Modern Science and Its Philosophy

Philipp Frank

CHAPTER
5

is there a trend today toward idealism in physics?

It is generally recognized that modern exact science, the creation of which in the age of Galileo and Newton led to the great expansion of our technical civilization, is distinguished from ancient and medieval science by the fact that the psychic, anthropomorphic elements are being eliminated more and more from science. In place of the medieval doctrines of "the most perfect orbit," "the position appropriate to a body,” “the difference between celestial and terrestrial bodies," and the like, we now have the mathematically formulable laws of Newton's principles, in which only observable and measurable quantities occur. There is no doubt that the physics of Galileo and Newton has created a gulf between body and mind which did not exist in the anthropomorphic, animistic science of the Middle Ages. This separation of the two became unpleasant to those who were interested in a science that would account for the behavior not only of inanimate bodies, but of all bodies in nature, including the human body. Thus arose the problem of explaining the mind on the basis of mechanistic physics, a problem which many have discussed, but without ever making any real advance, and which is actually only an apparent problem. Its insolubility in this form, which was really obvious to all, led many scholars to have a deep dislike for mechanistic physics and to derive a malicious pleasure from every difficulty that the latter encountered. R. Ruyer is quite right when he says:

That, basically, many scholars are tortured by the burden of this original sin, mechanistic physics, is strikingly shown by their reaction whenever the mechanistic or quantitative conception of physics appears to suffer a setback. The most philosophical minds, far from being disturbed by this setback, hope every time to find in it an opportunity for introducing once more the subjective. This was the case with the discovery of potential energy, gravitation, and the degradation of energy, as well as chemical affinity. [1]

One need not be surprised, therefore, if the recent revolutions in the field of theoretical physics, the creation of the relativity and quantum theories, were received by the scholars whom Ruyer called "the most philosophical minds" with the same feelings as the preceding theoretical overturnings, such as the degradation of energy. Indeed, today one can hardly open a periodical or book dealing with the development of our general scientific ideas without meeting such expressions as "the end of the age of Galileo," "the failure of mechanistic physics," "the end of the hostility of science toward the spirit,” “the reconciliation between religion and science." There is even a book on modern physics, by Bernhard Bavink, entitled "Natural Science on the Path to Religion.” [2]

Some are of the opinion that the new physical theories of the twentieth century have brought about a change in the general conception of the world as important as that caused by the physics of Galileo, which replaced the animistic conception of the Middle Ages by the mechanistic one of modern times. In the same way, the new physics is supposed to form a bridge from the "mechanistic world conception” of the eighteenth and nineteenth centuries to the "mathematical conception" of the twentieth century. The latter is thought to be nearer, in a certain sense, to the medieval, animistic conception than to the mechanistic one, because in mathematics there resides an "ideal" or "spiritual" element, and a "mathematical world" is not as foreign to the spirit as is a mechanical world. This view was presented in all solemnity by General Smuts in his opening address at the celebration of the centenary of the British Association of Science on September 23, 1931. [3] He said, among other things:

There is the machine or mechanistic world view dominant since the time of Galileo and Newton, and now, since the coming of Einstein, being replaced by the mathematician's conception of the universe . . . If matter is essentially immaterial structure or organization, it cannot fundamentally be so different from organism or life . . . or from mind, which is an active organizer.

First, we must ask from the standpoint of the logic of science whether the physical theories of the twentieth century really contain any spiritualistic elements and, second, we must ask with what processes outside of physics the demand for a spiritualistic conception of nature is generally found to be associated. Let us begin by touching briefly on the second question in order to be able to consider the first in greater detail.

It is certainly no accident that the culmination of the mechanistic conception of nature, as it is found, say, in the work of Laplace, coincided with the triumph of the French Revolution. It is certainly no accident that since that time the struggle against the "ideas of 1789" has almost always coincided with a criticism of this conception of nature, a longing for a more idealistic or spiritualistic theory. The struggle against the "ideas of 1789" has been crowned in recent years by the fact that in a series of countries, especially in Italy and Germany, a directly opposite world conception has prevailed politically. This conception has a philosophic basis that is in sharp contradiction to the mechanistic conception of nature and urges a more "organismic" picture of the world, by which is meant a partial return to the spiritualistic or animistic doctrines of the Middle Ages, just as the new conception of a state is connected with that of the Middle Ages. The adherents of this antimechanistic, organismic conception of nature strive to show that in the exact sciences "spontaneously," "from purely scientific considerations," a revolution has taken place. According to their argument, on the basis of the relativity theory and quantum mechanics one can set up a conception of nature in which the mind again plays a role, and which is compatible with an "antimechanical, organismic, independent" biology.

