By 'ontology' here is meant the theoretical representation of scientific entities, such as light, electrons etc.; the relevant structures, on the other hand, are represented for Worrall by the appropriate mathematical equations - Snell's Laws are incorporated into Maxwell's Equations and so on. Thus whereas the ontological component of a theory may be subjected to a pessimistic meta-induction, as far as the structural component is concerned things look quite optimistic. This gives rise to a form of 'Structural Realism' (SR) which holds that one can, and should, adopt a realist attitude towards the well confirmed structural aspects of theories. As Ladyman has pointed out (1998), this should be regarded as an epistemic form of SR since it holds that all that we know are the structures, while the objects themselves remain epistemologically inaccessible. It is worth noting that in defending this position Worral draws on the history of structuralism in the form of those famous passages from Science and Hypothesis where Poincaré writes that theoretical terms '... are merely names of the images we substituted for the real objects which Nature will hide forever from our eyes. The true relations between these real objects are the only reality we can ever obtain.'  However, as Domski has emphasised, Poincaré may not be the most appropriate name to drop in order to give historical legitimacy to this version of SR, given his Kantian inclinations and rejection of truth as the aim of science. Such inclinations appear again and again through the history of structuralism and the issue arises as to whether the structural realist can neatly peel them off from the rest of structuralist programme. With regard to the former, both Cassirer and Eddington not only explicitly and famously tied their structuralism to the development of group theory but also drew on the quantum treatment in order to further their structuralist aims and I want to sketch the relevant history here with an eye on what lessons might be drawn.

About  Structures, Symmetry and Subjectivity

In a recent contribution to the on-going revival of interest in Cassirer's work, Ihmig identifies the central theme running through Cassirer's writings in the philosophy of science as the analysis of the concept of object (Ihmig 1999). The fundamental perspective from which this analysis should proceed is epistemological:'... epistemological reflection leads us everywhere to the insight that what the various sciences call the "object" is nothing in itself, fixed once for all, but that it is first determined by some standpoint of knowledge.' (Cassirer 1953, p. 356)

As is well known, Cassirer's interest in this issue can be traced back to his reflections on then nature of space and the influence of Klein's Erlanger programme. And the crucial insight offered by the latter concerned the introduction of the concept of group. What this yields, of course, is a structural conception of geometrical objects which shifts the focus from individual geometrical figures, grasped intuitively, to the relevant geometrical transformations and the associated laws. This shift is then manifested in Cassirer's neo-Kantian assertion of 'the priority of the concept of law over the concept of object.' This assertion in turn forms an integral component of Cassirer's interpretation of the Kantian understanding of objectivity: 'For objectivity itself - following the critical analysis and interpretation of this concept - is only another label for the validity of certain connective relations that have to be ascertained separately and examined in terms of their structure. The tasks of the criticism of knowledge ("Erkenntniskritik") is to work backwards from the unity of the general object concept to the manifold of the necessary and sufficient conditions that constitute it. In this sense, that which knowledge calls its "object" breaks down into a web of relations that are held together in themselves through the highest rules and principles.'

These 'highest rules and principles' are the symmetry principles which represent that which is invariant in the web of relations itself. And these principles, in turn, are represented group-theoretically; thus the relevant group effectively lays down the general conditions in terms of which something can be viewed as an object. Cassirer's 'application' of this framework to the foundations of relativity theory is well known. According to Ihmig, what it does is restore the unity of the concept of object which is apparently undermined by the relativistic transformations. From the structuralist perspective, this unity is 'reinstated on a higher level.' (ibid., p. 525) via the 'lawful unity' of inertial systems offered by the Lorentz transformations. The process of abstraction from a substantivalist conception of objects to a structuralist one is furthered by the General Theory of Relativity and what we are left with is an understanding of the objects of a theory as defined by those  transformations which leave the relevant physical magnitudes invariant. Thus Cassirer saw General Relativity as the natural conclusion of the structuralist tendency. Cassirer's understanding of the foundations of GR has been further pursued by Ryckman (1999), who points to the central importance of the principle of general covariance in this understanding. According to Ryckman, Cassirer viewed general covariance as a principle of objectivity which offers a 'deanthropomorphized' conception of a physical object. Furthermore, he (Ryckman) claims, this view of Cassirer's meshed with Einstein's own and underpinned the latter's objections to quantum mechanics through its implementation in the separability principle. As the requirement that the laws of nature be formulated so that they remain valid in any frame of reference, general covariance '... is a further manifestation of the guiding methodological principle of "synthetic unity" necessary to the concept of the object of physical knowledge.' (ibid., p. 604). Regarded as a synthetic requirement, general covariance comes to be seen as both a formal restriction and a heuristic guide for the discovery of general laws of nature (ibid.). Physical objectivity - apparently lost by space and time themselves - re-emerges in deanthropomorphised form in terms of the functional forms of connection and coexistence. There has been comparatively little discussion of Cassirer's analysis of the other major revolution of the twentieth century, namely quantum mechanics3. Ihmig states in a footnote that Cassirer assessed the results of this revolution in the same way as the above with regard to the objects of science. However, there is the additional element of the loss of individuality of the particles themselves which was apparently implied by the new quantum physics. The overall framework is the same, encompassing as it does a shift from things-assubstances to relations as the ground of objectivity in science; or as Cassirer put it, '[w]e are concerned not so much with the existence of things as with the objective validity of relations; and all our knowledge of atoms can be led back to, and depends on, this validity'. In classical mechanics objectivity rests on the spatiotemporal persistence of individual objects and here, '"[o]bjective" denotes a being which can be  ecognized as the same in spite of all changes in its individual determinations, and this recognition is possible only if we posit a spatial substratum.'