 1830 and the
end is not yet in side - this period can be seen as the third golden age of
mathematics.
Discovery of Non-Euclidean geometry around 830 by Lebesgue’s, that gave rise to
enormous development of various geometries with such strong impacts on whole
mathematics and physics was one of the starting points.
Mathematics during that period can be characterized as follows
Ever greater generality and abstraction and ever sharper self-criticism.
Ever stronger emphases on probability and approximation methods needed to solve
huge and complex problems.
Ever growing use of randomness and probability tools.
A switch of emphases from solving particular problems to the development theories as
tools to attack a class of problems.
Another key tool developed by mathematics is the theory of differential equations
because 99% of all important laws of physical sciences are in terms of differential
equations or of their systems.
In spite of the fact that during first 100 years of this period enormous successes have
been obtained in mathematics, mathematics was not prepared enough to deal with a
variety of complex problems war needed to deal with.
For example, exact solution of various such problems were out of question and a
variety of approximate methods have to be developed that in turn influenced also
pure mathematics.
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 111/125
In some departments of mathematics more was
discovered in less than decade of war than it would be
discovered in 50 years of piece.
In spite of that around 1955 it was guessed that if
mathematical physics be annexed as a province of
mathematics, a detailed, professional mastery of the
whole domain of modern mathematics , would require
life long toil of 20 or more richly gifted men.
Successes of mathematics in WW2 so much speed
development of mathematics that today an order or
two more mathematicians would be needed to achieve
the same.
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HIGHLIGHTS in HISTORY of MATHEMATICS
Position number systems
Euclid geometry as abstract deductive theory
Heliocentric theory of heavenly bodies which at that time had only mathematical
advantages
Analytic geometry
Analysis
Differential equations
Differential geometry
Non-euclidean geometry
Topology
Linear algebra
Abstract algebra
Number theory
Discrete mathematics
Set theory and mathematical logic
Probability theory and statistics
Game theory and
Category theory
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TRUE ENDS od MATHEMATICS
Before 17th century mathematics brought large amount of still valid knowledge,
beautiful and scientifically powerful, but its impact on the development
of society has been next to zero.
17th century True end of mathematics is the greater glory of God.
18th century True end of mathematics is the greater glory of human mind.
19th century True end of mathematics is to be the queen of sciences
20th century True end of mathematics is to be main servant of natural sciences and
technology fields
20th century Through outcomes of industrial revolution mathematics, at that time very
closed inter vined with physics, stated to have large impact on society.
21th century True end of mathematics is to be the queen of informatics and one of its
main servants.
20th century WW 2 demonstrated enormous potential of physics and mathematics to
influence development of society.
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GREAT METHODOLOGIES of MATHEMATICS
Abstractions - to study problem of physical world by transforming them into the
(idealized) problems of the informatio processing world. By Plato: the world
perceived by the senses is confused and deceptive and in any case imperfect and
impermanent; He believed that physical knowledge is unimportant, because material
objects change and decay; therefore a direct study of nature and purely physical
investigations are worthless.
Greek deductive mathematics based on qualitative axioms as a way to understand
nature, They tried to understand why things happened - to unearth the cause of the
phenomena.
Descartes and especially Galileo’s deductive mathematics based on quantitative
axioms as an essence of science and a systematic way to develop science. The goal
was to describe (using simple quantitative laws) how things happen, to describe
some properties of phenomena (and not why they are as they are). Using this
methodology scientists produced during next two centuries deep and sweeping laws
of nature on the basis of very few, almost trivial, experiments and observations. One
can say that Galileo’s idea was the most fruitful idea that anyone has had about
scientific methodology. Newton’s quantitative characterization how gravitation
acted, whose action could not be explained in physical terms, was an excellent (and
much convincing) example of the power of new methodology.
Generalisation (abstract algebras, mathematical logic, topology, metric spaces,
category theory).