As a typical example, a work of B. Bavink may be quoted. Bavink has a thorough knowledge of physics and biology and is an outstanding representative of the "organismic conception of nature." He maintains that

today there reigns within the circle of the sciences a willingness to tie once more the threads of science to all the higher values of human life, to God and the soul, freedom of will, etc.—threads that seemed almost completely severed; it is a willingness the like of which has not been present for centuries. That this change should take place at the present time is a coincidence almost bordering on the miraculous, for this change has in itself nothing to do with political and social transformations; it manifestly arose from purely scientific motives. [4]

Whether or not the last sentence is correct is exactly the question we are trying to answer.

On the other hand, in Russia, since the founding of the Soviet Union, a system has been established that seeks its philosophic basis in the "dialectical materialism" of Karl Marx as adapted by Lenin. I do not wish here to discuss the relation between this "dialectical materialism" and what one is accustomed to call "materialism" in Germany and France. I want only to call attention to the fact that in countless articles in the philosophical and political journals of present‑day Russia, the tendency toward spiritualism which is often found as an accompaniment of modern physical theories is interpreted as one of the "phenomena of decadence" of science in capitalist countries. [5] In these articles the following line of thought often appears: in western Europe science, to be sure, is still making progress on individual problems, such as the formulation of laws of atomic processes, just as capitalist economy is still progressing technically. However, just as the life of the industrial population is shaken more and more by crises which finally make a generally acceptable solution impossible, so science, in spite of its progress in details, cannot produce a satisfying general picture of the processes in nature. In working on such a general picture it no longer proceeds scientifically, in the modern sense. Rather, it borrows from the animistic, spiritualistic physics of the Middle Ages and interprets modern theories in this light, because the intellectual trends dominant in political life obscure the scientific theories by wrapping them in a spiritualistic fog.

In spite of the tremendous conflict between the "materialistic" Soviet Union and the states based on the organismic conception of the world, they all agree on the fact that the tendency toward spiritualism in modern physics corresponds to the ideology of the new organismic state. By some this tendency is welcomed as a necessary consequence of modem physics; by others it is condemned as an adulteration of it. That the representatives of both groups comprehend physics in this way is a fact that is as well established empirically as the best observations of experimental physics, a fact which we must therefore take into account in any consideration of modern physical theories.

I wish to say at once that the result of our investigation will be as follows: In the process of eliminating the "animistic," nothing has been changed in the slightest by the modern physical theories. This process continues irresistibly forward as before. He who would interpret physics by means of "psychic factors" had at the time of the physics of Galileo and Newton the same justification as today. The role of the “psychic" has remained exactly the same. Hence, if there does exist today a greater tendency toward spiritualistic interpretation, it is connected only with processes that have nothing at all to do with the progress of physics.

The arguments that are supposed to show that psychic factors play a greater role in modern physics than in the physics of Newton are of various kinds. In one group it is claimed that the role of the "observing subject" in the relativity and quantum theories can no longer be eliminated from physical statements, as was still the case in "classical physics." This argument is often put as follows: Whereas in classical physics expressions like "length of a rod" or "time interval between two events" assert something about "objective" facts, in the Einstein relativity theory such expressions have a meaning only if the observer to whom they refer is specified. One can only say, for instance, "This body has a length of one meter with respect to this particular observer." It appears, therefore, that every physical statement possesses a psychological constituent. In popular literature on the relativity theory writers often even go so far as to compare the various lengths of a rod for various observers with the optical illusion that arises when one draws two straight lines of equal length but places different ornaments at their ends, producing the illusion of different lengths.

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This conception, in its "scientific" as well as its "popular" form, is based on a complete misunderstanding of the relativity theory. Wherever in the theory of relativity reference is made to an observer, a physical measuring instrument can be substituted. It is asserted only that the results of the measurement will be different according as the motion of the measuring instrument is different. But in this there is nothing psychological, at any rate not any more than in classical physics. The role of the observer is in both cases exactly the same: he merely substantiates the fact that in a certain instrument a pointer coincides with a division mark on a scale. For this purpose the state of motion of the observer himself is quite immaterial. In the theory of relativity, as well as in classical physics, it is assumed that such a substantiation is "objective," that there can never arise any difference of opinion in connection with it. Naturally, it remains "subjective" in the sense that some observer is necessary for it. Here "objective" means “the same for all subjects," or "intersubjective."

