Part 3
Instances of articles, mathematical problems and the other works of Dr. Hashtroudi
1
In each era, mathematics is the mirror of its civilization. Technical, industrial, scientific and
art progress in any century can be realized in worldview of its mathematics.
The history of evolutions in cultures and modernity of thoughts is simultaneous with the
history of progress in mathematics.
In this century that empirical development in natural sciences and biology and evolution in
observational sciences such as astronomy and cosmology with wave knowledge has made far
scientifical research possible, mathematics as an abstract inductiveand discursive knowledge
has led to where abstract mathematics has been introduced without resorting to the study
of classifications in human sciences and on the other hand evolution of mathematical logic
and modern science semantics has made abstract thought search and research regardless of
statement and words quality and has founded the statement of other abstract sciences.
In this brief article, short mention to the quality of mathematical thought evolution will be
accomplished in particular in the 20th century so that readers would get informed more or less
perfectly in this issue.
The basis foundation of abstract mathematics begins with the set theory. This theory that
had been considered more or less in the 18th century, occupied thoughts of mathematicians
at the beginning of the 20th century and made clear coordination and integration of logic and
mathematics. The basic of this theory, later with other generalities that are always required in
mathematics, founded eventually the discursive school in mathematics.
Antinomies that arose in the set theory, was considered by logicians and the coordination
between logic and mathematics (the set theory and categories) led to the building of
semantics.exclude
In order to clarify the issue according to the historical development, we talk about appearance
of the set theory and the existence of antinomies so that we lead eventually to the situation
and establishment of abstract modern mathematics.
***
Category and union of elements in a category is always arbitrarily possible, but the set that is
obtained in this way, is an indeterminate and more or less an ambiguous set. With a particular
definition that includes all members of the set and excludes non members, it can be defined
and determined (Note that here an inclusive and exclusive logical definition is not necessary,
any definition that is containing the result, suffices and such definitions are equivalent).
Clearly, sets can be defined that the own set is apparently also a member of itself. For
instance, if we form the set of all Persian names and we name this set a Persian name, it will
seem so that the set itself is a member of the set. More logical antinomies in the set theory
begins with this point. Resolving such problems that are now so normal, made difficulties at
the beginning for mathematicians that domination them seemed abstained. Later, it became
clear that such antinomies are not confined to the area of mathematics and it occurs in the area
of logic too. For instance, some cases of these antinomies in the area of logic are mentioned
here:
First. The singular sentence antinomy – This antinomy has been known from ancient
and it is quoted from Aristotle. Ghiyath al-Din Jamshid Kashani says that Kashanians are
mendacious. Obviously, this sentence makes a cycle because this quotation has been said by
a Kashanian thus it is included by the sentence itself and that sentence is not true, namely
Kashanians are truthful. And since the narrator is from Kashan (Kashanian), the sentence will
be true and the cycle becomes evident.
Should note that this sentence actually is not a singular sentence and it has been formed
by two sentences: a – Ghiyath al-Din Jamshid Kashani is from Kashan; b – He claims that
Kashanians are mendacious. Hence, the antinomy appears and makes a cycle, since lie
does not form a group, because lie of lie is equivalent to true but not lie, if Ghiyath al-Din
Jamshid Kashani said that Kashanians are truthful, no antinomy arises, because true of true is
equivalent to true and it forms a group.
Second. The Antinomy of Sentences based on Definition – For every number, it can
be mentioned either mathematical definition or another non mathematical definition that
2
determines that number. For instance, the number (a) is defined such that „the smallest
number that its definition requires less than ten words.“. Obviously, the numbers that are
defined by less than ten words, form a set (finite or infinite) that among them, by comparing,
we determine their smallest and the number (a) is specified. For instance, the number π
is defined such that „the ratio of a circle's circumference to its diameter“. This statement
(or sentence) has 9 words, namely, it is a sentence that has less than ten words, thus the
number „Pi“ is such as these numbers. As well, the number 9 (the number of planets in the
solar system), since the sentence „the number of planets in the solar system“ has less than ten
words, it is such as thsese sentences. If it is contented with these two numbers, the number (a)
will be π that is the smallest between two numbers 9 and π. However, if other definitions are
also mentioned for these such numbers, the number (a) can be always determined. Now, we
see that the number (a) itself has been defined by the statement „the smallest number that its
definition requires less than ten words.“ that has eleven words and here antinomy happens,
because the number (a) should be defined by a statement (or a sentence) that has less than
ten words, namely, according to the result of the above definition, the number (a) exists and
according to the sentence form, this number is not contained in the set of numbers which are
determined by this defintion. With a little attention, it can be found that the set of numbers
which definition needs less than ten words, is a sequence of numbers (finite or infinite,
in terms of numbers) and the defintion of the number (a) is indeed such that „the smallest
number of this set“ and in this case, there would be no antinomy, because the last sentence has
less than ten words.
