Dedicated to the
Dedicated also to
Buckminster Fuller whom I quote here:
Buckminster Fuller worked on a minimum
system of concepts and processes contemporary information, in many disciplines,
that would lead people think, and act so as to have a satisfatory physical and
metaphysical life on the planet earth and avoid extinction of life on the planet. The individuals integrity was a
crucial part for this to have success. The interconnected disciplines were,
astronomy, mathematics, philosophy and metaphysics, engineering, design ,
architecture, poleodomia, arts, economics, politics, sociology, education etc.
See also a Lecture in the National
Technical University of Athens during June 2011.
Digital_mathematics_lectureNTUA.ppt
And a special blog about the present ideas
at
http://thedigitalmathematics.blogspot.gr/2012/06/1-draft-view-of-contents-of-book.html
THE NEW
MILLENNIUM UNIVERSE OF MATHEMATICS WITHOUT THE INFINITE (NATURAL MATHEMATICS)
The next three
axiomatic systems of the classical historical and foundational logical systems
in mathematics have many finite models of various sizes! This immediately proves
the consistency and non-vain character of the axiomatic systems. The smaller
size layer is always the observable, visible or phenomenological, or logical
and expressional interface-layer of the system. The larger layer is the
ontological or hidden or invisible, an analogue to the hidden by the
programmer, file and procedures structure of a software from the user. The
axioms are stated sometimes for the one layer other times for the other,
and also in combination of the two layers. These choices give a totally new
way of thinking, observing with our senses, feeling, and
handling, our creations, compared to the traditional mathematical
thought! It is a new experience. For the first time after 2000 years e.g.
reasoning in Euclidean geometry, may acquire new consistency of mind
and observation with the human senses. For example an axiom that
between two different points there is always a third, is not consistent with
observation with the senses. In the new present mode of the axioms, there is no
such an axiom! Also the points do have a finite size, as it is confirmed by the
senses, and are not of no size at all!
In the mathematics of
this layer the complete elimination of the concept of infinite, requires a new
technique in observing with the senses, feeling, acting and reasoning. This
technique is more sophisticated than the usual techniques in classical
mathematics. The reasoning and observing with the senses, (this applies
especially to geometry or disciplines that the object of study is pictorial) is
in a complete consistency with the reasoning for the smaller or external
or visible layer, while there is also reasoning for a layer which is
completely invisible or non-observable , larger or internal. Still the internal
invisible layer which is invisible under normal conditions can become visible
and observable too, with pictorial representation, under
exceptional enhanced conditions. The latter guarantees the validity of the
reasoning for the non-observable layer.
They are natural too
after the human-computer interface interaction in the technology of multimedia.
From this point of view e.g. the concept of the abstractness of the infinite is
also, a measure of how limited are the logical expressional means and
informational handling of the system of mathematical entities, which can always
be assumed finite in the ordinary sense at the ontological layer. Thus some
arguments became simple and elegant in such systems (like a nice and friendly
interface of a complicated software system) , but this should not be pushed to
its limits, and other types of properties of the same finite system require
different axiomatization which has to be devised or updated from time to
time! We follow here the simple interplay of the observable layer with the
non-observable layer, that is suggested by the senses (audio, visual and
touching thresholds) , the physics of handling objects, and the logic of
creating them by software in the computer screen. This suggests specific axioms
for the interplay of the two layers. Other means of observing, like for example
those defined by specific system of physical experimental devices for the
micro-world, would result in to different choice of axioms. But in this first
approach we use the definition of "observable" by the direct sense-inspection
of a statistical standard human being. Thus the only abstraction of the
axioms refers, to the size of each layer. Even the concept of being finite is
layer depending. If in such a system of axioms are added some axioms that
specify the exact finite cardinality, of each layer, and their mutual relation,
called the key-axioms then it becomes a categorical axiomatic system,
as it has one and only one finite model that satisfies it! The
intended relation of these axiomatic systems with the corresponding classical
axiomatic systems is mainly described with the next facts:
1) They are
interpretable and consistent: All these new axiomatic systems have finite
models definable within the corresponding old and familiar axiomatic system. In
their full version of all (also the key) axioms, they can have only finite
models. But by eliminating some axioms, they can also have models described in
the old-familiar axiomatic systems as infinite in one of two layers, the
hidden or the observable layer, or in both.
2)If possible they
should be logically adequate and finite-wise complete relative
to the old mathematics in the next sense: Any proposition provable in the
old-familiar axiomatic systems as holding for all the finite models of the new
systems, is also provable within the new axiomatic systems. This property may
make some researchers to think of the completeness of the 1st order Logic or
the Forcing method in set theory. It is not sure that we may have this property
without using also axiom-schemes in the new axiomatic systems. A different
version would be that any property provable in the old-familiar axiomatic
systems as holding for all models that the observable layer is finite set
in the model of the new systems, is also provable within the new axiomatic
systems.
3) If possible they
should be logically adequate and complete as categorical: After
adding the key-axioms that specify one and only one finite model for each of
them, then any proposition provable in the old-classical axiomatic system, for
this specific finite model is also provable in the new categorical axiomatic
system. This completeness does not seem to require to make use of axiom-schemes
in the new axiomatic-systems as the categorical versions with the key axioms do
not require axiom-schemes.
We notice here that these
relations are way different than those used by non-standard analysis and
mathematics. There is no way, that any theorem of the old-standard mathematics,
can be transferred in the new mathematics! At first the very axioms of the
old-standard mathematics are abandoned in the new mathematics! But all the
finite and substantially practical content of the old-mathematics can indeed be
saved in the new mathematics!
4) If possible all the
axioms of the axiomatic systems are stated within 1st order formal languages,
and Logic! It is possible, that in their categorical version with the key
axioms, there are no axiom-schemes, or 2nd and higher order
formulae for the axioms. Axiom schemes as that of induction in the natural
numbers, or of the supremum (continuity) for the real numbers, or replacements
etc for set theory etc, are easily provable theorems in the new axiomatic
systems, deducible from simple 1st order logic axioms . This is possible as the
new categorical axiomatic systems have only finite models! Thus the advantages
of the completeness of first order logic, is fully used in logical reasoning.
But we may not have the property 2) above for them when stated without the key
axioms.
5) The relation and
sizes of natural numbers used in the meta-mathematical level of the Formal
Language and Logic with the natural numbers used in the objective
language of the axiomatic theory is a carefully discriminated
role, is discussed and specified, and is responsible for the creation of
the concept of "operational infinite" within the finite universe of
layer 1 of mathematics.
It is standard practice
that we make use of the natural numbers in the meta-mathematical level of say
the mathematics of the formal language of the theory (e.g. counting
propositions formulae etc) and the natural numbers inside the objects of the
theory (e.g. counting rational numbers or , sets, or straight lines). We may
think of the meta-mathematical level as the thinking and speaking actions,
while the objective mathematical level as the writing and hand-handling actions
too. The meta-level in mathematics corresponds in the computer science to the
hardware, or to the operating system, while the objective level of the theory
to the particular software programmed within the operating system. As the
axiomatic system of natural numbers, when postulated in the new mathematics,
may have a maximum element, the new mathematics are sensitive in the
relation of the maximum natural number available in the meta-mathematics of the
formal language, and the maximum number available in the mathematics of the
objective language. E.g. If the former is smaller than the latter we may have a
perfectly plausible concept of infinite of the objective natural numbers
compared to the natural numbers available in Logic! Their relation also
determines if we need axiom-schemes like that of Induction in
the natural numbers, or not. These concepts are also familiar in computer
science and the theory of algorithms (computation) as the available memory, and
time resources in the operating system, the ram memory , and the input data
memory of a procedure. Proofs may be also procedures. We may not
only think of the maximum possible natural number in the objective language of
the theory and the maximum possible natural number of the meta-language of the
Logic of the axiomatic theory as unknown constants, but also as unknown
variables, that furthermore may be linked by parallel processing algorithms if
postulated so. The same with the maximum possible natural number in the
observable layer and the hidden layer. This gives the interpretations
of the infinite as a constraint which requires a transcendence of the available
space and time resources of the cognition system in representing and reasoning
about the ontology, of another system. The various grades of the infinite, are
interpreted as (possibly parallel) processing complexity measures between
these inaccessible to the cognitive system, finite numbers of the
ontological system . In the present approach for the concept of the
infinite in the next axiomatic systems we assume that it is always possible
with sufficient enhancement of the resources of the cognition system, to
"turn" or "reveal" the infinite of the ontology of
the studied system in to finite, while still we describe a new concept of
"operational infinite" that corresponds the the classical concept of
infinite , and entirely within finite systems.! The dynamic
interpretation of infinite, as an algorithm that increases the finite and also
as am ultraistic way of talking about the finite, has been used often in the
past as a way to keep a distance of the material, or human action, ontology of
the finite, especially when the thinker considers it undesirable or an
obstruction in his attempts to think about the situation. But when
the situation is tamed a different finitary approach is obviously better.
Boldly speaking we
consider a mathematical system as operationally infinite, if the
counting of its elements, with the maximum possible natural number of the
meta-mathematical formal system (e.g. Logic) does not exhaust it. We must
notice that both the meta-mathematical system, and the objective mathematical
system, are assumed, strictly speaking, finite. This in the world of computers
corresponds e.g. to that the size of the RAM memory, and the maximum acceptable
run time complexity of a procedure, cannot count, or scan the disc or
space-memory size of the data of the system. In the human situation it
corresponds to a situation where the cognitive powers of perception, and memory
(of the individual or collective theory-maker), are inadequate for scanning or
counting the elements of the objective entity to be studied. It may be
considered a large gap, between, minds powers, and hand's operational powers.
If on the other hand, The maximum possible natural number, of the formal
meta-mathematics (e.g. Logic, or formal axiomatic system) are comparable or
larger of the number of elements of the objective mathematical system, that is
to study, them it is considered operationally finite.
Thus in this approach of
the finite 1st layer mathematical universe, not only the continuum of
geometric lines or real numbers is created entirely from the finite, but also
the very concept of infinite is also created, within finite systems. The
finite and the infinite turns out to be two types of interplay, of
meta-mathematics with mathematics, or of Logic, and practice. It
is the first time after the time of ancient Pythagoras, and ancient Euclid,
that public mathematics can create in a rational clear , Logically
complete and practically sound way, within finite systems, the
concepts or "realities" of "continuum",
"infinite" and "irrational numbers". For the first
time irrational numbers do not really exist, the continuum is created
entirely from the finite after a logical representation of the phenomenology of
the senses, and the infinite is a type of interplay of the finite of Logic,
with the finite of mathematical procedures.
At least for the new
axiomatic systems of integers, and real numbers, we could chose
equivalent axioms that are stated only for the hidden layer, while
the observable phenomenological layer, could be simply defined over the hidden
layer. This is because the interplay of the visible and the hidden layer is
quite standard. We cannot say the same nevertheless, for geometry or sets. For
example a line segment would appear in the phenomenological layer as line
segment either if it was made my a square lattice of evenly spaced pixels in the
computer screen, or by an unevenly spaces system of atoms, of say glass
in a palpable ruler of glass. Although each axiomatic system refers to two
layers, we may define new ones, where for example the observable external layer
of the second is the hidden layer of the first, according to the needs of
logical arguments. E.g. in the standard practice of computer images,
a marked point by the user in a computer image has, observable
size, chosen from a pallet by the user, it is made from a connected set of
software-resolution pixels, of the image as a bitmap, while each
software-resolution pixels is in its turn a connected rectangular set of
monitor-resolution pixels! Thus if the user shall mark a most dense rectangular
set of points, then three resolutions shall participate in this situation.
Research ideas under development
A
For the natural numbers the
two layers resemble the discrimination of two categories of positive integer
numbers represented in a computer language, the simple integers and the long
integers. Or the integers available to the user of a software, and the integers
available to the programmer of it, and to the operating system. The operations
in the small or interface layer may have results in the larger layer,
which is the only closure property of the operations postulated. Nevertheless,
a subset of the axioms of this system has also as model the transfinite numbers
like, the ordinal natural numbers. The next axioms are crucial
modifications of the Peano's axioms of natural numbers. This system of numbers
can also be formulated as two Galois finite Fields, each endowed with a
natural order, and with obvious modifications of the cyclic operations so that
the circles open to linear segments, and the operations of the smaller field
have output to the larger field.
