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Reference: The Nature of the Physical World

This paper presents Chapter XI (section 3) from the book THE NATURE OF THE PHYSICAL WORLD by A. S. EDDINGTON. The contents of this book are based on the lectures that Eddington delivered at the University of Edinburgh in January to March 1927.

The paragraphs of original material are accompanied by brief comments in color, based on the present understanding.  Feedback on these comments is appreciated.

The heading below links to the original materials.

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Selective Influence of the Mind

This brings us very near to the problem of bridging the gulf between the scientific world and the world of everyday experience. The simpler elements of the scientific world have no immediate counterparts in everyday experience; we use them to build things which have counterparts. Energy, momentum and stress in the scientific world shadow well-known features of the familiar world. I feel stress in my muscles; one form of energy gives me the sensation of warmth; the ratio of momentum to mass is velocity, which generally enters into my experience as change of position of objects. When I say that I feel these things I must not forget that the feeling, in so far as it is located in the physical world at all, is not in the things themselves but in a certain corner of my brain. In fact, the mind has also invented a craft of world-building; its familiar world is built not from the distribution of relata and relations but by its own peculiar interpretation of the code messages transmitted along the nerves into its sanctum.

Accordingly we must not lose sight of the fact that the world which physics attempts to describe arises from the convergence of two schemes of world-building. If we look at it only from the physical side there is inevitably an arbitrariness about the building. Given the bricks—the 16 measures of world-structure—there are all sorts of things we might build. Or we might take up again some of the rejected lumber and build a still wider variety of things. But we do not build arbitrarily; we build to order. The things we build have certain remarkable properties; they have these properties in virtue of the way they are built, but they also have them because such properties were ordered. There is a general description which covers at any rate most of the building operations needed in the construction of the physical world; in mathematical language the operation consists in Hamiltonian differentiation of an invariant function of the 16 measures of structure. I do not think that there is anything in the basal relation-structure that cries out for this special kind of combination; the significance of this process is not in inorganic nature. Its significance is that it corresponds to an outlook adopted by the mind for its own reasons; and any other building process would not converge to the mental scheme of world-building. The Hamiltonian derivative has just that kind of quality which makes it stand out in our minds as an active agent against a passive extension of space and time; and Hamiltonian differentiation is virtually the symbol for creation of an active world out of the formless background. Not once in the dim past, but continuously by conscious mind is the miracle of the Creation wrought.

The role that mind plays in world building is bringing the criterion of continuum of substance by ensuring consistency, harmony and continuity.

By following this particular plan of building we construct things which satisfy the law of conservation, that is to say things which are permanent. The law of conservation is a truism for the things which satisfy it; but its prominence in the scheme of law of the physical world is due to the mind having demanded permanence. We might have built things which do not satisfy this law. In fact we do build one very important thing “action” which is not permanent; in respect to “action” physics has taken the bit in her teeth, and has insisted on recognising this as the most fundamental thing of all, although the mind has not thought it worthy of a place in the familiar world and has not vivified it by any mental image or conception. You will understand that the building to which I refer is not a shifting about of material; it is like building constellations out of stars. The things which we might have built but did not, are there just as much as those we did build. What we have called building is rather a selection from the patterns that weave themselves.

This criterion brings about the element of permanence and the law of conservation of substance.

The element of permanence in the physical world, which is familiarly represented by the conception of substance, is essentially a contribution of the mind to the plan of building or selection. We can see this selective tendency at work in a comparatively simple problem, viz. the hydrodynamical theory of the ocean. At first sight the problem of what happens when the water is given some initial disturbance depends solely on inorganic laws; nothing could be more remote from the intervention of conscious mind. In a sense this is true; the laws of matter enable us to work out the motion and progress of the different portions of the water; and there, so far as the inorganic world is concerned, the problem might be deemed to end. But actually in hydrodynamical textbooks the investigation is diverted in a different direction, viz. to the study of the motions of waves and wave-groups. The progress of a wave is not progress of any material mass of water, but of a form which travels over the surface as the water heaves up and down; again the progress of a wave-group is not the progress of a wave. These forms have a certain degree of permanence amid the shifting particles of water. Anything permanent tends to become dignified with an attribute of substantiality. An ocean traveller has even more vividly the impression that the ocean is made of waves than that it is made of water.* Ultimately it is this innate hunger for permanence in our minds which directs the course of development of hydrodynamics, and likewise directs the world-building out of the sixteen measures of structure.