In the set theory, these antinomies had been already seen and their dependence on the logical
area became later clear. The base and the fundamental of the mathematical logic is on the
basis of sentences and theorems and category theory.
If among members of a set (finite or infinite), there are relations such that the result of using
these relations for two members of that set gives another member of the set, that set is called
structu For instance, if we consider the set of integers and the addition operation, since the
sum of two integers is also an integer number, the set of integers with addition opertaion is
called a structure. Without mention to the generalities of structure in mathematics, we content
ourselves with this instance and equivalence of this structure to the structure of integers with
multiplication operation, because the set of integers with multiplication operation forms also
a structure. (In addition operation of the first structure, the number zero is as identity element,
because adding it to every number keeps preserved that number, but in multiplication of the
second structure, the identity element is the number one, because multiplying it by every
number keeps preserved that number). It is possible that two relations are defined between
elements of a set (like the set of integers with multiplication and addition operations), in
this case this such structure is called ring. The equivalence between rings and structures
with two multiplication and addition operations is always obvious. The equivalent structure
to the structure with addition is called module and the equivalent structure to the structure
with multiplication is called group. Thus, ring is indeed a structure both a module and a
group. Field of numbers is a set that the result of the four basic arithmetic operations over
its elements keeps preserved that set (except divisionon zero of module which becomes
exception). For instance, the set of rationals forms a field of numbers, because the result of
the four basic operations on rationals is a rational number. The set of irrationals is also a field
of numbers. If an irrational which integer and decimal parts are integers, is called an integer
irrational number, it is observed that the set of integer irrationals is not a field of numbers, but
rather is just a ring. So, the main classification mathematical structures are:
1- Modules that is also called an additive group and as well group that means multiplicative
group which is called in short group. If in such structure: a . b = b . a, the group is called
abelian group. The equivalence between module and group is obvious, namely, these two
structures are indeed equivalent.
2- Ring is a structure that is either module and also group, namely, the three operations:
addition, subtraction and multiplication are possible in it.
3- Field of numbers that the four basic arithmetic operations are possible in it.
Structures are divided into two fundamental parts: algebraic structures (what are so far
mentioned, are algebraic structures) and logical structures. For every logical structure, there
is a particular algebraic computations that is called correspondence structure. For instance,
Boolean algebra is the fundamental of computations of logical Aristotelian structure (i.e.
binary logic). The importance of this structure is so much in modern mathematics and the
fundamental of electronical machines or automatic systems is based on it.
***
In the history of mathematics, a is well-known, geometry has a distinguished situation. The
evolution of scientific ideas in this area is the oldest and the most acquainted history of
mathematics. Axiomatic geometry that argues about shapes with particular methods, is the
oldest geometrical system that has been ever known (Euclid's elements known as Euclid's
axioms). By applying the principles of algebraic computations in geometry, Descartes has
been known as the founder of analytic geometry (Khayyam's studies and inquiries, without
applying system of algebraic equations is indeed the same analytic geometrical method that
Descartes has laters used). And this method has been finally led to affine and projective
geomtries. Laters, with Poncelet's studies and inquiries, these geometries became
independently and anew the method of analytic geometry was founded much more firmly and
logically than before as axoimatic method. Extensive efforts that have been done in this
method, generalized gradually to other mathematical desiplines and made the basic of other
branches in mathematics to the axiomatic method. And this issue is also researched and
inquired now by mathematicians. By founding the axiomatic method in mathematics and
renewed independence of geometry from algebra, algebraists proceeded anew to establish a
branch called algebraic geometry that is now researched and inquired. Applying the principles
of computations in structures and in particular logical structures in definitions and theorems
in geometry helps suficiently to make clear the basic of geometry. For instance, in 3-
dimensional Euclidean geometry, we denote points (the set of points) with capital Latin letters
such that A, B, C, D, … are the different members of the point set. As well, we denote lines
with small Latin letters like a, b, c and surfaces with Greek letters like α, β, γ, … Now, we