. As Cassirer points out, 'The entire axiomatic system of classical mechanics is based on this presupposition.' As is well known, this presupposition features explicitly in Boltzmann's axioms for example and it forms the basis of the 'world-view' of classical (particle) physics in which we have individual objects possessing at all times well-defined properties and traversing well-defined spatio-temporal trajectories. It is this world-view that is apparently overturned by quantum mechanics (at least under the orthodox interpretation) and in the new situation in which we find ourselves, we cannot say that the particles unambiguously possess definite properties at all times, even beyond measurement interactions, or that they travel along well-defined trajectories. It is at this juncture that Cassirer asks a pair of crucial questions: '... what are these electrons whose path we can no longer follow? Is there any sense in ascribing to them a definite, strictly determined existence, which, however, is only incompletely accessible to us?'. In answering these questions, Cassirer makes the fundamental demand of the ontic form of structural realism, namely that we take the 'conditions of accessibility' as 'conditions of the objects of experience'. If we do that, then '... there will no longer exist an empirical object that in principle can be designated as utterly inaccessible; and there may be classes of presumed objects which we will have to exclude from the domain of empirical existence.  There are no epistemically inaccessible objects laying behind the structures which we can know. What is an electron then? Not, Cassirer insists, an individual object and he cites Born's comment  that from the perspective of quantum statistics, the particles cannot be identified as individuals at all. Of course, this is to follow the 'received view' regarding the indistinguishability of quantum particles which draws the conclusion that they are non-individuals in some sense. Nevertheless, Cassirer takes it to further support the shift away from particles as substantival 'things'. If we want to continue to talk, in everyday language, about electrons as objects - because we lack the logico-linguistic resources to do otherwise - then we can do so 'only indirectly', '... not insofar as they themselves, as individuals, are given, but so far as they are describable as "points of intersection" of certain relations' (ibid.). And this relational conception of an object is taken straight from Kant himself: 'All we know in matter is merely relations ... but among these relations some are self-subsistent and permanent, and through these we are given a determinate object.' Charge, understood as an intrinsic or state-independent property of particles, is just such a 'self-subsistent and permanent relation' but as Cassirer points out, in an acute rebuttal of the assumption made by the 'standard' realist, ' ... the constancy of a certain relation is not at all sufficient for the inference of a constant carrier' (ibid.). The permanence of charge justifies our regarding the electron, say, as a 'determinate object', where the scare quotes indicate that the sense is that of an entity prior to reconceptualisation in structural terms, but it is does not justify what Cassirer calls the 'substantialization and hypostasis' of the electron in the sense of an entity which is not so reconceptualised. Charge, like the other intrinsic properties, features in the relevant laws of physics and according to Cassirer, what we have here is a reversal of the classical relationship between the concepts of object and law (ibid., pp. 131-132): instead of beginning with a 'definitely determined entity' which possess certain properties and which then enters into definite relations with other entities, where these relations are expressed as laws of nature, what we now begin with are the laws which express the relations in terms of which the 'entities' are constituted. From the structuralist perspective, the entity '... constitutes no longer the self-evident starting point but the final goal and end of the considerations: the terminus a quo has become a terminus ad quem.' (ibid., p. 131) Objectivity, therefore, is determinable through law, which is prior to it (ibid., p. 176) and the boundaries of law mark the boundaries of objective knowledge (ibid., p. 132). As already indicated, Cassirer saw these developments in physics as confirming a neo-Kantian epistemology according to which the laws of physics - in particular those of quantum mechanics and relativity theory - provide the sole basis for our integration of experience. In this integration, a crucial role is played by the 'principle' of causality, regarded not as a proposition pertaining to events themselves, but, rather, '... a stipulation concerning the means through which things and events are constituted in experience.' (ibid., p. 789). As such, the principle is not undermined by quantum mechanics; indeed, Cassirer insists, understood as a demand for strict functional dependence, the essence of causality remains untouched. The significance of quantum physics for epistemology lies precisely with the above consideration regarding the nature of objects. The retention of causality provides of course a further connection between Cassirer and Einstein. As I mentioned above, Ryckman also argues that general covariance underpins Einstein's criterion of observer independent objectivity in terms of his principle of separability. The connection is provided by Schlick who claimed that only general covariance can adequately satisfy the Maxwellian requirement that causal differences between two events should not depend upon the particular spatio-temporal locations of the events. This further requires a way of distinguishing causal occurrences so that they may be regarded as similar but not identical and this is what the principle of separability allows. As is now well known, the latter is central to the EPR objection and by '... distinguishing physical systems by virtue of causal independence of measurement interactions, [it] serves as a principle of individuation in lieu of the usual identification of physical systems by reference to a fixed background of space and time ...' (ibid., p. 615). The lesson drawn by Ryckman is that Einstein's criterion of 'observer objectivity' is not the expression of a 'simple minded realism', '... but rather a presupposition for the application of causal laws in the physical description of the world.' (ibid., p. 616). Howard understood separability both in spatio-temporal terms and as a sufficient condition for the individuality of physical systems.  