The Chair which I have been elected to occupy, in succession to Professor Darwin, is associated with the name of a great scholar of our fathers' generation, Peter Guthrie Tait. This name has been familiar to me from the time when I first began to study mathematical physics. At that time Felix Klein was the leading figure in a group of outstanding mathematicians at Göttingen, amongst them Hilbert and Minkowski. I remember how Klein, ever eager to link physics with mathematics, missed no opportunity of pointing out to us students the importance of studying carefully the celebrated Treatise on Natural Philosophy of Thomson and Tait, which became a sort of Bible of mathematical science for us.
Today theoretical physics has advanced in very different directions, and "Thomson and Tait" is perhaps almost unknown to the younger generation. But such is the fate of all scientific achievement; for it cannot claim eternal validity like the products of great artists, but has served well if it has served its time. For myself this book has a special attraction by reason of its title. The subject known everywhere else in the world by the dull name "Physics" appears here under the noble title of "Natural Philosophy," the same title as is given to the two Chairs of Physics in this University. Our science acquires by virtue of this name a dignity of its own. Occupied by his tedious work of routine measurement and calculation, the physicist remembers that all this is done for a higher task: the foundation of a philosophy of nature. I have always tried to think of my own work as a modest contribution to this task; and in entering on the tenure of the Tait Chair of Natural Philosophy at this University, though far from my fatherland, I feel intellectually at home.
The justification for considering this special branch of science as a philosophical doctrine is not so much its immense object, the universe from the atom to the cosmic spheres, as the fact that the study of this object in its totality is confronted at every step by logical and epistemological difficulties; and although the material of the physical sciences is only a restricted section of knowledge, neglecting the phenomena of life and consciousness, the solution of these logical and epistemological problems is an urgent need of reason.
For describing the historical development it is a convenient coincidence that the beginning of the new century marks the separation of two distinct periods, of the older physics which we usually call classical, and modern physics. Einstein's theory of relativity of 1905 can be considered as being at once the culmination of classical ideas and the starting-point of the new ones. But during the preceding decade research on radiation and atoms, associated with the names of Röntgen, J J Thomson, the Curies, Rutherford, and many others, had accumulated a, great number of new facts which did not fit into the classical ideas at all. The new conception of the quantum of action which helped to elucidate them was first put forward by Planck in 1900. The most important consequences of this conception were deduced by Einstein, who laid the foundations of the quantum theory of light in 1905, the year in which he published his relativity theory, and by Niels Bohr in 1913, when he applied the idea of the quantum to the structure of atoms.
Every scientific period is in interaction with the philosophical systems of its time, providing them with facts of observation and receiving from them methods of thinking. The philosophy of the nineteenth century on which classical physics relied is deeply rooted in the ideas of David Hume. From his philosophy there developed the two systems which dominated science during the latter part of the classical period, critical philosophy and empiricism.
The difference between these systems concerns the problem of the a priori. The idea that a science can be logically reduced to a small number of postulates or axioms is due to the great Greek mathematicians, who first tried to formulate the axioms of geometry and to derive the complete system of theorems from them. Since then the question of what are the reasons for accepting just these axioms has perpetually occupied the interest of mathematicians and philosophers. Kant's work can be considered as a kind of enormous generalisation of this question; he attempted to formulate the postulates, which he called categories a Priori, necessary to build up experience in general, and, he discussed the roots of their validity. The result was the classification of the a priori principles into two classes, which he called analytic and synthetic, the former being the rules of pure logical thinking, including arithmetic, the latter containing the laws of space and time, of substance, causality, and other general conceptions of this kind. Kant believed that the root of the validity of the first kind was "pure reason" itself, whereas the second kind came from a special ability of our brain, differing from reason, which he called "pure intuition" (reine Anschauung). So mathematics was classified as a science founded on a priori principles, properties of our brain and therefore unchangeable; and the same was assumed for some of the most general laws of physics, as formulated by Newton.