define the classification ring of these factors: the addition operation between two factors of
the universal set S (A, B, C, …, a, b, c, ..., α, β, γ, …) is defined as follows: between two
factors of the members, we define a relation that we show with the sign +, such that the sum
of two factors of members is another factors of members with the least dimension. The set S
contains the subset: S (A, B, C, …), S (a, b, c, …), S (α, β, γ, …) such that every subset has a
dimension which is shown by the index of the subset, namely, the dimension of A, B, …. is
zero, the dimension of a, b, … one and of α, β, γ, … two. The sum of two factors of the
members in S is the other factor of the universal set S such that both given factors lie on it and
this factor has a minimal dimension. For instance, if [P] and [Q] are two indeterminant factors
of S (we denote unknown factors with the capital Latin letters), then [P] + [Q] = [R] is
defined so that R is chosen from the subsets of points, lines or surfaces so that P and Q lie on
it and the index of R (the index of correspondence subset that [R] belongs to it) has a minimal
amount. For example, if we sum a factor like a (line) with a factor like B (point), then we
have: a + B = γ in which the index of a, B and γ equal respectively to one, zero and two,
namely, the sum of a line and a point is a surface which passes through them. Obviously, if
the point B lies on a line a, the sum of them is the same line a, because in this case, the point
lies on the line and all of points of the line a lie also on the line a, so the factor with minimal
dimension which contains both of them, is the same line a. In this case, a + A = a (on the
contrary to usual algebraic addition, the sum of two factors is one of those factors without one
of factors is zero). In order to complete these operations, we need also another definition. We
denote the universal set S (in terms of content) with U (i.e. the whole of geometrical space
which contains all of points, lines and surfaces) and the null set which has no member, with N
(which is like zero in arithmetic); Adding these two concepts generalizes additive and
multiplicative operations (which are later defined). In this case, we observe that the sum of
two lines, a + b, if they are not parallel or intersecting, equals to U, namely, a + b = U. And if
these two lines are parallel or intersecting (i.e. they are in the same surface), the sum of them
is the surface which contains them, namely, a + b = α.
The multiplication of two factors in the universal set S is defined as: the multiplication
of two unknown factors [P] and [Q] as the members of the set S is another factor of S
which is common in both factors [P] and [Q] and has maximum dimension. We denote the
multiplicative operation with the sign × and define as [P] × [Q] = [R]. The index of R is the
greatest index which has the mentioned property in the definition of multiplication in the
subsets. For instance, the multiplication of two intersecting lines is their common point: a × b
= C, and the multiplication of two skew lines is the null set N: a × b = N.
Now, we consider the solution of equation [P] × [X] = N in which P is a point and X is a
factor of shapes (point, line or surface) and N is the null factor or set. Obviously, one of
solution is the same null set N, because the multiplication of each factor in the null factor
will be the null set, the other solution is the same point P, because a geometrical shape
which contains the point P and has the minimum dimension, is the same point P, namely,
the solutions of above equation are: X = [P] and X = N. We consider if the factors of the left
side in the above equation belong to P, then we will see that the point P has no other member
except the point P itself and the null factor, namely in other words, the part and whole of a
point are equal to each other and it is the same Euclidean definition of a point.
The above instance is an example from algebraic geometry which is a subject of mathematical
logic and is necessary for comparison, development and extension of geometrical definitions.
***
The fundamental structures of modern mathematics are logical, algebraic and topological
structures that are requirements in the basic of all mathematical issues. Mathematical analysis
is a particular discipline that applies different methods in real and complex functions (it
means functions of real and complex variable). In the late 19th and at the beginning of the 20th
century, some functions were considered that are not differentiable in any points of interval
continuity and furthermore, there are continous functions that their graph covers completely
inside a square. The concepts of derivation and dimension were needed new inquiries and in
topology, the theory of dimension and measure were defined anewly.
The existence of functions with real variables that all rational points are their disconnected
points and all of these points are also limit (accumulation) points, made to review in basic
of mathematical analysis. In other hand, analysing of harmonic functions and integrals, in
one side, and the convergence theory of Fourier series, in other side, opened a new issue in
mathematics that eventually led to axiomatize potential functions.
