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 (gμν). 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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