But I doubt whether Kant would have maintained this view if he had lived a little longer. The discovery of non-Euclidean geometry by Lobachevsky and Bolyai shook the a priori standpoint. Gauss has frankly expressed his opinion that the axioms of geometry have no superior position as compared with the laws of physics, both being formulations of experience, the former stating the general rules of the mobility of rigid bodies and giving the conditions for measurements in space. Gradually most of the physicists have been converted to the empirical standpoint. This standpoint denies the existence of a priori principles in the shape of laws of pure reason and pure intuition; and it declares that the validity of every statement of science (including geometry as applied to nature) is based on experience. It is necessary to be very careful in this formulation. For it is of course not meant that every fundamental statement-as, for instance, the Euclidean axioms of geometry-is directly based on special observations. Only the totality of a logically coherent field of knowledge is the object of empirical examination, and if a sufficient set of statements is confirmed by experiment, we can consider this as a confirmation of the whole system, including the axioms which are the shortest logical expression of the system.
I do not think that there is any objection to this form of empiricism. It has the virtue of being free from the petrifying tendency which systems of a priori philosophy have. It gives the necessary freedom to research, and as a matter of fact modern physics has made ample use of this freedom. It has not only doubted the a priori validity of Euclidean geometry as the great mathematicians did a hundred years ago, but has really replaced it by new forms of geometry; it has even made geometry depend on physical forces, gravitation, and it has revolutionised in the same way nearly all categories a priori, concerning time, substance, and causality.
This liberation from the idea of the a priori was certainly important for the development of science, but it already took place during the last century, and does not represent the deciding difference between classical and modern physics. This difference lies in the attitude to the objective world. Classical physics took it for granted that there is such an objective world, which not only exists independently of any observer, but can also be studied by this observer without disturbing it. Of course every measurement is a disturbance of the phenomenon observed; but it was assumed that by skilful arrangement this disturbance can be reduced to a negligible amount. It is this assumption which modern physics has shown to be wrong. The philosophical problem connected with it arises from the difficulty in speaking of the state of an objective world if this state depends on what the observer does. It leads to a critical examination of what we mean by the expression "objective world."
The fact that statements of observations depend on the standpoint of the observer is as old as science. The orbit of the earth round the sun is an ellipse only for an observer standing just at the centre of mass of the two bodies. Relativity gave the first example in which the intrusion of the observer into the description of facts is not so simple, and leads to a new conception to conserve the idea of an objective world Einstein has acknowledged that his studies on this problem were deeply influenced by the ideas of Ernst Mach, a Viennese physicist who developed more and more into a philosopher. From his writings sprang a new philosophical system, positivism, which is much in favour today. Traces of it can be seen in fundamental papers of Heisenberg on quantum theory; but it has also met with strenuous opposition, for instance, from Planck. In any case, positivism is a living force in science. It is also the only modern system of philosophy which by its own rules is bound to keep pace with the progress of science. We are obliged to define our attitude towards it.
The characteristic feature of this system is the sharp distinction it draws between real and apparent problems, and correspondingly between those conceptions which have a real meaning and those which have not. Now it is evident and trivial that not every grammatically correct question is reasonable; take, for instance, the well-known conundrum: Given the length, beam, and horse-power of a steamer, how old is the captain? - or the remark of a listener to a popular astronomical lecture: "I think I grasp everything, how to measure the distances of the stars and so on, but how did they find out that the name of this star is Sirius?" Primitive people are convinced that knowing the "correct" name of a thing is real knowledge, giving mystical power over it, and there are many instances of the survival of such word-fetishism in our modern world. But let us now take an example from physics in which the thing is not so obvious. Everybody believes he knows what the expression "simultaneous events" means, and he supposes as a matter of course that it means the same for any other individual. This is quite in order for neighbours on this little planet. Even when science made the step of imagining an individual of similar brain-power on another star there seemed to be nothing problematical. The problem appeared only when the imagination was driven so far as to ask how an observer on the earth and another on, say, Mars could compare their observations about simultaneous events. It was then necessary to take into account the fact that we are compelled to use signals for this comparison. The fastest signal at our disposal is a flash of light. In using light, or even only thinking about it, we are no longer permitted to rely on our brainpower, our intuition. We have to consider facts revealed by experiments. We have not only the fact of the finite velocity of light, but another most important fact, disclosed by Michelson's celebrated experiment: that light on this earth travels with the same speed in all directions, independently of the motion of the earth round the sun. One usually expresses this by saying that these experiments disprove the existence of an ether-wind which we would expect from the analogy of the wind felt in a moving car.