00) New axioms for (double layer)
finite systems of natural numbers.
B
The Logic with
the most standard techniques is reformulated but in a finite mode. This means
that all entities like, terms, constants, variables , relations,
operations etc are not only finite in their internal composition, but also all
of them together are finite in cardinality (finite resources) , and the
size is very critical, to what can proved or not. As Logic itself is the
object of study here, it is required also a system of numbers as the above
double layer system of natural numbers to count its objects. The relative
size of the system of numbers used and the cardinality of the objects of Logic,
is very important to questions like consistency, completeness, decidability
etc. As the Meta-Logic for this Logic can be assumed a finite Logic too, the
size of the meta-Logic, and the acceptable length, and various types of
complexity for the proofs of Meta-Logic, is critical about what is
considered valid for Logic as object of study. Troubles like those introduced
by Goedel, concerning, the definition of "Propositions" by some kind
of "diagonal" argument of the Meta-Logic, are completely resolved,
and controlled ( if they fall inside the system at all) by the relative
balance of the complexity and resources of Meta-Logic, and natural
number system used, and Logic as the object of study, The reverse
of all the negative and celebrated theorems in Logic (e.g. Goedels theorems)
can be proved to hold too, with a right choice of the parameters of complexity
and resources.
In the modern
computer science it is discriminated the input-size complexity from, the
RAM-size complexity, the run-time complexity, the code-length of the algorithm
complexity etc. All of the above have applications not only in programming
languages but in formal languages too (as programming languages may be
considered specifications of formal languages). Although it is a celebrated
theorem that there are problems that do not admit algorithmic solution
e.g. the labyrinth problem, the word problem (Post theorem ) etc , it is
because the input complexity ("All possible algorithms") is too large
for the complexity of only and the same algorithm e.g. the "all
labyrinths" is an unspecified complexity. Nevertheless if the
input complexity is balanced to: "Labyrinths of upper bounded size, and
the upper bound is not greater than so much", then it may very well exist
one algorithms that solves the problem. In the same way many impossibility
theorems in Logic simply disappear, and are valid in the converse way, if the
size of the acceptable propositions is only finite and upper bounded by an
appropriate number for the resources of the used system of natural numbers and
Login in the meta-theory (context). Although it may seem, that it is
created a circle-reference between , Logic and Numbers, it is all resolved by a
recursive discrimination, of Logic, or Numbers as specific systems of a level,
for the other systems, and not as one only, standalone entity .
For more
sophisticated approaches of the new finite resources Logic, we may discriminate
between two layers of propositions in the system, the explicit
propositions (like the interface accessible to a user in a software)
and the hidden or implicit or invisible propositions
(like the inaccessible to the user Logic in a software, which is accessible
though to the programmer. The human-machines interaction is abundant in such an
experience. E.g. the courses HCI [human-computer-interaction], in computer
science). Usually the former system has a model and interpretation in the
latter system. If the Logic as a system is also attached to a axiomatic
system, then the hidden or implicit layer defines a categorical
axiomatic system (with unique up to isomorphism finite model), while the explicit
layer of propositions defines an abstract non-categorical axiomatic
theory (with possibly many non-isomorphic finite models).
01) New axioms for, finite and bounded
resources, systems of Logic
C
For the real numbers, the
two layers resample the discrimination of two categories of rational
numbers: the single precision and double precision numbers represented in
a computer language. They have also a human visual discrimination, as
the visible external or phenomenological layer represents the
significant part of the (rational) quantity, while the invisible or
hidden ontological, the higher, decimals produced by the operations and
required for a stable definition of the phenomenological visible layer. Or the
discrimination may not be relevant to the visual discrimination of the human
eye when the applications are not geometric. In such cases it is relevant to
the relative size of the accuracy level of measurements of observable
physical quantities and the atomic structure of the physical
system. The only closure of the operations postulated is one that the
operations in numbers from the phenomenological layer may lead to results in
the hidden ontological layer. Bounds similar to the overflows in the computer
handle of the quantities are crucial for the definition of the system.
Again the hidden ontological layer is a finite system of rational numbers.
Concepts like dense subsets, Borer sets, and other concepts of descriptive sets
of real numbers are all finite sets! Obviously any function or
distribution on them can be represented with a finite dimensional vector
and even more with a finite list of rational numbers. The operations at each of
the two layers are defined for each granulation pixel (bin) from the ordinary
definitions of operations on numbers, and the centers of the granulation
pixels that are the canonical representatives. The coarse external layer
defines an equivalence relation of rounding for the granulation pixels (bins)
of the finer internal layer. This system of numbers could
also be formulated as three Galois finite Fields (over the
same prime base) , each endowed with a natural order, and with obvious
modifications of the cyclic operations so that the circles open to linear
segments, and the operations of the smaller field have output to the larger
field. The larger Galois field has invisible pixels as seem from the human eye
and a standard distance, and corresponds to the non-accountable (still
finite!) part of the calculations (what calculations although possible due
to the ontology are not to be included in the description of the phenomenon) .
If in addition its size is a significant larger number, from the maximum
possible natural number of the formal Logical axiomatic system for these real
numbers, then also the invisible part of the real numbers (largest Galois
field) is also operationally infinite, as defined previously (although still
strictly speaking finite!), otherwise even the non-countable hidden part,
is operationally finite. The middle size Galois field defines the boundary of
the phenomenological to the invisible ontology, and mainly corresponds to the
human visual discrimination threshold according to the set conditions. It has
the visible of finite size points of a line, and represents the part of the countable
part of the calculations that is significant in the final representation of
a phenomenon.. The smaller Galois field corresponds to the natural numbers
within the real numbers. In this setting natural numbers, rational numbers and
real numbers do not really have any essential difference. The fact that Galois
fields are always powers of prime numbers is convenient, as also in
Multi-Resolution wavelet Analysis, the sizes of the pixels of the
different resolutions are chosen as a sequence of powers of a prime number.
Multi-Resolution wavelet Analysis has remarkable applications in the efficient
representation of the continuum of digital images, and sound. The choice that
the pixel sizes in different resolutions increase as a powers of a prime, is in
accordance with the law of Fechner in psychology and physiology of the
senses which states that if the input of the senses is in multiplicative
progression, the bio-ware representation is in additive progression. This is an
economy of nature and our condition as human beings, which defines, too, the
type of our cognition . The same principle is met also in music and design of
the scales but also in the decimal base of the system of numbers , where
the measuring units are in multiplicative progression (powers of 10). The characteristic
of the larger Galois field defines also the resolution of the number
system. Instead of Galois fields we may also use, simply, finite rings modulo a
power of 10. We reserve the symbols Rm,n for such
finite systems of real numbers , where m is the power of base 10 till the
smallest visible point or mark (or cell) and n the additional orders of 10 till
the smallest indivisible of the resolution or invisible point. We assume
symmetry in the powers of 10 for larger to 1 sizes. The n represents in
other words the depth of the continuum. For example we can have as
reference system, the R4,4 where there are 4 orders of
10 (decimal digits) of invisible or non-observable or non-significant in
measurement sizes after the last significant decimal digit, which is at the
forth place after the point. (It is instructive to compare it with a
corresponding term and infinite aleph in the infinite real numbers . As it is
known through the forcing method of Cohen it was resolved that the continuum
hypothesis, and also that infinite real numbers can have any depth. This is one
more case in my way of thinking where classical mathematics of the infinite
prove to be too a kind of early encryption of facts of the finite
universe of real numbers) We must remark that the ancient Greek word "άρρητος αριθμός" was translated in Europe as
"Irrational number" but a translation closer to truth is
"classified number" . A classified number does not have to be
something different from a finite number. What is classified in this system of
numbers is what is the size of the invisible Galois Field, in other words the resolution.
That it was classified, was an element of abstractness or transcendence
from the material of the objects of application. At that ancient age (the time
of Pythagoras) it was not widely know that material objects are made of
indivisible atoms, therefore any one believing on this should consider such an
assumption in the mathematical ontology too, as classified or άρρητο . The analogue concept of the άρρητος αριθμός as depth of the resolution, in physics and
chemistry is the Avogadro's number 6.022*10^23 , that essentially defined the
physical "resolution" the tangible continuum (e.g. gaseous) matter
that was familiar to them in the experiments. The first measurement of this
number which was essentially the universally accepted birth of the atomic
physics, was by Loschmidt and was based on the formula of sample variance in
statistics of Brownian motion, which is not invariant to the sample size
(of particles) , thus giving the necessary clue. It is therefore obvious that
the finite real numbers have to introduce, besides the standard of unit of 1 meter,
also a new standard for the resolution. For the natural sciences we may
use as ratio of the unit to the invisible pixel of the invisible layer the
number 10^(-12) or 12 decimal digits, as this is close the the electron
The next axioms are slight
but crucial and critical modifications of the usual complete ordered
commutative field 's axioms of real numbers.
02) New axioms for (double layer) finite
system of real numbers.
D
For the sets of set theory,
the two layers (the external or sets and the internal or classes) resample the discrimination
in to classes and sets in the Bernays-Goedel axiomatic system of Cantorian set
theory. The axioms are slight but crucial modifications of the Bernays-Goedel
axioms, and as I pointed out previously they do have many simple, finite models
for them! The set operations in the small or interface layer may have
results in the larger layer, which is the only closure property of the set
operations postulated. If Cantor had studied subsets of such finite systems of
real numbers as the above, he would not have to present his famous set theory,
but he would have resulted in a set theory as the one below, where all sets are
finite. In addition Goedel would need not introduce the ideas of Cantorian
infinite in metamathematics too! And quite probably, to my instinct, Cantor and
Goedel too, might not have resulted in the mental sanatorium in their late
years, in such an unfortunate way. I consider, the present suggested
developments or updates in the science of mathematics, an a significant cure of
the incompatibilities of the long range and beyond, collective mind in the
sciences. From the computer science point of view, they resemble, say, in a
data-base software system, the external observable layer (sets) with the
tables and queries, that a user can perform in the data-base, while the hidden
layer (classes) with the tables, and queries, that a programmer with
SQL-statements can perform in the data base! Furthermore it could be
compared with the modern theory of "objects" of object-oriented
programming languages , with their inheritance (belonging) ,
classes and other relations.
There is a closest concept
to that of Cantorian (or should we say Zermelo-Frankel’s) infinite in this finite world of
finite sets. And this concept is the protocol of collective , daily updated
maximal finite sets. E.g. for the real numbers we may imagine the densest
or highest resolution finite system of real numbers, based on any real number
defined by any creative worker in mathematics in this planet from an
initial day till yesterday. Such totalitarian large and socially agreed
finite sets A (till last update of date n) , have the property that for any
element a in A , the {a} U a belongs in A not today but not earlier than the
next morning. As we can easily see such a concept goes beyond the
traditional initial concepts of set theory, as it involves a collective
agreement and protocol in the social scientific and mathematical communities,
and time unit, concepts like planetary days. It is a concept much like e.g. weather or other social news and economic data that are daily
updated in the data bases of the Internet, in a globalize civilization. The
Internet is both a democratized and also globalizes. It is readily realized
though that such maximal till yesterday sets (e.g. of real numbers) are very
clumsy totalitarian entities to be used in the mathematical arguments
(arguments that might be so that hold for any future update, and fore any
refinement of the resolution), while smart and elegant arguments would require
only “democratic” finite versions, till a fixed resolution of the real numbers.
A very important feature of
this set theory (which also resolves part b) of the initial remarks of this
page, as motivations for this work) is that it is not a new gap in abstraction,
and does not really introduces new ontology that numbers and logic cannot
derive! This is achieved by defining axiomatically all sets as finite sets
of numbers, and derived only through the means of formal logic of order n
(n-order formal languages and Logic as described for example initially by the
theory of types and predicates of Russell or even simpler by Hilbert etc)
when included in the objective level of the theory. Thus we apply here a simple
philosophical equation Sets=Logic + Numbers. This restores the property
that all mathematical entities are arithmogenic (=generated by numbers) and
logical operations! After all, the initial concept of belonging which is
denoted by € is also the first letter of the word
"epomenos" that in ancient and modern Greek means "next",
and the only difference of the initial concept of belonging € in sets
from the initial concept of next € in natural numbers, is that in
the latter every entity can have at most one next and one previous, while in
the former, it can have many previous, which leads naturally in to tree
structures. Tree structure is the basic pattern nevertheless of the types
of formal propositions and predicates in Logic. This assumption in the present
set theory, is in conformance with natural sciences, (where from atoms are made
not only the lattice structures of metals, but also the tree-like structures of
chemical compounds etc) and also with computer science, where all data-base
tables are made after all from sequences of bits. This keeps the continuity in
the genealogy of the ontology of all mathematical entities. Thus from this
point of view the hidden layer is the numbers from which sets are made , and
the external observable layer may be the Logical structure in defining
structures upon them. The external layer is also only a part of the
n-logical layers of the n-order formal Logic. As part of the Logic is already
in the objective language in this way, the meta-level for this theory
introduces further levels of Logic that are not included in the objects for
study of this theory, and for many purposes it can be kept as only 1st order
logic. Thus we strictly separate the axioms, that refer to the numbers,
from the axioms that refer, to the higher order predicates and relations over
numbers. Axioms like that of replacement or comprehension are already part of
the facts of Logic rather and lead again in to entities of logical character.