How is it to be understood then? And if, as Ryckman suggests, it does function as some kind of principle of individuation, how does this mesh with Cassirer's apparent realisation that quantum particles should not be regarded as individuals? What I would like to suggest (and this needs further elaboration, I know) is that it acts as a principle of 'pseudo-individuality' which allows us to distinguish systems - in a limited and localized way - in terms of their independent causal effects but does not give us licence to effectively import this principle beyond the observable effects and regard the systems as full-blown individual objects7. Citing Heisenberg, Cassirer writes, 'The process of observation cannot be simply objectified; its results cannot be turned immediately into real objects.' (1937, p. 142). The apparent failure of separability in EPR situations should then be read, not as a failure of the principle as a 'Principle of Pseudo-Individuality' but as a failure of the attempt to regard it as a Principle of (Full-Blown) Individuality and import it beyond the immediate measurement situation. In line with Ladyman's ontic structural realism, how this failure in turn should be understood is not in terms of the systems being non-individual objects, but in terms of their not being objects at all. Thus structuralism may offer a different ontological perspective on the implications of the Bell/EPR results. Finally, it is interesting that both Ihmig and Ryckman mention Eddington in the context of Cassirer's structuralism; Ihmig in particular notes Eddington's emphasis on the importance of group theory in fleshing out the structural approach to knowledge. I'd like to turn to Eddington's form of structuralism now, as here the implications of quantum mechanics for the view of physical objects as individuals played an even more important role in its development. In the preface to his later philosophical work, The Philosophy of Physical Science (Eddington 1939), Eddington remarks that in giving a name to his philosophy he hesitates between 'Selective Subjectivism' and 'Structuralism' (ibid., p. viii)8. Both can be traced back to his early reflections on the significance of relativity theory as presented in his Mind papers of 19209. In the first (Eddington 1920a), Eddington rejects the standard approach of beginning with intervals as measured by clocks and rods and then obtaining the field equations, since the introduction of such clocks and rods before one has introduced the matter out of which they are supposedly composed would be 'inconvenient' when one is in the business of constructing the world in a 'strict analytical development' (ibid., p. 152). We shall encounter this attitude again when we come to his view of particle indistinguishability. Instead, Eddington begins with point events, the aggregate of which constitute 'the World' (ibid., p. 147) and which is postulated to be four-dimensional. Between any two neighbouring point events one can then define the interval, as a quantitative relation, and comparison of intervals leads to a 'rule of connexion' (ibid., p. 148) which expresses a 'quality of the World' as measured by the usual coefficients g. By an 'exceedingly complicated combination' of operations on the gone obtains the G (ibid., pp. 149-150) and voilá, Eddington introduces the field equations for the case of empty space and for when matter is present10. These equations, he insists, should be read from left to right, not as laws of the World relating the continuum of points events and matter, since that leads to a kind of dualism (between the continuum of point events and matter) but as mathematical identifications denoting 'definite and absolute' conditions of the world (ibid., p. 151) which give us the perceptions of emptiness and of matter respectively. The field equation with non-vanishing stressenergy tensor describes how the theoretical quality represented by the left-hand side is 'appreciated' by the mind. Hence, 'Matter does not cause an unevenness in the gravitational field; the unevenness is matter.' (ibid., p. 152) There are two important aspects to this, which relate to the structuralist and subjectivist components of Eddington's thought respectively. By matter as the putative cause of irregularities in the field, Eddington means matter as substance and thus this construction is seen as eliminating substance from our ontology in favour of relational structures. Secondly, 'matter', in this new sense, becomes dependent on the mind, since 'Matter is but one of a thousand relations between the constituents of the World, and it will be our task to show why one particular relation has a special value for the mind.' (ibid., p. 153). In his later paper in the same volume (Eddington 1920b; contributed to the 1920 International Congress of Philosophy), Eddington draws an analogy with the construction of constellations out of the distribution of the stars: Here we see how the structuralism and the selective subjectivism mesh. Not all laws are subjective, however. What we have learned from relativity theory, according to Eddington, is that there is a certain quality which distinguishes substantial matter from mere emptiness. We have not yet discovered why the quality formerly known as matter comes in lumps; hence the 'law of atomicity' may be a law of the World itself. It is this 'particularising' of the structure that is described by quantum mechanics, of course, and here Eddington is quite explicit that in order to understand how it is that the same quality which is chosen by the mind as substantial matter is singled out by Nature for the property of atomicity, we must understand how relativity theory and quantum physics can be related. This, of course, is the aim of the programme pursued in his later work. It is important to note that Eddington's structuralism is limited both globally and locally. It is limited globally in that structuralism is appropriate only for metrical (or as he later calls it 'symbolic') knowledge, such as we obtain through physics, and not non-metrical (or 'intimate') knowledge (ibid., p. 322), which would include biology as well as theology13. This is not the place to discuss Eddington's religious beliefs but it is interesting to note that Dingle, in his critique (Dingle 1954), characterises the difference between the metrical and non-metrical in terms of that between structure and nature; thus non-metrical knowledge is knowledge of the nature of things. If Dingle is correct, the Eddington would count as an epistemic structuralist (of a rather peculiar, religious stripe, perhaps).I shall return to this point shortly. His structuralism is also limited locally in the way already indicated, namely with regard to the lack of a good theory of matter itself. As he wrote, 'The possibility of the existence of an electron in space is a remarkable phenomenon which we do not yet understand. As Kilmister notes, Eddington concludes, again, that atomicity may be a reflection of nonsubjective laws of the World and hence represents non-structural substance. By the time of his 1927 Gifford lectures, published the following year as The Nature of the Physical World, Eddington was able to say rather more about quantum physics than his earlier rather brief and simplistic remarks concerning the quantum of action14. Nevertheless, he was not able to say enough to be able to incorporate atomicity within his 'world building'. This is presented even more clearly than in the 1920 paper but with regard to the structural 'building material', Eddington quite clearly does not take the relata as the basic building blocks: '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.'

