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Update 29.07.2002

AXIOMATICS OF NATURAL SCIENCES

# Ph.M. Kanarev

The Kuban State Agrarian University, Department of Theoretical Mechanics

13, Kalinin St., 350044 Krasnodar, Russia

E-mail: kanphil@mail.kuban.ru

## ABSTRACT

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”)

### INTRODUCTION

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].

### Postulates

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.”

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]

### Definitions

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].

### WHICH CHARACTERIZE THE PRIMARY ELEMENTS OF UNIVERSE

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.

# 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.

# POSTULATES OF NATURAL SCIENCES

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.

# DISSCUSION OF RESULTS

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.

# CONCLUSION

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.

## REFERENCES

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.

http//axioms.innoplaza.net

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