Notice also that as we make use of formal language of order up-to n, the
"sets" of numbers have "height" or "deepness"
only up to n . But furthermore and in a similar way the "width" of
any set or tree of €, is at most an other integer number m, which
is not revealed in the axioms as particular sequence of decimal digits, but is
stated as existing for any set. This says a lot more than what says
the usual well-foundation axiom, and thus permits many more wonderful
structures and ways of reasoning and proving in this new set theory!
This should be so as in the Layer 1 of the 7-layers of mathematics, the sets,
are not only of finite many elements but also a finite system of sets. This is
a basic requirement for all of mathematics when existing in the Layer 1! The
axiom of infinity, has a different meaning in this setting, and refers to the
closure of sets of the external , observable layer to some operations within
the larger hidden layer. There is always again the issue if the external layer,
is finite or infinite compared to the available natural numbers, of the
meta-language for this theory. But this point has already been met, in the
discussion of the new axiomatic system of natural numbers and is resolved and
chosen in the same way. The probable ideas of the ancient Pythagorean
philosophers, that partially ordered entities (figured numbers) are the source
of everything in maths, has a proof here as we re-create all of the mathematics
at Layer 1, from such entities. My preference here is to define the whole of
the maths of Layer 1, from the concept of tree-action, which is nothing
more than the old concept of program or algorithm , described as a flow-chart,
which when opened is a Tree, and over one elementary operation which is the
unit-counting in natural numbers, and elementary decisions, that are
identification or comparison of such a counting. It is quite spectacular that
we can prove that :
1)
Tree-actions (a modern correspondent to the ancient Pythagorean figured
numbers)
2) Markov
Normal Algorithms
3) Turing
Machines
4) Recursive
functions
5) Free-ring
polynomials (algebra)
6) Finite
sets of finite sets etc. of natural numbers (Cantor, Peano)
7) Finite,
logical types (Russel)
are
entities essentially equivalent, as each one can be transformed in the
other, and thus contain equivalent information. Their systems, as finite
systems are isomorphic up-to defined relations and operations.
A tree-action is a geometric
entity together with an action or flow on it defined axiomatically. It is
similar in some sense to the ancient Pythagorean figured numbers. It
corresponds here to an algorithm not upon the strings of a formal language but,
upon unit numeric counting, and deciding about the
equality of natural numbers. That any number is created by
repetitive counting of the unit is of course what we are familiar with.
But the counting that creates the natural numbers is sequential. Here we introduce also the concurrent counting. Sequential counting creates
the branches of the tree, and are the natural numbers (height of the tree). But
the concurrent counting creates new entities that are not the natural number,
and are the branching of the node of the tree to many parallel branches (width
of the tree). If it was musical action,
sequential counting would correspond to a melody,
while concurrent counting to a chord. We
introduce a symbolic writing of the tree action , where 1 is the unit counting,
o is the sequential composition (repetition) of unit counting, and U (union) is
the concurrent repetition of unit counting. Thus while 1o1=2 , 1U1 is not the
number 2. We introduce also symbols and operation for elementary logical decisions
that are necessary in any algorithm, and we put them as exponents or operators of a 3rd external operation.
These two operations make a
tree action in to a free-ring polynomial with associativety, distributivity
etc. While union or concurrent composition is commuttative, sequential is not.
We notice that the the o corresponds to the usual addition, of numbers, The
usual multiplication of numbers is defined from the addition. Notice
nevertheless that the union is an operation where the addition of natural
numbers is distributive! Notice also that the sequential composition o is a
wider operation than addition of numbers as it applies to pairs of
tree-actions, which makes it in general non-commutative. Natural numbers are
only a special category of tree-actions, where the tree is only one branch.
The free-ring polynomials
is a very elegant and simple way to write in lines, all the information,
of a program, and its flow-chart either as a graph or as a tree, and either if
it is sequential or of parallel computation. We enhance the entities of such
free-ring polynomials (thus of non-commutative multiplication) with
exponents (e.g. from a finite Boolean algebra of events) corresponding to the
elementary decision that has to be taken before we execute the next commands
(the base of the power). The multiplication of such polynomials is the
sequential composition (or call) of commands, and the addition (or union) the
parallel or concurrent execution of two commands. It is remarkable that the
elements of a finite set as in 5) are interpretable as concurrent (parallel
run) of commands. Thus the above equivalence of the basic entities of Logic,
Sets, Algebra, and Computer programs, proves that at Layer 1 it is the same
from the technical point of view and in some sense conceptually too, if we base
Mathematics on Logic, or on Numbers, or on Geometry, or on Algebra
(operationalism), or on Computer Science!
03) New axioms for (double layer)
systems of sets.
E
For the Euclidean geometry the
two layers correspond on the visible phenomenological interface , e.g. of
a finite strait line segment on the computer screen, while the larger
ontological layer corresponds to the invisible programmers pixels of the the
lines over some finite resolution. Even in the computer image processing, there
is the discrimination between the hardware's screen or monitor resolution, and
the software resolution of the image as bitmap. In the case of the computer of
course, according to the user's choices maybe only the latter is visible,
or none or both. Usually, the monitor's resolution is invisible while the
software defined resolution of the image as bitmap can be visible. In addition
the user's marks or points are still of different size, (always made from a software
resolution connected set of points or pixels) and their visible size, as
corresponding to the thickness of the marking tool (pencil etc) cab be chosen
from a pallet! Thus the points are of at least these two types and so are the
lines, planes etc. A phenomenological unique point may contain many ontological
pixel points! In this geometry the concept of accuracy level is very
important. The identity of entities in the layers is defined according to he
accuracy level. As in the ancient mode in
The necessity of the discrimination
between a phenomenological and ontological layer, is more plausible if it is
identified through the discrimination threshold of the average human visual
ability at a standard distance, thus as the visible points layer,
and the invisible points layer.
It becomes even more
important, in the theory of curves and surfaces, as the infinitesimal or
tangent space at any observable point is exactly the partition or
equivalence class of the observable point over the non-observable points of the
hidden layer. The whole setting is by far non-equivalent to that of the
old mathematics of differential geometry, as continuity and differentiability
as we remarked may or may not hold and may switch in holding over the same
surface and at different resolutions or space-scales. So, many different
systems of differential equations may be written for the same set of points or
manifold, at different resolutions, representing different geometric properties
of the same entity at different resolutions! Much of the work of
digitalized image processing in computer science is of great help, for
the appropriate concepts in this new geometry. An other spectacular deviation
from the ancient Euclidean geometry is e.g. the definition of the ratio of the
length of the circumference of a circle to its diameter. This number that in
classical mathematics is the irrational number pi (π), is here a rational number. The answer to the question which rational
number is this ratio , is that it is a different rational number for geometries
of different resolution. Thus for circles of the same diameter this
number is different. E.g. if we are talking for a circle from material
copper wire, the size of the copper atom defines this number. If we are
talking for a circle drawn by say Archimedes on the sand, then the average size
of the granule of the sand defines this number, and if we are talking for a
circle on the screen of a computer, the size of pixels of the screen and bitmap
image resolutions, defines this number. In this 1st-layer geometry endowed with
a resolution specification, the "irrational" pi (π) as an algorithm of increasing the digits of
3.14, is meaningless as a non-terminable algorithm, and meaningful only as a
numeric, rational number, final output. We must not forget that the
concepts of infinite and irrationality (of numbers) refer and are
properties not so much of the ontology of the entities of study but
rather of the states of consciousness (individual and social or civilization
collective) of the subjects that study and make the knowledge. It is the
evolution of culture and societies, that a century may come that this
may substituted by the finite and rational, in a glorius new setting.
04) New axioms for (double layer) entities
of finite Euclidean geometry.
F
To clarify how
powerful are these new lines of reasoning, observing and handling of the
mathematical entities, in the layer 1 of the 7-layers, we may present a direct
proof of the famous and still unsolved (as far as I new) Poincare
conjecture in the topology of three-dimensional manifolds (actually what has
survived after the refutation of an important part of it). The topology on
systems over finite resolutions is based on the initial concept of two
observable points in contact, or on the concept of topological closure (or
1st-order continuation) of a set (all points in contact with the points of the
set). As the observable points are cell of the resolution lattice, being in
contact or being in zero distance (over the observable number system) means the
obvious, either being identical cells or having a point or side in common. This
topological closure operator is not like the closure operator in classical
topology, and actually it is not even a closure operator as defined in algebra.
The difference is that the closure of the closure maybe a strictly larger set,
and we may define the n-order closure of a set. There is no valuable
distinction in open and closed sets, as in the topologies of interest all
can be both. The information required for the topological arguments is
defined by the set of all 1st-order closures (or continuations) of any set, or
equivalently by the binary relation on all observable points of being in
contact or not. There is though the definition of interior point of a set.
The proof can by enhanced with induction on the number of points for any
finite model of the geometry! (Here we take all models of the geometry
that have finite many points, on both layers and even the concept of finite is
ramified to observable finite or not). The completeness of 1st order logic, the
adequacy and completeness of these axiomatic systems as stated in 2) and the
fact that all axioms are within 1st order logic, give the validity of the
proposition for the axiomatic system. If we do not want to make use of a
completeness as in 2) but a completeness as in 3), that for sure does not
require axiom-schemes in the new mathematics, then we must formulate, the new
axiomatic systems appropriately for this and we get a different proof for each
one categorical such system which is as we remarked a decidable system.. The
previous proof shows how indeed the "naive" belief of Hilbert,
that practically most of the mathematical problems are solvable, as
integration of his formalism program, and after they have been formulated, is
indeed almost so, through the decidability property, but only at the
appropriate layer and with the appropriate assumptions about the available
resources in logic and numbers.
In the same way we may as
well obtain a relatively simple proof of the Riemann Hypothesis, of the zeros
of the zeta function, over the (finite resolution) complex numbers, which cab
be also essentially the proof of
The above approach shows a
different mentality and philosophy, about many unsolved problems in
mathematics, which may also prove that such problems are essentially
meaningless to try to prove in the old setting of infinite mathematics (layer
7), but naturally solved in the practical and finite approach (layer 1) of
mathematics
05) A direct proof of
the Poincare Conjecture in the new double layer and finite resolution geometry
for small depth resolutions.
06) A direct proof
of the Riemann hypothesis in the new double layer and finite resolution
complex numbers for small depth resolutions.
07) A direct proof of
the Goldbach hypothesis, in (finite) systems of natural numbers for small
size of maximum number ω.
A tour in Mathematics as seen at
the Layer 1, of mathematics without the infinite
Let us make a tour in
classical Historical mathematics (of layer 7) as they could be re-written at
layer 1, after founding them without the concept of infinite. With new
techniques of visible and invisible elements and resolution, the infinite
disappears. Many unsolvable problems become solvable. The feeling of such
mathematics is more positive, and optimistic. Some of the known complexities in
concepts and cases completely disappear too, but other complexities of
revealing importance appear. Let us not forget, that neither
I take as an example the
three volumes book with title MATHEMATICS , Its Content, Methods, and Meaning,
edited by A.D. Aleksabdrov, A.N. Kolmogorov, M.A. Lavrent'ev and translated in
English by K. Hirsh. The 3-volumes collective work, has been published by The
MIT Press , Copyright 1963 American Mathematical Society. We shall make a tour
among the chapters of the book, and make some sketchy remarks of how the new
finite foundations affect, improve , simplify, or make more sophisticated the
various areas of mathematics.
