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Eddington's Search for a Fundamental Theory: A Key to the Universe.

Cambridge, Cambridge University Press, 1994

Elie Zahar Department of History and Philosophy of Science, University of Cambridge

Clive Kilmister's book is only partly about Eddington's attempts to unify or bring together Relativity Theory and Quantum Mechanics. It is also a masterly examination of the main themes which dominated the development of twentieth-century physics.

Eddington's Kantianism

Eddington is correctly described as a quasi-Kantian according to whom our most important theories are informed not by the world-conditions but by freely adopted conventions, that is, by our methodological decisions. This thesis finds its most radical expression in the claim that 'There is nothing in the whole system of physics that cannot be deduced unambiguously from epistemological considerations' (quoted p. 216, Kilmister's italics).

Eddington's position is quasi rather than strictly Kantian because of his contention that we possess several perspectives from which to view physical nature; a thesis encapsulated in the two principles of 'selective subjectivism' and 'descriptive tolerance' (pp. 48 and 49). According to Kant, we have one rigid framework which, being systematically imposed on all appearances, does not reflect the nature of things-in-themselves. The breakdown both of Euclidian geometry and of determinism as presuppositions of science put an end to this uniqueness thesis. Like Poincare, Eddington consequently held the view that there exist several legitimate frameworks, each appropriate to a particular situation. But whereas Poincare believed that a maximally convenient and empirically adequate system would reveal true relations between the noumena, Eddington seems to have given up such a hope and even to have welcomed its demise. In his MTR (Mathematical Theory of Relativity) he wrote:

According to the new point of view, Einstein's law of gravitation does not impose any limitation on the basal structure of the world. [G.sub.[Mu][Nu]] may vanish or it may not. If it vanishes we say that space is empty; if it does not vanish we say that momentum or energy is present ([section]54).

However, as pointed out by Kilmister, Eddington remained a metaphysical realist who at one time subscribed to the falsifiability criterion (p. 49). But he was also prepared to accept the frameworks both of Relativity and of Quantum Mechanics without requiring them to be globally compatible with each other. After all, these frameworks usually apply to very different situations: to the infinitely large in one case, and to the indefinitely small in the other. They none the less occasionally overlap, as is evidenced by Dirac's quantum-relativistic theory of the electron. Dirac had conjoined the Lorentz-covariance requirement with a quantization condition based on the rules:

[p.sub.x] [approaches] -i(h/2[Pi])[Delta]/[Delta]x, [p.sub.y] [approaches] -i(h/2[Pi])[Delta]/[Delta]y, [p.sub.z] [approaches] -i(h/2[Pi])[Delta]/[Delta]z, H [approaches] -i(h/2[Pi])[Delta]/[Delta]t.

Eddington felt that Dirac had not gone far enough: a comparison of the two methods should yield the values of certain constants which enable the two frameworks to overlap consistently. Eddington was in effect emulating Maxwell who, by unifying electricity and magnetism, had theoretically determined the value of the speed of light c. It should be noted that c can also be directly measured, thus providing a check on the coherence of Maxwell's system. Through applying the covariance condition and his own quantization procedure to the state of the hydrogen atom, Eddington maintained he could account for the occurrence of an extra term, namely of 2[Pi][e.sup.2]/hcr, in Dirac's equation. 2[Pi][e.sup.2]/hcr represents the Coulomb potential between proton and electron. We should remark that 2[Pi][e.sup.2]/hc, also called the fine-structure constant, can be empirically determined. Despite Kilmister's real tour de force in recasting Eddington's hypotheses, Eddington's performance does not match up to Maxwell's. In Part 2 of his book, Kilmister does succeed in giving a coherent reconstruction of many, though not of all of Eddington's 'proofs'. The most arduous and convincing arguments, however, clearly stem from Kilmister himself, not from Eddington. The author admits to being incapable of making sense of some of Eddington's moves and of rebutting a charge of intellectual dishonesty levelled at Eddington by J. Merleau-Ponty (pp. 230-1).