An admirable logical analysis of these facts led Einstein to the result that the question of simultaneity of two distant events is almost as absurd as that regarding the age of the captain. Just as this question would become significant by adding some data, say about his life insurance, the problem of simultaneity becomes reasonable by adding data about the motion of the observer. In this way the conception of time loses its absolute character, and space becomes involved in this revolution. For it becomes meaningless to speak about "space at this moment"; if we assume two observers in relative motion just passing one another, then each has his own "space at this moment," but the events contained in this space are different for the two observers.
What has now become of the idea of a world independent of the observer? If one sticks to the meaning of a static assembly of things at one moment, this idea of an objective world is lost. But it can be saved by considering as the world the assembly of events, each having not only a given position in space but also a given time of occurrence. Minkowski has shown that it is possible to get a description of the connection of all events which is independent of the observer, or invariant, as the mathematicians say, by considering them as points in a four-dimensional continuum with a quasi-Euclidean geometry. But the division of this four-dimensional world into space and time depends on the observer.
When I wrote a popular book on relativity in 1920 I was so impressed by this wonderful construction that I represented this method of objectivation as the central achievement of science. I did not then realise that we were soon to be confronted with a new empirical situation which would compel us to undertake a much deeper critical review of the conception of an objective world.
I have here used the phrase "new empirical situation," following Niels Bohr, the founder of modern atomic theory, and the deepest thinker in physical science. He has coined this expression to indicate that the birth of new and strange ideas in physics is not the result of free or even frivolous speculation, but of the critical analysis of an enormous and complicated body of collected experience. Physicists are not revolutionaries but rather conservative, and inclined to yield only to strong evidence before sacrificing an established idea. In the case of relativity this evidence was strong indeed, but consisted to a large extent of negative statements, such as that mentioned above regarding the absence of an ether-wind. The generalisation which was conceived by Einstein in 1915 combining the geometry of the space-time world with gravitation rested, and still rests, on a rather slender empirical basis.
The second revolution of physics, called quantum theory, is, however, built on an enormous accumulation of experience, which is still growing from day to day. It is much more difficult to talk about these matters, because they have a much more technical character. The problem is the constitution of matter and radiation, which can be adequately treated only in laboratories with refined instruments. The evidence provided there consists of photographic plates, and of tables and curves representing measurements. They are collected in enormous numbers all over the world, but known only to the experts. I cannot suppose that you are acquainted with these experiments. In spite of this difficulty, I shall try to outline the problem and its solution, called quantum mechanics.
Let us start with the old problem of the constitution of light. At the beginning of the scientific epoch two rival theories were proposed: the corpuscular theory by Newton, the wave theory by Huygens. About a hundred years elapsed before experiments were found deciding in favour of one of them, the wave theory, by the discovery of interference. When two trains of waves are superposed, and a crest of one wave coincides with a valley of the other, they annihilate one another; this effect creates the well-known patterns which you can observe on any pond on which swimming ducks or gulls excite water-waves. Exactly the same kind of pattern can be observed when two beams of light cross one another, the only difference being that you need a magnifying-lens to see them; the inference is that a beam of light is a train of waves of short wave-length. This conclusion has been supported by innumerable experiments.
But about a hundred years later, during my student days, another set of observations began to indicate with equal cogency that light consists of corpuscles. This type of evidence can best be explained by analogy with two types of instruments of war, mines and guns. When a mine explodes you will be killed if you are near it, by the energy transferred to you as a wave of compressed air. But if you are some hundred yards away you are absolutely safe; the explosion-wave has lost its dangerous energy by continuously spreading out over a large area. Now imagine that the same amount of explosive is used as the propellant in a machinegun which is rapidly fired, turning round in all directions. If you are near it you will almost certainly be shot, unless you hastily run away. When you have reached a distance of some hundred yards you will feel much safer, but certainly not quite safe. The probability of being hit has dropped enormously, but if you are hit the effect is just as fatal as before.
Here you have the difference between energy spread out from a centre in the form of a continuous wave-motion, and a discontinuous rain of particles. Planck discovered, in 1900, the first indication of this discontinuity of light in the laws governing the heat radiated from hot bodies. In his celebrated paper of 1905, mentioned already, Einstein pointed out that experiments on the energetic effect of light, the so-called photoelectric effect, can be interpreted in the way indicated as showing unambiguously the corpuscular constitution of light. These corpuscles are called quanta of light or photons.
This dual aspect of the luminous phenomenon has been confirmed by many observations of various types. The most important step was made by Bohr, who showed that the enormous amount of observations on spectra collected by the experimentalists could be interpreted and understood with the help of the conception of light-quanta. For this purpose he had also to apply the idea of discontinuous behaviour to the motion of material particles, the atoms, which are the source of light.
