The quantum mechanical description of an assembly of particles of the same type ( a fermion gas, for example) includes nothing to individuate or differentiate these particles from each other. One option was to postulate  individuating factors (thisness, haecceity) which do not enter the quantum state -- an empirically superfluous hidden variable. Another option was to say that this very question reveals an inadequacy: the proper formulation will be in terms of fields, denying the existence of individual or individuatable particles. A still further option was deny the identity of indiscernibles, and say that these particles are distinct without being in any way different. This 'problem of identical particles' and the options it allows for interpretation can be used as paradigm to motivate Ladyman's exploration, following Weyl, of invariance as definitive clue to structure. In the above 'identical particle' debate, the salient scientific fact is that the quantum mechanical state is permutation invariant. This may be glossed as : only those features which are permutation invariant belong to structure -- if there are other features they belong to 'content'. Any such other features there may be characterize what bears that structure, in a way which goes beyond what science describes. Reading the scientists who write about this, we certainly see ample precedent for this gloss, though it remains generally quite unclear which options are being taken up. Thus Weyl: Objectivity means invariance with respect to the group of automorphisms James Ladyman (1998a, 1998b) argues that the guiding idea for structural realism was equivocal between two positions. These positions are sharpened forms of what I called reification and radical structuralism. Ladyman sharpens these into: an epistemological position: 'ordinary' scientific realism plus the view that all we can come to know (have reason to believe, can hypothesize?) about is the structure of nature alone. An ontological position: that the structure of nature described by scientific theories is really -- properly understood -- all there is to nature. The former requires that sense be made of two dichotomies: epistemic dichotomy: what we have epistemic access to (can come to know, can theorize about in a scientifically significant way) and what we have no such access to. The structure of nature and those features which are not structure (matter or content) We certainly have examples of views for which we can draw such distinctions.  Which topic and it consequences Weyl places in historical perspective: the founders of modern science [...] discarded the sense qualities, on account of their subjectivity, as building material of the objective world which our perceptions reflect.

But they clung to the objectivity of space, time, matter, and hence of motion and the corresponding geometric and kinematic concepts [....] But soon the objectivity of space and time also became suspect.  But this is treacherous ground, for "objectivity" has many antonyms. "Objective" can contrast with "subjective", "relative", perspectival", "unreal" and "scientifically insignificant". Thus, if electrons have 'haeccity' to individuate them, that is not invariant, hence (pace Weyl) not objective -- but it is neither subjective, nor relative, nor unreal in that case. On the other hand, length (being not invariant under Lorentz transformations) is relative, but not subjective, unreal, or even scientifically insignificant. And so forth. Let me suggest a specious argument, of which no writer on this subject is guilty, but which might lurk behind an unsuspecting reader's comprehension: A symmetry of an object is a transformation that leaves the object the same -- identical to itself. The paradigm examples are reflection through an axis of bilateral ('mirror image') symmetry, or a rotation through 360 degrees. But in physics we keep as many symmetries as are required by the dynamics. As this group gets larger, there is much in the original representation that is no longer invariant; then it appears as if the symmetries do not preserve all the structure there was in the object. But a symmetry of an object is a transformation that leaves the object the same -- identical to itself. So this appearance must be illusory or deceptive: there is nothing to the object except what is invariant. The representation 'clothed' the reality with appearances. Stated so bluntly, this sophistry will not take in anyone. The correct response is of course to note that a symmetry is a transformation that leaves the object the same in all relevant respects. What are the relevant respects -- what are the inessential aspects, the irrelevant parameters that symmetries can vary -- is equivalent to the question of which transformations are the symmetries. But relevance is contextual. A parameter may be relevant in the solution of one problem and not in another. Two isomorphic groups can differ from each other; they just do not differ as groups, there is no difference if we take only the group operations into consideration. Symmetry, isomorphism, relevant sameness are all context-dependent notions 

Is radical structuralism coherent? 