* This was not intended to allude to certain consequential effects of the waves; it is true, I think, of the happier impressions of the voyage.

The element of permanence is expressed through the process of quantization.

Perhaps it will be objected that other things besides mind can appreciate a permanent entity such as mass; a weighing machine can appreciate it and move a pointer to indicate how much mass there is. I do not think that is a valid objection. In building the physical world we must of course build the measuring appliances which are part of it; and the measuring appliances result from the plan of building in the same way as the entities which they measure. If, for example, we had used some of the “lumber” to build an entity x, we could presumably construct from the same lumber an appliance for measuring x. The difference is this—if the pointer of the weighing machine is reading 5 lbs. a human consciousness is in a mysterious way (not yet completely traced) aware of the fact, whereas if the measuring appliance for x reads 5 units no human mind is aware of it. Neither x nor the appliance for measuring x have any interaction with consciousness. Thus the responsibility for the fact that the scheme of the scientific world includes mass but excludes x rests ultimately with the phenomena of consciousness.

The above is an example of maintaining consistency.

Perhaps a better way of expressing this selective influence of mind on the laws of Nature is to say that values are created by the mind. All the “light and shade” in our conception of the world of physics comes in this way from the mind, and cannot be explained without reference to the characteristics of consciousness.

The world which we have built from the relation-structure is no doubt doomed to be pulled about a good deal as our knowledge progresses. The quantum theory shows that some radical change is impending. But I think that our building exercise has at any rate widened our minds to the possibilities and has given us a different orientation towards the idea of physical law. The points which I stress are:

Firstly, a strictly quantitative science can arise from a basis which is purely qualitative. The comparability that has to be assumed axiomatically is a merely qualitative discrimination of likeness and unlikeness.

Secondly, the laws which we have hitherto regarded as the most typical natural laws are of the nature of truisms, and the ultimate controlling laws of the basal structure (if there are any) are likely to be of a different type from any yet conceived.

Thirdly, the mind has by its selective power fitted the processes of Nature into a frame of law of a pattern largely of its own choosing; and in the discovery of this system of law the mind may be regarded as regaining from Nature that which the mind has put into Nature.

Mind is part of the same system as rest of Nature. So the laws of Nature are not arbitrarily influenced by the mind. Consciousness in Man is the product of evolution. It is subject to the laws of Nature. We have not yet become fully aware of the laws that apply to consciousness.

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Reference: The Nature of the Physical World

This paper presents Chapter XI (section 1) from the book THE NATURE OF THE PHYSICAL WORLD by A. S. EDDINGTON. The contents of this book are based on the lectures that Eddington delivered at the University of Edinburgh in January to March 1927.

The paragraphs of original material are accompanied by brief comments in color, based on the present understanding.  Feedback on these comments is appreciated.

The heading below links to the original materials.

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We have an intricate task before us. We are going to build a World—a physical world which will give a shadow performance of the drama enacted in the world of experience. We are not very expert builders as yet; and you must not expect the performance to go off without a hitch or to have the richness of detail which a critical audience might require. But the method about to be described seems to give the bold outlines; doubtless we have yet to learn other secrets of the craft of world building before we can complete the design.

The first problem is the building material. I remember that as an impecunious schoolboy I used to read attractive articles on how to construct wonderful contrivances out of mere odds and ends. Unfortunately these generally included the works of an old clock, a few superfluous telephones, the quicksilver from a broken barometer, and other oddments which happened not to be forthcoming in my lumber room. I will try not to let you down like that. I cannot make the world out of nothing, but I will demand as little specialised material as possible. Success in the game of World Building consists in the greatness of the contrast between the specialised properties of the completed structure and the unspecialised nature of the basal material.

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Relation Structure

We take as building material relations and relata. The relations unite the relata; the relata are the meeting points of the relations. The one is unthinkable apart from the other. I do not think that a more general starting-point of structure could be conceived.

To distinguish the relata from one another we assign to them monomarks. The monomark consists of four numbers ultimately to be called “co-ordinates”. But co-ordinates suggest space and geometry and as yet there is no such thing in our scheme; hence for the present we shall regard the four identification numbers as no more than an arbitrary monomark. Why four numbers? We use four because it turns out that ultimately the structure can be brought into better order that way; but we do not know why this should be so. We have got so far as to understand that if the relations insisted on a threefold or a fivefold ordering it would be much more difficult to build anything interesting out of them; but that is perhaps an insufficient excuse for the special assumption of fourfold order in the primitive material.