Chapter I General view of
Mathematics
No comments
Chapter II Analysis
The infinitesimals dx as discussed above, are simply rational numbers in decimal form, but below the accuracy level (referred as resolution)
of the implementation or
instance of Analysis (e.g. 0.00001000). There is no need of limits, that would
correspond in the present approach to a sensitivity analysis among different
resolutions. All become simple , comprehensible, transparent, as far as a
single invisible resolution is concerned. It is a perfect restoration of
ideas of
The integral is also unique (Cauchy, Riemann, Lebesque, Daniel-scheme etc are all become the same integral). According to the depth of the continuum and the resolution, the definite integrals exist and are finite sums provided the integration limits are apart by a threshold distance.
Nevertheless a
multi-resolution analysis, is by far more comprehensive and sophisticated, as a
function, may be differentiable in up to a resolution, but non-differentiable
in finer resolutions, or vice versa, The same applies to
continuity. Functions are always defined both on the visible layer, and a
1st invisible layer (finite resolution) , and two can be equal at the visible
layer but not equal at the invisible. Going to finer resolutions require always
an extension of the function, as a functions for analysis, continuity,
differentiability, topology etc are definable only till a 1st invisible
resolution.
Chapter III Analytic Geometry
The marvelous idea of
coordinates of Desquartes, may be considered a forerunner of the present
digitalization of images, and sound in computer software. The methods of
Analytic geometry become even more transparent and effective, as now the
visible and invisible points of a line segment are finite and so are the system
of numbers used, as the coordinates. So it is required a double
correspondence of numbers to points: at the visible layer and at the
invisible layer. Their relations is the depth of the resolution. For more
details see above the remarks in the paragraph, about double layer Euclidean
geometry, without the concept of infinite. Only one invisible resolution is
required, even if differentiable curves are included in classical
Analytic geometry. In other words the pixels of invisible points (that
constitute a permanent invisible simplicilization of space, of a cubic or
parallelogram lattice form) , and the grid of visible points. The accuracy
threshold level, here, has to be identified as a fixed visual discrimination
level, at a fixed standard distance from the geometric figures. An other
interesting situation is with the concepts of length, area, and volume. The
Hilbert's 3rd problem holds with a converse way in the this realm of
mathematics: Any two polyhedral that are of equal volume are also
equidecomposable. The tools of Dehn invariant and of equal volume but
non-equidecomposble polyhedra of infinite Euclidean space are of null
interpretation in the finite resolution Euclidean geometry. This is not strange
as difference between the Euclidean geometry based on the infinite Cantorian
sets, and Euclidean geometry of finite resolution. As we mentioned above in
this page, the "surgery" of taking one Euclidean ball, and by
the axiom of choice cutting it in to finite set of pieces and then
reassembling them to give two (!) balls, although it exists in the Euclidean
geometry of infinite Cantorian sets, it does not exists in the finite
resolution Euclidean geometry. It exist only as giving two balls but of lower or
coarser resolution to the resolution of the initial ball. Conversely there are
phenomena of perfect symmetry in the Euclidean geometry of finite resolution
that do not exist in the Euclidean geometry of infinite Cantorian sets. We
already mentioned e.g. the case that orthogonal triangles have always sides
that are rational numbers, and the Pythagorean theorem holds exactly too. An
other example is that we can tessellate a spherical surface with a large number
of congruent spherical squares. (or a large number of congruent spherical
equilateral triangles). (To see how this is possible, take e.g. the projection
of an inscribed cube on the spherical surface to give an initial tessellation
of 6 congruent spherical squares. This is the best you can
have in the Euclidean geometry of infinite Cantorian sets. But in the Euclidean
geometry of finite resolution we can have massive number of perfects spherical
tessellations. We divide e.g. the sides of the inscribed cube with a sufficient
large number of squares, so that when projected, on to the spherical surface,
their difference. is below the discriminating threshold of the congruence
up-to-the space resolution. Therefore they are all perfectly congruent in the
space's resolution although there is of course a finer than the initial
resolution of the space, in which they are not congruent.). In general in the
Euclidean geometry of finite resolution the conceptual order and perfection is
higher, while many of the logical difficulties of the Euclidean geometry of infinite
Cantorian sets, seem as if of tricks to impress and make people spent time as
academic researchers in to matters that are not of real value in applications,
or in the physical ontology of the world.
Chapter IV Theory of Algebraic
Equations
Here we have many changes.
The methods of solving equations , with symbolic calculations, closed formulae,
radicals, four operations etc, loose a lot of their significance and interest,
after realizing, that :
Because all real numbers
are a finite system of rational numbers, represented in decimal form, and an
equivalence relation, up to an accuracy level, even the perfect algebraic
solutions, have to be exact rational numbers, up to some decimal. Therefore,
the computer algorithms to solve an arbitrary equation, in most cases are not
worse methods or approximation methods but exact methods.
Nevertheless if we change
the reference resolution we get different solutions, which brings a whole new
realm of new sophistication in the domain.
Of course the
classical techniques of symbolic calculations to solve particular types of them
are not lost.
The celebrated theories of
Galois and others, the theorem of Abel on the non possibility of solution of 5th
order polynomial equations with radicals etc turn out to be much trouble just
for the sake of restricting the methods to find solutions.
In the present universe of
mathematics where the Cantor’s infinite does not exist, it is possible to have
simple and powerful results:
For every polynomial
equation of any degree (2nd , 3rd, 4rth, 5th
,nth etc) , there is an algorithm that factors it in to first order and second
order polynomials, with real coefficients, thus also an algorithm to find all
its complex and real roots. Furthermore the algorithm can be chosen so that the
only operations used are addition subtraction, multiplication, division, and
integer power!
Chapter V Ordinary Differential
equations
All differential equations
, are solvable with almost invariably the same algorithm, up to a resolution.
All functions are finite many in this universe of mathematics. Of course the
classical techniques of symbolic calculations to solve particular types of them
are not lost. In the present new universe of mathematics without the infinite,
the previous mentioned algorithm has complexity depending on the depth of the
resolution and can be therefore much too high. Algorithms of radical less
complexity can solve any first-order differential equation , and in general any
differential equation , that can be solved in one side with the values variable
and in the second side all other variables, can certainly be solved, as it is essentially a difference equation , with
the additional specification of the equivalence relation of accuracy level,
which simplifies a lot all recursive calculations (thus simpler both from a
differential equation in classical mathematics of the infinite that requires
convergence issues, and simpler than difference equations in classical
mathematics, as the equivalence relation of accuracy level of the finite number
system, saves redundant calculations)
No need of limits or speeds
of convergence etc, We do not talk about "approximation" as the
mathematical ontology, and definition of functions etc are always up-to-a
resolution therefore all solutions are exact . Thus there is no need to have
two courses, one "differential equations" and a second
"numerical analysis" to practice the solutions with computers. The
very first course is the same time the second.
Various physical, or
social, or financial phenomena, have systems of differential equations, up-to-a
resolution. Nevertheless of we change the resolution, and accept smaller and
larger numbers relative to the unit, and different levels of significant level
of accuracy, he we may have to add some new equations to handle new areas of
quantities and mutual relations. In addition the same system of equations,
although with a unique solution in the classical sense , may have different
solutions, at different resolutions!
This formulates for the
first time as far as I know, the old idea that "causality is a block of
flats", or a many organization layers hierarchical system of
contingencies, in a purely deterministic way. A complicated phenomenon in many
levels of time, space, material realm or other informational realm, might
be formulated as a function defined in many resolutions, by differential
equations that are supposed to describe its "causal law". It might be
thought that the unique necessary causality is the form of the differential
equations at the finest resolution . But it seems that the truth is, that there
are separate causalities , with different form of differential equations
(describing the causality in two consecutive resolutions) at each resolution.
The causality at the finest resolution is significant only at the finest
resolution. So although the whole phenomenon may function as a totality, the
causalities are many and different at each resolution. An example of this
approach is the many different explanations, of the same e.g. social events,
that different groups of people give (e.g. sociologist, politicians,
psychologists, religious people, astrologers, economists, biologists etc) Very
often they reflect contingencies in different layers of organization of society
and the world, relevant to the events.
Of course even if we
restrict to a single only invisible layer, which I do recommend, we get the
usual formulations of the "causality" of phenomena in the form of
familiar differential equations, with the advantage that they have a more
clear-cut practical numerical interpretation, without limits, easier to solve
and to teach in students.
If in the deterministic
mode the idea of hierarchical specification of the causality may seem
surprising, at the non-determinist stochastic mode it is certainly already a
quite familiar technique, as we can for example see in the discipline of
Hierarchical Linear Models (HLM) of time series and stochastic processes.
Chapter VI Partial
Differential equations
The same remarks made above
for ordinary differential equations apply for partial differential equations
too.
Chapter VII Curves and
surfaces
The remarks made above for
Analysis, and analytic geometry apply in combination here.
Again we must remark that
although the classical calculus, and differential geometry of curves and
surfaces, at single resolutions is a lot simpler and more realistic than the
classical approach, we have now a new source of complexity and sophistication:
The Curves and Surfaces, if defined simultaneously in many resolutions, may
possess different differential (or topological) structure at different
resolutions! But for repeating all the good results of differential geometry
only one invisible resolution is adequate. The "tangent space"
is literally the geometry by the invisible pixels, inside a visible
pixel, and parallel connections, curvature, metrics etc are definable, without
limits, in a transparent charming, clear way, that even an a accountant that
does not use variables can understand!
Chapter VIII The Calculus of
Variations
The remarks made above for
Analysis, apply here too.
Of course the classical
techniques of symbolic calculations to solve particular types of variations
problems are not lost. But any function is now equal to another up to an
accuracy level, (or up-to a resolution) thus numerical techniques are simpler
uniformly the same for different problems, and thus more effective.
Chapter IX Functions of a
Complex Variable
This traditional beautiful
subject does not loose its beauty, but becomes even more beautiful and transparent.
The same remarks made above
for Analysis, apply here. All series of analytic functions are up-to-a
resolution, therefore they have a finite number of terms. The concept of
conformal or analytic function is only up to a resolution. If they are defined
simultaneously in many resolutions, they may possess different differential (or
topological) structure at different resolutions! Thus the analytic properties
of complex functions become layered. This is source of new sophistication (the
cost of eliminating infinite), but a single resolution the whole theory is
simpler, more realistic, and more beautiful. For small depth resolutions, a
computer, or smart proofs, can easily answer old celebrated problems, like
Riemman's hypothesis etc
Chapter X Prime Numbers
Almost nothing changes
here. I should remark though, that even the system of natural numbers is only
up to a maximum number ω (omega).
This number may be unknown
or hidden, or explicit. Nevertheless it exists, and affects, not only the
ontology of the theory, but also the arguments. E.g. We may have simple proof
of the Goldbach hypothesis, or Fermat's theorem, for all numbers (of
course up to ω), if we assume a particular form of
ω, based on its prime number
decomposition, or number of its decimal digits.
The classical proof that
the square root of 2 is an irrational number cannot exist in the present
approach, as the equality of natural numbers is not to me confused with the
equality of rational (real) numbers up-to-a resolution. The square
root of 2 is easily proved to be a rational number in a particular resolution!
Chapter XI The Theory of
Probability
This very important subject
changes in the same way as the Analysis. All distributions are up to a
resolution. All the moments of distribution are finite many. The characteristic
function of distribution is a series of finite many terms. The continuous
random variables are those that the distance of two possible values may
be less than the visual threshold! Again all become simpler and transparent at
a single resolution. But we may have also multi-resolution probability and
statistics.
Various physical, or
social, or financial phenomena, have systems of stochastic equations, up-to-a
resolution. Nevertheless of we change the resolution, and accept smaller and
larger numbers or probabilities relative to the unit, he we may have to add
some new equations to handle new areas of quantities and mutual relations. In
addition the same system of equations, although with a unique solution in the
classical sense , may have different solutions, at different resolutions!
In particular the paradoxes
of geometric probability (Bertrand's, Buffon's needle etc) are better
understood why they are met, after the specification of resolution in geometry
and resolution in the quantities of probabilities. The probability sample
spaces are finite, and all the paradoxes are resolved in to crystal-clear terms
in a unique way, that most would find almost obvious due to new details of the
geometric ontology that were not existing in the classical geometry of infinite
point sets.
Although the next remark is
not directed related with the changes that the ontology of finite resolution
makes in maths, it is of significance to mention if we want to avoid
tactics that breed intentions to almost lie with sophisticated scientific way.
