It is to Max Born and Albert Einstein that we owe the "statistical interpretation" of quantum mechanics that Born proposed as a third interpretation of quantum mechanics, in addition to Niels Bohr and Werner Heisenberg's matrix mechanics with its emphasis on particles and (the Copenhagen Interpretation) Ernst Schrödinger's wave mechanics with its emphasis on waves. Louis de Broglie was a critical link between the work of Albert Einstein and Max Born's statistical interpretation of quantum mechanics. It was de Broglie who gets credit for arguing that if light, which was thought to consist of waves, is actually discrete particles (Einstein' called light quanta, later called photons), then matter, which is thought to consist of discrete particles, might also have a wave nature. But he was inspired by Einstein.
The fundamental idea of [my 1924 thesis] was the following: The fact that, following Einstein's introduction of photons in light waves, one knew that light contains particles which are concentrations of energy incorporated into the wave, suggests that all particles, like the electron, must be transported by a wave into which it is incorporated... My essential idea was to extend to all particles the coexistence of waves and particles discovered by Einstein in 1905 in the case of light and photons."Einstein had said that the light wave at some position is a measure of the probability of finding a light particle there, that is, the intensity of the light wave is proportional to the number of photons there. It may have been implicit in his 1905 light quantum hypothesis, as de Broglie seems to think. Although Einstein had given this opinion to colleagues as early as 1921, and although he accepted the statistical nature of quantum mechanics (despite his claim that "God does not play dice), we don't have specific quotes from Einstein until 1927 at the fifth Solvay conference, when he said.
|ψ|2 expresses the probability that there exists at the point considered a particular particle of the cloud, for example at a given point on the screen.Over a year earlier than that Solvay conference, in July of 1926, Max Born used de Broglie's matter waves, which he calculated using Erwin Schrödinger's wave equation, to interpret the wave as the probability of finding an electron going off in a specific collision direction as proportional to the square of the probability amplitude wave function. Born gave full credit to Einstein, de Broglie, and Schrödinger for the idea, although the "statistical interpretation" itself is pure Einstein..
Collision processes not only yield the most convincing experimental proof of the basic assumptions of quantum theory, but also seem suitable for explaining the physical meaning of the formal laws of the so-called “quantum mechanics.”Indeed, as it seems, it always produces the correct term values of the stationary states and the correct amplitudes for the oscillations that are radiated by the transitions, but opinions are divided regarding the physical interpretation of the formulas. The matrix form of quantum mechanics that was founded by Heisenberg and developed by him and the author of this article starts from the thought that an exact representation of processes in space and time is quite impossible and that one must then content oneself with presenting the relations between the observed quantities, which can only be interpreted as properties of the motions in the limiting classical cases. On the other hand, Schrödinger (3) seems to have ascribed a reality of the same kind that light waves possessed to the waves that he regards as the carriers of atomic processes by using the de Broglie procedure; he attempts “to construct wave packets that have relatively small dimensions in all directions,” and which can obviously represent the moving corpuscle directly. Neither of these viewpoints seems satisfactory to me. Here, I would like to try to give a third interpretation and probe its utility in collision processes. I shall recall a remark that Einstein made about the behavior of the wave field and light quanta. He said that perhaps the waves only have to be wherever one needs to know the path of the corpuscular light quanta, and in that sense, he spoke of a “ghost field.” It determines the probability that a light quantum - viz., the carrier of energy and impulse – follows a certain path; however, the field itself is ascribed no energy and no impulse. One would do better to postpone these thoughts, when coupled directly to quantum mechanics, until the place of the electromagnetic field in the formalism has been established. However, from the complete