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p Knowledge acquires its most developed and perfected form, as we know, in science (its model being physics), which ensures adequate cognition of nature. Systems and structural approaches are inherent in science as the highest form of knowledge, unlike other forms. Their presence means that a set of concepts considered outside the theoretical construction is not yet science.

p From this point of view, the axiomatic method plays a most important role in science since it was developed historically as a method of theoretical construction of science and, therefore, as the method determining its architecture. The axiomatic method helps cognise the most general laws operating in the sphere of phenomena covered by any one science; axioms arise, or principles of an (axiomatically) created scientific system, which unite the set of interconnected phenomena under study into a single structure.

p In an axiomatically constructed theory its statements are deduced from axioms. It would seem to follow from this that the axiomatic method, which ensures exactness of the concepts employed, and certainty, consistency, and conclusiveness in the argument, excludes the idea of flexibility of concepts, recognition of the variability of scientific propositions, and a transition from certain scientific theories .to other, deeper ones. Hence the conclusion can easily be reached that this method simply serves in science for the full logical substantiation and final shaping of the content of scientific cognition, and lhat, by its nature, it is alien to dialectical thinking.

p This assertion, however, is undoubtedly an extreme one. Introducing order into the language of scientific or 336 theoretical concepts is an essential task of the axiomatic method, but it cannot be reduced simply to such putting into order. As we showed with examples in the preceding section, the axiomatic method allows one to find the new elements in physics not only in the sense that concepts or statements that exist potentially in a given theory are brought out and made explicit in the course of deduction, but also in the deeper sense that the method makes it possible to find new principles and fundamental concepts that are used as the logical foundation of a new theory. We find the same thing also in mathematics, to understand the essence of which is the major goal of the axiomatic method. According to Bourbaki, ’where the superficial observer sees only two, or several, quite distinct theories, lending one another "unexpected support"... through the intervention of a mathematician of genius, the axiomatic method teaches us to look for deeplying reasons for such a discovery, to find the common ideas of these theories, buried under the accumulation of details properly belonging to each of them, to bring these ideas forward and to put them in their proper light’.^^10^^

p Bourbaki’s point of view on the axiomatic method in mathematics reflects the content and spirit of modern mathematics. It seems to us that the axiomatics in modern physics is similar to that about which Bourbaki writes; we shall try to demonstrate in what follows that that is the position.

p How do physicists themselves deal with the question of axiomatics in their science? We shall briefly discuss the views of Einstein and Feynman.

p Einstein believed it is possible, through the use of purely mathematical constructions, to find those concepts and regular connections between them that provide the key to understanding the phenomena of nature. The corresponding mathematical concepts could be prompted by experiment but they could not in any case be deduced from it. Einstein dwelt more than once on the point that experiment remains the sole criterion of the suitability of the mathematical constructions of physics. ’But,’ he emphasised, ’the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed.’^^11^^

p Feynman, it would seem, disagrees with Einstein. So long as physics is not complete and we are trying to discover new laws, he says, ’we must always keep all the alternative 337 ways of looking at a thing in our heads, so physicists do Babylonian mathematics, and pay little attention to the precise reasoning from fixed axioms’.^^12^^

p According to Feynman, this situation will change when physicists know all the laws of nature, i. e. when physics becomes complete, and he believes that quite probable.^^13^^

p Let us consider Einstein’s and Feynman’s statements ia greater detail.

p Both of them stress the explanatory and predictive functions of the axiomatic method as most important for any theory, but which do not lead beyond the context of this theory. While Einstein accepts this without reservation, Feynman acknowledges their greatest significance for theory in principle, i. e. for the time when physics has cognised all the laws of nature, which is quite probable (on the other hand, physics in the incomplete state it is in today needs a Babylonian, i. e. empirical, method when much is known but it is not completely realised that this known can be deduced from a set of axioms).

p Thus, both Einstein and Feynman deny the function to the axiomatic method that we would call heuristic, i. e. the function of searching for a new fundamental theory (and, therefore, new axioms) that was discussed in general at the beginning of this section; in Einstein’s statements this can be seen directly, but in Feynman’s understanding the function of the search for new fundamental propositions belongs to the Babylonian method.