There is one striking point about the 'ontological position', regardless of whether or not it is cashed out with the concept of invariance.  Instead it must imply: what has looked like the structure of something with unknown qualitative features is actually all there is to nature. But with this, the contrast between structure and what is not structure has disappeared. Thus, once the position is adopted, any difference between it and 'ordinary' scientific realism also disappears. It should, once adopted, not be called structuralism at all! If there is no non-structure, there is no structure either. But for those who do not adopt the view, it remains startling: from an external or prior point of view, it seems to tell us that nature needs to be entirely re-conceived, with the appearances classified as pure illusion.  This point is not just striking, it is paradoxical. Indeed, as far as my exposition here goes, it comes to us in barely coherent guise. Let me try to suggest a sense for this 'ontological position' (radical structuralism). First, there is the ordinary or old-fashioned way of thinking which has a domain of entities and a family of properties and relations. The properties are instantiated by some entities, the relations by sequences of entities in that domain. We may suppose that the properties and relations are divided into structural and non-structural ones. Some of the properties may of course be described in a rather indirect way -- for example, one structural property might be invariance under certain sorts of operations, such as rigid motions. Second on the road to abstraction is the representation of all those properties and relations by means of a single structure, a logical space (state space, phase space, ...). When that is done, all the properties and relations instantiated by a given entity or sequence of entities can be summed up by assigning that entity or sequence a location (point or cell) in that space. But this is still not very abstract, for we can then suppose that there are further properties, not present in this representation, which characterize those entities. We might suspect or even insist that there must be such 'hidden variables' if, for example, we notice that more than one entity can have the same location in the logical space. The identity of indiscernibles would imply that there must then be distinguishing unrepresented properties. So, third step, the radical move: we eliminate the domain of entities in our representation of nature. Instead of saying that entities 17 and 19 both have location q, we say that q is doubly instantiated. A location in a logical space stands for a complex of properties of all sorts which can together be the complete specification of an individual, in the old way of thinking. Retain the idea that this location is in effect a  composite property, and allow for instantiation as a primitive notion, a property of properties so to speak. An old-fashioned thinker can accept this as a façon de parler: the statement that COWHOOD is multiply instantiated can just be read as meaning that there are more than one cow, for example. But now comes the 'ontological position', if I understand it correctly: only the multiple instantiation talk (assignment of occupation numbers to the locations or cells of the logical space) is literally accurate. The "there are two ..." way of speaking is not to be taken all that seriously, it is the façon de parler to adopt when speaking with the old-fashioned. This assertion eliminates the question concerning what might differentiate or individuate distinct occupants of a single cell or location. There are no occupants; there is only multiple occupancy. It is rather easy to see how well this goes with the 'identical particle' problem and one of its solutions -- in fact, Fock space is probably the simplest and most direct scientific illustration of this line of thinking. But there are more venerable precedents in philosophy, under the heading of anti-essentialism.  Now, however, we may have lost touch with concrete reality altogether! There are many familiar examples in which we attribute properties to properties. The statement "Orthogonality is symmetric" and "Orthogonality is invariant under Euclidean transformations" are good examples. Such statements do not imply the existence of anything but abstract entities: properties or relations like orthogonality and properties of properties like symmetry or invariance. So if God had -- so to speak -- decided not to create nature at all, nothing at all that belongs to the proper domain of physics, those statements would still have been true. The statement "X is multiply instantiated", where X is some property or relation like orthogonality, must be different from this. If God had decided not to create anything concrete, then that statement would have been false. Therefore, taking the contrapositive, if such a statement is true, then there exist entities other than properties and relations. Mutatis mutandis for logical space, configuration space, Fock space .... If the occupation number of any cell or location is more than zero, then there exist entities other than this space. To put it differently, retracing our steps of abstraction: if structure is not just there as mathematical or abstract entity, then it is not true that structure is all there is. I do not see any way out of this. The radical form of structuralism seems to me to lead right back to reification: the whatever it is that bears this structure may be denied other properties perhaps, but not existence. This does not mean, of course, that there have to be distinguishable particles. What it does mean is  that we must take as at best metaphorical any attempt to equate particle talk, of any sort, with descriptions of structure. Following Weyl and Wigner, we can say that the classification of elementary particles is via e.g. the irreducible representations of a given symmetry group, or with Mackey via systems of imprimitivity. But identification via is not identification with! This discussion is not closed, clearly.  But we' ll now return from the most radical form to structuralism about science in general. 

How is structural realism supported? 