analogy between light quanta and electrons, one might consider formulating the laws of electron motion in a similar manner. This is closely related to regarding the de Broglie-Schrödinger waves as “ghost fields,” or better yet, “guiding fields.” [Einstein had used both "Gespensterfeld" and "Führungsfeld."] I would then like to pursue the following idea heuristically: The guiding field, which is represented by a scalar function ψ of the coordinates of all particles that are involved and time, propagates according to Schrödinger’s differential equation. However, impulse and energy will be carried along as when corpuscles (i.e., electrons) are actually flying around. The paths of these corpuscles are determined only to the extent that they are constrained by the law of energy and impulse; moreover, only a probability that a certain path will be followed will be determined by the function ψ. One can perhaps summarize this, somewhat paradoxically, as: The motion of the particle follows the laws of probability, but the probability itself propagates in accord with causal laws. Probability and statistics were very important in the two centuries before Born's work, but most physicists and philosophers saw the implied randomness to be the consequence of human ignorance. They denied any underlying absolute chance, with the exception of a few thinkers like Franz S.Exner and (at least until the late 1920's) his student Erwin Schrödinger. The random distributions were thought to be completely deterministic at the particle level, with individual atoms following Newton's dynamical laws. Max Planck solved the great problem of blackbody radiation by applying the statistical mechanics of the Maxwell-Boltzmann velocity distribution law to the distribution of radiating oscillators, which he described as limited to quantized amounts of energy, ε, 2ε, 3ε, etc. Planck did not quantize the radiation field. Five years later, Einstein saw that Planck's work implies that light itself comes in quanta with hν units of energy, where h is Planck's constant and ν is the frequency of the light wave. Planck did not believe, and for years denied, that light existed as light quanta. His quantization assumption was a mathematical device to make the distribution of light as a function of frequency (and thus energy) resemble the Maxwell-Boltzmann distribution of molecular velocities in a gas as a function of velocity (and thus energy). Albert Einstein explained the photoelectric effect with Planck's discrete units of light energy, later called photons. Since the momentum of a particle is the energy divided by velocity of a particle, the momentum p of a photon is p = hν/c, where c is the velocity of light. To make the dual aspect of light as both waves and particles (photons) more plausible, Einstein interpreted the square of the light wave amplitude as the probable density of photons. Schrödinger's creation of his quantum mechanical wave function Ψ followed a suggestion by Louis De Broglie that a wave could be associated with a particle of matter - by analogy with the particle of energy that was associated with an optical wave. De Broglie predicted that the wavelength λ of a matter particle wave would be λ = h/p, since the wavelength of a photon is related to its frequency by λ = c/ν. Note that Born's interpretation of the quantum mechanical wave function of a material particle as the probability (amplitude) of finding the material particle somewhere is a direct extension of Einstein's interpretation of the connection between light waves and photons. In the history of science it is hard to find ears more likely to be sympathetic to a new idea than these three great scientists should have been for Max Born's suggestion that the square of the amplitude of Schrödinger's wave function |Ψ2| should be interpreted statistically as the likelihood of finding the particle. Yet they all objected strenuously, not so much to the probability and statistics as to the conviction of Born and his brilliant student Heisenberg that quantum phenomena, like quantum jumps between atomic energy levels, were only predictable statistically, and that there was a fundamental indeterminacy in the classical idea that particles have knowable positions and velocities (momenta). Born, Heisenberg, and Bohr had declared classical determinism untrue of the physical world. Indeterminism and absolute chance had reappeared in the atomic world twenty-two centuries after Epicurus had called for atoms to swerve to provide room for free will.
The Natural Philosophy of Cause and Chance, (The Waynflete Lectures), 1949
From the Introduction, pp.1-4.