p These considerations of theirs about the axiomatic method need, however, to be understood cum grano sails. Both of them draw attention to the fact that one cannot be content just with employing mathematics in physics. Mathematics, Feynman says, prepares the abstract reasoning that the physicist can use if he has ’a set of axioms about the real world’; but the physicist should not forget about the meaning to all his phrases, and ’it is necessary at the end to translate’ the conclusions into the language of nature. ’Only in that way can (the physicist) find out whether the consequences are true. This is a problem which is not a problem of mathematics at all.’^^14^^

p But the physicists’ reasoning, he continues, is frequently useful to mathematicians; one science helps another. Without dwelling on this, let us look at Feynman’s final thought: ’To those who do not know mathematics it is difficult to 338 get across a real feeling as to the beauty, the deepest beauty, of nature.’^^15^^

p Einstein said approximately the same thing, but unlike Feynman he put forward an additional idea that is quite essential on the plane of our problems.

p Einstein more than once developed the idea that ’the axiomatic basis of theoretical physics cannot be extracted from experience but must be freely invented’.^^16^^ That, however, does not mean at all that he adhered to Plato’s philosophy or defended the a priori approach or questioned the possibility of finding the right way to understand nature.

p Einstein’s statements have to be considered as a whole so as to judge his philosophical and methodological ideas in physics properly. He maintained that ’experience is the alpha and omega of all our knowledge of reality’,^^17^^ and that ’there is ... a right way, and that we are capable of finding it’.^^18^^ He took his stand on the many-sided nature of cognition; this dialectical feature of his epistemological views was discussed in the first chapters of this book.’¹⁹ Here we would like to draw the reader’s attention simply to the following point.

One has to agree with Einstein when he stated, with the formal logic in mind, that the axioms of physics cannot be deduced logically from the empirical data. The axioms of physical theories, he noted, could not be reached by the ’logical path’ but only by that of ’intuition based on penetration into the essence of experience’.^^20^^ The term ‘intuition’, it seems to us, should be replaced by ‘fantasy’; the most rigorous science cannot do without fantasy, as Lenin aptly said in his Philosophical Notebooks.*^^1^^ And that is not far from the idea that scientific creative work and dialectics are always in harmony. In Section 3 of this chapter we shall consider matters related to this more fully.

p Einstein considered that there were shortcomings to some extent in Newton’s views on the principles of mechanics, which consisted in the fundamental concepts and principles of his system, in the belief of the author of the Principia, being deduced logically from experience. The same idea about the basic laws and fundamental concepts of physics permeated the views of most scientists of the eighteenth and nineteenth centuries.

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p According to Einstein, as we have seen, such an understanding is erroneous; it was only the general theory of relativity that brought clear recognition of its erroneousness. The general theory of relativity, he said, ’showed that one could take account of a wider range of empirical facts, and that, too, in a more satisfactory and complete manner, on a foundation quite different from the Newtonian.’^^22^^

p We do not think that the views of Newton and the scientists about whom Einstein spoke, do deserve such a characterisation. The axioms of classical mechanics were, in fact, deduced from the data of experience, but that does not mean at all that they were logically deduced (i. e. by means borrowed from formal logic). The principles both of classical mechanics and of other classical theories are generalised facts of experience, and the corresponding generalisations are made at the level of experimental data. As regards generalisation, it is not simply an operation in formal logic. ’The approach of the (human) mind to a particular thing,’ Lenin wrote, ’the taking of a copy (a concept) of it i s not a simple, immediate act. a dead mirroring, but one which is complex, split into two, zig-zag-like, which includes in it the possibility of the flight of fantasy from life__ For even in the simplest generalisation, in the most elementary general idea (“table” in general), there is a certain bit of / a n t a s jr.”^^23^^ In the principles of classical theories this is expressed through their content’s possessing elements that cannot be deduced by the logical means of the given system.