I cannot follow out here the historical development of the quantum idea which led step by step to the recognition that we have here to do with a much more general conception. Light is not the only "radiation" we know; I may remind you of the cathode rays which appear when electric currents pass through evacuated bulbs, or the rays emitted by radium and other radioactive substances. These rays are certainly not light. They are beams of fast-moving electrons, i.e. atoms of electricity, or ordinary atoms of matter like helium. In the latter case this has been proved directly by Rutherford, who caught the beam (a so-called a-ray of radium) in an evacuated glass vessel and showed that it was finally filled with helium gas. Today one can actually photograph the tracks of these particles of radiating matter in their passage through other substances.
In this case the corpuscular evidence was primary. But in 1924 de Broglie, from theoretical reasoning, suggested the idea that these radiations should show interference and behave like waves under proper conditions. This idea was actually confirmed by experiments a short time later. Not only electrons, but real atoms of ordinary matter like hydrogen or helium have all the properties of waves if brought into the form of rays by giving them a rapid motion.
This is a most exciting result, revolutionising all our ideas of matter and motion. But when it became known, theoretical physics was already prepared to treat it by proper mathematical methods, the so-called quantum mechanics, initiated by Heisenberg, worked out in collaboration with Jordan and myself, and quite independently by Dirac; and another form of the same theory, the wave-mechanics, worked out by Schrödinger in close connection with de Broglie's suggestion. The mathematical formalism is a wonderful invention for describing complicated things. But it does not help much towards a real understanding. It took several years before this understanding was reached, even to a limited extent. But it leads right amidst philosophy, and this is the point about which I have to speak.
The difficulty arises if we consider the fundamental discrepancy in describing one and the same process sometimes as a rain of particles, and at other times as a wave. One is bound to ask, What is it really? You see here the question of reality appears. The reason why it appears is that we are talking about particles or waves, things considered as well known; but which expression is adequate depends on the method of observation. We thus meet a situation similar to that in relativity, but much more complicated. For here the two representations of the same phenomenon are not only different but contradictory. I think everyone feels that a wave and a particle are two types of motion which cannot easily be reconciled. But if we take into account the simple quantitative law relating energy and frequency already discovered by Planck, the case becomes very serious. It is clear that the properties of a given ray when appearing as a rain of particles must be connected with its properties when appearing as a train of waves. This is indeed the case, and the connecting law is extremely simple when all the particles of the beam have exactly the same velocity. Experiment then shows that the corresponding train of waves has the simplest form possible, which is called harmonic, and is characterised by a definite sharp frequency and wave-length. The law of Planck states that the kinetic energy of the particles is exactly proportional to the frequency of vibration of the wave; the factor of proportionality, called Planck's constant, and denoted by the letter h, has a definite numerical value which is known from experiment with fair accuracy.
There you have the logical difficulty; a particle with a given velocity is, qua particle, a point, existing at any instant without extension in space. A train of waves is by definition harmonic only if it fills the whole of space and lasts from eternity to eternity! [The latter point may not appear so evident; but a mathematical analysis made by Fourier more than a hundred years ago has clearly shown that every train of waves finite in space and time has to be considered as a superposition of many infinite harmonic waves of different frequencies and wave-lengths which are arranged in such a way that the outer parts destroy one another by interference; and it can be shown that every finite wave can be decomposed into its harmonic components.] Bohr has emphasised this point by saying that Planck's principle introduces an irrational feature into the description of nature.
Indeed the difficulty cannot be solved unless we are prepared to sacrifice one or other of those principles which were assumed as fundamental for science. The principle to be abandoned now is that of causality as it has been understood ever since it could be formulated exactly. I can indicate this point only very shortly. The laws of mechanics as developed by Galileo and Newton allow us to predict the future motion of a particle if we know its position and velocity at a given instant. More generally, the future behaviour of a system can be predicted from a knowledge of proper initial conditions. The world from the standpoint of mechanics is an automaton, without any freedom, determined from the beginning. I never liked this extreme determinism, and I am glad that modern physics has abandoned it. But other people do not share this view.