The notions of cause and chance which I propose to deal with in these lectures are not specifically physical concepts but have a much wider meaning and application. They are used, more or less vaguely, in everyday life. They appear, not only in all branches of science, but also in history, psychology, philosophy, and theology; everywhere with a different shade of meaning. It would be far beyond my abilities to give an account of all these usages, or to attempt an analysis of the exact significance of the words 'cause' and 'chance' in each of them. However, it is obvious that there must be a common feature in the use of these notions, like the theme in a set of variations. Indeed, cause expresses the idea of necessity in the relation of events, while chance means just the opposite, complete randomness. Nature, as well as human affairs, seems to be subject to both necessity and accident. Yet even accident is not completely arbitrary, for there are laws of chance, formulated in the mathematical theory of probability, nor can the cause—effect relation be used for predicting the future with certainty, as this would require a complete knowledge of the relevant circumstances, present, past, or both together, which is not available. There seems to be a hopeless tangle of ideas. In fact, if you look through the literature on this problem you will find no satisfactory solution, no general agreement. Only in physics has a systematic attempt been made to use the notions of cause and chance in a way free from contradictions. Physicists form their notions through the interpretation of experiments. This method may rightly be called Natural Philosophy, a word still used for physics at the Scottish universities. In this sense I shall attempt to investigate the concepts of cause and chance in these lectures. My material will be taken mainly from physics, but I shall try to regard it with the attitude of the philosopher, and I hope that the results obtained will be of use wherever the concepts of cause and chance are applied. I know that such an attempt will not find favour with some philosophers, who maintain that science teaches only a narrow aspect of the world, and one which is of no great importance to man's mind. It is true that many scientists are not philosophically minded and have hitherto shown much skill and ingenuity but little wisdom. I need hardly to enlarge on this subject. The practical applications of science have given us the means of a fuller and richer life, but also the means of destruction and devastation on a vast scale. Wise men would have considered the consequences of their activities before starting on them; scientists have failed to do so, and only recently have they become conscious of their responsibilities to society. They have gained prestige as men of action, but they have lost credit as philosophers. Yet history shows that science has played a leading part in the development of human thought. It has not only supplied raw material to philosophy by gathering facts, but also evolved the fundamental concepts on how to deal with them. It suffices to mention the Copernican system of the universe, and the Newtonian dynamics which sprang from it. These originated the conceptions of space, time, matter, force, and motion for a long time to come, and had a mighty influence on many philosophical systems. It has been said that the metaphysics of any period is the offspring of the of the preceding period. If this is true, it puts us physicists under the obligation to explain our ideas in a not-too-technical language. This is the purpose of the following lectures on a restricted though important field. I have made an attempt to avoid mathematics entirely, but failed. It would have meant an unbearable clumsiness of expression and loss of clarity. A way out would have been the reduction of all higher mathematics to elementary methods in Euclidean style — following the celebrated example of Newton's Principia. But this would even have increased the clumsiness and destroyed what there is of aesthetic appeal. I personally think that more than 200 years after Newton there should be some progress in the assimilation of mathematics by those who are interested in natural philosophy. So I shall use ordinary language and formulae in a suitable mixture ; but I shall not give proofs of theorems (they are collected in the Appendix). In this way I hope to explain how physics may throw some light on a problem which is not only important for abstract knowledge but also for the behaviour of man. An unrestricted belief in causality leads necessarily to the idea that the world is .an automaton of which we ourselves are only little cog-wheels. This means materialistic determinism. It resembles very much that religious determinism accepted by different creeds, where the actions of men are believed to be determined from the beginning by a ruling of God. I cannot enlarge on the difficulties to which this idea leads if considered from the standpoint of ethical responsibility. The notion of divine predestination clashes with the notion of free will, in the same way as the assumption of an endless chain of natural causes. On the other hand, an unrestricted belief in chance is impossible, as it cannot be denied that there are a great many regularities in the world; hence there can be, at most, 'regulated accident'. One has to postulate laws of chance which assume the appearance of laws of nature or laws for human behaviour. Such a philosophy would give ample space for free will, or even for the willed actions of gods and demons. In fact, all primitive polytheistic religions seem to be based on such a conception of nature: things happening in a haphazard way, except where some spirit interferes with a purpose. We reject today this demonological philosophy, but admit chance into the realm of exact science. Our philosophy is dualistic in this respect; nature is ruled by laws of cause and laws of chance in a certain mixture. How is this possible? Are there no logical contradictions? Can this mixture of ideas be cast into a consistent system in which all phenomena can be adequately described or explained? What do we mean by such an explanation if the feature of chance is involved? What are the irreducible or metaphysical principles involved? Is there any room in this system for free will or for the interference of deity? These and many other questions can be asked. I shall try to answer some of them from the standpoint of the physicist, others from my philosophical convictions which are not much more than common sense improved by sporadic reading. The statement, frequently made, that modern physics has given up causality is entirely unfounded. Modern physics,it is true, has given up or modified many traditional ideas; but it would cease to be a science if it had given up the search for the causes of phenomena. I found it necessary, therefore, to formulate the different aspects of the fundamental notions by giving definitions of terms which seem to me in agreement with ordinary language. With the help of these concepts, I shall survey the development of physical thought, dwelling here and there on special points of interest, and I shall try to apply the results to philosophy in general.