Eddington's numerological programme

Before going into any details about Kilmister's critical discussion of Eddington's scientific work, I think it helpful to try and characterize the latter's heuristic in general terms. Eddington appears to have followed a method which can broadly be described as 'numerological'. Kilmister writes:

The essence of numerology is the identification of certain numbers derived without any theoretical reason known for them. In this situation, the only interest in a numerological result must lie in the closeness of agreement of the numbers (p.213).

Although I agree with the author about a numerological law being judged by the close agreement between some measured and the corresponding theoretical quantity, I should like to widen the above definition of 'numerology'. All reference to the absence of theoretical reasons for the derivation of certain numbers will thereby be avoided; which is desirable since such a caveat clearly poses a problem of regress. At some point of any scientific explanation, one has to posit - without further justification - some thesis or other in order to account for a given state-of-affairs. The numerologist might thus claim that his numbers are obtained through purely mathematical speculations, which he could spell out without feeling any need to justify them. His methodology would be irreproachable.

There is another reason for my wanting to broaden the definition of numerology. It seems to me that as author of the 'Cosmic Mystery', Kepler qualifies as a numerologist, though he dealt mainly in geometrical rather than in numerical figures. The numerological or pure-mathematical heuristic will be characterized as follows: one starts by examining the properties of some mathematical structure independently of any physical considerations; this does not necessarily mean that the latter played no part in choosing the structure in question; but once the selection has been made, the programme proceeds - in its middle course - without regard to any empirical results; only at the end of this formal process are some features or components of the system identified with certain physical entities. Ideally, this last step ought also to be taken without ad-hoc manoeuvering. Note that the Principle of Identification, to which we shall return, plays a central part in the final stages of a numerological programme.

Let us now compare this heuristic with the role which is normally assigned to mathematics in research programmes. This role can be described as follows. Start from some scientific conjecture. Formulate it in a mathematical 'language', thus giving it some surplus structure and possibly some extra empirical content. Test the additional content. A new confirmation of the hypothesis could thus be achieved; after which the testing will normally be repeated and then extended until fresh difficulties arise. A new conjecture might be put forward and the same process will start all over again. Here the interaction between mathematics and physics is dialectical; it consists of a to-and-fro movement between the a priori and the empirical. By contrast, in the case of numerology, there ought to be a parallel development of mathematics on the one hand and of experimental science on the other - a development which is intermittently punctuated by operations of identification.

Eddington's Principle of Identification

As explained above, numerologists develop mathematical schemes, initially without the intrusion of any physical considerations. These schemes are none the less intended to be scientific, that is, testable. Hence certain constructs have to be identified with physical entities. Two questions arise. First: what kind of entities must these be? Secondly: what principles guide the identification? As correctly pointed by Kilmister, Eddington often conflated 'physical magnitude' with 'observable quantity'. These ought, however, to be two distinct concepts (p. 213). In his MTR, Eddington speaks of the tensors of his pure geometry being identified with the tensors 'summarizing the mass, momentum and stress' possessed by matter, that is, with highly theoretical entities (MTR, [section]54). Noting that the experimental determination of constants must also be underwritten by physical laws, it can be concluded that identifications usually occur between new constructs and some theoretical terms of older hypotheses. This is in fact illustrated by Eddington himself who identified ([G.sub.[Mu][Nu]] + [G.sub.[Nu][Mu]])/2 with the space-time metric [g.sub.[Mu][Nu]], and ([G.sub.[Mu][Nu]] - [G.sub.[Nu][Mu]])/2 with the tensor [F.sub.[Mu][Nu]] representing the electromagnetic field; where [G.sub.[Mu][Nu]] is the Ricci tensor of Eddington's purely affine geometry (MTR, [section]96). As for the principles informing these identifications, they are based on both structural and simplicity considerations. For example: [G.sub.[Mu][Nu]] is obtained from the Riemann-Christoffel tensor, that is, from the quantity which largely determines the structure of space-time, by the most straightforward method, namely by the contraction of two indices. [G.sub.[Mu][Nu]] then naturally splits into a symmetric part [R.sub.[Mu][Nu]] = ([G.sub.[Mu][Nu]] + [G.sub.[Nu][Mu]])/2, and an anti-symmetric part [K.sub.[Mu][Nu]] = ([G.sub.[Mu][Nu]] - [G.sub.[Nu][Mu]])/2. This is why [R.sub.[Mu][Nu]] and [K.sub.[Mu][Nu]] were subsequently identified with [g.sub.[Mu][Nu]] and [F.sub.[Mu][Nu]] which, thanks to previous theories, were known to be symmetric and anti-symmetric quantities respectively. The mathematical constructs are thus required to be the simplest entities bearing to one another relations, and possessing properties similar to those of their 'physical' counterparts, that is, of certain terms defined by previous scientific hypotheses. These older conjectures might, however, either be false or else stand in need of corrections; which immediately poses the problem of excess testability. In other words: although Kilmister rightly distinguishes the identification of mathematical terms with physical notions from that with directly calculable quantities, there remains the necessity of testing the proposed numerological theory. Thus one is still left with the task of somehow linking the identified constructs with certain observables.