***
The applied mathematics, in particular mathematical and theoretical physics, made to appear
especial functions that admit just particular values and thus from theory of Dirac function
to distribution theory, there was just one step to go. To apply the calculation axioms of
probability in applied and theoretical problems led to appear information theory and after that
difficulty or complexity theory that is even now researched and inquired. The appearance of
particular functions in physic such as Dirac function that was mentioned, led mathematicians
in other hand to the problem of eigenvalues or proper values that are necessarily applied in
statistical and Quantum physics and eventually ergodic theory to solve differential equations
brought into existence a general and exact discipline in theoretical physic that classified most
problems under a method.
***
In applied problems and difficulties that arise in emperical sciences, empirical formula and
rules are mostly required and until now mentioned empirical issues were frequently inquired
by convergent extension with appropriate series and validity of formula. To apply method and
principles of dificulties and complexity theory, appropriate functions are before determined
for extension of empirical formulas by analytical criteria and thus applied or empirical
formula is obtained with a high validity.
Applied mathematics and calculus, for instance in numerical solutions of algebraic systems
or differential equations, have developed greatly by electronic computing machines or simple
electrical machines and have evolved much more with respect to saving time. In applied
technical problems and in particular in cosmic problems, determination of ranges of orbital
position and stability and movings have developed considerably by using exact computational
methods known as Lyapunov method. The asymptotic (Poincaré) integration method has
attained maturity and encommpassed much more problems.
Geoemtric interpretation of differential equations (or differential systems) predetermines
solutions of equations in long distances by using global geometry and it made possible to
change movings in distant times and far distances by using topology.
Mathematical analysis has attained considerable development under coordination of different
methods that were mentioned and many unsolved problems have now been in the area of
elementary mathematical knowledge.
The concept of finite and infinite groups and its applications in analysis and geometry has
made coordinated interpretation for various problems in one side and in other side, by transfer
data from a close area to wider, it makes solvable many problems that were not verifiable
in analysis area, in geometry. Hence, the various solutions of Schrödinger equation are
determined by using rotation groups of 3- and 4-dimensional spaces.
***
To apply the concept of connection in infinitesimal geometry and generalize the concept to
existence of general groups and correspond the space of parameters in groups to the space
of geometry of group has solved many problems of differential geometry and mechanics
with a new interpretation so that it was not possible before these concepts. Some particular
differential equations are solved without using quadratic forms in their general solutions.
Combination of eigenvalues with the concepts of finite groups in Quantum and statistical
physics and the problems relating to electrons and the other particles of material to face
with differential equations bases the theory of spinors in Euclidean space that is applied
considerably in relativity theory to solve problems.
***
3
4
The above introductions makes possible to found operator theory in physics and mechanics
and the combination of this concept with the concept of mathematical transformations
introduces the concepts such as homomorphism and isomorphism for general mathematical
and physical problems that can be also applied with a little generalization in problems of
human sciences (such as problems in transportations, sociology, psychology). Gradually,
biology introduces and sets forth its specific problems in mathematics and the abstract
mathematical organizations are applied in this area as well.
***
The history of science has made clear since long that attempt and endeavour for solving some
problems sets forth some other problems that make to introduce some other new sciences,
fields and branches. The 20th century is not also excluded from this general rule. In solving
problems and difficulties of human sciences, though success and evolutions have been
obtained, the other problems have been introduced that no step has been yet taken to solve
them or if a few steps have been gone ahead, it is not yet clear how problem can be solved.
For instance in physics, will gravity like electromagnetics become particle and will quantum
of gravity be discoverd? It is a mystery that
has not been yet expressed any opinion. In spite of all, the 20th century is one of the centuries
that human activities have been efficient and undoubtedly, the continuation of researches that
have been begun in this century, will make generally to change technologies.
As it has been mentioned in introduction that mathematics in an era is the mirror of its
civilization, mathematics in the 20th century is the mirror of development and modernization
of thoughts and ideas. The astonishing result of this development and modernization that are
just now more or less conceivable, is openning gates of unknown skies, whether the sky of
this material universe or whether sky of thought universe.
1The first part of this article, by Dr. Mohsen Hashtroudi, has been firstly published in Yekan magazine, No. 1
(January 19??) and then all of article in Yekan magazine (March 19??).
2Ghiyath al-Din Jamshid Kashani (1380-1429) Persian astronomer and mathematician
3Aleksandr Mikhailovich Lyapunov (1857-1918): Russian mathematician, physicist and mechanician.
