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.
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