Similar considerations have also been associated with the quantum theory. According to this theory, as Heisenberg showed, the position and the velocity of a given particle can never be exactly determined simultaneously. If one makes use of an experimental arrangement that allows the position to be measured very accurately, the exact measurement of velocity by means of the same experimental arrangement is impossible. It is then held that whereas in classical physics a statement about the position and velocity of a particle was a statement about an objective fact, without any psychologic elements, in the quantum theory one cannot speak of the position and velocity of a particle, but only of what is given by a certain measurement. Thus every statement about particles involves the observer himself, and hence contains a psychologic element. To this argument we must reply as I have just indicated in the discussion of the relativity theory, pointing out that in quantum mechanics, too, what matters is never the observer, but only the instruments of observation. The role of the human being as observer is limited here again to establishing whether or not a pointer on a scale coincides with a division mark. This observation, however, is regarded here, just as in classical physics, as something "objective," or better, as something "intersubjective." What one can learn from the relativity and quantum theories in this connection is only what is also given by a consistent presentation of classical physics: every physical principle is, in the final analysis, a summary of statements concerning observations, or, if one wishes to speak in a particularly physical way, concerning pointer readings.

We have thus seen that the new role of "observer" in physics cannot be used in favor of a tendency toward a more spiritualistic conception of physics. There exists, however, a whole series of other arguments which are customarily used to establish the approach or “return" of physics to the "organic, idealistic" conception of nature. Such arguments run somewhat like this: "Quantum mechanics contains a teleological element," or "The indeterministic interpretation of the quantum theory makes room for free will." Here we shall not consider these questions, but shall speak of a still more general argument for the "spiritualistic character" of modem physics. This is an argument which in recent years has been repeated so often and by scholars of such prominence that there is danger that many, through becoming accustomed to such lines of thought, will accept them as justified—indeed, as obvious. These ideas have perhaps received the most extensive dissemination in 150,000 copies of a book by the outstanding physicist and astrophysicist, J. H. Jeans. Jeans depicts the present situation in physics as follows:

Today there is a wide measure of agreement, which on the physical side of science approaches almost to unanimity, that the stream of knowledge is heading towards a nonmechanical reality; the universe begins to look more like a great thought than like a great machine. Mind no longer appears as an accidental intruder into the realm of matter. [6]

Jeans bases his opinion that nature is to be regarded as something “spiritual" essentially on the assertion that modern physics has shown that one cannot give any mechanical representation of natural processes, although one can give a mathematical representation. He says:

The efforts of our nearer ancestors to interpret nature on engineering lines proved equally inadequate . . . On the other hand, our efforts to interpret nature in terms of the concepts of pure mathematics have, so far, proved brilliantly successful. [7]

In the laws of mathematics, in contrast to those of mechanics, of machinery, Jeans sees, however, a spiritual element. If nature behaves according to mathematical laws, it must be the work of a mind which can create mathematics, like the human mind, but is more comprehensive. Jeans is so strongly convinced that the movement toward idealism is connected with the present state of theoretical physics that he keeps in view the possibility that, with a change of the theories of physics, there may again develop a movement away from idealism. Thus he says in a later book:

So far the pendulum shows no signs of swinging back, and the law and order which we find in the universe are most easily described—and also, I think, most easily explained—in the language of idealism. Thus, subject to the reservations already mentioned, we may say that present‑day science is favorable to idealism . . . Yet who shall say what we may find awaiting us round the next corner? [8]

Similar views are presented by Sir Arthur Stanley Eddington, in his book The Nature of the Physical World. [9] While there is in this book a great deal that is beneficial in furthering the understanding of modern physics and in bringing its results to a wide circle of readers through a concrete and lucid presentation, yet it has numerous sections which Eddington himself regards as bold interpretation of present‑day physics to which many will perhaps take exception, and which, in my opinion, form obstacles to the task of fitting physics into a self‑consistent picture of the processes of the whole of nature. Eddington, like Jeans, believes that these views are matters of faith and that it is impossible to force one by proof to accept them. That is certainly true. However, what can be shown clearly, in my opinion, is that these idealistic views have nothing at all to do with modern physics. If anyone wanted to accept them, he could have done so just as well in connection with the physics of Galileo and Newton, which is not less "mathematical" than twentieth‑century physics.