4Erwin Schrödinger (1887- 1961) Austrian physicist
Translated by Fariba Elliee
Instances of articles, mathematical problems and the other works of Dr. Hashtroudi
1
In each era, mathematics is the mirror of its civilization. Technical, industrial, scientific and
art progress in any century can be realized in worldview of its mathematics.
The history of evolutions in cultures and modernity of thoughts is simultaneous with the
history of progress in mathematics.
In this century that empirical development in natural sciences and biology and evolution in
observational sciences such as astronomy and cosmology with wave knowledge has made far
scientifical research possible, mathematics as an abstract inductiveand discursive knowledge
has led to where abstract mathematics has been introduced without resorting to the study
of classifications in human sciences and on the other hand evolution of mathematical logic
and modern science semantics has made abstract thought search and research regardless of
statement and words quality and has founded the statement of other abstract sciences.
In this brief article, short mention to the quality of mathematical thought evolution will be
accomplished in particular in the 20th century so that readers would get informed more or less
perfectly in this issue.
The basis foundation of abstract mathematics begins with the set theory. This theory that
had been considered more or less in the 18th century, occupied thoughts of mathematicians
at the beginning of the 20th century and made clear coordination and integration of logic and
mathematics. The basic of this theory, later with other generalities that are always required in
mathematics, founded eventually the discursive school in mathematics.
Antinomies that arose in the set theory, was considered by logicians and the coordination
between logic and mathematics (the set theory and categories) led to the building of
semantics.exclude
In order to clarify the issue according to the historical development, we talk about appearance
of the set theory and the existence of antinomies so that we lead eventually to the situation
and establishment of abstract modern mathematics.
***
Category and union of elements in a category is always arbitrarily possible, but the set that is
obtained in this way, is an indeterminate and more or less an ambiguous set. With a particular
definition that includes all members of the set and excludes non members, it can be defined
and determined (Note that here an inclusive and exclusive logical definition is not necessary,
any definition that is containing the result, suffices and such definitions are equivalent).
Clearly, sets can be defined that the own set is apparently also a member of itself. For
instance, if we form the set of all Persian names and we name this set a Persian name, it will
seem so that the set itself is a member of the set. More logical antinomies in the set theory
begins with this point. Resolving such problems that are now so normal, made difficulties at
the beginning for mathematicians that domination them seemed abstained. Later, it became
clear that such antinomies are not confined to the area of mathematics and it occurs in the area
of logic too. For instance, some cases of these antinomies in the area of logic are mentioned
here:
First. The singular sentence antinomy – This antinomy has been known from ancient
and it is quoted from Aristotle. Ghiyath al-Din Jamshid Kashani says that Kashanians are
mendacious. Obviously, this sentence makes a cycle because this quotation has been said by
a Kashanian thus it is included by the sentence itself and that sentence is not true, namely
Kashanians are truthful. And since the narrator is from Kashan (Kashanian), the sentence will
be true and the cycle becomes evident.
Should note that this sentence actually is not a singular sentence and it has been formed
by two sentences: a – Ghiyath al-Din Jamshid Kashani is from Kashan; b – He claims that
Kashanians are mendacious. Hence, the antinomy appears and makes a cycle, since lie
does not form a group, because lie of lie is equivalent to true but not lie, if Ghiyath al-Din
Jamshid Kashani said that Kashanians are truthful, no antinomy arises, because true of true is
equivalent to true and it forms a group.
Second. The Antinomy of Sentences based on Definition – For every number, it can
be mentioned either mathematical definition or another non mathematical definition that
2
determines that number. For instance, the number (a) is defined such that „the smallest
number that its definition requires less than ten words.“. Obviously, the numbers that are
defined by less than ten words, form a set (finite or infinite) that among them, by comparing,
we determine their smallest and the number (a) is specified. For instance, the number π
is defined such that „the ratio of a circle's circumference to its diameter“. This statement
(or sentence) has 9 words, namely, it is a sentence that has less than ten words, thus the
number „Pi“ is such as these numbers. As well, the number 9 (the number of planets in the
solar system), since the sentence „the number of planets in the solar system“ has less than ten
words, it is such as thsese sentences. If it is contented with these two numbers, the number (a)
will be π that is the smallest between two numbers 9 and π. However, if other definitions are
also mentioned for these such numbers, the number (a) can be always determined. Now, we
see that the number (a) itself has been defined by the statement „the smallest number that its
definition requires less than ten words.“ that has eleven words and here antinomy happens,
because the number (a) should be defined by a statement (or a sentence) that has less than
ten words, namely, according to the result of the above definition, the number (a) exists and
according to the sentence form, this number is not contained in the set of numbers which are
determined by this defintion. With a little attention, it can be found that the set of numbers
which definition needs less than ten words, is a sequence of numbers (finite or infinite,
in terms of numbers) and the defintion of the number (a) is indeed such that „the smallest
number of this set“ and in this case, there would be no antinomy, because the last sentence has
less than ten words.
