相對性理論の辨證法

Abstract

In his recent work, “An Inquiry into a New Methodology of Theoretical Physics”, the writer of the present paper argued in the first place, that the system of complex numbers, unlike the system of real numbers, has the moment of self-negation in itself, so that a function in the former system is to be interpreted not in terms of the external relation of correspondence between the variables, as in the case of the latter, but in terms of an internal self-development. Complex numbers themselves as a whole are thus understood to develop dynamically into functions of the complex variables. The writer asserted also that the analytic continuation of the functions of complex variables in circles of convergence, based on the so-called Cauchy-Riemann's differential equation, represents the structure of spatio-temporal union of the “world” of relativity theory, and that the clue to the solution of the present-day problem of the theoretical physics of uniting the relativity theory and the quantum theory is to be found on the very side of the relativity theory. This contention is founded upon the fact that time in the relativity theory is conceived to be the unity of contradictories, i. e. the unity in the present, of the past and the future which are opposed to each other by mutual negation, and thus it is represented on imaginary axis, while space is represented on real axes. Now this structure of the “world” of relativity theory, as represented by the system of complex numbers, was what had already been revealed by the special theory of relativity itself or the theory concerning the relativity of uniform translatory motion, without the arrival of the later-developed so-called general theory of relativity which includes the theory of universal gravitation. This is the reason why the present writer treated only the special theory of relativity, and not the general theory, in the said work. The relation of these two theories, however, is not so simple as the terms “special” and “general” might suggest. The usual passage of the special (the particular) to the general is what is called generalization, i. e. the “extension” of species to genus performed through the removal of the restriction on the former. But the “expansion” of the special theory of relativity to the general theory of relativity is neither a mere generalization nor an extension. It requires the reversion of the standpoint. The expansion of the relativity theory is therefore considered to have been possible, not with the analytic logic in which species can be formally extended to genus, but only by following the line of dialectic as a logic of the unity of contradictories. But then, might not the above assertion that the structure of the world of the special theory of relativity is adequately represented by complex numbers be denied by this reversion of the standpoint? Besides, the relativity theory which present-day physicists seek to integrate with the quantum theory is not limited to the special theory, but is expanded to the general theory. Thus the said assertion of the present writer might seem to be brought to a dilemma. It is the aim of this paper to try to reveal the dialectical structure of the relativity theory itself and thus to break through this apparent difficulty.

This task comprises three stages or three themes. The first theme to be maintained is that the relation of the special and the general theories of relativity is, as explained above, not an identity-logical relation between species and genus, but that it pre-supposes a reversion of the standpoint ; it is not a subsumption but a dialectial relation and the unity of contradictories. Of course the rotation which is treated in the general theory of relativity is but a sort of motion ; but it is something which corresponds as the equivalent to the circular motion with acceleration, in distinction from the uniform translatory motion treated in the specical theory of relativity. The former is thus regarded as a universal which comprehends the latter in itself as a particular, and so it would be natural that the special theory of relativity is expanded to the general. But what the writer means by the term “natural” is the “psychological naturalness” in Einstein's sense of the term and not the “logical necessity”. This expansion could only be accomplished through a volitional decision. Namely, it needs a volitional act to overcome the difficulty of the reversion of the standpoint. To speak more concretely, while the “world” of the special theory of relativity is the so-called hyperbolic space whose space curvature is negative and in which it is possible to let pass through a point not on a given straight line an infinite number of lines not intersecting the given line, the “world” of the general theory of relativity is the elliptic space whose space curvature is positive and where there is no parallel line and straight lines always intersect. These two make indeed a contrary opposition. This is why these two standpoints are said to be contradictory to each other. To clarify this point makes the first stage of the discussion. In the second stage, this opposition is examined more thoroughly. Why, we should ask, is it that, though the uniform translatory motion treated in the special theory of relativity and the accelerated motion or rotatory motion treated in the general theory of relativity seem apparently to have the difference of the particular and the general, these two motions should be explained from the opposed standpoints as stated above? It is because, the writer asserts, while in the former the relativity of motion is self-evident by experience and we need not go to the back or the depth of experience, in the latter force which can not be directly experienced is supposed to be in the back or the depth of experience as the cause of the change of the motion which is called acceleration. Moreover, that the existence of force, in spite of the fact that it can not be directly experienced, is the reason why we understand the rotatory or circular motion as the absolute, unlike the uniform translatory motion, leads us inevitably to the antinomy between our demand of regarding force as relative and that of admitting its absoluteness. In the general theory of relativity, seeking to expand the theory from the relativity of the uniform translatory motion of the special theory of relativity to the relativity of force, we must run on the rock of the absoluteness of force. The theory of relativity can establish itself only when it breaks through this difficulty.

