Update 29.07.2002
AXIOMATICS OF NATURAL SCIENCES
The Kuban State Agrarian University,
Department of Theoretical Mechanics
13, Kalinin St., 350044 Krasnodar, Russia
E-mail: kanphil@mail.kuban.ru
The ideas expressed by the author in
numerous previous publications are
generalized. As correctness of the chosen research direction can be determined only by
means of comparison of correspondence of this direction to the fundamental axioms of
natural science, it is necessary to systematize these axioms. An attempt to systematize
the axioms of natural science according to the level of their generalized sense and
importance for scientific investigations has been made.
(Chapter 3 from the book “The Foundations of Physical
– Chemistry of Microworld”)
The Euclidean axioms are known to be the fundamental axioms of exact sciences. First of all, Euclid gives the definition to those notions, which he uses during formulations of postulates and axioms. We’ll not adduce all these definitions, we’ll list a number of notions, which have been determined by Euclid [1].
The famous definition of “a point” notion occupies the first place.
“A point is that which has no part”. Then the following definitions of the
notions are given: a line, a straight line, a surface, an angle and the notions of various
geometrical figures. After that Euclid gives postulates, but he has failed to define the
notion “postulate” itself [1].
Let the following be postulated:
1. To draw a straight line from any point
to any point.
2. To produce a finite straight line
continuously in a straight line.
3. To describe a circle with any centre
and radius.
4. (Axiom 10) That all right angles equal
one another.
5. (Axiom 11) That, if a straight line
falling on two straight lines makes the interior angles on the same side less than two
right angles, the two straight lines, if produced indefinitely, meet on that side on which
are the angles less that two right angles.”
Then there is the headline
Common Notions
(Axioms)
1. Things which equal the same thing also
equal one another.
2. If equals are added to equals, then the wholes are equal.
3. If equals are subtracted from equals,
then the remainders are equal.
4. If equals are added to the unequals,
then the wholes are unequal.
5. The duplicates of one and the same
thing equal one another.
6. The halves of one and the same thing
equal one another.
7. Things which coincide with one another
equal one another.
8. The whole is greater than the part.
9. Two straight lines do not contain
space.
It is unbelievable, but it is so. This information serves as a foundation for all
exact sciences. Let us pay attention to the fourth postulate. In the parenthesis, it is
given as the tenth axiom, and the fifth postulate – as the eleventh axiom. We do not
know why the fourth and the fifth postulated statements are considered to be axioms. Or
one should suppose that they can be simultaneously considered as the postulates and the
axioms. If Euclid managed to define the notions “a postulate” and “an
axiom”, the fourth and the fifth postulates could be in the list of axioms.
The disputes of the scientists in relation to correctness of wording of the fifth
postulate of Euclid are known [11]. They have taken place due to the lack of definitions
of the notions “a postulate” and “a axiom”. Further definitions of
these notions have not acquired significance in consciousness of the scientists, which
could be given to them if they were in “Euclid’s Foundations”.
Nevertheless, we should treat this drawback as a natural one without infringement of
genius of Euclid [3], [4].
Nearly two thousand years after Euclid, “Mathematical Foundations of Natural
Philosophy” by Isaac Newton appeared. As Euclid, he paid great attention to the
definition of the new notions, on which his laws are based. His mathematical principles
begin from the headline [2]
Definition 1
The quantity of matter (mass) is its
measure of the same, arising from its density and bulk conjunctly”.
Then Newton determines the notions
“the quantity of motion”, “an innate force”, “an impressed
force”, “a centripetal force”, etc.
After it Newton describes his notion of
absolute space and absolute time without application of axiomatic meaning to these
notions. His main ideas are given under the headline [2]
Axioms, or laws of motion
Law 1
Every body continues in its state of
rest, or of uniform motion in a straight line, unless it is compelled to change this state
by forces impressed upon it.
Law 2
The change of motion is proportional to
the motive force impressed; and is made in the direction of the straight line in which
that force is impressed.
Law 3
To every action there is always opposed
and equal action; or, the mutual actions of two bodies upon each other are always equal,
and directed to contrary parts”.
Then Isaac Newton formulates the effects originating from these laws.
The above-mentioned laws deal with mechanical motion of the bodies. Their
trustworthiness has been confirmed by experiments completely. After these laws, many other
laws have been discovered, which describe electrical, magnetic, electromagnetic and other
properties of bodies, gases, liquids and various physical phenomena. We’ll not
enumerate and analyse them. The main thing for us is that their trustworthiness has been
confirmed by experiments.
When we analyse the postulates of Euclid and the axioms or laws of Newton, we see
that they were the first to attach importance to the necessity to determine the notions,
which they used. It was done for the purpose to get uniformity in understanding the
essence of these notions, because no mutual understanding was possible without it.