It is here that we see elements of Eddington's 'numerological' tendencies, as he expressed the hope that from a 'structural interlocking' of relations, one might derive the desired physical properties of the world. In particular, by applying symmetry constraints to the structure, Eddington claimed that we could construct geometry and mechanics, on the one hand, and electromagnetism on the other15. Again, the relevant laws are described as 'mathematical identities', whose violation is 'unthinkable'.And again, the construction obtained is too coarse to accommodate the microscopic structure of the world. Here Eddington acknowledges that he 'scarcely knows' what to think. Perhaps the laws of quantum physics will also come to be seen as mathematical identities, arising 'only in the presentation of the world to us'; or perhaps they will be acknowledged as genuine 'laws of control' of an external world. According to Kilmister, the summary given in this work of the state of play in quantum mechanics around 1926-7 essentially cemented into place Eddington's understanding of the theory, with the exception of the Dirac equation to be mentioned below. Here too the selective subjectivism comes into play. Why are the properties of the building we obtain ordered the way they are? The answer is that the theoretical world building must converge to the mental world building which gives us familiar experiences: '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.' (ibid., p. 241)

In particular, these familiar experiences are subject to the mind's demand for permanence and it is this which underpins the conservation laws and also gives rise to the illusion of substantiality. Again, Eddington makes an allusion to picking out constellations from the stars, so that this world building actually amounts to '... a selection from the patterns that weave themselves' (ibid.). As far as Eddington was concerned, the most suitable representation of the world structure was through the tensor calculus. Indeed, he wrote that, 'I do not think it is too extravagant to claim that the method of the tensor calculus, which presents all physical equations in a form independent of the choice of measure-code, is the only possible means of studying the conditions of the world which are at the basis of physical phenomena.'

It is not surprising, then, that Dirac's equation had a dramatic impact, expressed as it was in terms of spinors (see Kilmister op. cit., Ch. 5). It prompted Eddington to elaborate and investigate a new set of algebraic structures, described by what he called the 'wave-tensor' calculus, which, he believed, would provide the bridge between relativity and quantum theory. Furthermore, he maintained, the laws constituting this bridge have the above form of mathematical identities and thus the construction of the bridge proceeds on the same analytic basis. In particular, and famously, manipulation of this wave-tensor calculus appeared to give the values of certain fundamental physical constants, such as the fine structure constant. As far as Eddington's contemporaries were concerned, this was nothing more than a form of numerology which transformed into necessities numbers which were only contingent. Dingle's criticism is representative: even supposing that Eddington's mathematics is correct, it does not follow that his conclusions are strictly 'epistemological' since they depend on the choice of certain postulates and this choice is ultimately guided, at least in part, by experience. There is the further question whether his mathematics is correct and answering this is partly the aim of Kilmister's project (Kilmister op. cit.). It is also to show that Eddington's manipulations, although apparently bizarre and poorly motivated, are perfectly plausible from the perspective of his own philosophy. I'm not going to go through Kilmister's courageous reconstruction here; all I want to do is emphasise the crucial role played by considerations of particle indistinguishability in quantum mechanics.