From Part IX, Chance
There is no doubt that the formalism of quantum mechanics and its statistical interpretation are extremely successful in ordering and predicting physical experiences. But can our desire of understanding, our wish to explain things, be satisfied by a theory which is frankly and shamelessly statistical and indeterministic? Can we be content with accepting chance, not cause, as the supreme law of the physical world? To this last question I answer that not causality, properly understood, is eliminated, but only a traditional interpretation of it, consisting in its identification with determinism. I have taken pains to show that these two concepts are not identical. Causality in my definition is the postulate that one physical situation depends on the other, and causal research means the discovery of such dependence. This is still true in quantum physics, though the objects of observation for which a dependence is claimed are different: they are the probabilities of elementary events, not those single events themselves.
Part X, Metaphysical Conclusions
The statistical interpretation which I have presented in the last section is now generally accepted by physicists all over the world, with a few exceptions, amongst them a most remarkable one. As I have mentioned before, Einstein does not accept it, but still believes in and works on a return to a deterministic theory. To illustrate his opinion, let me quote passages from two letters. The first is dated 7 November 1944, and contains these lines:`In unserer wissenschaftlichen Erwartung haben wir uns zu Antipoden entwickelt. Du glaubst an den würfelnden Gott und ich an volle Gesetzlichkeit in einer Welt von etwas objektiv Seiendem, das ich auf wild spekulativem Weg zu erhaschen suche. Ich hoffe, dass einer einen mehr realistischen Weg, bezw. eine mehr greifbare Unterlage für eine solche Auffassung finden wird, als es mir gegeben ist. Der grosse anfängliche Erfolg der Quantentheorie kann mich doch nicht zum Glauben an das fundamentale Würfelspiel bringen. (In our scientific expectations we have progressed towards antipodes. You believe in the dice-playing god, and I in the perfect rule of law in a world of something objectively existing which I try to catch in a wildly speculative way. I hope that somebody will find a more realistic way, or a more tangible foundation for such a conception than that which is given to me. The great initial success of quantum theory cannot convert me to believe in that fundamental game of dice.)The second letter, which arrived just when I was writing these pages (dated 3 December 1947), contains this passage:`Meine physikalische Haltung kann ich Dir nicht so begründen, dass Du sie irgendwie vernünftig finden würdest. Ich sehe natürlich ein, dass die principiell statistische Behandlungsweise, deren Notwendigkeit im Rahmen des bestehenden Formalismus ja zuerst von Dir klar erkannt wurde, einen bedeutenden Wahrheitsgehalt hat. Ich kann aber deshalb nicht ernsthaft daran glauben, weil die Theorie mit dem Grundsatz unvereinbar ist, dass die Physik eine Wirklichkeit in Zeit und Raum darstellen soll, ohne spukhafte Fernwirkungen.... Davon bin ich fest überzeugt, dass man schliesslich bei einer Theorie landen wird, deren gesetzmässig verbundene Dinge nicht Wahrscheinlichkeiten, sondern gedachte Tatbestände sind, wie man es bis vor kurzem als selbstverständlich betrachtet hat. Zur Begründung dieser Überzeugung kann ich aber nicht logische Gründe, sondern nur meinen kleinen Finger als Zeugen beibringen, also keine Autorität, die ausserhalb meiner Haut irgendwelchen Respekt einflössen kann. (I cannot substantiate my attitude to physics in such a manner that you would find it in any way rational. I see of course that the statistical interpretation (the necessity of which in the frame of the existing formalism has been first clearly recognized by yourself) has a considerable content of truth. Yet I cannot seriously believe it because the theory is inconsistent with the principle that physics has to represent a reality in space and time without phantom actions over distances.... I am absolutely convinced that one will eventually arrive at a theory in which the objects connected by laws are not probabilities, but conceived facts, as one took for granted only a short time ago. However, I cannot provide logical arguments for my conviction, but can only call on my little finger as a witness, which cannot claim any authority to be respected outside my own skin.)I have quoted these letters because I think that the opinion of the greatest living physicist, who has done more than anybody else to establish modern ideas, must not be by-passed. Einstein does not share the opinion held by most of us that there is over-whelming evidence for quantum mechanics. Yet he concedes 'initial success' and 'a considerable degree of truth'. He obviously agrees that we have at present nothing better, but he hopes that this will be achieved later, for he rejects the 'dice-playing god'. I have discussed the chances of a return to determinism and found them slight. I have tried to show that classical physics is involved in no less formidable conceptional difficulties and had eventually to incorporate chance in its system. We mortals have to play dice anyhow if we wish to deal with atomic systems. Einstein's principle of the