The standard way that statisticians or applied scientists "fit" or
estimate a stochastic processes or time series over a one-element sample of
paths, (a single only observed path) must be definitely be avoided! We can
"fit" in this way plenty many radical different stochastic processes
with high degree of classical "goodness of fit" measures and
practically claim all different and opposite assertions ! Statistics requires
repetition, and large samples, and this applies in the case of stochastic
processes to paths and not points. So at best, a way to cut and make many
elements sample of paths, is required when only only path is observed!
Remark about stochastic differentiation and stochastic
integration.
Many interesting
changes occur at the theory of continuous time stochastic processes. E.g. the
ITO's stochastic calculus is entirely simplified as the stochastic Integral is
after all a finite sum of random variables. The role of accuracy level
threshold, is the key to simplify all the troubles of the stochastic
convergence, and different types of stochastic limits in defining the
derivative and the Integral. The new stochastic calculus based on a resolution,
is not only mathematically more robust but also easily comprehensible. On the
other hand other types of continuous time stochastic calculi, like this used in
the signal processing, sound and image filters etc is also entirely simplified
and in complete correspondence to actual implementation with computer software,
without the traditional concept limits and of "approximation".
Furthermore the continuous time stochastic processes used in quantum mechanics,
became also better understood. The latter stochastic processes are Markovian
and are usually described, rather, by higher order partial differential
equations that rule the time evolution of the probability distribution,
than direct equations of the state random variables. An interesting remark is
that, if such PDE of the probability densities are higher than first order, and
as any higher order Delta involves more than two terms, then at the invisible
grid of points (fixed resolution) the process as discrete time series may
be non-Markovian with many steps memory, while at the grid of visible
points appears as Markovian with one step memory only.
Definitely the stochastic
differential equations are vastly simplified as a subject, and again
practically all stochastic differential equations are solvable almost with the
same algorithm.
Chapter XII Approximation
of Functions
As functions are always
defined only up to a resolution, both at the visible and at the invisible
layer, there is no approximation, but exact ontology. This subject is
already included in the other subjects.
Chapter XIII Approximation
methods and computing techniques
As functions are always
defined only up to a resolution, there is no approximation, but exact ontology.
This subject is already included in the other subjects. Here it is apparent
that all of the mathematics of layer 1 have a direct implementation in
computers.
Chapter XIV Electronic
Computing Machines
The mathematics of zero
level without infinite may be called of course computer mathematics , as
realizing them in computers , is entirely more easy and appropriate, and may be
a good emotional motivation to work them out. But we should not miss the point
that can be developed entirely on paper, and a teaching white or blackboard,
without computers at all. They can also be used as an extensive manual in paper
form for any implementation of mathematics in computers.
Chapter XV Theory of
functions of a real variable
The remarks made above for
Analysis apply here too. The system of real numbers is a finite set of rational
numbers in decimal representation, thus all sets to be used in measure theory
are finite many and with finite elements. This makes all simpler , to simplify
arguments, and make new more powerful theorems. There is no discrimination of
Cauchy Integral, Riemann, Integral, Lebesque Integral, Daniel Integral etc All
are the same for a single resolution. But if functions are defined
simultaneously in many resolutions, then the same Integral, may have different
values at different resolutions!
Chapter XVI Linear Algebra
The mechanisms of
solving linear systems of equations, and the basic theorems of Linear Algebra
and Linear vector spaces remain, with an important point in mind: The very
concept of Field of numbers and vector space, strictly speaking do not exist
here.
The number system is not
a field. To define a kind of closeness (or it should be called openness!) to
addition, and multiplication or scalar multiplication, it is required to refer
and discriminate between the visible and invisible points. The specification of
resolutions is again here the new key.
Chapter XVII Non-Euclidean
Geometry
The same remarks made above
for Euclidean geometry apply here too. Because the congruence of
lines, triangles etc is only up-to-a resolution and accuracy level, and have
always a finite length (they are segments) , we may have even in Euclidean
geometry non-congruent lines (segments), from a point outside a line,
that still they never intersect within the geometric space (which is of course
bounded and finite size "window")
Thus the very definition of
parallel lines has to be done carefully (e.g. based on angles to a common
intersecting third line)
Chapter XVIII Topology
As it is known topology
comes from the continuity properties of functional and geometric entities. But
as continuity is always here up to a resolution, topology too, reflects
continuity properties only at single resolution. In addition all points are
finite many. It can be based on the concept of two points being neighbor or in
contact. We must also discriminate between visible and invisible points! These
issues is new source of sophistication, as we have eliminated the infinite. The
closure of a (finite of course) set of invisible points may be a set of visible
points. Thus indepondency (closure of a closure is the same closure) of the
closure operator may be here by definition. But if we define a closure operator
on the visible points (to include all visible points that are in contact ) then
the resulting operator is not indemponent! Therefore the axioms and initial
concepts of of topology are different here!
Many arguments of topology
become simpler, and many different types of topological spaces that make sense
in classical mathematics of the infinite, do not make sense here.
Chapter XIX Functional Analysis
This is a course that many
things change completely! The remarks made above for analysis, differential
equations, probabilities , topology etc apply here. Two functions are equal
only up to an accuracy level. An the functions are defined only up to a
resolution.
To define Dirac's Delta
(e.g. at zero) , we need two number systems, at two different resolutions. In
the coarser resolution the function seems zero every were, and with value not
definable at zero, while at the finer resolution, is not zero, everywhere, and
finite at zero with a value larger than the largest number of the system of
numbers at the coarser resolution .If integrating at the finer
resolution, it gives a number, existing, in the coarser resolution too and
equal to 1. All are simple and there is no need for the twisted functional
definition of Schwartz, neither of sequential definition of Shilov! Engineers
would recognize in this definition of Dirac's Delta, what always had in mind
but was never formulated and defined in mathematics, as the concept of finite
system of quantities at a specific resolution, had never before been defined in
mathematics
All the functions of
functional space are finite many, and the linear space dimension is finite too!
Thus the complications of Unbounded, and bounded operators in Hilbert spaces do
not exist here. All arguments become easier, and many new theorems of
remarkable power can be proved. Although the functions are finite many, they
may be large even for a computer according to the depth for a resolution. But
making then so many so as to permit a computer to scan them, is always an
optimistic attitude as far as trying to prove a theorem in functional analysis.
The interesting theorems
are of different nature in the finite resolution functional analysis. E.g.
Instead of proving that any "almost periodic" function is a limit of
series of purely periodic functions, the interest here is similar to the
Shannon-type theorems, concerning the size of the required information:
How large has to me a base
of periodic functions to derive exactly at a resolution, and the visible layer
a function? And similar many more questions relating the information at the
side of base of functions, and at the side of visible accuracy level and depth
of resolution.
The complications of the
axiom of choice in set theory , disappear too! Let us look to the
celebrated arguments that proves with the axiom of choice in Euclidean geometry
that we can cut a finite spherical ball in to finite many pieces, and to
resemble them to make two spherical balls of equal radius with the original. In
the light of finite resolutions in geometry, the arguments is essentially
equivalent to that we can indeed do that but the derived new spherical balls
are of lower resolution (so that the sum of the finite many points of the
resulting new balls make in total the points of the original ball in higher
resolution!)
Chapter XX
Groups and other Algebraic Systems
Only the finite groups , of
classical mathematics of infinite, and finite algebraic structure exist in
exactly the same way here.
The infinite algebraic
structures of classical mathematics do not exist in literal way of definition.
In their place exist new entities, defined at two layers of elements so as to
define a kind of "closure" or "openness" to the algebraic
operations. An example of how this is treated and handled here is the systems
of real numbers, at a resolutions, that has only a finite number of elements
eventually. Of course most of the techniques of morphisms, categories, automorphisms,
isomorphisms, inductive limits, systems of generators, free algebras, and other
concepts of universal algebra, can essentially survive here too.
THE OLD 19TH CENTURY EUROPEAN UNIVERSE OF MATHEMATICS
WITH THE INFINITE
OLD
INFINITE MATHEMATICS
HOLY GRAIL COULD BRING DISASTER IN THE COMPUTERS OF THE INTERNET ?
These papers were
written by the author in order to resolve two important situations in
mathematics, as he noticed while lecturing in the
a) For 2 or more centuries
mathematicians and physicists were writing equations where the infinitesimals
were treated separately and together but as different than ordinary quantities.
The present Differential Calculus makes use only of their quotients which are
ordinary real numbers.
b) Till the 19th century
all mathematical entities were generated by numbers (arithmogenic). Including
all geometric shapes curves ,surfaces etc , after the arithmetisation by
coordinates by Desquartes , Riemann etc. During 20th century , and after Cantor
, the ontology of mathematics changed and all entities are created by sets
instead of numbers. So, the question arises: Is it possible for entities like
numbers or computer procedures to gain back the power to create all entities in
mathematics?
The present work
resolve mainly the part a). But in order for the results of them to
have fruitful applications , the part b) is also resolved, which is also
relevant with a major turning point in the history of mathematical
thought, on his planet.
The true resolving of issue
a) is through a redefinition of the system of numbers , as finite systems of
finite (rational) numbers, with a finite resolution, (exactly as the computer
represents the numbers but not only with single and double precision but also
with many degrees of precision ) where the phenomena of "orders of
magnitudes" can be formulated by concepts almost the same as the concepts
of non-Archimedean or pre-emptive orders of infinitesimals , finite and
infinite numbers etc. As in every creation there is a phase of
preliminary artistic design prior to developments, here also the artistic and
phenomenological abstraction of infinite numbers is a prelude to it. The
discrimination between finite and infinite or various grades of infinite is
simply a discrimination of transcendentally separated (meaning with large gap)
areas of the finite that may have also different informational and logical
determination. In addition the concept of the abstractness of the
infinite is also, a measure of how detailed and specific are the logical
expression means and informational handling of the system of mathematical
entities, which can always be assumed finite in the ordinary sense.
The simplest concept of
infinite arises when the cognitive resources of space or time in representing
numbers or information or data-objects (for an individual or a group of minds,
a standalone computer or a computer network etc), cannot represent a
number or data-object of the environment physical ontology, because it is
too large. Then the alternatives are not to represent it at all or to represent
it with an abstract object (e.g. a set) or with a symbol representing an
unknown constant or variable, as this number may also change while we do not
have the resources to count and determine it. This is the transcendence for the
infinite
The next papers
belong to the old mathematics and from the point of view of the 7-layers, in
layer 6 and above. As I see it at the present time (2005) the value of using
infinity (besides its historic role of an earlier phase in the evolution of
civilization), is mainly that it permits a distance from the finite material
ontology, and permits free thinking which is felt better . Otherwise
practicing is hardly separated from thinking anymore and hinders thinking with
many emotional traps too Therefore infinity should not be taken
seriously in its literal sense, as a new different ontology as this
would lead to a collective paranoiac dead end without hope for practical
applications or with more and more difficult practical applications. Not to
mention that it would become like a "computer worm" a mental
seduction that would consume more and more time and would lead more and more
away from reality and life. In my way of appreciating it today infinite
can be considered mainly as a metaphor or "encryption
code" of facts about the finite. Probably facts about the finite that
the civilization had not been entirely ready at its time of introduction.
Thus the concept of
infinite is related with the limited measure of the chances of a group of
human minds either as natural or artificially extended, in dealing with
the ontology and phenomenology of their environment world that surrounds
them. From this point of view we may consider as an early study of the
infinite in ancient Greek culture the book "Psamitis" (which means
"sand") of Archimedes about very very large numbers. Obviously what
is objectively infinite changes as the collective cognitive civilization
resources change. E.g. what is infinite when counting with pencil and paper, is
different with what is infinite when counting with a computer, and higher order
formal languages in Logic.
The dynamic concept of
infinite: as a procedure that e.g. computes the digits of the number π and is terminable, only by an external
artificial stopping time or length, and not intrinsically by the logic of the
algorithm, is analyzed in the present approach as follows: We see in this
concept two different elements, instead of one in the traditional mathematics
(Traditional mathematics sees the infinite number of digits, of the one
irrational number π). We see, at first the category of
finite entities, which are the finite instances of π, (rational numbers), on which the algorithm applies, and we see also, a
procedure or algorithm of a special type, and of finite syntax length too.