The issue is that of constructing a bridge between quantum mechanics and electromagnetism and, as the ratio between the Compton wavelength and classical electron radius, the fine structure constant was seen by Eddington as the capstone of the bridge (just as the speed of light was for the unification of electricity and magnetism; see Eddington 1936, p. 4). Originally, measurements of the reciprocal of this constant gave a value of 136. Beginning with four algebraic elements, related to Dirac's operators and decomposed into 3 + 1, Eddington generated a complex algebraic structure, applied to the case of two particles not just one as for Dirac. This was for two reasons: first, in the treatment of the hydrogen atom the electron and proton should be placed on an equal footing; secondly, according to the principle of relativity, if the electron was the 'object' particle, a 'comparison' particle also had to be introduced, representing, in idealised form, the environment. As Kilmister points out, the whole project gains a certain plausibility if it is viewed from a structuralist perspective: if physics is primarily the investigation of structures, then the most appropriate tool for this investigation will be forms of mathematics in which structure is paramount (ibid., p. 118). Kilmister somewhat downplays Eddington's selective subjectivism here (and emphasises his form of operationalism according to which the origin of a law is revealed by the 'unravelling' of the series of operations resulting in the relevant physical quantity) but I think he would admit that it is crucial in justifying the very basis of Eddington's algebraic manipulations. This is precisely the purpose of Eddington's 1928 discussion mentioned above. Finally, as Kilmister notes, although he uses the phrase 'algebraic structure' to describe the mathematics employed, what Eddington was primarily concerned with was group theory. With regard to the last point, we have already touched on Eddington's enthusiasm for tensors and, according to Kilmister, his conviction '... that here there is everything needed to describe the 'condition of the world' simply rested on the prolific character of this generation of [group-theoretic] representations ...' (ibid., p. 72). Eddington was also  explicit in his insistence that the structure of the world is of a kind defined and investigated by group theory (see Eddington 1936, Ch. XII and 1939, Ch. IX). However, it appears that by the late 1920s/early 1930s, this group-theoretic structuralism was also motivated by the implications of quantum mechanics for particle indistinguishability. These played a crucial role in the rescue of his structuralist derivation of the fine structure constant when experimental results revised the value of the latter from 136 to 137. Eddington was well aware of the philosophical implications of the new quantum statistics and understood the non-classical indistinguishability of the particles to be the logically prior notion. However, his understanding went beyond that of other physicists in shaping his notion of 'interchange'. Following a geometric analogy with rotation, Eddington considered the conditions under which an interchange of particles made no difference from an algebraic perspective. Such conditions provide a representation of indistinguishability which he can then effectively feed into his algebraic programme and by a great detail of jiggery-pokery obtain the desired result. The details are once again given by Kilmister (ibid., Chs. 8 and 9) but there are two curious, not to say bizarre, features of Eddington's use of indistinguishability which throw further light on his form of structuralism. The first concerns a technical issue: Eddington not only had to account for the revised value of the fine structure constant, but also had to accommodate the fact that as applied to the hydrogen atom, Dirac's equation gave an extra term 1/137r, where the 1/r acts like the Coulomb potential. Eddington's response was to argue that the Coulomb force could be identified with Fermi-Dirac exclusion and hence was a consequence of particle indistinguishability for fermions. The second feature is that, with regard to the hydrogen atom, Eddington regarded the proton as indistinguishable from the electron (Kilmister's gloss on this is uncharacteristically not as helpful as one would wish). In order to get a grip on these claims, we need to start with a principle which Eddington himself identifies as the fundamental epistemological principle of this 1936 work, the 'Principle of the Blank Sheet'.  Thus we begin with intrinsically indistinguishable particles and space-time frames (ibid., p. 33 and p. 56) in order that the relevant physical differences are introduced openly rather than smuggled in via the initial assumptions23. Such differences include mass and charge, of course, neither of which are regarded by Eddington as intrinsic properties of the particles. Rest mass is nothing more than the energy of the particle in an assembly of particles in statistical equilibrium. Similarly charge has its origins in the permutation of indistinguishable particles. This is what Eddington claims to have demonstrated through the identification of the Coulomb force with the results of the Exclusion Principle. The heuristic origins of this identification are not entirely clear and I shall simply note that the basis of Eddington's demonstration is permutation invariance. As already indicated, Eddington saw this as a 'new kind of relativity transformation' (ibid., p. 283) in which the interchanges of indistinguishable particles is represented by a rotation of the system in configuration space (this foreshadows more recent configuration space approaches to particle indistinguishability)24. This allows the permutation to be represented as a continuous transformation, as probability is gradually transferred from one 'identification' to the opposite one (ibid., p. 284). The interchange energy is then the momentum conjugate to the permutation co-ordinate and it is this that Eddington claims is equal to the observable value of the Coulomb energy. Furthermore, and crucially, it is the addition of this permutation co-ordinate which requires the extension of his original 136-dimension phase space by a further dimension to provide the algebraic foundation of the  the fine structure constant (ibid., p. 286). Thus from the point of view of Eddington's analysis, protons and electrons begin life, as it were, as completely indistinguishable units, to which various attributes are added as the analysis proceeds: Thus the mass cannot be used to distinguish a proton from an electron because it is represented within quantum mechanics by an appropriate operator and this cannot be applied to a given particle until we have first determined how that particle should be identified at different times. To use mass as a criterion to distinguish particles presupposes that they have already been distinguished (ibid.). More generally, the identification of particles is always relative since a change in attributes such as position or colour (Eddington gives several illustrative examples involving different coloured balls throughout the discussion) can always be effected by a shift in reference frame. Even the permutation co-ordinate is not absolute in terms of observational meaning and all we can do is adopt some conventional criterion for determining the constancy of the co-ordinate and measure changes in it relative to this standard (ibid., p. 291).