It seems to me that, given its method of construction, a numerological theory can be empirically supported in essentially one of two possible ways. Either some construct q coincides - to within observational error - with a quantity [q.sub.0] which has already been experimentally determined; in which case one identifies q with [q.sub.0] and then adduces the improbability of the coincidence in order reasonably to claim that the hypothesis has been corroborated; or the new term q differs sensibly from the [q.sub.0] with which it is meant to be identified; q could then be taken to replace and hence to correct [q.sub.0]. In this second case, there ought to be further testing and the new experiments could prove crucial. Should the theory withstand these tests, then it must be regarded as having been strongly confirmed since it corrects previous results. Should it, however be refuted, then manoeuvering so as to bring it into line with known facts might well mark the beginning of a degenerating phase. This move would not only be ad hoc in the ordinary sense; it would also violate the essence of the numerological heuristic. And this is really what happened to Kepler's original programme: having realized that the five regular solids do not fit into and around the celestial spheres, Kepler gave the latter enough thickness to accommodate the planets' orbits; but despite this ad-hoc move, the system could never be made to work.

One of the great merits of Kilmister's book is to have demonstrated that as long as the canons of testability are adhered to, numerological laws - in the amended sense above - need be neither metaphysical nor ad hoc. Kilmister also points out that Eddington initially subscribed to the refutability criterion (p.49). However, by the time Eddington wrote RTPE (Relativity Theory of Protons and Electrons):

Falsifiability is not important because it has taken a back seat. Eddington allows himself such freedom to change to a new model half-way through an argument that the ostensible falsifiability of an exact numerical prediction is a hoax (p.207).

Kilmister's masterly reconstruction of Eddington's attempts to determine the fine-structure constant shows how and why Eddington's programme, like Kepler's, finally degenerated.

The fine-structure constant

In Chapter 5, Kilmister describes the shock experienced by Eddington in 1928 when Dirac proposed his relativistic theory of the electron. Until then, Eddington had taken the heuristic of Relativity to be coextensive with the general methods of the tensor calculus. But although Dirac's equation is Lorentz-covariant, his state-function is a 4x1 matrix which behaves, not like a 4-vector, but like a spinor whose transformation is fixed by a change of frame only to within a factor of [+ or -]1. (By the way, these technical details are beautifully explained in Chapters 5-7).

Dirac's equation moreover involves four anti-commuting matrices [[Gamma].sup.a] (a = 0, 1, 2, 3) which satisfy: [[Gamma].sup.a][[Gamma].sup.b] + [[Gamma].sup.b][[Gamma].sup.a] = 2[[Eta].sup.ab]I, where [[Eta].sup.ik] = 0 for j [not equal to] k and 1 = [[Eta].sup.00] = -[[Eta].sup.11] = -[[Eta].sup.22] = -[[Eta].sup.33].