The arguments of both Jeans and Eddington depend on the contrast between a physics that reduces everything to mechanics (that of Galileo and Newton) and one that bases everything on mathematical formulas (the physics of Einstein and the quantum theory). How can one formulate clearly the distinction between a "mechanical" and a "mathematical" basis for the processes of nature? Newtonian physics reduces all phenomena to the equations of motion for mass points between which there act central forces, that is, to a system of differential equations. The mechanics of Einstein changes these differential equations in a few respects which give essential differences only for very high velocities, and points out that the equations so changed have mathematically the same form as the geodesics in a curved (non‑Euclidean, Riemannian) space. In place of one system of differential equations, another occurs. Why then is one theory called “mathematical," the other "mechanistic"? Surely similarity to the geodesics cannot be the only reason, for Newtonian physics can also be brought into this form without any difficulty.

Adhering to the concrete interpretation of physics as a representation of observable facts, we can try to summarize the difference between "mechanical". and "nonmechanical, mathematical" physics approximately as follows: By means of Newtonian mechanics we can describe the motions of bodies with which we deal in everyday life, so long as they have also the velocities that are encountered in daily experience. To this class of bodies belong the ordinary tools such as hammers and tongs, but also such things as steam engines, automobiles, and airplanes. During the reign of the physics of Galileo and Newton it was believed that in time it would be possible by means of these same equations to describe also the motions of the smallest particles of matter, such as atoms and ions, as well as the motions of celestial bodies during arbitrarily long time intervals and with arbitrarily high velocities. In other words, it was believed that all processes of nature, in the large and in the small, could be covered by the same laws that had  been established for the motions of "bodies of average size with moderate velocities." This belief has been shaken by the development of physics in the twentieth century. We know today that the motions of bodies with velocities comparable to that of light can be described only with the help of the relativity theory of Einstein, the motions of the smallest particles in the atoms only with the help of quantum and wave mechanics.

If we understand by mechanics the doctrine of the motion of "bodies of average size with moderate velocities," then we can rightly say that modern physics has established the impossibility of a mechanical basis for the processes of nature. If we say, however, that the mechanical foundation has been replaced by a mathematical one, it is, in my opinion, a very inappropriate mode of expression. We ought to say, rather, that the place of a special mathematical theory, that of Newton, has been taken by more general theories, the relativity and quantum theories. The opinion that a special mathematical theory could represent all the processes of nature has turned out to be false; that is all. But from this fact no contrast between the propositions "Newtonian physics = mechanics = materialism," on the one hand, and “modern physics = mathematics = idealism," on the other hand, can be deduced.

Newton, in his Mathematical Principles of Natural Philosophy, [10] replaced the matter filling the world and acting through pressure, collisions, and fluid vortices, as pictured by the Cartesians, by small masses, almost lost in vast empty space and acting on each other only through forces at a distance. When this work was published, the new theory was hailed by many of his followers as a triumph over the materialism of the "Epicureans."

As proof, one need only read the famous controversy between Leibniz and Clarke, [11] in which Clarke defends Newton's teachings against the attacks of Leibniz. Clarke says in his first reply:

Next to the corruptible dispositions of human beings, it [the disavowal of religion] is to be ascribed first of all to the false philosophy of the materialists, who oppose the mathematical principles of philosophy [i.e., Newton's] . . . These principles, and indeed only they, show matter and the body as the smallest and most insignificant part of the universe.

Hence at that time Newton's followers, in so far as they were adherents of spiritualistic metaphysics, extolled his teachings as "mathematical" and "spiritual" in contrast to materialism. Today those with analogous philosophic inclinations say that Newtonian physics was “materialistic," but that Einstein has again brought in a "mathematical," “spiritual" element in place of the mechanical one.

We have already seen that the assertion that the laws of nature are not "mechanical" but "mathematical" means only that the laws are expressed not by means of the special mathematical formulas of Newton, but by means of the more general formulas of the relativity and quantum theories. When, however, we say, not that the formulas used to describe nature are mathematical, but that the world is mathematical, it is difficult to say what we mean. By mathematics, considered concretely, we can only understand a system of formulas or propositions. With these formulas are to be correlated the observations that we make of the processes of nature, if the formulas are to represent physical theories. The processes themselves, however, do not consist of these formulas. In an assertion such as "The world is basically mathematics," the word "is" can only be used in a mystic sense, as it occurs perhaps in the sentence "This architecture or this music is pure mathematics."