The introduction of observation here and in the quote above suggests a role for Eddington's selective subjectivism in his philosophical attitude towards indistinguishability. As he puts it, there is nothing 'mystical' about the effects of indistinguishability (ibid., p. 285), in the sense that they arise from some ontological peculiarity of the particles, such as 'non-individuality', say. A being 'more gifted than ourselves' could identify individual particles and apply the ordinary equations relevant for distinguishable particles, but, crucially, the results obtained would be of no interest for, or use to, us because we have no access to the relevant observational data (ibid.). We are unable to identify particles at different spatio-temporal locations and thus for Eddington quantum indistinguishability is ultimately observational in origin. This is a familiar position among physicists, but Eddington understands it within the framework of his epistemology according to which this observational limitation is subjective. It may be asked, why should the statistical behaviour of particles be affected by our inability to distinguish them? As he says, this would be absurd or incredible, '... unless we bear in mind the subjectivity of the world described by physics and of all that it is said to contain.' (1939, p. 37). The question would be a legitimate one to ask with regard to wholly objective particles displaying wholly objective behaviour but '... our generalisations about their behaviour ... describe properties imposed by our procedure of observation ...' (ibid.) Indistinguishability for Eddington is thus a form of epistemological principle; one can imagine it being tested but the test would be perfunctory 'like the experimental verification of propositions of Euclid' (ibid. - significant analogy?). There is a further distinction to be made. If the reduction of the Coulomb force to Fermi-Dirac statistics is extended to cover all interaction forces, as Eddington believed it could be (1939, p. 128), then interaction has a subjective origin due to indistinguishability. There is a further subjectivity attached to the 'ultimate particles' which is strictly independent of the former27. That these ultimate particles are 'identical structural units' arises as a 'specialisation' of the concept of analysis, viewed as an ingrained form or 'frame' of thought (ibid., p. 122). Apparently intrinsic attributes can be resolved into relational ones, so that 'All the variety in the world, all that is observable, comes from the variety of relations between entities.' (ibid.) but the entities themselves are precisely alike. And this is not because the objective universe is built of such units but rather that our knowledge is 'impressed' by a fundamental form of thought (ibid., p. 123 and p. 125). Thus, the laws of atomicity, which Eddington had earlier speculated might be objective, are brought within the subjectivist fold, thanks, at least in part, to indistinguishability and permutation invariance. From this perspective, such a unit cannot be taken as separate or disassociated from the system of analysis of which it is a part. Taken as it stands, this is quite general but it becomes more precise when it isexpressed mathematically, so that the relevant frame of thought is transformed into a mathematical frame28. And the appropriate mathematics, of course, is group theory. Here it is interesting that Eddington doesn't tie the introduction of group theory explicitly to the presence of his identical structural units. It comes in as a way of expressing the relationships between relations and the important point is that whatever the nature of the entities, the use of group theory allows us to abstract away the 'pattern' or structure of relations between the entities. Knowledge of structure, therefore, is communicable whereas other forms of knowledge (my knowledge of what something tastes like, for example) are not. Hence, it is through structure that we can have inter-subjective knowledge and Eddington proposes group theory as the answer of modern physics to the old philosophical question, 'what sort of thing is it that I know?'

The philosopher's perplexity arises from the assumption that knowledge of the external world must be based on sensations, which are mental and hence the problem arises as to how the mental can give us knowledge of the non-mental. The assumption is incorrect, however, as a single sensation tells us nothing about the physical world. The logical starting point of physical knowledge is '... knowledge of the group-structure of a set of
sensations in a consciousness.'. These fragments of structure are then collected together, represented through the fundamental forms of thought and completed by inference to unobservable structures to give the 'structure [formerly] known as the physical universe. Note that group theory enters both at the bottom level, as it were, in representing the structure of sensations, and at the top-most theoretical level, in representing the structure of the ultimate theoretical elements. That the sensations themselves are non-structural, and our knowledge of them is by direct awareness, might seem to resurrect a form a dualism but Eddington insists that this is a 'logical confusion' according to which we cannot give meaning to the notion of dualism without making certain presuppositions which undermine that very dualism. The idea is this: if we conceptually distinguish that part of the world of which we have direct awareness, namely that part which has to do with our sensations, from that part of which we have structural knowledge, then structurally the latter is no different from the former. Yet in order to give meaning to the dualism we would have to suppose that we have some non-structural knowledge of that part by which we could assess its difference from the sensational part. But that is to suppose that we could have direct awareness of the structural part which would show that it is non-sensational. But that is impossible, for if we had direct awareness of it, then it would be sensational; hence the very possibility of dualism is undermined. This issue, of the relationship between the structural and non-structural components of our knowledge, is obviously a fundamental one. Thus the ordinary 'frames of thought' which, as indicated above, are transformed by mathematics feature nonstructural, 'general' concepts, from which structural concepts are obtained by eliminating everything which is not essential to the role the concept plays in a group-structure. If the structural concept becomes a mere element, whose properties are those of a mathematical symbol, then a general concept '... is our conception of what the symbol