Eddington was thus led to consider a multiplicative algebra generated by four entities [E.sub.a] (a = 1, 2, 3, 4) such that: [Mathematical Expression Omitted], and [E.sub.j][E.sub.k] = -[E.sub.k][E.sub.j] for all j [not equal to] k. To these four elements Eddington added a fifth defined by: [E.sub.5] = [E.sub.1][E.sub.2][E.sub.3][E.sub.4]. It can be easily established that: [Mathematical Expression Omitted] and [E.sub.a][E.sub.5] = -[E.sub.5][E.sub.a] for all a = 1, 2, 3, 4. Putting [E.sub.ab] = [E.sub.a][E.sub.b] for all a [not equal to] b, we have: [E.sub.ab] = -[] for all a [not equal to] b, where a, b [element of] {1, 2, 3, 4, 5}. It therefore suffices to consider [E.sub.ab] for a [less than] b.

Define G = {I, [E.sub.a], [E.sub.ab]: a = 1, 2, 3, 4, 5; b = 1, 2, 3, 4, 5; a [less than] b}. Thus G has 16 elements and it can easily be checked that [X.sup.2] = [+ or -]I for all X [element of] G. Moreover: ([X.sup.2] = +I) [if and only if] (X [element of] S), where S = {I,[E.sub.4],[E.sub.14],[E.sub.24],[E.sub.34],[E.sub.45]}. S has therefore 6 elements and it follows from the above that: ([Y.sup.2] = -I) [if and only if] (Y [element of] G\S), where G\S consists of 16-6 = 10 elements.

Note that G, not being closed under multiplication, does not form a group. However, if we define H = {[+ or -]X: X [element of] G}, then we can verify that H - which consists of 32 as opposed to 16 elements - does constitute a group. In what follows, G will nonetheless be referred to as an E-algebra.

Eddington associates an E-algebra with every elementary particle, more particularly with each proton and with each electron. Furthermore, according to his conception of the Relativity Principle, a quantum-mechanical system should consist not only of an object but also of a comparison particle relative to which positions and momenta are observed. A minimal such system comprises two particles and should therefore have, associated with it, the direct product G x G of two E-algebras. Let (X, Y) [element of] G x G. By definition, [(X, Y).sup.2] = (X, Y)[multiplied by](X, Y) = ([X.sup.2], [Y.sup.2]) = ([+ or -]I, [+ or -]I). Eddington calculated the number of elements (X, Y) of G such that [X.sup.2] = [Y.sup.2]; i.e such that: either [X.sup.2] = I = [Y.sup.2] or [X.sup.2] = -I = [Y.sup.2]. By the above, X and Y must both belong either to S or to G\S. This yields [6.sup.2] + [10.sup.2] = 136 possibilities (p. 113).

This first stage of Eddington's numerological programme can be regarded as a near-success. Proceeding in a priori fashion, he had determined a pure number: 136, which ought then to have been identified with some physical magnitude, for example, with the fine-structure constant hc/2[Pi][e.sup.2]. When measured, hc/2[Pi][e.sup.2] was however found to be equal not to 136, but to something like 137.036 (p. 198). Eddington had therefore to find some convincing way of replacing 136 by 137. At this point, he changed his heuristic by directly appealing to physics in order first to construct, then gradually to modify a series of mathematical models. These moves might have been judged acceptable if he had not resorted to ad-hoc methods, that is, to methods whose only function was to bring his conjectures into line with well-known results. For example: he considered the hydrogen atom, that is, a system consisting of one proton and of one electron. He knew that an equation which adequately - but incomprehensibly - describes this system can be obtained by adding the term [e.sup.2]/r to the energy operator -i(h/2[Pi])[Delta]/[Delta]t in Dirac's equation for the free electron. Dividing by hc/2[Pi], we find that, in the equation as usually written, there occurs the extra operator (2[Pi][e.sup.2]/hc)(1/r)(pp.124-5). Eddington set himself the task of proving from purely theoretical assumptions that Dirac's equation must actually be augmented by the term (1/137)(1/r). Through identifying 137 with hc/2[Pi][e.sup.2], he could then plausibly maintain that his numerological approach had been empirically vindicated. The genuineness of this claim would, however, depend on whether or not the methods used were ad hoc.