The relation between two human individuals in its broadest sense comprises every kind of connection or comparison between them—consanguinity, business transactions, comparative stature, skill at golf—any kind of description in which both are involved. For generality we shall suppose that the relations in our world-material are likewise composite and in no way expressible in numerical measure. Nevertheless there must be some kind of comparability or likeness of relations, as there is in the relations of human individuals; otherwise there would be nothing more to be said about the world than that everything in it was utterly unlike everything else. To put it another way, we must postulate not only relations between the relata but some kind of relation of likeness between some of the relations. The slightest concession in this direction will enable us to link the whole into a structure.

The starting point is “continuum of substance”. The fundamental element is SUBSTANCE. The substance starts out as an undifferentiated continuum. But as time proceeds a gradual differentiation sets in. But, at a fundamental level, the continuum is maintained.

We assume then that, considering a relation between two relata, it will in general be possible to pick out two other relata close at hand which stand to one another in a “like” relation. By “like” I do not mean “like in every respect”, but like in respect to one of the aspects of the composite relation. How is the particular aspect selected? If our relata were human individuals different judgments of likeness would be made by the genealogist, the economist, the psychologist, the sportsman, etc.; and the building of structure would here diverge along a number of different lines. Each could build his own world-structure from the common basal material of humanity. There is no reason to deny that a similar diversity of worlds could be built out of our postulated material. But all except one of these worlds will be stillborn. Our labour will be thrown away unless the world we have built is the one which the mind chooses to vivify into a world of experience. The only definition we can give of the aspect of the relations chosen for the criterion of likeness, is that it is the aspect which will ultimately be concerned in the getting into touch of mind with the physical world. But that is beyond the province of physics.

The differentiation is never 100% because the similarity remains that everything is substance. Hence, the continuum remains; but there is differentiation in terms of unlimited aspects. The substance can be material, field, or even thought.

This one-to-one correspondence of “likeness” is only supposed to be definite in the limit when the relations are very close together in the structure. Thus we avoid any kind of comparison at a distance which is as objectionable as action at a distance. Let me confess at once that I do not know what I mean here by “very close together”. As yet space and time have not been built. Perhaps we might say that only a few of the relata possess relations whose comparability to the first is definite, and take the definiteness of the comparability as the criterion of contiguity. I hardly know. The building at this point shows some cracks, but I think it should not be beyond the resources of the mathematical logician to cement them up. We should also arrange at this stage that the monomarks are so assigned as to give an indication of contiguity.

The substance lies in the “distance” also. There is no distance that is completely empty. So there is a gradient of substance even before any other aspects are added. This gradient gives us quantization.

Let us start with a relatum A and a relation AP radiating from it. Now step to a contiguous relatum B and pick out the “like” relation BQ. Go on to another contiguous relatum C and pick out the relation CR which is like BQ. (Note that since C is farther from A than from B} the relation at C which is like AP is not so definite as the relation which is like BQ.) Step by step we may make the comparison round a route AEFA which returns to the starting-point. There is nothing to ensure that the final relation AP’ which has, so to speak, been carried round the circuit will be the relation AP with which we originally started.

We have now two relations AP, AP’ radiating from the first relatum, their difference being connected with a certain circuit in the world AEFA. The loose ends of the relations P and P have their monomarks, and we can take the difference of the monomarks (i.e. the difference of the identification numbers comprised in them) as the code expression for the change introduced by carrying AP round the circuit. As we vary the circuit and the original relation, so the change PP’ varies; and the next step is to find a mathematical formula expressing this dependence. There are virtually four things to connect, the circuit counting double since, for example, a rectangular circuit would be described by specifying two sides. Each of them has to be specified by four identification numbers (either monomarks or derived from monomarks) ; consequently, to allow for all combinations, the required mathematical formula contains 4⁴ or 256 numerical coefficients. These coefficients give a numerical measure of the structure surrounding the initial relatum.