In order to make his views clear, Jeans must speak of the world architect; he represents him, not according to Newtonian physics as a kind of engineer, but according to modern physics as a kind of mathematician. Since engineers also produce their work according to mathematical formulas, Jeans has to indicate the distinction between the engineer and the world creator somewhat as follows: The engineer fits his formulas to the observations, whereas the creator invents formulas at will and then constructs the world according to them. Jeans brings in here the difference between "pure” and "applied" mathematics. The engineer is an applied mathematician, the world creator a pure one. Jeans tries to show it in this way: The man who works in pure mathematics invents formulas and propositions without any regard to the question of practical application; later, it turns out that the physicist or engineer, by means of the results obtained by the pure mathematician, can represent the processes of nature, of which the pure mathematician knew nothing when he devised his theory. This can only be explained by saying that the processes are themselves the work of a pure mathematician, and the theoretical physicist who finds these formulas for representing observations is only rediscovering the ideas of the pure mathematician who created the world. The creations of the demiurge must accordingly agree to a large extent with those of a human pure mathematician.

The assertion that the world is built according to the principles of "pure" mathematics is to be found not only in the works of Jeans. It is very often used in setting up mystical conceptions of the world. If one wants to be clear as to its meaning, one must first of all obtain clarity as to the meaning of the propositions of "pure" mathematics in general. According to the conception of B. Russell and L. Wittgenstein, which is also that of the Vienna Circle, the propositions of pure mathematics are not statements concerning natural processes, but are purely logical statements concerning the question of what assertions are equivalent to one another, which can be transformed into one another by formal transformations. The propositions of pure mathematics, therefore, remain correct, no matter what the natural processes may be; these propositions can be neither confirmed nor refuted by observation, since they state nothing concerning the real processes of nature. Mathematical theorems, as is often said, are of an analytic character.

For example, if I prove the theorem "The sum of the angles of a triangle is equal to 180º" as a proposition of pure mathematics, I prove only that from the axioms of Euclidean geometry, including the axiom of parallels, it follows by logical transformation that the sum of the angles of a triangle is equal to 180º if the straight lines and points of which it consists have all the properties ascribed to them by the Euclidean axioms. That is to say, if for a concrete physical triangle I can establish by observation the validity of the Euclidean axioms, then the sum of the angles is equal to 180º. In other words, the statements "The sum of the angles is 180º" and "The axioms are valid" are only two expressions of the same thing, two statements with the same content (where, of course, the proposition of the sum of the angles is only a part of the content of the whole system of axioms). Once this has become clear, the world, whatever it may be, will always obey the propositions of pure mathematics; the assertion that it obeys them says nothing at all about the real world. It says only what is self‑evident, that all statements about the world can be replaced by equivalent statements.

Something else must obviously be meant, however, when Jeans and so many others say that the world is constructed according to the principles of "pure" mathematics. As an example, the following is adduced: Mathematicians—Christoffel, Helmholtz, Ricci, Levi Civitá, and others—long ago built up the theory of the curvature properties of Riemannian space. When Einstein set up his general theory of relativity, he found this whole branch of mathematics ready for him. Although it was invented without any intention of its being used in physics, Einstein was able to apply it in his theory of gravitation and general relativity. One must therefore assume that the creator built the world according to those principles of pure mathematics. Otherwise it would be an inconceivable coincidence that such a complicated branch of mathematics, developed for quite other purposes, could be used for the theory of gravitation.

We have already seen that this assertion cannot mean that the world is built according to the propositions of the Riemannian curvature theory or of the absolute differential calculus invented by Ricci and Levi Civitá; for these propositions, like the proposition of the sum of the angles and all other propositions of pure mathematics, are only statements of how one can express the same thing in different ways. The assertion, therefore, can only mean that the concepts and definitions of pure mathematics—the geometry of Riemannian spaces—created certain structures—the Christoffel three‑index symbols, the Riemannian curvature tensor—which could be utilized in the Einstein theory of gravitation. This, however, is the same, although perhaps on a higher level, as saying: "The concepts of the square or the square root or the logarithm have come out of pure mathematics; it is therefore amazing that they also occur in the formulas of physics." If we now use the possibility of representing the world according to Einstein, with the help of the Riemannian curvature tensors, as proof that the world was created by a mathematician, we might have said with the same justification, back in the time of Newton, that the world must have been created by a mathematician; for in Newton's formulas the chief role is played by the "square of the distance," and the concept of the square of a number originated in geometry and was introduced without any regard for physics. If we consider the matter from this standpoint, that is, if we speak not of mathematical theorems but of mathematical concepts, a little reflection shows that the distinction drawn by Jeans between engineer and mathematician, or between “applied" and "pure" mathematics, cannot be maintained.