Then one should pay attention to the fact that the fundamental notions, which serve
as the basis for the rest proofs. Euclid divided into two classes: the postulates and the
axioms. Form his “Foundations” it is difficult to see, what principles he was
guided by when he attributes some statements to the class of postulated and other
statements to the class of axioms [1]. Newton did not give any definition in this respect
as well. He called his laws axioms [2].
The followers of Euclid and Newton attached no importance to this issue as well,
that’s why the process of attributing the fundamental scientific statements to the
class of axioms or to the class of postulates has become a chaotic one. Each scientist had
no exact criterion concerning evaluation of the essence of his fundamental scientific
statements and attributed them either to the class of postulates or the class of axioms.
There was no exact notion of the fact that in order to strengthen significance of various
axioms in scientific research it is necessary to arrange them according to the level of
community and importance. There is an impression that we have understood this necessity
only when the features of crisis of theoretical physics have been exposed. We cannot
overcome it if we fail to put in order the fundamental scientific notions, which we use.
The task, which should be solved, is not a simple one. First of all, it is
necessary to find its beginning. Without it we’ll fail to systematize our fundamental
scientific statements and establish their completeness. We see that it is necessary to
begin with the analysis of the essence of the main properties of the notions, which we use
now. This area of investigations belongs to the theory of knowledge. We should begin from
it [5], [6], [10].
Probably, the process of knowledge has begun when separate sounds uttered by human
beings have started to form the words, which have led to the formation of images, which
correspond to sense content of these words. The range of the things and the phenomena
formulated as words has widened. Now a man uses so many words, which have various
meanings, that uniform understanding of the essence of this content has become one of the
most complicated problems of communication between people, including between scientists.
The notions are created by people in order to understand each other. On what does
this understanding depend? The main thing, which determines uniformity of understanding of
any notion, is its sense capacity. Let us pay attention to sense capacity of such notions
as “a point”, “a line”, “a triangle”, “a number”,
“the world”, “nature”, “matter”, “the universe”,
“happiness”, “love”, etc. [3].
“A point” notion has the smallest sense capacity. The majority of people
attribute one and the same sense to this notion. Nevertheless, there is no uniform
definition of this notion. Why? Because in order to determine “the point”
notion, other notions with large sense content are attracted.
Thus, as sense capacity of the notion is increased, the difficulties with its
uniform definition are increased. For example, let us take “happiness” notion
and try to define it. We see at once that it is impossible to do it, because it is
connected with person’s perception of the world round him. A person, who has lost an
expensive thing, feels unhappy. A person, who has found this thing, feels happy.
We’ll not go into details in this analysis, but we should note an importance
of sense capacity for their uniform understanding, without which science cannot exist. Now
we understand why Euclid and Newton, geniuses of the mankind, have begun from the
definition of the notions being the basis for their proofs.
It is natural that not all scientific notions have similar generalized sense and,
as a result, similar significance for scientific knowledge. It means that it is important
to arrange the fundamental scientific notions according to the level of generalized sense
and scientific importance.
What notions do we use when we recognize the world around us? The answer is simple:
we use the notions, which determine the fundamental or primary elements of the universe.
Can the world exist outside the space? Certainly, not. That’s why “space”
notion is attributed to the primary element of the universe, without which existence is
impossible. Thus, “space” notion occupies the first place due to the level of
significance for scientific cognition of the world [4], [5].
If we put
“space” notion on the first place due to the level of significance for
scientific cognition of the world, we should define it. But it is simple to do it, because
“space” notion belongs to the notions with large sense capacity. Nevertheless,
the majority of people have formed the like or similar notions concerning the essence or
the sense content of this notion. We’ll take advantage of it. For us, the definition
of “space” notion is of less importance than the fact that it is the receptacle
of all main points, that’s why we put it on the first place due to its significance
for the scientific cognition [9].
Now it is necessary to define the main features of space, on which precision of our
knowledge depends concerning everything that exists in this space. The first and foremost
feature of space is its absoluteness. What is it? How can absoluteness be determined?
Modern level of knowledge allows us to consider space as absolute one, because there are
no phenomena in Nature, which could influence space: compress, expand or distort it [9].
The statement concerning relativity of space, on which theoretical physics of the
20th century was based, has no uniform experimental proof, that’s why we do not take
it into consideration [4].
What scientific notion is the second due to significance? Matter. Without it, space
would be empty. Now we understand that extremely large sense capacity of this notion
excludes the possibility of its simple definition. Essence, which is reflected by this
notion, has such large quantity of various features that we cannot find the sign of this
essence, which could give us the reason to consider matter is don’t an absolute one. We can be guided by more or less
similar comprehension of the essence of “matter” notion by the scientists, and
it is enough for us at the given stage of scientific knowledge development.