To understand how the quantum idea and causality are connected, we must explain the second fundamental law relating particles and waves. This can be readily understood with the help of our example of the exploding mine and the machine-gun. If the latter fires not only horizontally but equally in all directions, the number of bullets and therefore the probability of being hit, will decrease with distance in exactly the same ratio as the surface of the concentric spheres, over which the bullets are equally distributed, increases. But this corresponds exactly to the decrease of energy of the expanding wave of the exploding mine. If we now consider light spreading out from a small source, we see immediately that in the corpuscular aspect the number of photons will decrease with the distance in exactly the same way as does the energy of the wave in the undulatory aspect. I have generalised this idea for electrons and any other kind of particles by the statement that we have to do with "waves of probability" guiding the particles in such a way that the intensity of the wave at a point is always proportional to the probability of finding a particle at that point. This suggestion has been confirmed by a great number of direct and indirect experiments. It has to be modified if the particles do not move independently, but act on one another; for our purpose, however, the simple case is sufficient.
Now we can analyse the connection between the quantum laws and causality.
Determining the position of a particle means restricting it physically to a small part of space. The corresponding probability wave must also be restricted to this small part of space, according to our second quantum law. But we have seen that by Fourier's analysis such a wave is a superposition of a great number of simple harmonic waves with wavelengths and frequencies spread over a wide region. Using now the first quantum law stating the proportionality of frequency and energy, we see that this geometrically well-defined state must contain a wide range of energies. The opposite holds just as well. We have derived qualitatively the celebrated uncertainty law of Heisenberg: exact determination of position and velocity exclude one another; if one is determined accurately the other becomes indefinite.
The quantitative law found by Heisenberg states that for each direction in space the product of the uncertainty interval of space and that of momentum (equal to mass times velocity) is always the same, being given by Planck's quantum constant h.
Here we have the real meaning of this constant as an absolute limit of simultaneous measurement of position and velocity. For more complicated systems there are other pairs or groups of physical quantities which are not measureable at the same instant.
Now we remember that the knowledge of position and velocity at one given time was the supposition of classical mechanics for determining the future motion. The quantum laws contradict this supposition, and this means the break-down of causality and determinism. We may say that these propositions are not just wrong, but empty: the premise is never fulfilled.
The result that the discovery of the quantum laws puts an end to the strict determinism which was unavoidable in the classical period is of great philosophical importance by itself. After relativity has changed the ideas of space and time, another of Kant's categories, causality, has to be modified. The a priori character of these categories cannot be maintained. But of course there is not a vacuum now where these principles were previously; they are replaced by new formulations. In the case of space and time these are the laws of the four-dimensional geometry of Minkowski. In the case of causality there also exists a more general conception, that of probability. Necessity is a special case of probability; it is a probability of one hundred per cent. Physics is becoming a fundamentally statistical science. The mathematical theory called quantum mechanics which expresses these ideas in a precise form is a most wonderful structure, not only comparable with, but superior to, classical mechanics. The existence of this mathematical theory shows that the whole structure is logically coherent. But this proof is rather indirect, and convincing only for those who understand the mathematical formalism. It is therefore an urgent task to show directly for a number of important cases why, in spite of the use of two such different pictures as particles and waves, a contradiction can never arise. This can be done by discussing special experimental arrangements with the help of Heisenberg's uncertainty relation. In complicated cases this sometimes leads to rather puzzling and paradoxical results, which have been carefully worked out by Heisenberg, Bohr, and Darwin, my predecessor in this Chair.
I shall mention only one case. Looking through a microscope I can see a microbe and follow its motion. Why should it not be possible to do the same with atoms or electrons, simply by using more powerful microscopes? The answer is that "looking through" the microscope means sending a beam of light, of photons, through it. These collide with the particles to be observed. If these are heavy like a microbe or even an atom they will not be essentially influenced by the photons, and the deflected photons collected by the lenses give an image of the object. But if this is an electron, which is very light, it will recoil on colliding with the photon, an effect first directly observed by Compton. The change of velocity of the electron is to some extent indeterminate, and depends on the physical conditions in such a way that Heisenberg's uncertainty relation is exactly fulfilled in this case also.
Bohr has introduced the expression "complementarity" for the two aspects of particles and waves. just as all colours which we see can be arranged in pairs of complementary colours giving white when mixed, so all physical quantities can be arranged in two groups, one belonging to the particle aspect, the other to the wave aspect, which never lead to contradictions, but are both necessary to represent the full aspect of nature.