existence of an objective real world is therefore rather academic. On the other hand, his contention that quantum theory has given up this principle is not justified, if the conception of reality is properly understood. Of this I shall say more presently. Einstein's letters teach us impressively the fact that even an exact science like physics is based on fundamental beliefs. The words ich glaube appear repeatedly, and once they are underlined. I shall not further discuss the difference between Einstein's principles and those which I have tried to extract from the history of physics up to the present day. But I wish to collect some of the fundamental assumptions which cannot be further reduced but have to be accepted by an act of faith. Causality is such a principle, if it is defined as the belief in the existence of mutual physical dependence of observable situations. However, all specifications of this dependence in regard to space and time (contiguity, antecedence) and to the infinite sharpness of observation (determinism) seem to me not fundamental, but consequences of the actual empirical laws. Another metaphysical principle is incorporated in the notion of probability. It is the belief that the predictions of statistical calculations are more than an exercise of the brain, that they can be trusted in the real world. This holds just as well for ordinary probability as for the more refined mixture of probability and mechanics formulated by quantum theory. The two metaphysical conceptions of causality and probability have been our main theme. Others, concerning logic, arithmetic, space, and time, are quite beyond the frame of these lectures. But let me add a few more which have occasionally occurred, though I am sure that my list will be quite incomplete. One is the belief in harmony in nature, which is something distinct from causality, as it can be circumscribed by words like beauty, elegance, simplicity applied to certain formulations of natural laws. This belief has played a considerable part in the development of theoretical physics — remember Maxwell's equations of the electromagnetic field, or Einstein's relativity — but how far it is a real guide in the search of the unknown or just the expression of our satisfaction to have discovered a significant relation, I do not venture to say. For I have on occasion made the sad discovery that a theory which seemed to me very lovely nevertheless did not work. And in regard to simplicity, opinions will differ in many cases. Is Einstein's law of gravitation simpler than Newton's? Trained mathematicians will answer Yes, meaning the logical simplicity of the foundations, while others will say emphatically No, because of the horrible complication of the formalism. However this may be, this kind of belief may help some specially gifted men in their research; for the validity of the result it has little importance. The last belief I wish to discuss may be called the principle of objectivity. It provides a criterion to distinguish subjective impressions from objective facts, namely by substituting for given sense-data others which can be checked by other individuals. I have spoken about this method when I had to define temperature: the subjective feeling of hot and cold is replaced by the reading of a thermometer, which can be done by any person without a sensation of hot or cold. It is perhaps the most important rule of the code of natural science of which innumerable examples can be given, and it is obviously closely related to the conception of scientific reality. For if reality is understood to mean the sum of observational invariants — and I cannot see any other reasonable interpretation of this word in physics — the elimination of sense qualities is a necessary step to discover them. Here I must refer to the previous Waynflete Lectures given by Professor E. D. Adrian, on The Physical Background of Perception, because the results of physiological investigations seem to me in perfect agreement with my suggestion about the meaning of reality in physics. The messages which the brain receives have not the least similarity with the stimuli. They consist in pulses of given intensities and frequencies, characteristic for the transmitting nerve-fibre, which ends at a definite place of the cortex. All the brain 'learns' (I use here the objectionable language of the 'disquieting figure of a little hobgoblin sitting up aloft in the cerebral hemisphere') is a distribution or 'map' of pulses. From this information it produces the image of the world by a process which can metaphorically be called a consummate piece of combinatorial mathematics: it sorts out of the maze of indifferent and varying signals invariant shapes and relations which form the world of ordinary experience. This unconscious process breaks down for scientific ultra-experience, obtained by magnifying instruments. But then it is continued in the full light of consciousness, by mathematical reasoning. The result is the reality offered by theoretical physics. The principle of objectivity can, I think, be applied to every human experience, but is often quite out of place. For instance: What is a fugue by Bach ? Is it the invariant cross-section, or the common content of all printed or written copies, gramophone records, sound waves at performances, etc., of this piece of music? As