“Time” notion is the next one due to importance for scientific cognition
of the world round us. Essence, which is present in this notion, has manifested when
matter has taken place in space. There was no time in empty space. The experience
accumulated by mankind in the process of understanding the essence of “time”
notion shows importance of its main feature: irreversibility. It goes only in one
direction. Constant rate of its course is another important feature of time. That’s
why we have every reason to believe that time is absolute, and we can define this feature
in the following way. Time is absolute, because there are no phenomena in Nature, which
could influence the rate of its course: increase or decrease this rate [5], [9].
The statement concerning relativity of time, on which theoretical physics of the
20th century was based, has no direct experimental proof of its trustworthiness. The
change of the rate of the course of time registered with the help of various devices
reflects the features of the devices themselves, but not the fact of the change of the
rate of the course of time. That’s why we think that this delusion will disappear
from the field of the actual activities of the scientists and become history.
Thus, we have determined three primary elements of the universe, on which it has
been based since the day of its creation if the one existed.
Now we should pay attention to the thing, which has remained unnoticed by Euclid,
Newton and their followers and which plays such important role in cognition of the world
by us as the notions “space”, “matter” and “time”. How are
the essences, which are reflected in these notions, connected with each other?
First of all, matter cannot exist outside space. Time passes only in space, which
contains matter. All three primary elements of the universe are inseparable. As this
important property remained unnoticed, the theories took place, in which a spatial value
of a moving object seems to be independent of time. It has turned out that time can be
separated from space as it is done in Lorentz transforms, and regularity of the passing of
time can be analysed separately.
As space cannot be separated from time and it is impossible to imagine existence of
matter outside space, inseparability of these three primary elements of the universe is an
axiom. This is the third important axiom of exact sciences.
Now, when we address to Euclid’s postulates and axioms, we feel that it is
necessary to determine these notions.
An axiom is an obvious statement, which
requires no experimental check and has no exceptions [9].
A postulate is a non-obvious statement,
its reliability being proved in the way of experiment and results from the experiments
[5], [9].
Certainly, one can challenge the accuracy of these statements. But these statements
are enough in order to divide all fundamental statements of exact sciences into two
classes: the axioms and the postulates.
Taking into consideration these definitions of the notions “a postulate”
and “an axiom”, Euclid’s postulates and axioms can be considered as axioms
with some correction of their content. Newton’s axioms or laws become postulates
automatically, because the essence reflected in them is not obvious, and reliability of
his statements requires experimental check.
As we have decided to systematize the axioms of exact sciences, and to be more
precise of knowledge of nature, and to arrange them according to the level of significance
and general sense, let us give an updated list of the axioms of natural science.
1- space is absolute;
2 - time is absolute;
3 - space, matter and time are
inseparable;
4 - it is possible to draw only one
straight line between two point;
5 - it is possible to produce a finite
straight line in both directions;
6 - it is possible to describe a circle
with any centre and radius;
7 - all right angles equal one another;
8 - if a straight line falling on two
straight lines makes the right interior angles or the sum of two right angles on the same
side, the two straight lines, if produced indefinitely, will never meet;
9 - things which equal the same thing
also equal one another;
10 - if equals are added to equals, then
the wholes are equal;
11- if equals are subtracted from equals,
then the remainders are equal;
12 - if equals are added to the unequals,
then the wholes are unequal;
13 - the duplicates of one and the same
thing equal one another;
14 -
the halves of one and the same thing equal one another;
15 - things which coincide with one
another equal one another;
16 - the whole is greater than the part.
As it can be seen, we have added three new axioms to Euclid’s axioms, but as
far as the level of general sense and significance for natural science is concerned, they
are on the first place.
We put Newton postulate on the first place:
1 - Law 1.
Every body continues in its state of rest, or of uniform motion in a straight line, unless
it is compelled to change this state by forces impressed upon it.
2 - Law 2. The
change of motion is proportional to the motive force impressed; and is made in the
direction of the straight line in which that force is impressed.
3 - Law 3. To
every action there is always opposed and equal action; or, the mutual actions of two
bodies upon each other are always equal, and directed to contrary parts.
4 - Law of
gravitation. Every object
in the Universe attracts every other object with a force directed along the line of
centers for the two objects that is proportional to the product of their masses and
inversely proportional to the square of the separation between the two objects.
Let us give the formulation of the second postulate of A. Einstein, on which
theoretical physics of the 20th century was based. “2. Any ray of light moves in the stationary system of
co-ordinates with the determined velocity, whether the ray be emitted by a stationary or a
moving body.
Modern level of knowledge allows us to give more exact formulation of this
postulate.
5 - Velocity of
electromagnetic radiation (photons) in the stationary system of co-ordinates in relation
to space is constant and does not depend on the direction of the source, which emits the
photons [8].