Such a short expression for a complicated and difficult situation is very useful, for instance, with respect to the naive question: Now, what is a beam of light or a material substance "really," a set of particles or a wave? Anybody who has understood the meaning of complementarity will reject this question as too much simplified and missing the point. But this rejection does not solve the problem whether the new theory is consistent with the idea of an objective world, existing independently of the observer. The difficulty is not the two aspects, but the fact that no description of any natural phenomenon in the atomistic domain is possible without referring to the observer, not only to his velocity as in relativity, but to all his activities in performing the observation, setting up the instruments, and so on. The observation itself changes the order of events. How then can we speak of an objective world?
Some theoretical physicists, among them Dirac, give a short and simple answer to this question. They say: the existence of a mathematically consistent theory is all we want. It represents everything that can be said about the empirical world; we can predict with its help unobserved phenomena, and that is all we wish. What you mean by an objective world we don't know and don't care.
There is nothing to be objected against this standpoint-except one thing, that it is restricted to a small circle of experts. I cannot share this l'art pour l'art standpoint. I think that scientific results should be interpreted in terms intelligible to every thinking man. To do this is precisely the task of natural philosophy.
The philosophers today concentrate their interest on other questions, more important for human life than the troubles arising from a refined study of atomistic processes. Only the positivists, who claim to have a purely scientific philosophy, have answered our question. Their standpoint (Jordan, 1936) is even more radical than that of Dirac mentioned above. Whereas he declares himself content with the formulae and uninterested in the question of an objective world, positivism declares the question to be meaningless.
Positivism considers every question as meaningless which cannot be decided by experimental test. As I said before, this standpoint has proved itself productive by inducing physicists to adopt a critical attitude towards traditional assumptions, and has helped in the building of relativity and quantum theory. But I cannot agree with the application made by the positivists to the general problem of reality. If all the notions we use in a science had their origin in this science, the positivists would be right. But then science would not exist. Although it may be possible to exclude from the internal activity of science all reference to other domains of thinking, this certainly does not hold for its philosophical interpretation. The problem of the objective world belongs to this chapter.
Positivism assumes that the only primary statements which are immediately evident are those describing direct sensual impressions. All other statements are indirect, theoretical constructions to describe in short terms the connections and relations of the primary experiences. Only these have the character of reality. The secondary statements do not correspond to anything real, and have nothing to do with an existing external world; they are conventions invented artificially to arrange and simplify "economically" the flood of sensual impressions
This standpoint has no foundation in science itself; nobody can prove by scientific methods that it is correct. I would say that its origin is metaphysical were I not afraid of hurting the feelings of the positivists, who claim to have an entirely unmetaphysical philosophy. But I may safely say that this standpoint rests on psychology, only it is not a sound psychology. Let us consider it applied to examples of everyday life. If I look at this table or this chair I receive innumerable sense-impressions - patches of colour - and when I move my head these impressions change. I can touch the objects and get a great variety of new sense-impressions, of varying resistance, roughness, warmth, and so on. But if we are honest, it is not these uncoordinated impressions that we observe, but the total object "table" or "chair." There is a process of unconscious combination, and what we really observe is a totality which is not the sum of the single impressions, not more or less than this sum, but something new. What I mean will perhaps become clearer if I mention an acoustical phenomenon. A melody is certainly something else than the sum of the tones of which it is composed; it is a new entity.
Modern psychology is fully aware of this fact. I allude to the Gestalt-psychology of v. Ehrenfels, Köhler, and Wertheimer. The word Gestalt, which seems to have no adequate English translation, means not only shape, but the totality which is really perceived. I cannot explain it better than by referring again to the example of melody. These Gestalten are formed unconsciously; when they are considered by the conscious mind they become conceptions and are provided with words. The unsophisticated mind is convinced that they are not arbitrary products of his mind, but impressions of an external world on his mind. I cannot see any argument for abandoning this conviction in the scientific sphere. Science is nothing else than common sense applied under unaccustomed conditions. The positivists say that this assumption of an external world is a step into metaphysics, and meaningless, since we can never know anything about it except by the perceptions of our senses. This is evident. Kant has expressed the same point by distinguishing between the empirical thing and the "thing in itself" (Ding an sich) which lies behind it. If the positivists go on to say that all our assertions regarding the external world are only symbolical, that their meaning is conventional, then I protest. For then every single sentence would be symbolical, conventional; even if I merely say, "Here I am sitting on a chair." The "chair" is no primary sensual impression, but a notion connected with a Gestalt, an unconscious integration of the impressions to a new unit which is independent of changes in the impressions. For if I move my body, my hands, my eyes, the sensual impressions change in the most complicated way, but the "chair" remains. The chair is invariant with respect to changes of myself, and of other things or persons, perceived as Gestalten. This fact, a very obtrusive fact, of "invariance" is what we mean by saying that there is "really" a chair. It can be submitted to test, not by physical experiment, but by the wonderful methods of unconscious mind, which is able to distinguish between a "real" and a painted chair by merely moving the head a little. The question of reality is therefore not meaningless, and its use not merely symbolic or conventional.