In the set theory, these antinomies had been already seen and their dependence on the logical
area became later clear. The base and the fundamental of the mathematical logic is on the
basis of sentences and theorems and category theory.
If among members of a set (finite or infinite), there are relations such that the result of using
these relations for two members of that set gives another member of the set, that set is called
structu For instance, if we consider the set of integers and the addition operation, since the
sum of two integers is also an integer number, the set of integers with addition opertaion is
called a structure. Without mention to the generalities of structure in mathematics, we content
ourselves with this instance and equivalence of this structure to the structure of integers with
multiplication operation, because the set of integers with multiplication operation forms also
a structure. (In addition operation of the first structure, the number zero is as identity element,
because adding it to every number keeps preserved that number, but in multiplication of the
second structure, the identity element is the number one, because multiplying it by every
number keeps preserved that number). It is possible that two relations are defined between
elements of a set (like the set of integers with multiplication and addition operations), in
this case this such structure is called ring. The equivalence between rings and structures
with two multiplication and addition operations is always obvious. The equivalent structure
to the structure with addition is called module and the equivalent structure to the structure
with multiplication is called group. Thus, ring is indeed a structure both a module and a
group. Field of numbers is a set that the result of the four basic arithmetic operations over
its elements keeps preserved that set (except divisionon zero of module which becomes
exception). For instance, the set of rationals forms a field of numbers, because the result of
the four basic operations on rationals is a rational number. The set of irrationals is also a field
of numbers. If an irrational which integer and decimal parts are integers, is called an integer
irrational number, it is observed that the set of integer irrationals is not a field of numbers, but
rather is just a ring. So, the main classification mathematical structures are:
1- Modules that is also called an additive group and as well group that means multiplicative
group which is called in short group. If in such structure: a . b = b . a, the group is called
abelian group. The equivalence between module and group is obvious, namely, these two
structures are indeed equivalent.
2- Ring is a structure that is either module and also group, namely, the three operations:
addition, subtraction and multiplication are possible in it.
3- Field of numbers that the four basic arithmetic operations are possible in it.
Structures are divided into two fundamental parts: algebraic structures (what are so far
mentioned, are algebraic structures) and logical structures. For every logical structure, there
is a particular algebraic computations that is called correspondence structure. For instance,
Boolean algebra is the fundamental of computations of logical Aristotelian structure (i.e.
binary logic). The importance of this structure is so much in modern mathematics and the
fundamental of electronical machines or automatic systems is based on it.
***
In the history of mathematics, a is well-known, geometry has a distinguished situation. The
evolution of scientific ideas in this area is the oldest and the most acquainted history of
mathematics. Axiomatic geometry that argues about shapes with particular methods, is the
oldest geometrical system that has been ever known (Euclid's elements known as Euclid's
axioms). By applying the principles of algebraic computations in geometry, Descartes has
been known as the founder of analytic geometry (Khayyam's studies and inquiries, without
applying system of algebraic equations is indeed the same analytic geometrical method that
Descartes has laters used). And this method has been finally led to affine and projective
geomtries. Laters, with Poncelet's studies and inquiries, these geometries became
independently and anew the method of analytic geometry was founded much more firmly and
logically than before as axoimatic method. Extensive efforts that have been done in this
method, generalized gradually to other mathematical desiplines and made the basic of other
branches in mathematics to the axiomatic method. And this issue is also researched and
inquired now by mathematicians. By founding the axiomatic method in mathematics and
renewed independence of geometry from algebra, algebraists proceeded anew to establish a
branch called algebraic geometry that is now researched and inquired. Applying the principles
of computations in structures and in particular logical structures in definitions and theorems
in geometry helps suficiently to make clear the basic of geometry. For instance, in 3-
dimensional Euclidean geometry, we denote points (the set of points) with capital Latin letters
such that A, B, C, D, … are the different members of the point set. As well, we denote lines
with small Latin letters like a, b, c and surfaces with Greek letters like α, β, γ, … Now, we