p The fundamental laws of many of the theories of classical physics were discovered in a similar manner (by the method of principles). Maxwell’s electromagnetic theory was an exception since its principles were obtained by the method of mathematical hypothesis. This theoretical method has become widely used in non-classical physics; it visibly demonstrates the correctness of Einstein’s idea that the fundamental laws and concepts of physics are free creations of the human mind. This method has great significance in finding from nature the principles of quantum mechanics and quantum electrodynamics. The method of fundamental observability had the same importance in obtaining the principles of quantum mechanics and of the special and general theory of relativity. Let us consider the general 340 theory of relativity in the light of Einstein’s remark above about it.

p In Einstein’s view the fact that one can point to two, essentially different theoretical foundations (classical theory and the general theory of relativity) that explain the appropriate set of experiments reveals the speculative nature of the principles that underlie the theory. But is the example of the general theory of relativity convincing in this respect?

p Newton’s theory of gravitation and Einstein’s gravitational theory were in fact built and constructed as a generalisation of the same data of experience. The predictive function of Einstein’s theory, however, as became clear, proved to be broader than the corresponding function of Newton’s theory: the general theory of relativity predicted and explained phenomena which were obstacles to Newton’s theory of gravitation (the motion of Mercury’s perihelion; the deflection of light in the Sun’s gravitational field). In addition, both theories also differ in certain respects, which precludes their comparison on one and the same logical plane, proposed by Einstein. The general theory of relativity could not have been created in Newton’s time; furthermore, this theory itself would not have been constructed if there had not been the special theory of relativity, and the latter would also have not been formulated if classical mechanics had not existed.

p In other words, Einstein’s example is too abstract, although it can be used to illustrate his idea of the speculative nature of the basic principles of a theory in the sense above. If this example is translated to the plane of reality, it speaks not simply of principles being just suggested by the experiment (as was assumed by Einstein) but of the formation of the principles of a theory being dependent on circumstances appertaining to the level of development of physics and science as a whole, including philosophy, and also on the state of spiritual and material culture. This is the basis of the answer to the question why a theoretical system is almost unambiguously determined by the sphere of observations, although there is no logical way from the observations to the theory’s fundamental principles and concepts.

p Does all this mean that the principles of classical theory are generalised facts of experience, and the principles of 341 non-classical theories something else? Let us note for the present (it will be discussed in greater detail in Section 3) that the principles of non-classical theories are also generalised facts of experience. Unlike classical theories, however, in which the generalisations are made at the level of experimental data, in non-classical theories the generalisation is made at the level of theory: the point is that, for example, classical mechanics was necessary for the special theory of relativity and quantum mechanics, as the basis for the description of experiments. With this is associated that function of the axiomatic method in physics whose existence could not be even suggested in eighteenth and nineteenth century science, and which took shape as non- classical theories developed. We shall now pass to questions relating to this.

p We have several times pointed above to two functions of the axiomatic method. One of them, the ordering function (which unites the explanatory and predictive functions) expresses the tendency of a theory (as a certain system) toward logical completeness, and in this sense to the completion of its development as a certain theoretical system. The other function, the heuristic one (in which the axiomatic method finds ways of resolving the paradoxes that have arisen during the development of a theory), expresses the theory’s tendency to go beyond the context of its system that makes it precisely such-and-such a theory and not another one. Below, to the end of this section, we shall discuss the axiomatic method from the aspect of its ordering function (the heuristic function being considered in Section 3). The axiomatic method arose in physics, as we know, together with classical mechanics. Like Newton’s mechanics, it is in some respect a product of Newton’s Rules of Reasoning in Philosophy that he included in the third book of the Principia. These Rules have something in common with Descartes’ Regies pour la direction de 1’esprit. In the literature one more often meets a stressing of their differences, determined by the personal philosophical positions of Descartes and Newton (for example, Newton’s sharply negative attitude to Descartes’ theory of vortices is well known) than any mention of their similarity. In our view it is essential, in order to understand the philosophical essence of classical science when it was being built, to pay attention rather to the common element in Descartes’ Regies and 342 Newton’s Rules. It constitutes a part, as it were, of the ’spirit of the times’ that also put its stamp on the content and laws of development of the science of the time, thereby promoting its very comprehension. Let us compare the two sets of rules in this connection.

p Descartes:

p 1. Only that should be regarded as true which appears before the mind in such clear and lucid form that it does not provoke any doubt.

p 2. The difficulties that one encounters should be divided into parts so that they may be overcome.