represents in our ordinary non-mathematical form of thought.' (ibid., p. 144). However, with the exception of those general concepts concerning things of which we are directly aware, such concepts - albeit ingrained as ordinary forms of thought- may be no more than forms of 'selfdeception' which persuade us '... we have an apprehension of something which we cannot apprehend.' (ibid., p. 144). I mention this because it can be made to relate to the motivations for ontic structural realism mentioned above: we have a general concept of an object as an individual, which is so ingrained as a form of thought that we export it from the classical to quantum realm and are persuaded that we have an apprehension of that which we cannot apprehend. All that we can apprehend, following Eddington, is the relevant group-theoretic structure, of course, as it is represented in terms of symmetric and anti-symmetric state functions. Bizarrely perhaps, Eddington applies his distinction between 'general' and 'structural' concepts to the issue of what we mean by the term 'exist'. He rejects '... any metaphysical concept of "real existence"' (ibid., p. 162) and introduces in its place a 'structural concept' of existence according to which it only makes sense to ask if a given entity exists in the given structure or not. Since there are only two possibilities, existence and non-existence (of course), 'The structural concept of existence is represented by an idempotent symbol.' (1939, p. 162; Eddington's emphasis). Embodying the simplest possible structure, an entity represented by such a symbol must have no parts and the entity in physics corresponding to this element of analysis is, of course, the elementary particle. Now an interesting question arises: from the perspective of individuality and indistinguishability how are these particles to be regarded? As Eddington notes, this mathematical representation in terms of idempotent quantities encourages a treatment of the particles as 'pseudo-individuals': 'It will not be surprising if in our gropings into the structure of things a legend of individuality has attached itself to the carrier of an idempotent variate. In statistically grounded theory it is the closest counterpart of the obsolete classical particle. We now know that matter cannot be analysed into elements having the individual distinctness that classical particles were supposed to have; but in the carriers of idempotent variates we reach elements which, though not less statistical than other carriers, do not betray their statistical character in the ordinary calculations of dynamics.' (ibid.)

According to Eddington, the association of this notion with idempotency has 'profound implications' for the logical structure of physical science, involving, as it does, the transfer of a metaphysics appropriate to the macroscopic realm, to the microscopic. What we observe are macroscopic (Eddington uses the term 'molar') phenomena and underlying this realm '... we are accustomed to picture a microscopic world populated by individuals ... and it is further supposed that protons and electrons are such individuals.' (ibid.). Such a picture encourages the view that the process of analysis has a terminus (in the individuals) but if the only kind of individuality is actually this pseudo- form conferred by idempotency, then there is no reason to suppose that the process will ever have to stop for metaphysical reasons. We may decide to stop once we have achieved our analytical aimsbut that is another matter entirely. Furthermore, once we realise that the analysis of macroscopic objects into microscopic carriers has a goal that is mathematically, rather than objectively, defined in this way, the numerological efforts of Eddington's programme may not seem so implausible (or so he hopes; ibid., p. 132). From this perspective, then, 'the elementary particle is a product of analysis of ... group structure.' (ibid., p. 164). This programme was subjected to vigourous criticism by Braithwaite in his 1941 review of Philosophy of Physical Science (Braithwaite 1941). What's interesting here is to note the similarity between Braithwaite's objections to Eddington's structuralism and Psillos' more recent concerns, as cited above, particularly with regard to the structureobject distinction. Thus Braithwaite rejects as invalid what he sees as Eddington's dichotomy between structure and the incommunicable 'Erkenntnis' of the content of experience (1940, p. 462). Focusing on the claimed group-theoretic nature of the structure, Braithwaite insists that such groups are defined only with respect to given 'modes of combination', so that the group structrure is in fact less abstract than Eddington supposes. In other words, the group-structure is only given once the relevant transformations have been specified (i.e. whether we're talking about rotations or permutations, for example), but to do this is to supply content and so we no longer have pure structure. This appears to be analogous to Psillos' argument which, the latter claims, leads to the collapse of epistemic structural realism (op. cit.). Even more interesting, in this latter context, is that in a footnote to the above passage, Braithwaite refers to Newman's famous result that for any collection of objects of a given cardinality, the claim that there exists a particular structure, expressed in terms of the relevant relations, defined over this set, can be trivially satisfied (see, for example, the discussions in Demopolous and Friedman (1995) and Psillos. Eddinsgton's reply is revealing. With regard to Braithwaite's claim that a group is only defined with respect to a particular 'mode of combination', Eddington points out that what Braithwaite appears to have in mind here are group representations, whereas in order to represent the 'pattern of interweaving', he has been talking about abstract groups. And it is precisely the abstract aspect that renders the concept of a group so useful in the philosophy of physics. From this perspective we lose the distinction between the nature of the element and the nature of the combining relaiton which makes it an element of the group: 'The element is what it is because of its relation to the group structure.'  