As is well known, the Coulomb field between two electrons gives rise to a repulsion which prevents the two particles from coming too close to each other. Eddington looked upon this repulsive force as the outward manifestation of Pauli's exclusion principle, which can be roughly formulated as follows: in certain circumstances, such as within the same atom, no two identical particles with half-integral spin, such as no two electrons, can be in exactly the same state. Eddington however needed to apply this role to the proton and the electron which, needless to say, are far from being identical objects; not only do they differ in mass and charge, they also attract rather than repel each other. So Eddington invoked a (Wittgensteinian) principle according to which the notion of distinguishability is theory-language dependent: since the electron differs from the proton only as regards mass and charge, the two particles must be pronounced indistinguishable in a language possessing neither mass nor charge predicates. Eddington furthermore claimed that:

mass can never be used as a criterion for distinguishing particles; it presupposes that they have already been distinguished (quoted pp. 132-3).

I admit to finding such arguments very specious, not to say appalling. Of course, no grammar of one theory-language can legitimately be used in order to criticize that of another; but this applies only as long as the two languages remain disjoint. Eddington, however, starts by noting the indiscernibility, within a simple language, of the proton from the electron, in order then to show that these two particles can after all be told apart in a more complex language. We have here a transition from a rudimentary to a more comprehensive grammatical structure. In ordinary logic this would yield - within the more complex system - the inconsistent statement that the proton both is and is not distinguishable from the electron. Eddington's arguments might be considered acceptable and even illuminating, had he constructed some coherent dialectical logic; but he did not. He simply went on to use the superposition principle in order to add to Dirac's equation an expression in which the matrix [F.sub.23] occurs; whereas the correct extra term should contain, not [F.sub.23], but the imaginary number i. Without further ado, he substituted i for [F.sub.23] by fiat. At this point, Kilmister understandably loses patience. He writes:

That is to say, he [Eddington] simply replaces it [[F.sub.r4]] by i, so as to agree with the usual equation. For [F.sub.r4] one should read [F.sub.23] as explained above. That error is not serious. But I would argue that the remaining point is a major hiatus in the argument, perhaps the only one. Eddington's last sentence really will not do (p. 148).


The author succeeds in showing that numerological theories - in the modified sense above - need be neither absurd nor even unreasonable. But despite the undoubted genius of thinkers like Kepler and Eddington, numerology - as a programme - appears always to have ended in failure. Putting it in Eddingtonian language: there is a strong suggestion that the 'world-conditions' do not bring forth beings equipped with enough true a priori knowledge to be able to anticipate the workings of nature. Science seems to be in need of continually checking its conjectures against sense-experience.

As already mentioned, this book provides far more than an account of Eddington's search for a fundamental theory. Not only does it contain a panorama, together with a discussion in depth, of the regulative ideas which dominated the development of modern physics; namely: operationalism, subjectivism, falsifiability, conventionalism and the Principles of Identification, Relativity and Quantization; but the reader can also gain from it an insight into physics which is to be found in no ordinary textbook. Scientific theories are presented as creative responses to concrete problems, not as posits for which problems provide mere illustrations. The reader interested exclusively in the philosophy of science will naturally be able to follow the main text without having to consult the Notes. But with a modicum of mathematical knowledge, one draws maximum profit from reading the text in conjunction or in parallel with the Notes; these ought to be treated as examples and they are provided with so many helpful hints that they constitute a natural and easily accessible complement to the main text. In short: C.W.Kilmister's Eddington's Search for a Fundamental Theory is a classic which ought strongly to be recommended to all those seriously interested both in physics and in its philosophy.
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Author:Zahar, Elie
Publication:The British Journal for the Philosophy of Science
Article Type:Book Review
Date:Mar 1, 1997
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