These two entities are not to be confused, as one entity. Numbers are to remain
always finite and rational, while the "cardinalities" or
"ordinalities" of "infinite" is to be defined and analyzed,
as complexity structures of these special types, of (externally terminable)
algorithms. Thus the dynamic concept of infinite is to be reformulated
separately, as finite length data, entities, and finite algorithms of a special
type. The concept of algorithm has many variations in computer science,
but the basic alternatives of them have been proved equivalent. There are of
course some logically non-equivalent gradations of the concept of algorithm
based on the mutual combinations of the of their syntax-size , input-data
memory size and run time complexity bounds. And also we may think of new
concept of algorithms that computer science has not formulated and study yet
(e.g. algorithms that do not have a fixed syntax. They might have a fixed
nucleus of syntax that itself can reproduce the rest of the syntax in a
variable way, always of course with an fixed upper limit to its length), Still
the discrimination between the number as a finite data element (a rational
number) and a well defined finite algorithm that acts on it and increases
its information, should always be made. The diagonal arguments of the hierarchy
of cardinalities and ordinals, would correspond to the diagonal arguments of
"non-computability" of some decisions by some type of algorithms.
This "non-computability"
is an effect that is almost always converted to that the complexity of an
algorithm, that enumerates, what a system of other algorithms does, and
themselves too, should have higher complexity that all of them, and if there is
a constraint to that complexity in its definition, then obviously it cannot be
written or run, as such a type of algorithm. The diagonal arguments of Goedel
in Logic about the "non-provability" of some formulae, are again
converted to such effect of an impossibility due to complexity constraint. E.g.
if we want a software to verify the consistency or inconsistency of all
possible paths and events of an other software system, then this has to
be of a very larger (memory space, and run time) complexity, compared to the software
that has to analyze. And if we have put an upper limit to this complexity,
which may be non-adequate, then we result with an impossibility. The
impossibility becomes always again a possibility of course, if we relax the
complexity constraints.
We must not forget that the
concepts of infinite and irrationality (of numbers) refer and are
properties not of the ontology of the entities of study but of the states
of consciousness (individual and social or collective) of the subjects that study
and make the knowledge. It is the evolution of culture and societies, that a
century and a day may come that this may change to finite and rational.
The true resolving of b)
should be again by an extension of the concept of natural number with that of
computer data-object, where not only a complete order but in a
more economic way, non-complete or partial orders between them are meaningful
and hold.
Pythagoras was referred to
believe that "figured numbers" is the key to universality. Part b) is
for the moment resolved with a 7-layers description of any mathematical entity.
The key is layer 5 which is countable model of ZFC-set theory where each set is
computable and consists of computable sequences of finite trees created
by algorithms. In addition this model (or set theory is unique
up-to-isomorphism of the Î relation
of belonging (categorical axiomatic system with stable semantics).The next
papers contain the details, in relation to part a) . But there is a hint
in their introduction about a model of ZFC-set theory with sequences of finite
sets of cardinality at most of the continuum. This model can be improved to be
of computable sequences of growing finite trees, and the trees can be defined
to consists of numerical digits or alphabetic characters. Thus much like the
world created by a computer with itis 0,1 bits. The details of how this models
ZFC-set theory resolves part b), together with some new concepts like that of
internal and external consistency and 1st and 2nd completeness of 3-valued
Logic, is something that the author might present in
detail in the future. These results are not contradictory with the first or
second incompleteness theorems of Gödel , but rather they complete Gödel. With
this I mean that instead of assertions like "It is not possible to prove
within the formalism of a system that it is consistent" we are interested
rather to assertions like "It is not possible to prove within the
formalism of a system that it is inconsistent" [internal consistency] or
"It is possible to prove and we prove outside the formalism of
a system that it is consistent" [external consistency]. And instead to
assertions like "There is a statement of the formalism of the system that
cannot be proved from the axioms of the system" [1st incompleteness within
2-valued Logic] we are interested to assertions like "For any statement of
the system, it is decidable if it is independent from the axioms or not,
although this does not mean that that it is decidable if it is true or not
" [3-valued Logic completeness]. A 3-valued Logic complete system need not
be a (2-valued logic) decidable system. In a decidable system you know
everything, but is a 3-valued complete system you know what you do not know.
The 7-layers approach to the creation of
mathematics proves that all of the known mathematics can be created with a
succession of enhancements of layers such that each layer has as logical model
by interpretation the previous layer.
(As a visualization I
prefer a horizontal , image of the 7 successive enhancements of mathematics,
than a vertical building-like image, that may unavoidably make some think of
the old tale of the building of
Layer 1: It is a finite entity (like a
microprocessor in computer science) or a finite set of finite entities
(trees) , with only finite many propositions in the formal language and
the system is decidable [2-valued logic complete] (like the bounded resources
of disk and ram in a computer). In this layer everything is decidable and
known. We can have alternative set theories in layer 1 , so that not only all
sets are finite (and of finite rank) created from the empty set, but also
there is an upper bound on their cardinality (horizontal size) and rank
(vertical size) which gives that there are finite only different (finite) sets.
Such models of such set theories have a representation inside the computer as
say file-folders with the belonging relation € corresponding to the obvious
relation of a file-folder belonging to an other file-folder or at a
higher layer as tables of records of a finite memory size Data Base, where a
record represents already the information of an other table.. We may call such
set theories "Computer Data-Structures Set Theories"
Layer 2: It
is again a decidable system of finite many finite entities (trees) but
the acceptable formal propositions , and proofs, are computable many (maybe
infinite. Thus the infinite enters here but in the language first).(An analogy
with board ram and hard disk in the computer)
Layer 3: It
is a system of countable many finite entities (trees) with computable
many formal propositions and proofs such that although it is not decidable, it
is decidable if a propositions is not provable [3-valued Logic complete, much
like the user of a software that has been granted less responsibilities
relative to the programmer but he knows what he cannot do], So although in this
layer, it is not everything decidable known it is decidable known what it is
not known. (It is like the machine language and the operating system in the
computer).
Layer 4: It is a countable undecidable
system (the, as I call them, Pythagorean sets or trees, like the 1st-order
Logic or Natural numbers. Pythagorean sets are finite sets (finite horizontal
size) and of finite rank (finite vertical size)) , and the acceptable formal
language is also countable and undecidable. (like a programming language
computer science). In such a system there are not any infinite sets except of
the empty set which "looks" like an infinite set! It can be
considered that the ancient mathematical though is described till the layer 4.
The modern scientific and mathematical thought is described from layer 5 and
on.
Layer 5: This
is a very critical layer. It is a countable system of "weakly
computable" sets and the acceptable language is also countable
and undecidable. (It is like the RDBMS of a programming language). In this
layer for the first time each object of the system can be an infinite
object (in spite the fact that computer programs can have only finite lenght).
As it is computable, the syntax of the program to compute it is again finite.
It is not at all easy to prove that such a "simple" system can be a
countable model of the Zermelo-Frankel ZFC-set theory or BGC-set theory that
accepts infinite sets. Again, how a computer would represent sets, is the
guiding idea. This model of set theory has semantical stability in
the sense that it is a categorical theory (all models of it are isomorphic
up-to-the initial concepts, a property that the theory of real numbers and Euclidean
geometry has too). It is important though to notice that this
"weakly computability" is a triple of 1) an algorithm or timer
that counts time steps of a clock, 2) an general algorithm or an algorithm that
is a "perpetual program" (in other words an ordinary algorithm
that is terminating not internally but by the constraint that it has run so
much time till now) and 3) an initial input Pythagorean finite set. Such
a triple is assumed to represent a set e.g. like the sequence that computes the
irrational number, square root of 2. The perpetual program may change the
initial Pythagorean set and produce finite instances, till it stops. The timer
may vary the time that the algorithm is applying. Nevertheless if we try to
represent the overall process of increasing also the time duration of the
perpetual program we result with nothing else than a virus, that consumes the
resources of the computer till we close it! Such a model of ZfC may be called a
"Computer Processes Set Theory"
Layer 6:
Which is the well known universe of Zermelo-Frankel or Bernays-Goedel
set theory with the axiom of choice. (like a data-base application from
the point of view of a user).
Layer 7:
Which is a classical theory like real numbers or geometry or geometric
manifolds or functional spaces, or random variables and processes etc. (It is
like an application software).
The analysis of layers,
itself is assumed in to use as context, the layer 5. This proves that any
mathematical entity, can have many different dimensions of existence according
to its view from a corresponding layer of the creation of mathematics, and
level of externality of the user of it. There is no contradiction at all
in its different properties at each layer. At each new layer because of reasons
of abstraction we eliminate some of the information (axiom) that specifies the
objects, which still holds but only in the lower hidden layer. This approach is
an elaboration and a more synthetic and organized system of concepts that the
classical two-part theory-model. So any mathematical entity is at the same
time: 1) a unique finite entity, 2) a finite entity of a decidable system,3) an
entity of 3-valued logic complete system ,4) simply a finite entity of an
infinite universe, 5) an infinite but computable entity of an countable
universe 6) a set of ZFC-set theory of some cardinality possibly very large ,7)
a classical mathematical entity like an irrational number or a line of
Euclidean geometry or anything else. From layer 3 or 5 and beyond the gap between
"thinking-deciding", "acting" (constructability) and
"having" (ontology) increases and we come to the phenomenon of
mathematical theories that are more "oracular" or "mantic"
, (which in classical terminology of philosophy of mathematics means highly non-constructible,
based on existential assertions only and are not categorical axiomatic
systems.) than technological- rational and practical activities. The
non-categorical character of the axiomatic systems after layer 5 and their
"oracular" character resembles, the situation of the relation of the human
consciousness and the reality of its physical existence. The
way that layer 6 is modeled in layer 5 is critical, as layer 6 is not
categorical while layer 5 is and it shows how the infinite can be modeled in
the finite so that "the ontology of the infinite is the phenomenology
of encrypted changes of the finite where information about the changes is
missing (abstractness)" Or to put it in an even more clear statement:
The discrimination between the ontology of the finite and infinite or various
grades of infinite is simply a phenomenological encrypted discrimination of
transcendentally separated (meaning with large gap) areas of the finite that
may have also different informational and logical determination. In
addition the concept of the abstractness of the infinite is also, a
measure of how detailed and specific are the logical expression means and
informational handling of the system of mathematical entities, which can always
be assumed finite in the ordinary sense in the ontological layer. The simplest
concept of infinite arises when the cognitive resources of space or time in
representing numbers or information or data-objects (for an individual or a
group of minds, a standalone computer or a computer network etc), cannot
represent a number or data-object of the environment physical ontology,
because it is too large. Then the alternatives are not to represent it at all
or to represent it with an abstract object (e.g. a set) or with a symbol representing
an unknown constant or variable, as this number may also change while we do not
have the resources to count and determine it and while we reason about it. From
this point of view it is an abstractness, a lack of specification, and a
encryption approach too. This is the transcendence for the infinite
Thus the concept of
infinite is related with the limited measure of the chances of a group of
human minds either as natural or artificially extended, in dealing with
the ontology and phenomenology of their environment world that surrounds them.
From this point of view we may consider as an early study of the infinite in
ancient Greek culture the book "Psamitis" (which means
"sand") of Archimedes about very very large numbers. Obviously what
is objectively infinite changes as the collective cognitive civilization
resources change. E.g. what is infinite when counting with pencil and paper, is
different with what is infinite when counting with a computer, and higher order
formal languages in Logic.
The dynamic interpretation
of infinite, as an algorithm that increases the finite is different. It has
been used often as a way to keep a distance of the material, or human action,
ontology of the finite, especially when the thinker considers it undesirable or
an obstruction in his attempts to think about the situation.
Thus some arguments
became simple and elegant in such systems (like a nice and friendly interface
of a complicated software system) , but this should not be pushed to its
limits, and other types of properties of the same finite system require
different axiomatization which has to be devised or updated from time to time!
This 7-layers approach shows the coexistence of different philosophies and
mind-styles like ancient Pythagorean or Euclidean mathematical
thought , Intuitionism, Logicism, Formalism etc. In particular it becomes
apparent that the Hilbert-program of Formalism is possible to be integrated for
all of mathematics indeed ,but only till a lower hidden layer of them.
This approach shows also in full detail how classical mathematical object like
a Euclidean line or irrational number like pi etc are created by computers and
although infinite and possibly uncountable entities, still at some hidden lower
layer are computable or even finite.