There are a couple of things to note about this. First of all, it is important to bear in mind that the elements of the algebra should not be identified with the elements of the group, of course (more here?). But then the worry is that by shifting from the group to the associated algebra, Eddington might have evaded the issue somewhat. In stating that an element 'is what it is' by virtue of its relation to the group structure, Eddington is not quite offering a version of the ontic form of structural realism which sees particles as being what they are by virtue of their relation to the overall structure, since the 'element' for Eddington, here, is an operation, like rotation (see below). Secondly, however, there is a feature of this form of structural realism present in Eddington's remarks that there is no non-grouptheoretic content to 'lay hold of'. In this case, Eddington agrees that there is no structurecontent dichotomy, not because structure depends on content but rather because it is content - as represented in this case by Braithwaite's 'combining relations' - that depends on the structure. This dialectic is nicely mirrored in the present day differences between Psillos, in responding to Worrall by insisting that the latter's distinction between structure and content is untenable in the scientific context, and Ladyman in arguing that the structure-content distinction collapses because all the physical 'content' can be cashed out in structural terms. As an example of what he means by this 'pattern of interrelatedness of relations', Eddington presents the algebra of operators representing rotations acting on rotations, for which the 'pattern of interrelatedness' is manifested in the associated multiplication table. The information encoded in such a table is by no means trivial and hence Eddington feels able to conclude that there is no foundation to Braithwaite's contention that the Newman objection applies to the structure as described by this multiplication table; indeed, he accuses Braithwaite of not having grasped 'the main idea' of structuralism. A further indication of what Eddington had in mind is given in his example of spin, where the information encoded, as above, in the relevant structure gives all the information we can get (ibid., p. 279). What's particularly interesting here is the way in which Eddington deploys a certain structuralist strategy which amounts to assuming certain nonstructural elements in order to be able to articulate the structure in the first place, only to discard these elements once the structure has been constructed. Thus he notes that the components of spin can be specified in a set of mutually orthogonal planes and also that this represents non-trivial knowledge. Now of course, the Newman- Braithwaite objection would be that such knowledge is non-structural because we are acquainted with such orthogonal planes in the 'external' world. The way round it is to consider the set of operations represented by rotations through 90o in each of the planes. This yields a group-multiplication table which Eddington takes to define the relevant structure and now '[w]e need ... trouble no further about the planes;'  We initially associated the components of spin with the planes but we could equally as well have associated them with unit rotations in the plane, so that initial association was just a kind of heuristic move which takes us to the group-multiplication table which in turn represents what is important, namely the structure. The information encoded in the latter is definitely non-trivial, since it conflicts with other statements, some plausible, but the apparent non-structural knowledge acquired by our acquaintance with the planes is in fact 'non-existent'. Thus the appearance of a non-structural component is illusory, deriving from the heuristic role played by certain objects. There is much more to say of course, but I want to conclude this historical section, finally, by reflecting on the forms of structuralism proposed by Cassirer and Eddington39. In both cases, they appear to offer a strong dose of Kantian epistemology with their structuralism. Of course, we have to be a little careful with the labels here. Eddington himself wrote, 'We do not accept the Kantian label; but, as a matter of acknowledgment, it is right to say that Kant anticipated to a remarkable extent the ideas to which we are now being impelled by the modern development of physics.' . And as Ryckman notes, by historicizing the inquiry into the conditions of the possibility of knowledge, Cassirer moves away from Kant in conceiving of synthesis as a methodological requirement. Nevertheless, his account of objectivity and the constitution of objects possesses clear idealist characteristics. As indicated in French and Ladyman (forthcoming), the modern structural realist might want to avail herself of the structural analysis of objects, whilst articulating an alternative account of physical law, for example. Of course, as far as Cassirer is concerned, this would be to impale oneself on one horn of the 'old dilemma' of phenomenalism vs. naive realism which his structural understanding of objectivity is intended to avoid. Turning to Eddington, in one sense, his view is clearly epistemological: the world is not entirely structural. However, all our physical knowledge is knowledge only of structure and so we cannot have physical knowledge of these non-structural aspects. It is this perhaps that non-structuralists find so repugnant; in Eddington we see the Kantian noumena acquire a mystical resonance. But as we have seen, it is not quite that simple. The particles of physics are not to be found in the non-structural world; as far as Eddington is concerned, such a suggestion would be completely absurd. Thus the standard realist's cry that this epistemic structuralism leaves something beyond the reach of physics, belabouring as she is under the misapprehension that this something should be within the reach of physics, would be regarded with something approaching derision. There was never any possibility that this aspect could be grasped by physics because it could only be so if it were structural in the first place. Thus we cannot set Eddington beside Worrall in holding that when it comes to the particle of physics say, what we know are the structural relations they enter into but their 'natures' lie hidden. From the synthetic point of view, of course, we start with individual particles and combine them to form perceptible objects  but from the analytic perspective of Eddington's programme, it is the relation (between phenomena) which comes first and the elementary particle emerges as the product of analysis of the group structure. This aspect, of course, is closer to ontic structuralism but for Eddington to be placed next to Ladyman we would have to dispense with the selective subjectivism. Kilmister seems to think we can understand Eddington's structuralism without this but I'm not convinced. If we could cleanly excise it, however, we would lose the distinction between the structural and non-structural, or physical and 'external' worlds, leaving a form of ontic structuralism. 

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