Analytical geometry can be seen as a start of algebraist and quantification ofprof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 115/125
LESSONS LEARNED FROM HISTORY of CHEMISTRY
Chemistry has actually a longer continuous history than
physics.
Already in the 16th century, there had been remarkable
advances in the field that might nowadays be called
chemical technology.
In spite of that, chemistry, as a modern science, can be
sen as being born only in the second half of the 17th
century.
Moreover, only at the end of the 19th century can
chemistry actually be seen as having changed from
being a servant and adviser of industry to the new
position of being a creator of industry.
The history of chemistry seems to teach us that
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APPENDIX I
APPENDIX I
prof. Jozef Gruska IV054 2315. Future in the informatics era - Chapter 4 117/125
WHAT ARE (ULTIMATE) COMPUTERS?
History of attempts to have a full understanding of what a computer is is a nice
illustration of two ways physics and informatics look at the world and on interactions
between them.
In the golden era of Greek mathematics slaves, and not
free men, were considered as those to perform
computations.
In the 18-19th century the term computer was used for
girls doing computations in manufactures.
After the clock superparadigm got a momentum,
computers started to be seen and developed as physical
devices.
Turing and others liberated, and very successfully, the
concept of computer from its physical substance. (This
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WHAT ARE (ULTIMATE) COMPUTERS? - II.
By doing that Turing, Church and others were also able
to set up a bound/barrier, called Turing or
Church-Turing barrier, concerning the power of all
potential computers.
Development of modern computers and a need to get
deeper insides into computation processes and their
potentials and limitations brought an understanding
that a variety of models of ( universal) computers is
needed - all were liberated from the physical substance
though the laws of classical physics were kept in mind
when evaluating their potentials.
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WHAT ARE (ULTIMATE) COMPUTERS? - III.
Liberation of fundamental concepts of computation
from its physical origin has later been followed by the
hardware-independence (and later also by the softwareand
model-independence) of the key concepts of
computation.
Discovery that quantum computers could be more
powerful than classical brought back a need to pay
attention to the physical substance of computers.
At the same time the emergence of a global computer
(network) that can be seen as never terminating and
filled constantly with non-uniform inputs brought still
another ideas how an ”ultimate computer” can be seen.
All that brought back old quest to find model of
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WHAT ARE (ULTIMATE) COMPUTERS? - IV.
S. Lloyd calculated that an ultimate laptop could not
perform more than 1051
very elementary operations per
second.
Universe started to be considered as a (quantum)
information processing systems that has been able so
far to store, from its big bang, not more than 1092
bits
and perform not more than 10122
of elementary physical
operations (again by S. Lloyd).
How about computing at the Planck scale level?
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APPENDIX II.
APPENDIX II.
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GLOBAL TASKS of PHYSICS AND INFORMATICS
To develop a proper understanding of spacetime as well
as of the universe and its evolution.
How to transfer to physics such concepts us
universality, complexity, computability and feasibility?
To find out what is an ultimate computer?
To search for minimal universal physical information
processing phenomena.
To design powerful quantum computers
To make perfect security society in a way that
undetected physical crime is practically impossible.
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GLOBAL TASKS of PHYSICS, BIOLOGY and INFORMATICS
To get deeper insights into biological phenomena
through their simulation.
To get deeper insight into natural life phenomena by
designing and exploring artificial life phenomena.
To understand cells information processing
To understand the role information processing plays for
life and for evolution.
To understand life and evolution.
To develop a model of the brain and mind as well as of
conscious activities.
To develop individual patients models of human body
and its processes.
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CHEMISTRY and BIOLOGY
Chemistry is the study of the composition, properties and behaviours of matter.
Chemistry is concerned with atoms and their interactions with other atoms and
particular with the property of chemical bounds.
Chemistry is also seen as the central science bridging physics and biology.
Biology is the study of life and living organisms, including their structures, functions
growth, evolution, distribution and taxonomy.
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