This completes the first part of our task to introduce numerical measure of structure into the basal material. The method is not so artificial as it appears at first sight. Unless we shirk the problem by putting the desired physical properties of the world directly into the original relations and relata, we must derive them from the structural interlocking of the relations; and such interlocking is naturally traced by following circuits among the relations. The axiom of comparability of contiguous relations only discriminates between like and unlike, and does not initially afford any means of classifying various decrees and kinds of unlikeness; but we have found a means of specifying the kind of unlikeness of AP and AP’ by reference to a circuit which “transforms” one into the other. Thus we have built a quantitative study of diversity on a definition of similarity.

We seem to have some sort of binary scheme here to define the gradient.

The numerical measures of structure will be dependent on, and vary according to, the arbitrary code of monomarks used for the identification of relata. This, however, renders them especially suitable for building the ordinary quantities of physics. When the monomarks become co-ordinates of space and time the arbitrary choice of the code will be equivalent to the arbitrary choice of a frame of space and time; and it is in accordance with the theory of relativity that the measures of structure and the physical quantities to be built from them should vary with the frame of space and time. Physical quantities in general have no absolute value, but values relative to chosen frames of reference or codes of monomarks.

The absoluteness lies in the duality of SUBSTANCE—NO-SUBSTANCE. Space and time are two fundamental aspects of substance.

We have now fashioned our bricks from the primitive clay and the next job is to build with them. The 256 measures of structure varying from point to point of the world are somewhat reduced in number when duplicates are omitted; but even so they include a great deal of useless lumber which we do not require for the building. That seems to have worried a number of the most eminent physicists; but I do not quite see why. Ultimately it is the mind that decides what is lumber—which part of our building will shadow the things of common experience, and which has no such counterpart. It is no part of our function as purveyors of building material to anticipate what will be chosen for the palace of the mind. The lumber will now be dropped as irrelevant in the further operations, but I do not agree with those who think it a blemish on the theory that the lumber should ever have appeared in it.

As indicated earlier thought is also substance. The thought of continuum (as consistency, harmony and continuity) is a fundamental aspect of substance.

By adding together certain of the measures of structure in a symmetrical manner and by ignoring others we reduce the really important measures to 16.* These can be divided into 10 forming a symmetrical scheme and 6 forming an antisymmetrical scheme. This is the great point of bifurcation of the world.

* Mathematically we contract the original tensor of the fourth rank to one of the second rank.

Symmetrical coefficients (10). Out of these we find it possible to construct Geometry and Mechanics. They are the ten potentials of Einstein (g_μν). We derive from them space, time, and the world-curvatures representing the mechanical properties of matter, viz. momentum, energy, stress, etc.

Antisymmetrical coefficients (6). Out of these we construct Electromagnetism. They are the three components of electric intensity and three components of magnetic force. We derive electric and magnetic potential, electric charge and current, light and other electric waves.

We do not derive the laws and phenomena of atomicity. Our building operation has somehow been too coarse to furnish the microscopic structure of the world, so that atoms, electrons and quanta are at present beyond our skill.

The substance starts as field substance (electromagnetism), and quantizes into field-particles (quanta, electrons, sub-atomic particles), which form the structure of material-substance (atoms, molecules, etc.).

But in regard to what is called field-physics the construction is reasonably complete. The metrical, gravitational and electromagnetic fields are all included. We build the quantities enumerated above; and they obey the great laws of field-physics in virtue of the way in which they have been built. That is the special feature; the field laws—conservation of energy, mass, momentum and of electric charge, the law of gravitation, Maxwell’s equations—are not controlling laws.** They are truisms. Not truisms when approached in the way the mind looks out on the world, but truisms when we encounter them in a building up of the world from a basal structure. I must try to make clear our new attitude to these laws.

**One law commonly grouped with these, viz. the law of pondero-motive force of the electric field, is not included. It seems to be impossible to get at the origin of this law without tackling electron structure which is beyond the scope of our present exercise in world-building.

Charge and mass describe the substantiality of the substance. Gravitation, energy and momentum describe the interaction among substance.

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Reference: The Nature of the Physical World

This paper presents Chapter X (section 5) from the book THE NATURE OF THE PHYSICAL WORLD by A. S. EDDINGTON. The contents of this book are based on the lectures that Eddington delivered at the University of Edinburgh in January to March 1927.

The paragraphs of original material are accompanied by brief comments in color, based on the present understanding.  Feedback on these comments is appreciated.