a lover of music I say No! that is not what I mean by a fugue. It is something of another sphere where other notions apply, and the essence of it is not 'notions' at all, but the immediate impact on my soul of its beauty and greatness. In cases like this, the idea of scientific objective reality is obviously inadequate, almost absurd. This is trivial, but I have to refer to it if I have to make good my promise to discuss the bearing of modern physical thought on philosophical problems, in particular on the problem of free will. Since ancient times philosophers have been worried how free will can be reconciled with causality, and after the tremendous success of Newton's deterministic theory of nature, this problem seemed to be still more acute. Therefore, the advent of indeterministic quantum theory was welcomed as opening a possibility for the autonomy of the mind without a clash with the laws of nature. Free will is primarily a subjective phenomenon, the interpretation of a sensation which we experience, similar to a sense impression. We can and do, of course, project it into the minds of our fellow beings just as we do in the case of music. We can also correlate it with other phenomena in order to transform it into an objective relation, as the moralists, sociologists, lawyers do — but then it resembles the original sensation no more than an intensity curve in a spectral diagram resembles a colour which I see. After this transformation into a sociological concept, free will is a symbolic expression to describe the fact that the actions and reactions of human beings are conditioned by their internal mental structure and depend on their whole and unaccountable history. Whether we believe theoretically in strict determinism or not, we can make no use of this theory since a human being is too complicated, and we have to be content with a working hypothesis like that of spontaneity of decision and responsibility if action. If you feel that this clashes with determinism, you have now at your disposal the modern indeterministic philosophy of nature, you can assume a certain 'freedom', i.e. deviation from the deterministic laws, because these are only apparent and refer to averages. Yet if you believe in perfect freedom you will get into difficulties again, because you cannot neglect the laws of statistics which are laws of nature. I think that the philosophical treatment of the problem of free will suffers often (see Appendix, 36) from an insufficient distinction between the subjective and objective aspect. It is doubtless more difficult to keep these apart in the case of such sensations as free will, than in the case of colours, sounds, or temperatures. But the application of scientific conceptions to a subjective experience is an inadequate procedure in all such cases. You may call this an evasion of the problem, by means of dividing all experience into two categories, instead of trying to form one all-embracing picture of the world. This division is indeed what I suggest and consider to be unavoidable. If quantum theory has any philosophical importance at all, it lies in the fact that it demonstrates for a single, sharply defined science the necessity of dual aspects and complementary considerations. Niels Bohr has discussed this question with respect to many applications in physiology, psychology, and philosophy in general. According to the rule of indeterminacy, you cannot measure simultaneously position and velocity of particles, but you have to make your choice. The situation is similar if you wish, for instance, to determine the physico-chemical processes in the brain connected with a mental process: it cannot be done because the latter would be decidedly disturbed by the physical investigation. Complete knowledge of the physical situation is only obtainable by a dissection which would mean the death of the living organ or the whole creature, the destruction of the mental situation. This example may suffice; you can find more and subtler ones in Bohr's writings. They illustrate the limits of human understanding and direct the attention to the question of fixing the boundary line, as physics has done in a narrow field by discovering the quantum constant h. Much futile controversy could be avoided in this way. To show this by a final example, I wish to refer to these lectures themselves which deal only with one aspect of science, the theoretical one. There is a powerful school of eminent scientists who consider such things to be a futile and snobbish sport, and the people who spend their time on it drones. Science has undoubtedly two aspects: it can be regarded from the social standpoint as a practical collective endeavour for the improvement of human conditions, but it can also be regarded from the individualistic standpoint, as a pursuit of mental desires, the hunger for knowledge and understanding, a sister of art, philosophy, and religion. Both aspects are justified, necessary, and complementary. The collective enterprise of practical science consists in the end of individuals and cannot thrive without their devotion. But devotion does not suffice; nothing great can be achieved without the elementary curiosity of the philosopher. A proper balance is needed. I have chosen the way which seemed to me to harmonize best with the spirit of this ancient place of learning.