In view of the layer-5 of
the above approach , all of the transfinite real numbers , or surreal numbers,
or Ordinal real numbers , have an exact interpretation as size of concurrency
complexity of computer procedures! Concurrency complexity is a new concept,
it is not space or time , or resources complexity, but a measure of
dependence of procedures in parallel programming.
This restores the
link of human thoughts with practical human actions. A link that apart
from practical applications is important too, for the integration of the
human subjective state. Strange as it may seem, it holds that ,the creative
world of finite has more choices and freedom for the mathematician, than the
creative world of the infinite.
Although the ontology
of the infinite seems radically different than the ontology of the finite, it
may turn out without contradiction somehow, that there are logically valid
interpretations, where
the ontology of the infinite is created by some relations in the
parallel computations on the finite. It is also created by an abstraction on
the time states of an object that is gradually created. The
abstraction is that we are considering it as the same entity in all its
successive states of creation and the interruption or stopping of the
perpetual process is not internal from the procedure.. This is the way to
transcend from the dynamic concept of Aristotelian infinite to the Platonic
static infinite. Although we may accept creations of the mind as Cantor was
requiring, when it comes the moment to link thoughts with human actions
and operations , we have to supplement the abstraction with exactly
the missing information of it, that unlocks the concepts to
practical applications.
Strange as it may seem, it holds that ,the creative world of finite has more
choices and freedom for the mathematician, than the creative world of the
infinite. Although the initial impression was that G. Cantor was leading
mathematics to his paradise, it finally resulted to Cantor’s Hell. (Cantor himself, died mad in the sanatorium).
If we try to discover the closest concept to infinite in the world of
finite (as we shall see in the sequent) we immediately realize that infinite is
the totalitarianism in mathematics, while the world of finite permits real conceptual democracy of creativity. As any totalitarianism seems attractive and
might feel good at the beginning but sooner or later it results in to a totally
wrong and destructive role by its users.
From this point of view to abstract and transcend from the finite to the
infinite (or infinitesimal) is somehow the same with an encryption in
the sense of a logical formulation of the finite with missing information or as
the discrimination of the hidden (programmers control) and visible (users
control) part of a software application system. Thus to translate it in to the
science of informatics (computer science) the (Zermelo-Frankel) axioms of
set theory refer to how to create both data entities and procedures from
other data and procedures. The mental images of set theory, for an
infinite set, point not to the syntax of the procedure that creates it, but on
the fast changing successive states of data created by the procedure. So we
should not be surprised if after all the transfinite numbers, for many
practical applications, turn out to be interpreted in addition to concurrent
complexity also as ordinary rational numbers of significantly
very different resolutions of grids (orders of magnitude). The E.
Nelson's (at
We must remark of
course that if layer 7 (e.g. Real Numbers , or Euclidean geometry etc) have a model
in layer 6 , and layer 6 has a model in layer 5, then , transitively , real ,
numbers and Euclidean geometry have models within layer 5 too, where all sets
are computable (including special kind of non-terminating algorithms to the
definition of computability) ! This is how a computer scientists would try to
interpret the classical systems of real numbers and Euclidean geometry inside
the computer. To see how it would ever be possible to result with countable
continuous system of real numbers (model of real numbers in Layer 5), we just
notice that if in the definition of the real numbers as completion of the
rational numbers by Cauchy fundamental sequence we put the restriction that the
fundamental sequence is not any but a (weakly) computable one by a computer
program, then we get a countable system of numbers representing the continuum,
although by the diagonal argument non-computable anymore! But further-more we
do not really need in the definition "all computable sequences of
.....". We could as well define the real numbers by a an algorithm which
refines say a finite segment of the decimal lattice with finite digits. Then
the number system itself has an algorithm to derive it, thus it is also
computable. There is no need that the algorithms computing a number and
the algorithm computing the number system itself have any sequentially
dependence (sequential programming, an assumption corresponding to that the
real numbers have a higher cardinality than the cardinality of the digits of
one real number) as they can very well have concurrency dependence (parallel
programming ). In a similar way most of the infinite ontology of layer 6
when interpreted in layer 5, reflects this particular type of
"sequential" dependence of algorithms that define them. No other
computational dependence is permitted by the axioms of set theory (like
replacement axiom , power set etc) which is a serious drawback compared to what
can be defined between algorithms. "The simpler the computational
complexity the better" assumes the computer scientist, which is translated
in to that the creation processes available in set theory when translated in
layer 5 in to algorithms to create new entities do not correspond to good
enough practice and low enough complexity in the final procedures. The
available freedom in composing algorithms that increase finite sets is
lost. To derive pre-emptive or non-Archimedean order of the numbers that
corresponds to transfinite numbers as in surreal numbers, we should require a
modification of the equality of two numbers as fundamental Cauchy-sequences ,
and put other definitions that involve more information from the algorithm that
computes them. For example we may be interested not only "where" the
algorithm converges but mostly "how" it converges there. We may
notice that such ideas are close to the critique of the philosophical school of
intuitionism and neo-Pythagoreans to the rest of mathematics, at the beginning
of the 20th century, except that they are even more restrictive than the
intuitionistic techniques in to the next: We do not include "arbitrary
sequences by personal free will..." but only algorithms that can be
repeated by any mathematician or not. Obviously in geometry too, which is
defined after such a number system, only points defined by some algorithm are
included. This guarantees that the number system represent the geometric
continuum (at layer 5). Surprisingly enough in spite the similarities of this
approach with the critique of intuitionism, a reversed slogan seems more
appropriate: Instead of the slogan "Natural numbers are made by God,
all else by man" we should put it like " Only
finite entities are to be made by the activities and the mind of ordinary
present human beings. And from the land of finite only a
limited part is for the human mind, the human consciousness and the human
practice." . The land of finite is today certainly happier and of more
solution possibilities. The discrimination between finite and infinite or
various grades of infinite is simply a discrimination of transcendentally
separated (meaning with large gap) areas of the finite that may have also
different informational and logical determination. We should remark that the
previous example of practical countable model of the real numbers, is
absolutely distinct and different from the countable model of real numbers that
the Lewenheim-Scholem theorem predicts. But even this is not adequate as
necessary clarification and enhancement of ancient and medieval or
renaissance age, mathematics, to 21st century mathematics. In fact the
closure itself of the system can also be substituted with a concept of
successive closure of operations from a resolution-lattice or grid to a finer. In
other words we may as well define the system of real numbers as
finite set (model of real numbers in Layer 1, of course of different axioms and
algebraic structure!) almost exactly as real numbers are represented in the
computer, in any computer programming language , in single or double or higher
precision. To develop a finitary interpretation of the infinite, we must
define a new concept, that of Limited Model, or Instance of an
axiomatic theory . The definition of a limited Model or Instance, is as
the usual definition of a model of an axiomatic theory, in Logic , except that
at all universal quantifications on the set (like "for all natural
numbers"... , or "for all real numbers..." ) it is not
referred really to all the elements of the set, but to a limited subset of it.
So "for all natural numbers" may mean for all natural numbers less
than a limit number n0 which is used through out in all
logical arguments in the theory. After the concept of limited model or
instance, even large theories like the Zermelo-Frankel theory have
limited models, or instances that are finite sets of finite sets! All
cardinals, and ordinals are interpreted in this way as finite integers, while
the order between them as the order of natural numbers! Therefore the
axiomatic models of Natural numbers, Zermelo-Frankel set theory,
Cauchy-Dedekind real numbers, transfinite real numbers, surreal numbers,
ordinal real numbers etc have finite , limited models or instances that
consists of rational numbers, representable in the operating system, and a
programming language of a computer. In such systems of finite
limited models, or instances, all concepts like finite, countable infinite,
uncountable infinite, became logical grades of the usual order of finite
natural numbers. The discrimination between finite and infinite or various
grades of infinite is simply a discrimination of transcendentally separated
(meaning with large gap) areas of the finite and finite procedures of it, that
may have also different informational and logical determination.
It seems that we forget
that the classical axiomatic system of real numbers and Euclidean geometry as
well are not really realizable with terminating algorithms in a computer, and
any literal realization, produces simply computer viruses!
Should we start
thinking of the infinite as "viruses" in abstract thought too? If the
infinite is the inability of the cognition tools to specify the size of the
finite cardinality of the studied entity, or if this finite size is dynamically
changing during the cognition process, why should at all exist a knowledge or
description of it? And if there is, why should we accept it as adequate or try
to found everything in this imperfect cognition under such an unfavorable
situation? Propositions about "all the natural numbers" might be
meaningful only if this "all natural numbers" is always a finite
system, and in addition of size close to the human reality. To demand
"proofs" while this system may change size, and during the processes
of the proof, is an additional difficulty for the arguments. Why should we
accept as "proofs" only such restricted types of logical
acrobatically dangerous process? Take for example a non possible to prove yet, proposition in the
natural numbers. Why should we look for "proofs" that are logical
procedures, for a finite system of numbers, that we have hidden, its exact
size? Or in addition for a system of natural numbers that although finite, and
of unknown size, during the logical argument of the proof, may change size and
increase? Is it an appropriate human spiritual habit, to accept the proposition
proved, only if we can device and argument under such restricted and
unfavorable circumstances? Would it be more perfect for the cognition, and of
honest human interest, to require for a proof, of its truth or not, only if we
refer, to a fixed not-changing finite system of natural numbers and of size
that makes sense for the human world, and only if we specify the upper bound of
the finite size of the proof according to the symbolic tools?
This is not something
that can be overcome in a roundabout method by some say "numerical
analysis" reduction of calculus etc, as this does not really solves the
problem. And it is not simply a gap between two sciences that of mathematics
and that of computer science. It is rather a gap of the present phase of the
sciences and older, preserved till now, phases of the sciences. Its resolution
means a thorough re-examination of the very foundations of mathematics, and an
update with the present state of the art in thinking and manufacturing.
For a
requirement of classical computability with terminating algorithms, the real
line, and any figure of the Euclidean geometry must be endowed with at
least specific highest resolution. In fact for most of the
(elementary) mathematics all such entities must have at least two resolution
layers: one ontological or hidden and finer , and one phenomenological or
observable and coarser, and both are finite sets. The equality of the
phenomenological layer is only an equivalence relation over the hidden
ontological layer, with partition in to equivalent classes, or rounding
classes. This requires that all geometric equations (e.g. the Pythagorean
theorem) are never equalities in the classical sense but always an equivalence
relation after rounding up-to-a resolution not finer than the ultimate ontology
of the figure (or image ) in the computer graphical interface (display screen).
This is not an "imperfection" or "approximation" of
computation, but a new information property of the ontology of the mathematical
entity and exact relations. Therefore there is the requirement of new
axioms of the real numbers and new axioms of Euclidean geometry . It is of
course obvious that the traditional algebraic structures of group ,ring
,field, vector space etc are not appropriate any more and new modifications of them
are required. Happily modern universal algebra studies practically all types of
algebraic structures. If ancient Euclidean geometry as the Hilbert axiomatic
system was a historical phase 1, and Cartesian analytic geometry, together with
rational numbers was phase 2, the phase 3 requires new axioms of real
numbers and Euclidean geometry to account for the fact that we always mean that
the mathematical entity (number or figure or function ) has a finite
resolution. Thus Euclidean geometry and the system of real numbers should be
reformulated in a new finitary way, so that each line segment (e.g. in the
Hilbert axiomatic system, or in the Cartesian analytic geometry or vector space
definition) and the number system always have at least two layers: one ontological
or hidden and finer , and one phenomenological or observable and coarser, and
both are finite sets. The equality of the phenomenological layer is only an
equivalence relation over the hidden ontological layer, with partition
equivalent classes, or rounding classes. Relative to two such resolutions , the
differential calculus has an exact interpretation ,(where equality is the
rounding equivalence relation, defined appropriately so as to be a transitive
relation too, e.g as partition of the fine resolution in to balls around the
points of the coarse resolution), and we may talk about multi-resolution
Differential Calculus. Thus the old Newtonian symbolisms of the fluxions ox
of the number x can very well be interpreted by appropriate rounding relation
in appropriate finite resolution(s).The numbers would be at an observable
resolution, while the fluxions ox (or Leibniz infinitesimals dx) would be at
the hidden finite resolution. Each layer has its exact operations and the
observable induces rounded ones in the hidden layer. And similar simple and
transparent interpretations can be made for the Leibniz symbolism dx (and dx
is finite rational number of course but when geometrically represented it
is below the threshold of the human visual discrimination) with many
advantages over classical definitions with limits. There was an important
reason that the great
in his creative work
, if he would had met an earlier sympathetic appraisal and accompaniment
by sufficient many in his creative work plus most probably a direct support of
a personal rather than creative character he would probably not suffer as he
did in his late 40s and later.