The heading below links to the original materials.

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Principle of Indeterminacy

My apprehension lest a fourth version of the new quantum theory should appear before the lectures were delivered was not fulfilled; but a few months later the theory definitely entered on a new phase. It was Heisenberg again who set in motion the new development in the summer of 1927, and the consequences were further elucidated by Bohr. The outcome of it is a fundamental general principle which seems to rank in importance with the principle of relativity. I shall here call it the “principle of indeterminacy”.

The gist of it can be stated as follows : a particle may have position or it may have velocity but it cannot in any exact sense have both.

Both the principles of relativity and indeterminacy come about because of quantization as discussed earlier. Material-substance is highly quantized. The field-substance in “empty” space and within the atom is quantized to a much lesser degree. As quantization becomes less, the substance, space and time become less substantial. The space and time expand and become more diffused.

If we are content with a certain margin of inaccuracy and if we are content with statements that claim no certainty but only high probability, then it is possible to ascribe both position and velocity to a particle. But if we strive after a more accurate specification of position a very remarkable thing happens; the greater accuracy can be attained, but it is compensated by a greater inaccuracy in the specification of the velocity. Similarly if the specification of the velocity is made more accurate the position becomes less determinate.

Science addresses this diffusion of space by means of probability of location.

Suppose for example that we wish to know the position and velocity of an electron at a given moment. Theoretically it would be possible to fix the position with a probable error of about 1/1000 of a millimetre and the velocity with a probable error of 1 kilometre per second. But an error of 1/1000 of a millimetre is large compared with that of some of our space measurements; is there no conceivable way of fixing the position to 1/10,000 of a millimetre? Certainly; but in that case it will only be possible to fix the velocity with an error of 10 kilometres per second.

The error comes about because material units of highly quantized compact space are being used to express measurement.

The conditions of our exploration of the secrets of Nature are such that the more we bring to light the secret of position the more the secret of velocity is hidden. They are like the old man and woman in the weather-glass; as one comes out of one door, the other retires behind the other door. When we encounter unexpected obstacles in finding out something which we wish to know, there are two possible courses to take. It may be that the right course is to treat the obstacle as a spur to further efforts; but there is a second possibility—that we have been trying to find something which does not exist. You will remember that that was how the relativity theory accounted for the apparent concealment of our velocity through the aether.

The hidden influence is that of quantization.

When the concealment is found to be perfectly systematic, then we must banish the corresponding entity from the physical world. There is really no option. The link with our consciousness is completely broken. When we cannot point to any causal effect on anything that comes into our experience, the entity merely becomes part of the unknown—undifferentiated from the rest of the vast unknown. From time to time physical discoveries are made; and new entities, coming out of the unknown, become connected to our experience and are duly named. But to leave a lot of unattached labels floating in the as yet undifferentiated unknown in the hope that they may come in useful later on, is no particular sign of prescience and is not helpful to science. From this point of view we assert that the description of the position and velocity of an electron beyond a limited number of places of decimals is an attempt to describe something that does not exist; although curiously enough the description of position or of velocity if it had stood alone might have been allowable.

The “electron” is used to describe the lesser quantized substance within the atom. The space and time within the atom is diffused compared to the material space and time. Both position and velocity within the atom cannot be measured precisely using material units.

Ever since Einstein’s theory showed the importance of securing that the physical quantities which we talk about are actually connected to our experience, we have been on our guard to some extent against meaningless terms. Thus distance is defined by certain operations of measurement and not with reference to nonsensical conceptions such as the “amount of emptiness” between two points. The minute distances referred to in atomic physics naturally aroused some suspicion, since it is not always easy to say how the postulated measurements could be imagined to be carried out. I would not like to assert that this point has been cleared up; but at any rate it did not seem possible to make a clean sweep of all minute distances, because cases could be cited in which there seemed no natural limit to the accuracy of determination of position. Similarly there are ways of determining momentum apparently unlimited in accuracy. What escaped notice was that the two measurements interfere with one another in a systematic way, so that the combination of position with momentum, legitimate on the large scale, becomes indefinable on the small scale. The principle of indeterminacy is scientifically stated as follows: if q is a co-ordinate and p the corresponding momentum, the necessary uncertainty of our knowledge of q multiplied by the uncertainty of p is of the order of magnitude of the quantum constant h.