As a matter of fact, concepts such as those of Riemannian curvature have always been invented for the purpose of representing some problem of concrete reality, for describing processes of nature. The concepts of Riemannian geometry all go back to the problem of describing the motion of a real rigid body in general coordinates; one need only recall the work of Helmholtz on the facts at the basis of geometry. [12] Riemann, Christoffel, and Helmholtz set up certain mathematical expressions which are equal to zero in the case of the motion of a rigid body, according to the usual laws of physics. When Einstein proceeded to formulate the deviations from these laws, it was clear that he had to begin with the expressions that gave the properties of rigid bodies, according to classical physics, in a form valid for all coordinate systems. If there existed any deviations expressible independently of the coordinate system, it had to be possible to express them so that the quantities which in the old physics had the value zero were now different from zero and took on values depending in a simple manner on the distribution of matter. If such a simple dependence did not exist, then there could exist no laws independent of the coordinate system, as Einstein required. If such laws did exist, it had to be possible to express them through the concepts that were at hand for representing the motion of a rigid body. But this gives no evidence that the world creator was a "pure mathematician." The only thing that must be regarded as a real and astonishing characteristic of nature is the fact that there do exist, in general, simple laws for the description of nature. This, however, has nothing to do with the distinction between "mechanical" and "mathematical."

If today expressions with spiritualistic coloring are used to a greater extent than in the nineteenth century, this has no connection with any "crises in physics" or with any "new physical conception of nature." It is rather associated with a crisis in human society arising from quite different processes. In opposition to the materialistic social theories there have come into the foreground movements based on an "idealistic" picture of the world. These movements seek support in an idealistic or spiritualistic conception of nature. Just as at the end of the nineteenth century analogous movements made use of energetics, the electromagnetic picture of matter, and so on, to prophesy the end of  “materialistic" physics, so today the relativity and quantum theories are being used. All this, however, has no real connection with the progress of physics.

Notes

1  R. Ruyer, Revue de Synthèse 6, 167 (1933). [—> main text]

2  B. Bavink, Die Naturwissenschaft auf dem Wege zur Religion (Frankfurt am Main: M. Diesterweg, 1933). [—> main text]

3  Nature 128, 521 (1931). [—> main text]

4  B. Bavink, "The Sciences in the Third Reich" (in German), Unsere Welt 25, 225 (1933). [—> main text]

5  As a recent example may be cited A. K. Timiriazew, "The Wave of Idealism in Modern Physics in the West and in Our Country" (in Russian), Pod znamenem marksizma, 1933, no. 5. [—> main text]

6  J. H. Jeans, The Mysterious Universe (Cambridge: The University Press, 1930), p. 158. [—> main text]

7  Ibid., p. 143. [—> main text]

8  J. H. Jeans, The New Background of Science (Cambridge: The University Press, 1933), p. 296. [—> main text]

9  Cambridge: The University Press, 1928. [—> main text]

10  Newton, Philosophiae Naturalis Principia Mathematica (1687), tr. by A. Motte (1729), rev. by F. Cajori (University of California Press, 1934). [—> main text]

11  A Collection of Papers, which Passed between the Late Learned Mr. Leibnitz, and Dr. [Samuel] Clarke, in the Years 1715 and 1716 . . . (London: J. Knapton, 1717). [—> main text]

12  "Über den Ursprung und die Bedeutung der geometrische Axiome" (1870), translated as "On the Origin and Significance of Geometrical Axioms," in Helmholtz, Popular Lectures on Scientific Subjects, series 2, tr. by E. Atkinson (London: Longmans, Green, 1881). [—> main text]

SOURCE: Frank, Philipp. Modern Science and Its Philosophy. Cambridge, MA: Harvard University Press, 1949. Reprint: New York: George Braziller, 1955. Chapter 5, Is There a Trend Today Toward Idealism in Physics?, pp. 122-137.

Modern Science and Its Philosophy: Contents

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