We give the opportunity for other investigators to continue the list of the
postulates. It will be much longer than the list of the axioms. One should think that
mathematicians agree with the necessity to transfer many statements, which they considered
to be axiomatic ones and which do not correspond to the notion “axiom” now, to
the class of postulates.
Thus, we have a list of axioms, which are necessary for us in order to check the
connection of the existing physical theories with reality. If it turns out that a theory
contradicts one of the axioms of natural science, it is erroneous.
In our publications we have already shown how the axioms should be used for the
analysis of the connection of the existing theories with
reality and for elaboration of the new ones [3], [4], [5], [6], [7].
Now the statement that the parallel lines cross in infinity is not an axiom, it is
a postulate and requires experimental proof of reliability of this statement.
Thus, the first three given fundamental axioms of natural science act as
independent criteria for a check of reliability of mathematical models of various physical
theories. I’d like to inform those, who agree with obvious trustworthiness of three
given fundamental axioms of natural science, that they are realized only in Euclidean
geometry. It results from this that there is a connection of mathematical models of this
geometry with reality.
It is necessary to emphasize a role of the axiom of space-matter-time unity in
mathematical description of the motion process of any object in space. This axiom
established strict correspondence between motion of any object in space and the passing of
time during this motion. Mathematically, it is expressed by dependence of object position
coordinates in space on time.
It is impossible to separate matter from space. It is impossible to imagine the
passing of time outside space. Space, matter
and time are primary elements of the universe, they are
inseparable on no account. I think that trustworthiness of the statement concerning
unity of space, matter and time is obvious. It has no exceptions and contains all properties of an axiom. If we
acknowledge this fact, the axiom of space-matter-time unity become an independent judge of
reliability of mathematical models, which describe motion of material objects in space,
and the theories, to which these models belong.
Mathematical models of motion of material objects in space built in
pseudo-Euclidean geometries contradict the axiom of space-matter-time unity.
Four-dimensional Minkovky’s geometry will be the first to be rejected as well as his
idea of unity of space and time, because the mathematical model of four-dimensional
geometry postulated by him, in which his idea is realized, contradicts the axiom of
space-matter-time unity.
I’d like to emphasize the fact that scientists of exact sciences are eager to
call their scientific statements axioms, especially mathematicians. An axiom is an obvious
statement, which requires no experimental check and has no exceptions. The rest are
postulates. If a theory contradicts one of
the axioms of natural science or mutually accepted scientific postulate , it is erroneous.
It
is clear that the process of realization of the idea of observation of the given axioms of
natural science will be quicker and more fruitful if the world scientific community
understands that it is necessary to confer a status of obligation to the list of axioms.
Updated and systematized axiomatics of natural science consists of sixteen axioms
for the present. As far as the level of general sense and significance for knowledge of
nature is concerned, the axiom space is absolute occupies the first place, the
axiom time is absolute occupies the second place, and the axiom space, matter
and time are inseparable occupies the third place. Value of an axiom does not depend
on its acknowledgement [12].
In scientific investigations, an important role is played by the postulates - the
statements, their reliability being not obvious, but proved experimentally. The value of a
postulate is determined by the level of its reliability acknowledgement by the scientific
community.
1. Euclid. Euclid’s Foundations.
Books I-VI. M-L, 1948, 446 pages.
2. Isaac Newton. Mathematical Foundations
of Natural Philosophy. M. “Nauka”, 1987, 687 pages.
3. Ph.M. Kanarev. On the Way to the Physics of the 21st Century. Krasnodar,
1995. 269 pages (in English).
4. Ph.M. Kanarev. Crisis of theoretical physics. The third edition.
Krasnodar. 1998. 200 pages.
5. Ph.M. Kanarev. Water is new source of energy. The third edition.
Krasnodar. 2001. 194 pages (in English).
6. Ph.M. Kanarev. The Gravitational
Radius of a Black Hole. Journal of Theoretics. Vol. 4-1. http://www.journaloftheoretics.com
7. Ph.M. Kanarev. Modelling the Photon
and Analyzing its Electromagnetic and Physical Nature. Vol. 4-1. http://www.journaloftheoretics.com
8. Ph.M. Kanarev. New analysis of
fundamental problems of quantum mechanics. Krasnodar. 1990. 174 pages.
9. Ph.M. Kanarev, I.I. Artemov, S.A.
Zelensky. Abstract of lectures on theoretical mechanics. Krasnodar, 2001. 265 pages.
10. T.I. Hill. Modern Theories of
Cognition. M.: Progress. 1965. 530 pages.
11. M. Kline. Mathematics. Loss of Definiteness. M.: Mir. 1984. 446 pages.
12.
http://doppler.innoplaza.net
http//axioms.innoplaza.net
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