A general kind of reason for this can be seen without much difficulty. Suppose it is a question of knowing the position and momentum of an electron. So long as the electron is not interacting with the rest of the universe we cannot be aware of it. We must take our chance of obtaining knowledge of it at moments when it is interacting with something and thereby producing effects that can be observed. But in any such interaction a complete quantum is involved; and the passage of this quantum, altering to an important extent the conditions at the moment of our observation, makes the information out of date even as we obtain it.

Einstein’s theory is interpreted subjectively in terms of the experience of moving observer. This is unscientific and leads to errors. The same theory can be interpreted objectively in terms of quantization (substantialness of substance). This is the basis of Disturbance theory.

The quantum constant ‘h’ is the limiting energy per cycle for material substance. This is the accuracy we measure by. Material position is accurate within the material cycle of infinitesimal wavelength. The quantum of energy for an “electron” is larger and more spread out because wavelength increases at lower quantization. This leads to lesser accuracy in measurements.

Suppose that (ideally) an electron is observed under a powerful microscope in order to determine its position with great accuracy. For it to be seen at all it must be illuminated and scatter light to reach the eye. The least it can scatter is one quantum. In scattering this it receives from the light a kick of unpredictable amount; we can only state the respective probabilities of kicks of different amounts. Thus the condition of our ascertaining the position is that we disturb the electron in an incalculable way which will prevent our subsequently ascertaining how much momentum it had. However, we shall be able to ascertain the momentum with an uncertainty represented by the kick, and if the probable kick is small the probable error will be small. To keep the kick small we must use a quantum of small energy, that is to say, light of long wave-length. But to use long wave-length reduces the accuracy of our microscope. The longer the waves, the larger the diffraction images. And it must be remembered that it takes a great many quanta to outline the diffraction image; our one scattered quantum can only stimulate one atom in the retina of the eye, at some haphazard point within the theoretical diffraction image. Thus there will be an uncertainty in our determination of position of the electron proportional to the size of the diffraction image. We are in a dilemma. We can improve the determination of the position with the microscope by using light of shorter wave-length, but that gives the electron a greater kick and spoils the subsequent determination of momentum.

A picturesque illustration of the same dilemma is afforded if we imagine ourselves trying to see one of the electrons in an atom. For such finicking work it is no use employing ordinary light to see with; it is far too gross, its wave-length being greater than the whole atom. We must use fine-grained illumination and train our eyes to see with radiation of short wave-length— with X-rays in fact. It is well to remember that X-rays have a rather disastrous effect on atoms, so we had better use them sparingly. The least amount we can use is one quantum. Now, if we are ready, will you watch, whilst I flash one quantum of X-rays on to the atom? I may not hit the electron the first time; in that case, of course, you will not see it. Try again; this time my quantum has hit the electron. Look sharp, and notice where it is. Isn’t it there? Bother! I must have blown the electron out of the atom.

This is not a casual difficulty; it is a cunningly arranged plot—a plot to prevent you from seeing something that does not exist, viz. the locality of the electron within the atom. If I use longer waves which do no harm, they will not define the electron sharply enough for you to see where it is. In shortening the wavelength, just as the light becomes fine enough its quantum becomes too rough and knocks the electron out of the atom.

We cannot do direct experimentation, as proposed above, to see reactions within the atom.

Other examples of the reciprocal uncertainty have been given, and there seems to be no doubt that it is entirely general. The suggestion is that an association of exact position with exact momentum can never be discovered by us because there is no such thing in Nature. This is not inconceivable. Schrodinger’s model of the particle as a wave-group gives a good illustration of how it can happen. We have seen (p. 217) that as the position of a wave-group becomes more defined the energy (frequency) becomes more indeterminate, and vice versa. I think that that is the essential value of Schrodinger’s theory; it refrains from attributing to a particle a kind of determinacy which does not correspond to anything in Nature. But I would not regard the principle of indeterminacy as a result to be deduced from Schrodinger’s theory; it is the other way about. The principle of indeterminacy, like the principle of relativity, represents the abandonment of a mistaken assumption which we never had sufficient reason for making. Just as we were misled into untenable ideas of the aether through trusting to an analogy with the material ocean, so we have been misled into untenable ideas of the attributes of the microscopic elements of world-structure through trusting to analogy with gross particles.

The missing concept here is quantization.

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