The expression "invariant" which I have already used in speaking of relativity, and which appears here in a more general sense, is the link connecting these psychological considerations with exact science. It is a mathematical expression first used in analytical geometry to handle quantitatively spatial Gestalten, which are simple shapes of bodies or configurations of such. I can describe any geometrical form by giving a sufficient number of co-ordinates of its points; for instance, the perpendicular projections of its points on three orthogonal co-ordinate planes, But this is by far too much; it describes not only the form but the position relative to the three arbitrary planes, which is entirely irrelevant. Therefore one has to eliminate all the superfluous, uninteresting parts of the co-ordinate description by well-known mathematical processes; the result is the so-called invariants describing the intrinsic form considered.
Exactly the same holds if we have to do not only with size and shape, but also with colour, heat, and other physical properties. The methods of mathematical physics are just the same as those of geometry, starting with generalised co-ordinates and eliminating the accidental things. These are now not only situation in space, but motion, state of temperature or electrification, and so on. What remains are invariants describing things.
This method is the exact equivalent. of the formation of Gestalten by the unconscious mind of the man unspoiled by science. But science transcends the simple man's domain by using refined methods of research. Here unknown forms are found, for which the unconscious process does not work. We simply do not know what we see. We have to think about it, change conditions, speculate, measure, calculate. The result is a mathematical theory representing the new facts. The invariants of this theory have the right to be considered as representations of objects in the real world. The only difference between them and the objects of everyday life is that the latter are constructed by the unconscious mind, whereas the objects of science are constructed by conscious thinking. Living in a time in which Freud's ideas about the unconscious sphere are generally accepted, there seems to be no difficulty in considering this difference between common and scientific objects as of second order. This is also justified by the fact that the boundary between them is not at all sharp, and is continually changing. Conceptions which once were purely scientific have become real things. The stars were bright points on a spherical shell for the primitive man. Science discovered their geometrical relations and orbits. It met with furious opposition; Galileo himself became a martyr to truth. Today these mathematical abstractions are common knowledge of school-children, and have become part of the unconscious mind of the European. Something similar has happened with the conceptions of the electromagnetic field.
This idea that the invariant is the link between common sense and science occurred to me as quite natural. I was pleased when I found the same idea in the presentation of the Philosophy of Mathematics by Hermann Weyl (1926), the celebrated Princeton mathematician. I think it is also in conformity with Bohr's (1933) ideas. He insists on the point that our difficulties in physics come from the fact that we are compelled to use the words and conceptions of everyday life even if we are dealing with refined observations. We know no other way of describing a motion than either by particles or waves. We have to apply them also in those cases where observation shows that they do not fit completely, or that we really have to do with more general phenomena. We develop mathematically the invariants describing the new observations, and we learn step by step to handle them intuitively. This process is very slow, and it proceeds only in proportion as the phenomena become known in wider circles. Then the new conceptions sink down in the unconscious mind, they find adequate names, and are absorbed into the general knowledge of mankind.