Concluding remarks. I feel that any critical reference to philosophical literature ought to be based on quotations. Yet, as I have said before, my reading of philosophical books is sporadic and unsystematic, and what I say here is a mere general impression. A book which I have recently read with some care is E. Cassirer's Determinismus and Indeterminismus in der modernen Physik (Goteborg, Elanders, 1937), which gives an excellent account of the situation, not only in physics itself but also with regard to possible applications of the new physical ideas to other fields. There one finds references to and quotations from all great thinkers who have written about the problem. The last section contains Cassirer's opinion on the ethical consequences of physical indeterminism which is essentially the same as that expressed by myself. I quote his words (translated from p. 259):'From the significance of freedom, as a mere possibility limited by natural laws, there is no way to that "reality" of volition and freedom of decision with which ethics is concerned. To mistake the choice (Auswahl) which an electron, according to Bohr's theory has between different quantum orbits, with a choice (Wahl) in the ethical sense of this concept, would mean to become the victim of a purely linguistic equivocality. To speak of an ethical choice there must not only be different possibilities but a conscious distinction between them and a conscious decision about them. To attribute such acts to an electron would be a gross relapse into a form of anthropomorphism. Concerning the inverse problem whether the 'freedom' of the electron helps us to understand the freedom of volition he says this (p. 261): `It is of no avail whether causality in nature is regarded in the form of rigorous "dynamical" laws or of merely statistical laws.... In neither way does there remain open an access to that sphere of "freedom" which is claimed by ethics'.My short survey of these difficult problems cannot be compared with Cassirer's deep and thorough study. Yet it is a satisfaction to me that he also sees the philosophical importance of quantum theory not so much in the question of indeterminism but in the possibility of several complementary perspectives or aspects in the description of the same phenomena as soon as different standpoints of meaning are taken. There is no unique image of our whole world of experience. This last Appendix, added after delivering the lectures, gives me the opportunity to express my thanks to those among my audience who came to me to discuss problems and to raise objections. One of these was directed against my expression `observational invariants'; it was said that the conception of invariant presupposes the existence of a group of transformations which is lacking in this case. I do not think that this is right. The problem is, of course, a psychological one; what I call `observational invariants' corresponds roughly to the Gestalten of the psychologists. The essence of Gestalten theory is that the primary perceptions consist not in uncoordinated sense impressions but in total shapes or configurations which preserve their identity independently of their own movements and the changing standpoint of the observer. Now compare this with a mathematical example, say the definition of the group of rotations as those linear transformations of the coordinates x, y, z for which x2+y2+z2 is invariant. The latter condition can be interpreted geometrically as postulating the invariance of the shape of spheres. Hence the group is defined by assuming the existence of a definite invariant configuration or Gestalt, not the other way round. The situation in psychology seems to me quite analogous, though much less precise. Yet I think that this analogy is of some help in understanding what we mean by real things in the flow of perceptions. Another objection was raised against my use of the expression `metaphysical' because of its association with speculative systems of philosophy. I need hardly say that I do not like this kind of metaphysics, which pretends that there is a definite goal to be reached and often claims to have reached it. I am convinced that we are on a never-ending way; on a good and enjoyable way, but far from any goal. Metaphysical systematization means formalization and petrification. Yet there are metaphysical problems, which cannot be disposed of by declaring them meaningless, or by calling them with other names, like epistemology. For, as I have repeatedly said, they are 'beyond physics' indeed and demand an act of faith. We have to accept this fact to be honest. There are two objectionable types of believers: those who believe the incredible and those who believe that `belief' must be discarded and replaced by 'the scientific method'. Between these two extremes on the right and the left there is enough scope for believing the reasonable and reasoning on sound beliefs. Faith, imagination, and intuition are decisive factors in the progress of science as in any other human activity.