p 3. It is necessary to start with the simplest objects and to ascend gradually to the cognition of the complex, assuming the presence of order even where the objects of thinking are not given in their natural connection.

p 4. It is necessary to compile the lists and surveys of the objects under study as fully as possible.

p Newton:

p 1. ’We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearance.’

p 2. ’The same natural effects we must, as far as possible, assign the same causes.’

p 3. ’The qualities of bodies, which admit neither intensification nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies.’

p 4. ’In experimental philosophy we are to look upon propositions inferred by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions.’^^24^^

p Newton’s Rules served as the foundation that produced the axiomatic method in physics as an experimental science. Descartes’ Regies do not mention experiment at all; his position implied that the initial assumptions (axioms) of physics should be treated only as hypotheses, while clarity and obviousness were regarded as the criteria of truth.

p There is thus a sharp difference in the interpretation of axioms and truth in physics in Newton and Descartes. One 343 should remember, however, the hypothetical element in the content of the principles of Newton’s mechanics ( discussed above). According to Descartes and Newton, moreover, cognition developed from the simple to the complex and, if one allows for the fact that, as follows from Descartes’ Regies, clarity is a kind of synonym of simplicity, and that Newton stressed, in his explanations of his Rules, that nature is simple and did not affect ’the pomp of superfluous causes’, one can see that there is much in common between the two sets of rules in spite of certain serious differences.

p There is nothing surprising in that. Descartes, a great philosopher and the founder of analytical geometry, occupied himself with mechanics, optics, astronomy and acoustics from his youth, and made outstanding discoveries in them. And he saw in mathematics a general method for studying the physical world. His ideas had a great influence on the development of classical physics and have, to some extent, affected the development of physics to the present time. Huyghens’ wave theory of light, the analytical mechanics associated with the names of Euler, Lagrange, and Hamilton, Maxwell’s electromagnetic theory, the field theory, modern quantum mechanics—all these disciplines, in one form or another, and to some extent or another, emerged and became established under the influence of Descartes’ ideas. His methodological rules are a kind of spiritual ancestor of the method of mathematical hypothesis in modern physics, and of the modern view on the role of the mathematical apparatus in physical theory if one has in mind simply the mathematical form of the ‘hypothetical’ physics against which Newton fought. Let us, however, return to our theme.

p One can see from the content of Descartes’ Regies and Newton’s Rules that a constructed physical theory should satisfy the principles of completeness, independence, and consistency in so far as it remains a finished system. This idea was expressed explicitly much later, when the logical foundations of modern mathematical knowledge were laid.

p Let us touch on certain features of Newton’s methodology compared with Descartes’.

p Newton’s fourth rule says that the statements drawn from phenomena by means of induction are trustworthy so long as they are not disproved by new phenomena. This rule 344 directly points to the fact that ’experience is the alpha and the omega of all our knowledge of reality’; it would seem to be at a total variance with Descartes’ methodology. The words quoted, however, are Einstein’s^^26^^ who did not diverge at all from Descartes in saying: ’But the creative principle resides in mathematics.’^^26^^ We are consequently convinced once more that the opposition between Newton’s and Descartes’ Rules is not absolute, that the ‘empirical’ and ‘mathematical’ approaches in physics, when understood correctly, are not opposed to each other but complement one another, and form an inseparable unity.

p Finally, let us make the last comment on the ‘empirical’ and ‘mathematical’ methods in physics, whose founders were Newton and Descartes respectively. Classical theories arose and developed, for the most part, in such a way (we mentioned this in connection with a different point in Section 1) that determining the formulas for measuring quantities in them (the definition of physical quantities) preceded the search for equations (i.e. the propositions and, finally, the axioms) of the theory, while the content of the physical concepts itself appeared to be independent of the axioms’. Newton’s definitions of relative space, relative time, and relative motion  [344•*  were formulated independently of the axioms of mechanics; and these concepts figure in his axioms. In addition, he introduced definitions of absolute space, absolute time and absolute motion, but these concepts play the role in his theory rather of a certain purely philosophical supplement and not of physical concepts.  [344•**  Newton’s axiomatic method can be described as close to informal axiomatics.

p Modern physical theories, on the other hand, arose by another axiomatic path that is close to formal axiomatics. When a theory is created in modern physics, its mathematical apparatus is first found, the physical meaning ( content) of its concepts being still (totally or partially) unknown; their content is only revealed later as they become defined.