In quantum theory we are only at the beginning of this process. Therefore I cannot tell you in a few words of ordinary language what the reality is which quantum mechanics deals with. I can only develop the invariant features of this theory and try to describe them in ordinary language, inventing new expressions whenever a conception begins to appeal to intuition. This is what teaching of physics means. Well-trained youth takes things for granted which seemed to us horribly difficult, and later generations will be able to talk about atoms and quanta as easily as we are able to talk about this table and this chair, and about the stars in heaven. I do not, however, wish to belittle the gap between modern and classical physics. The idea that it is possible to think about the same phenomena with the help of two entirely different and mutually exclusive pictures without any danger of logical contradiction is certainly new in science. Bohr has pointed out that it may help to solve fundamental difficulties in biology and psychology. A living creature, plant, or animal is certainly a physico-chemical system. But it is also something more than this. There are apparently two aspects again. The time of materialism is over; we are convinced that the physico-chemical aspect is not in the least sufficient to represent the facts of life, to say nothing of the facts of mind. But there is the most intimate connection between both spheres; they overlap and are interwoven in the most complicated way. The processes of life and mind need other conceptions for their description than the physico-chemical processes with which they are coupled. Why do these differing languages never contradict each other? Bohr has suggested the idea that this is another case of complementarity, just as between particles and waves in physics. If you want to study a specific biological or psychological process by the methods of physics and chemistry, you have to apply all kinds of physical apparatus, which disturbs the process. The more you learn about the atoms and molecules during the process, the less you are sure that the process is that which you want to study. By the time you know everything about the atoms, the creature will be dead. This is briefly Bohr's suggestion of a new and deeper complementary relation between physics and life, life and mind.
The old desire to describe the whole world in one unique philosophical language cannot be fulfilled. Many have felt this, but to modern physics belongs the merit of having shown the exact logical relation of two apparently incompatible trends of thought, by uniting them into a higher unit.
But with this result physics has not come to rest. It is the achievement of a bygone period, and new difficulties have appeared since. Observations on nuclei, the innermost parts of the atoms, have revealed a new world of smallest dimensions, where strange laws hold. It has been shown that every kind of atom has a nucleus of definite structure, consisting of a very close packing of two kinds of particles, called protons and neutrons. The proton is the nucleus of the lightest atom, hydrogen, with a positive electric charge. The neutron is a particle of nearly the same weight, but uncharged. In the atom the nucleus is surrounded by a cloud of electrons, which we have mentioned several times. They are particles nearly 2000 times lighter than the proton or the neutron; they carry a negative charge equal and opposite to that of the proton. But recently positive electrons or "positrons" have also been discovered; in fact, their existence was predicted by Dirac on account of theoretical considerations. Hence we have four kinds of particles, two "heavy" ones, proton and neutron, and two "light" ones, the negative and positive electron, which can all move with any velocity less than that of light. But then there are the photons, which can move only with the velocity of light, and very likely another kind of particles called "neutrinos" the motion of which is restricted in the same way.
The question which modern physics raises is: Why just these particles? Of course a question put like, this is rather vague, but it has a definite meaning. There is, for instance, the ratio of the masses of proton and electron, the exact value of which has been found to be 1845. Then there is another dimensionless number 137, connecting the elementary charge, Planck's quantum constant, and the velocity of light. To derive these numbers from theory is an urgent problem-only a theory of this kind does not exist. It would have to deal with the relations between the four ultimate particles. There has been made the fundamental discovery that a positive and a negative electron can unite to nothing, disappear, the energy liberated in this process being emitted in the form of photons; and vice versa, such a pair can be born out of light. Processes of this type, transformations of ultimate particles including birth and death, seem to be the key to a deeper understanding of matter. We can produce these violent processes in the laboratory only on a very small scale, but nature provides us with plenty of material in the form of the so-called cosmic rays. In observing them we are witnesses of catastrophes in which by the impact of two particles large groups of new particles are generated, which have received the suggestive name of "showers." We seem here to be at the limit where the conception of matter as consisting of distinct particles loses its value, and we have the impression that we shall have to abandon some other accepted philosophical principle before we shall be able to develop a satisfying theory.
It would be attractive to analyse the indications which our present knowledge yields. But my time is over.
The purpose of my lecture has been to show you that physics, besides its importance in practical life, as the fundamental science of technical development, has something to say about abstract questions of philosophy. There is much scepticism today about technical progress. It has far outrun its proper use in life. The social world has lost its equilibrium through the application of scientific results. But Western man, unlike the contemplative Oriental, loves a dangerous life, and science is one of his adventures.
We cannot stop it, but we can try to fill it with a true philosophical spirit: the search of truth for its own sake.
References To Literature.
Bohr, N., 1933. "Licht und Leben," Naturw., vol. xxi, p. 245.
Jordan, P., 1936. A brilliant presentation of the positivistic standpoint is given in his book Anschauliche Quantentheorie, J. Springer, Berlin.
Weyl, H., 1926. "Philosophie der Mathematik und Naturwissenschaft," Handbuch der Philosophie, Abt. II, A, 11.
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