Thus the calculus with
limits could be a simplistic design which is a prelude to a really enhanced
calculus without limits in finite resolutions. If the system of numbers is a finite resolution
algebra, then even the epsilon-delta technique of Weirestrasse in calculus ,
does not have any longer , an interpretation as limits , but as equations
up-to-rounding. The claim of the usual Weirestrasse calculus by limits, and the
usual Cauchy-Dedekind real number system, for an ontology and properties on all
finite resolutions, in practice and in consciousness goes beyond the limits of
the place of the human practice and consciousness. As human beings we are
interested and we can control by our practice only a limited range of finite
resolutions. Therefore we should make our theories for a fixed, although maybe
variable finite resolution within some limits. A continuity and
differentiability in the usual calculus with limits gives continuity and
differentiability up to a pair of resolutions (roughly speaking
corresponding to the epsilon-delta choices). But not conversely. A
continuous or differentiable function up-to-a pair of resolutions is not
necessarily continuous, or differentiable in any other resolution!
I believe that we
must have a correct sense of the historical necessities in the evolution of
mathematics. A geometry created to be used with drawing on the sand or on
papyrus and when physics did not knew the atomic structure of matter, should be
as Euclidean geometry, but if it is to be realized with modern multimedia and
computer techniques of image processing must be a different axiomatic system. A
calculus created to be used for calculations of astronomy or physics in a time
that it was not realized in the sciences the atomic structure of matter, and
that would take place by hand and pen on paper should be as was suggested by
Newton and Leibniz or also as it was changed and developed in the usual system
of real numbers in 18th century, much artistic as it is, and of low symbolic
computation complexity. But a calculus or functional analysis to be computed
with modern computers and in a age where the atomic structure of matters is
realized in the sciences, should have a different axiomatic system for the real
numbers that should admit finite models of them too, and different definitions
and concepts for functions figures manifolds, random variables etc. In short
older quantitative mathematics intended for calculation on the paper and for a
time that chemistry and physics did not know the atomic structure of nature,
cannot be the same with modern quantitative mathematics intended for
computations in the computers and when the sciences have realized the atomic
structure of matter . The former were only a prelude to the latter.
In the construction of
the usual system of the field of real numbers, axioms like the closure of the
operations, and the continuity axioms of Dedekind or Cauchy, or Cantor, simply
reflect the early obscured stage of the theory of phases of matter in chemistry
and the natural sciences, that where still bisecting physical substances
without having yet found any first bottom, or realized the atomic structure of
it. No matter how brilliant and sophisticated might be for its time, it is like
a centuries old software, that has to be updated with an additional one that
the real numbers are rational , have finite bounded resolution and admit
finite models.
The dynamic interpretation
of infinite, as an algorithm that increases the finite has been used often in
the past as a way to keep a distance of the material, or human action, ontology
of the finite, especially when the thinker considers it undesirable or an
obstruction in his attempts to think about the situation.
In terms of the 7-layers
, the above remarks mean that we may reformulate almost all our fundamental
classical theories in mathematics, like geometry, numbers, analysis , set
theory etc, so that are modeled directly in layer 1 , where all sets are
finite! This has
tremendous control (ontological, logical and computational) advantages. E.g.
all functions, are of finite information, can be considered vectors of finite
dimensional spaces, all differential geometry, is of finite information, and
all of the functional analysis is of finite information and in finite
dimensional spaces! We must admit that the multimedia techniques in computer
science that create the sense- continuums of image, sound etc have already
suggested how, from, finite only, sets, we can get the behavior of
the classical continuous entities, in a logically different way from the
classical mathematical definitions that require them to be
"infinite"! This reformulation of basic mathematics is I believe
necessary and puts back the human cultural concepts in the true landscape
of the human mind which is meant to be linked with the works of practical
activities. The revealing missing information for this is the resolution
specification of the entity (number, figure, manifold, function , random variable,
or even set, etc) which can vary , but can always be only finite. That this
enhancement and simplification is a great advancement, can be realized from
the fact that it reflects the way in which the physical reality exists and is
evolving in its atomic particle structure. It is natural that the age of the
knowledge of atomic structure of matter, requires a similar finitary atomic
structure in the mathematical ontology too.
Thus not only the
differential equations of the calculus, may take a new meaning not through
limits, but as rounding equivalence relation up-to-a resolution, but also the
elementary algebraic equations of geometry. This apply even to the Pythagorean
theorem on a orthogonal triangle. Such equations are no longer
"exact" equations with no resolution specification (a
visual-phenomenological exactness based on the senses phenomenology rather than
on logical ontology), but are always exact as rounding equivalence relations
up-to-a resolution. Thus the number pi , as quotient of the circle length to
its diameter, is no longer unique, but depends on the resolution, that defines
the circle, and the diameter, and is always a rational number! There is no such
a thing as one "circle" that we "approximate" with rational
numbers or many polygons, but there are rather many finite-point circles always
up-to-a resolution, which are absolutely exact, and this is all that there is!
Maybe we must escape from the phenomenological and visual monarchy in the mind
to creational-practical ontological democracy of the mind, which is in harmony
with the work of practical activities.
Classical
mathematical ontology does have the property of atomic structure (lines e.g.
are sets of point) but in an artistic and visual style rather than practical
realistic which requires to be of finite only indivisibles too. The abstraction
of infinity as defined during 18th, 19th century and also by G. Cantror is also
an artistic phenomenological abstraction, therefore of an early phase of the
creation of the continuum and mathematical ontology, it has expiration
date, and must be replaced with the next phase in the creation which is
the present and future practical and creational abstractions, based on the
finite and the concept of finite resolution .
In the 20th century, the technology
and art of cinema, showed for almost a century to the collective mind, how the
continuum can me created by an invisible finite (the finite number of pictures
per a time unit , sufficient many as to create the senses effect of the
continuum). Then the computer multimedia extended and refined it to the
high degree of the present new millennium, sophistication and perfection. At
the same time in the science of physics, the concept of the atomic structure of
all continuous matter, became a widespread and well-explained concept. These
developments in sciences, technology and the arts, show how the early concept
of infinite, as an accounting of the invisible micro-structure, say of a
continuous geometric line, that was mainly created during the 18th, and
19th century, can be substituted and updated with a true invisible finite
structure. The gain from this switching from the infinite to the finite, in
mathematics and the sciences can be tremendous.
Maybe in the past
centuries when the classical theories where created and when computers where
not the usual widespread practice, this true land of human mind, mentioned above,
which is based on the finite, was inaccessible to tame spirit and was obscured
in intractable wilderness. Most of the thinkers that were sensitive enough
would refuse to create in the land of finite, as they would usually harm their
consciousness as creators. And those monopolizing it in the past, were
very often using it with an undesirable and injuring impact on the
consciousness of their audience. Therefore the infinite as more artistic and
phenomenological, was felt better! There is probably a good reason for
this situation as an early phase in collective thinking. At the beginning the
mathematical thinking is introverting rather than directed outwards to visible
reality. This creates an unconscious or subconscious realization of the
functions of the infinite at the very subjective or even bodily functions of
the thinker. Thus unconscious or subconscious the object of his thinking while
thinking of the infinite, is almost
himself! Therefore he puts unconsciously barriers to his own intellect,
as that of the concept of infinite where, the ontological status of the studied
may even dynamically change while an argument about it is carrier out! The very
specification of a magnitude as of limited size, or even changing by a law,
would automatically create an measurement intervention to the thinkers state,
that might reduce his freedom and clear power of reasoning. The reader might me
familiar to how the state of physical system in the microphysical world
(quantum mechanics) is influenced by a single measurement. E.g. the measurement
of the position in a quantum oscillator, automatically, practically, stops the
oscillator. The analogy to the subjective consciousness of the thinker in the
place of the physical system is clear. Thus the abstraction of the infinite
acts also not as an early imperfection but also as mature protective
encryption of the creative mind. This explains why the "infinite" was
"felt better" and preferable by the early thinkers. This was as
far as I know a most important factor for many creators, that preferred
to think and create in the land of infinite. But today, almost half century
after the spread of computers , the "magical background" of all this
situation has changed. It seems natural to proceed and re-found the
mathematics in the finite, the true land of the human mind, as kind and
sensitive spirit may have a home there too! It is not a restriction but rather,
an advantage for the cognitive process. Although the infinite as a way of
speaking about entities and changes, has added in the past psychologically to
the creators, its natural evolution is I think, to elaborate and transform to
sophisticated analysis of the ranks of the finite, and thus add to a better
link of thinking , feeling and acting (applications).
This in a historical
perspective of the evolution of mathematics means to proceed 1) from the
ontological specifications of the ancient Chinese, Egyptian and Greek
Mathematics directly to 2) the introduction of calculus in the 17th century and
then skipping the infinity tricks with series, of the 18th century, but also
the definitions of the infinite real numbers of the 19th century plus
also the infinite sets of the 20th century, to proceed directly to 3) the
21st century finitary techniques for all ontology and the continuum by
computer science.
This has nothing to do with becoming "mechanical" in the contrary it
may mean becoming elegant, simple and efficient but also honestly human ,
practical and detailed, while being implicit or explicit.
The next papers were created
when the previous philosophical ideas and choice of techniques had not yet been
integrated in my mind. Fortunately the next papers as well as many
from the mathematics of Layers 6 and 7 admit an interpretation or
"unlocking" in the layer 5 and 4 (as concurrency complexity in
representing parallel computations with rational numbers in the computer). In
addition they be a preliminary phase of the development of a
mulri-resolution Differential Calculus over finite systems of rational numbers,
which is mainly a finitary creation. How this may be so must be a
future creative task. There is no-doubt nevertheless that there are advantages
and the need for scale-sensitive description of natural and social phenomena.
To develop a finitary interpretation of the infinite, we must define a new
concept, that of Limited Model, or Instance. The definition of a limited Model
or Instance, is as the usual definition of a model , in Logic , except that at
all quantifications on set (like "for all natural numbers"... ,
or "for all real numbers..." ) it is not referred really to all
the elements of the set, but to a limited subset of it. So "for all
natural numbers" may mean for all natural numbers less than a limit number
n0 which is used through out in all logical arguments in the
theory. After the concept of limited model or instance, even large
theories like the Zermelo-Frankel theory have limited models, or instances that
are finite sets of finite sets! All cardinals, and ordinals are
interpreted in this way as finite integers, while the order between them as the
order of natural numbers! Therefore the axiomatic models of Natural numbers,
Zermelo-Frankel set theory, Cauchy-Dedekind real numbers, transfinite real
numbers, surreal numbers, ordinal real numbers etc have finite , limited models
or instances that consists of rational numbers, represent able in the operating
system, and a programming language of a computer. In such systems of finite
limited models, or instances, all concepts like finite, countable infinite,
uncountable infinite, became logical grades of the usual order of finite
natural numbers. The discrimination between finite and infinite or various
grades of infinite is simply a phenomenological discrimination of
transcendentally separated (meaning with large gap) areas of the finite that
may have also different informational and logical determination relative to the
resources of the cognitive system. For computer multimedia applications, the
parameters of the limited model or instance say of the real and ordinal real
numbers, maybe chosen so, that the infinitesimals are pixels, that fall below
the visual discriminating threshold, the time-infinitesimals must be of a
finite size lower the time-interval required to have the visual effect of
motion, etc. For applications in physics, the parameters that define the
discrimination of finite, infinitesimal, infinite etc, come from the structure
and function of the physical reality itself, that falls in to layers of
discretised material units, like Planets and stars (layer 1), protons
neutrons and electrons (layer 0) , even finer yet undiscovered permanent
particles (should we call them aetherons?) that make the known classical fields
like the electromagnetic , gravitational etc (layer -1) etc. Most probably the
actual sizes of the pixels of the layers of the physical reality (relative
sizes of aetherons to protons and to stars etc) follow a geometric progression.
(E.g. with a ratio that of a proton to the size of an average star!)
Even the mathematical axiomatic
systems have to be updates in the evolution of the civilization, as new
requirements appear in the societies.