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p Generally speaking, when an axiomatic theory is constructed, its fundamental concepts do not exist indepenpently of the axioms; being governed by the latter, they can be defined through them. This circumstance was indicated, as a matter of fact, by formal axiomatics. To take an example from classical theory, Newton’s first axiom can serve as an (implicit) definition of the inertial reference frame, one of the fundamental concepts of classical mechanics.^^27^^

Such are certain features of the axiomatic method in physics, when we consider its ordering function.

p In conclusion, let us consider briefly the principles that must be satisfied by the axioms of a logically complete ( axiomatised) physical theory.

p The axioms underlying the theory of a certain sphere of natural phenomena are elements of a system that has a certain structure. This means that they are connected through relations, which include the independence of axioms and their consistency and completeness. Such an aggregate of axioms is called the set of axioms.

p The independence of axioms expresses the fact that each axiom in a set is exactly a fundamental statement in a given theory; that is why it belongs to the set only of the fundamental propositions of a theory, in which no statement can be deduced from any other. If it is affirmed, for instance, that the exposition of a theory should begin with the simplest relations between its objects, it is the independence of the axioms in the content of the axiomatic method that expresses this statement.

p The consistency of axioms means that no axiom of the set can contradict any other. When there is such a contradiction, it is impossible to interpret the theory constructed on the axioms; as regards empirical interpretation this amounts to experiment not confirming the theory. The consistency of a system of axioms is thus a necessary requirement of their truth. By itself, however, the requirement is not sufficient to resolve the problem of the truth of a theory based on a consistent system of axioms. Here experiment, experience, and practice, of course, come to the fore. On the other hand, consistency of axioms is the necessary and sufficient condition for unity of the 346 propositions of a physical theory, if the latter is a deductive one. One has to take into account, however, that an axiomaliscd theory reflects adequately the sphere of phenomena corresponding to it, if one abstracts the connections between this sphere as a whole and others, and abstracts the transitions of the phenomena of this sphere to those of a broader sphere.

p For the purposes of our book, the principle of the completeness of the axiomatic system of a given theory can be expressed as follows: as regards the system of axioms, this requirement consists in the system’s being adequate for the theory of a certain sphere of phenomena to cover all the phenomena of that sphere (i.e. to explain all the known, and to predict all the unknown, phenomena of this sphere), linking them in a single chain of deductive reasoning.

p We shall not discuss the criteria of the completeness of an axiomatic system; in physics, however, when it is a matter of the completeness of such-and-such a theory, experience frequently plays the decisive role. The paradoxes at the junction of classical mechanics and classical electrodynamics, for instance, combined with the negative result of the Michelson-Morley experiment, led to the conclusion that neither the axiomatic system of classical mechanics nor that of classical electrodynamics was a complete system if each was assumed to cover phenomena appertaining to the electrodynamics of moving bodies. Einstein solved this problem, as we know, when he created his theory of relativity which has now become an engineering discipline.

p It is necessary to emphasise that only the totality of the requirements of independence, consistency, and completeness of the system of axioms forming the logical foundation of a theory ensures the deductive integrity of the latter, unity of its concepts and of the varied relations between them.

p As for the guarantee of a theory’s fullest reflection of its sphere of phenomena (i.e. of the phenomena plus their essence, laws, etc. ), the requirement of completeness is the most significant one in its axiomatic construction. The other two requirements (of independence and consistency) simply support completeness; without this ‘support’, however, completeness would be unable to do its job.

The objects of a given axiomatic system, connected through certain relations, are its initial fundamental concepts defined implicitly by axioms (which provide an 347 accurate, complete description of the relations between the system’s objects). The theory of such related objects is considered built when it is possible to deduce logical corollaries from the system of axioms (according to certain rules), abstracting all other assumptions (statements) with respect to the objects concerned. How is this axiomatic ideal realised in a physical theory (or in physics as a whole)? Analysis of the matters relating to that goes beyond the frame of this Section.