define the classification ring of these factors: the addition operation between two factors of
the universal set S (A, B, C, …, a, b, c, ..., α, β, γ, …) is defined as follows: between two
factors of the members, we define a relation that we show with the sign +, such that the sum
of two factors of members is another factors of members with the least dimension. The set S
contains the subset: S (A, B, C, …), S (a, b, c, …), S (α, β, γ, …) such that every subset has a
dimension which is shown by the index of the subset, namely, the dimension of A, B, …. is
zero, the dimension of a, b, … one and of α, β, γ, … two. The sum of two factors of the
members in S is the other factor of the universal set S such that both given factors lie on it and
this factor has a minimal dimension. For instance, if [P] and [Q] are two indeterminant factors
of S (we denote unknown factors with the capital Latin letters), then [P] + [Q] = [R] is
defined so that R is chosen from the subsets of points, lines or surfaces so that P and Q lie on
it and the index of R (the index of correspondence subset that [R] belongs to it) has a minimal
amount. For example, if we sum a factor like a (line) with a factor like B (point), then we
have: a + B = γ in which the index of a, B and γ equal respectively to one, zero and two,
namely, the sum of a line and a point is a surface which passes through them. Obviously, if
the point B lies on a line a, the sum of them is the same line a, because in this case, the point
lies on the line and all of points of the line a lie also on the line a, so the factor with minimal
dimension which contains both of them, is the same line a. In this case, a + A = a (on the
contrary to usual algebraic addition, the sum of two factors is one of those factors without one
of factors is zero). In order to complete these operations, we need also another definition. We
denote the universal set S (in terms of content) with U (i.e. the whole of geometrical space
which contains all of points, lines and surfaces) and the null set which has no member, with N
(which is like zero in arithmetic); Adding these two concepts generalizes additive and
multiplicative operations (which are later defined). In this case, we observe that the sum of
two lines, a + b, if they are not parallel or intersecting, equals to U, namely, a + b = U. And if
these two lines are parallel or intersecting (i.e. they are in the same surface), the sum of them
is the surface which contains them, namely, a + b = α.
The multiplication of two factors in the universal set S is defined as: the multiplication
of two unknown factors [P] and [Q] as the members of the set S is another factor of S
which is common in both factors [P] and [Q] and has maximum dimension. We denote the
multiplicative operation with the sign × and define as [P] × [Q] = [R]. The index of R is the
greatest index which has the mentioned property in the definition of multiplication in the
subsets. For instance, the multiplication of two intersecting lines is their common point: a × b
= C, and the multiplication of two skew lines is the null set N: a × b = N.
Now, we consider the solution of equation [P] × [X] = N in which P is a point and X is a
factor of shapes (point, line or surface) and N is the null factor or set. Obviously, one of
solution is the same null set N, because the multiplication of each factor in the null factor
will be the null set, the other solution is the same point P, because a geometrical shape
which contains the point P and has the minimum dimension, is the same point P, namely,
the solutions of above equation are: X = [P] and X = N. We consider if the factors of the left
side in the above equation belong to P, then we will see that the point P has no other member
except the point P itself and the null factor, namely in other words, the part and whole of a
point are equal to each other and it is the same Euclidean definition of a point.
The above instance is an example from algebraic geometry which is a subject of mathematical
logic and is necessary for comparison, development and extension of geometrical definitions.
***
The fundamental structures of modern mathematics are logical, algebraic and topological
structures that are requirements in the basic of all mathematical issues. Mathematical analysis
is a particular discipline that applies different methods in real and complex functions (it
means functions of real and complex variable). In the late 19th and at the beginning of the 20th
century, some functions were considered that are not differentiable in any points of interval
continuity and furthermore, there are continous functions that their graph covers completely
inside a square. The concepts of derivation and dimension were needed new inquiries and in
topology, the theory of dimension and measure were defined anewly.
The existence of functions with real variables that all rational points are their disconnected
points and all of these points are also limit (accumulation) points, made to review in basic
of mathematical analysis. In other hand, analysing of harmonic functions and integrals, in
one side, and the convergence theory of Fourier series, in other side, opened a new issue in
mathematics that eventually led to axiomatize potential functions.