The idea of force has been a stumbling block of modern dynamics. The 19th century methodology of dynamics has been developed with this antinomy as its motivation. Though the change of the state of motion can be directly experienced the demand to seek the cause of the change and to explain it can not be satisfied unless we assume the action of force which does not belong to experience. This is why the phenomenon of motion has been thought to have a close connection with the action of force. Especially when force was thought to act among all sorts of bodies as universal gravitation, and moreover when the magnitude of force was represented in a very simple proportion to mass and distance, the Newtonian classical dynamics was established as dynamics of the whole inorganic nature. It goes, however, against positivism, the chief assertion of modern science, to try to explain the phenomenon of motion in terms of force which can not be directly experienced. So in-the latter half of the 19th century, in the tendency towards the thoroughgoing purification of scientific thought, a movement arose, that of the Descriptive School as represented by Kirchhoff and Mach, in which they expelled the idea of force from dynamics, rejected the explanation by it, and regarded only the clear and distinct description of the experience of motion as the task of dynamics. Especially Mach, making wide and deep researches into epistemology and methodology, published the noted work on the history of mechanics which had a great influence on the succeeding thinkers. Einstein believed himself to be under the influence of Mach, and the relativity theory is regarded to be a development of Mach's positivism and empiricism. Against Newton, who took the rotation of the earth to be an absolute motion, Mach contended that it is but a motion relative to the fixed stars. And this idea was adopted into the general theory of relativity. But if we pursue this way of thinking still further, we will have to interpret the motion of the heavens of fixed stars as the relative motion to a matter outside of them, and therefore at last we will have to be led to a dilemma of considering its motion to be a motion relative to a matter which may be regarded non-existent lying at an infinite distance not to be directly experienced by us. This is an antinomy : in making the empiricism thorough-going, we have to transcend the experience and turn ourselves to an apriori conception. Dynamics is to be regarded as a unity of duality of experience and the apriori, which Kant considered fundamental in his criticism. Boltzmann's “Vorlesungen über die Prinzipe der Mechanik” (1897-1904) is a masterpiece representing this unity.

Boltzmann, who in mechanics is comparable to Kant in philosophy, thus overcame Mach's empiricism which was in the tradition of Hume. But as Kant could not get rid of the remains of dogmatism because of the assumption of things-in themselves, Boltzmann could not make his criticism thoroughgoing by reason of his assertion of the existence of atoms as the substance and of his assumption that their actions conform to the laws of classical dynamics. Hence mechanics needed to pass Hertz's monistic standpoint, on which the idea of force was rejected and replaced by the connection of masses, and ultimately reduce to the total compulsory motion-a standpoint comparable to Schelling's of intellectual intuition. Boltzmann's immanent dualism which is comparable to that of Kant is brought, by way of Hertz' system, to a dialectical unity. Comparable to Hegel, who is a synthesis of Kant and Schelling, is Einstein in the general theory of relativity, who can be understood to be a synthesis of Hertz and Boltzmann in the history of mechanics. There in Einstein, the apriori conception is not taken to be independent of experience and onesidedly to determine the latter, as in the case of Kant and Boltzmann, but is thought to presuppose the experience of the past as the medium, and to be determined by the act of the present which innovates the past experience towards the future. Here the apriori and experience determine each other historically, and mutually interpenetrate. Hegel's characteristic is found indeed in this historicistic dialectic. The general theory of relativity is near Hertz's mechanics in a way, but it is to be compared to Hegel's thinking in that Hertz's view of totality is not here taken and that, based on local observations, the self-consciousness of the act of observation in the historical connection is theorized. The development of that theory by the aid of the tensor analysis may be analogous to the development of Hegel's logic from his Phenomenology of Mind.

So far the outline of our second theme. Let us now state our third theme in brief. It concerns the problem how the structure of the “world” of the special theory of relativity represented by complex numbers, or its structure of hyperbolic space, is preserved and comprised in the “world” of the general theory of relativity, i. e. the elliptic space, which is contrary to the hyperbolic space. This problem would meet an unsolvable difficulty, if the relation of the special and the general theories of relativity were thought to be that of species and genus in the identity-logic. If so, in the “world” of the special theory of relativity, the structure represented by complex numbers would have to be abolished in order that that “world” might be comprehended in the “world” of the general theory of relativity. In other words, the elliptic space of the latter would not comprehend the hyperbolic space of the former without ruining itself. Thus from the view of the identity-logic, the special and the general theories of relativity contradict each other and can not be united. In order to make this unity possible, we must recognize a reversion of the standpoints of these two, convert the contradiction into an absolute negation, admit the contradiction as a contradiction, make it a moment of mutual tension, and bring it to a unity. This conversion-negation-resurrection-unification is just the act of the individual subject. This self-contradictory unity is indeed the individuality which unifies arbitrary paths of convergence as an open set into a circle of convergence, along the path of a function, as seen in the theory of functions of complex variables. As the relation of the special and the general theories of relativity is not equivalent to that of species and genus of the identity-logic, and contains a contradiction between them, the individual subject should be the medium to perform the negation and reversion. That is to say, only the dialectical trinity of genus, species and the individual can bring these theories into completion. Since the self-contradictory unity of the individual subject as the nucleus of this trinity is the discontinuous continuous unity of the circle of convergence, the medium to unite the special and the general theories of relativity should be considered to be the theory of functions of complex numbers. This is the point of the third theme of this paper.