***
The applied mathematics, in particular mathematical and theoretical physics, made to appear
especial functions that admit just particular values and thus from theory of Dirac function
to distribution theory, there was just one step to go. To apply the calculation axioms of
probability in applied and theoretical problems led to appear information theory and after that
difficulty or complexity theory that is even now researched and inquired. The appearance of
particular functions in physic such as Dirac function that was mentioned, led mathematicians
in other hand to the problem of eigenvalues or proper values that are necessarily applied in
statistical and Quantum physics and eventually ergodic theory to solve differential equations
brought into existence a general and exact discipline in theoretical physic that classified most
problems under a method.
***
In applied problems and difficulties that arise in emperical sciences, empirical formula and
rules are mostly required and until now mentioned empirical issues were frequently inquired
by convergent extension with appropriate series and validity of formula. To apply method and
principles of dificulties and complexity theory, appropriate functions are before determined
for extension of empirical formulas by analytical criteria and thus applied or empirical
formula is obtained with a high validity.
Applied mathematics and calculus, for instance in numerical solutions of algebraic systems
or differential equations, have developed greatly by electronic computing machines or simple
electrical machines and have evolved much more with respect to saving time. In applied
technical problems and in particular in cosmic problems, determination of ranges of orbital
position and stability and movings have developed considerably by using exact computational
methods known as Lyapunov method. The asymptotic (Poincaré) integration method has
attained maturity and encommpassed much more problems.
Geoemtric interpretation of differential equations (or differential systems) predetermines
solutions of equations in long distances by using global geometry and it made possible to
change movings in distant times and far distances by using topology.
Mathematical analysis has attained considerable development under coordination of different
methods that were mentioned and many unsolved problems have now been in the area of
elementary mathematical knowledge.
The concept of finite and infinite groups and its applications in analysis and geometry has
made coordinated interpretation for various problems in one side and in other side, by transfer
data from a close area to wider, it makes solvable many problems that were not verifiable
in analysis area, in geometry. Hence, the various solutions of Schrödinger equation are
determined by using rotation groups of 3- and 4-dimensional spaces.
***
To apply the concept of connection in infinitesimal geometry and generalize the concept to
existence of general groups and correspond the space of parameters in groups to the space
of geometry of group has solved many problems of differential geometry and mechanics
with a new interpretation so that it was not possible before these concepts. Some particular
differential equations are solved without using quadratic forms in their general solutions.
Combination of eigenvalues with the concepts of finite groups in Quantum and statistical
physics and the problems relating to electrons and the other particles of material to face
with differential equations bases the theory of spinors in Euclidean space that is applied
considerably in relativity theory to solve problems.
***
3
4
The above introductions makes possible to found operator theory in physics and mechanics
and the combination of this concept with the concept of mathematical transformations
introduces the concepts such as homomorphism and isomorphism for general mathematical
and physical problems that can be also applied with a little generalization in problems of
human sciences (such as problems in transportations, sociology, psychology). Gradually,
biology introduces and sets forth its specific problems in mathematics and the abstract
mathematical organizations are applied in this area as well.
***
The history of science has made clear since long that attempt and endeavour for solving some
problems sets forth some other problems that make to introduce some other new sciences,
fields and branches. The 20th century is not also excluded from this general rule. In solving
problems and difficulties of human sciences, though success and evolutions have been
obtained, the other problems have been introduced that no step has been yet taken to solve
them or if a few steps have been gone ahead, it is not yet clear how problem can be solved.
For instance in physics, will gravity like electromagnetics become particle and will quantum
of gravity be discoverd? It is a mystery that
has not been yet expressed any opinion. In spite of all, the 20th century is one of the centuries
that human activities have been efficient and undoubtedly, the continuation of researches that
have been begun in this century, will make generally to change technologies.
As it has been mentioned in introduction that mathematics in an era is the mirror of its
civilization, mathematics in the 20th century is the mirror of development and modernization
of thoughts and ideas. The astonishing result of this development and modernization that are
just now more or less conceivable, is openning gates of unknown skies, whether the sky of
this material universe or whether sky of thought universe.
1The first part of this article, by Dr. Mohsen Hashtroudi, has been firstly published in Yekan magazine, No. 1
(January 19??) and then all of article in Yekan magazine (March 19??).
2Ghiyath al-Din Jamshid Kashani (1380-1429) Persian astronomer and mathematician
3Aleksandr Mikhailovich Lyapunov (1857-1918): Russian mathematician, physicist and mechanician.
4Erwin Schrödinger (1887- 1961) Austrian physicist
Translated by Fariba Elliee
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