As further prolegomenon, we need to look briefly at how Worrall supports his structural realism. This support is, frankly, schizophrenic to empiricist eyes. Worrall needs to give support to three claims. The first is that contrary to the more radical 'revolutions and incommensurability' picture of science, scientific knowledge is cumulative in some important respect. The second is that the knowledge accumulated is of features of the world that transcend the empirical world disclosed in observation and experimentation -- features of the reality behind the phenomena. The third and last is that these features constitute structure rather than quality -- that in some significant sense, what we come to know is only the structure of that reality and not what it is like 'in itself', 'intrinsically'. The air of schizophrenia comes from the fact that all the support given for the first claim is explicitly and admittedly concerned only with an accumulation of empirical knowledge. The process at the empirical level (properly construed) was essentially cumulative .... Or take the Newton-Einstein case ... At the empirical level it does seem intuitively reasonable to say that Einstein's theory is a sort of "extension with modifications" of Newton's ... The picture of the development of science certainly seems, then, to be one of essential cumulativity at the empirical level, accompanied by sharp changes of an entirely non-cumulative kind at the top theoretical levels.  Very true! Now, what are we to make of that? If we take this at face value, we will not arrive at anything like a scientific realist position.  Worrall notes that himself, saying that these findings tend to lead us "into some sort of either pragmatic or 'constructive' anti-realism". But in opposition to this he now gives the great argument that will support his second claim: that despite appearances, the real accumulation of scientific knowledge is of some reality behind the phenomena: Such a position [anti-realism, with attention to the accumulation of empirical knowledge] restores a pleasing, cumulative (or quasi-cumulative) development to science ...; but it does so at the expense of sacrificing the 'no miracles' argument entirely. What is this 'no miracles' argument, and how is it supporting the second claim? It is the argument that the success of science at the empirical level would be a miracle. It would be too much of a coincidence or brute fact to be believed, unless we do think that science is pretty well true over all. That is, empirical success would be a miracle unless science was successfully latching onto the truth about what things and processes in nature are like below or behind the part that we can observe, the part that is not displayed in the data but which 'produces' those data. And what is the support for this contention? Unlike in the first part, no argument here draws on the history of science, details of actual scientific theories, relations between theories, structure of theories, nature of experiment, experimental design, .... It is as if science itself has been forgotten .... Only the scientists have not been forgotten: the support cited lies in the pronouncements of famous scientists in their popular writings, prefaces, and postscripts -- as well as in the expresssions of untutored intuitions putatively found in the common-sensical man in the street. But didn't appeal to authority disappear from respectable philosophical and scientific reasoning sometime during the Renaissance? This is what I had in mind when I referred to an air of schizophrenia: the tenor and content of argument changes startlingly as we go from the claim of accumulation to the one about observation transcendent reality.  Finally, what of the third claim, that the truth latched onto concerns solely structural features? We have already seen the two-fold motivation for this amendment which operates equally regardless of the level of description, empirical or theoretical. Today its main impetus comes from the insight that motorized modern mathematics, that the study of invariants, and the reduction modulo contextually irrelevant differences is the short, straight, royal road into the heart of the mathematical object. But in the case we are presently examining, that will fall very short of what Worrall needs. For his claim requires a context-independent, 'objective' division in nature between mere quality and the relational structure in which the qualities appear at the vertices, so to speak. This can be made good only by falling back on a (here unacknowledged) metaphysics in which such an intrinsic-extrinsic distinction makes sense. I rather doubt that today's structural realists in philosophy of science are very anxious to creep into that thicket ... but without that, and without a way of finessing the point, the burden of unacknowledged metaphysics rests heavily on their position. 

An Empiricist Structuralism 

I am going to argue that indeed there is a steady accumulation of knowledge in the sciences, and that this knowledge deserves to be called precisely knowledge of structure. But on the view I will present here, there are (both in individual experience and in science) only two sorts of things we deal with directly. These are the concrete, observable things, events, and processes in nature on the one hand and on the other hand, the abstract structures studied in mathematics. We characterize the structure of the former in terms of the latter. To show how this view differs from structural realism and to make it plausible, let us begin with a look at the problem of royal succession in science. When an older accepted theory falls to its unresolved anomalies, it is still respected as having been a highly successful theory in its time, in the domain where it was applied. In experiment, observation, and application it scored its empirical successes, all of them reasons for its earlier acceptance. The new theory must not only be able to duplicate those successes. We must in addition have an explanation of how and why  the older theory was in fact so successful. This is the relationship which took the place of the presumed reduction:  Requirement upon succession. The new theory is so related to the old that we can explain the empirical success of the old theory if we accept the new. This is a modest requirement, slanted toward empiricist thinking; and below I will defend its modesty against stronger alternatives. I will also give examples to illustrate how the superseding new theory explains those past successes. It is definitely not by explaining how essentially right the old theory was about the underlying structure of nature! It is instead, quite simply, by implying approximately the same predictions for the circumstances in which the older theories were confirmed and found adequately applicable. Thus the past empirical success can now also be counted as an empirical success for the new theory. It is not enough, but it is the initial set of credentials sine qua non for a victory. If there is a scientific method, this is part of it -- we see here a constraint on new candidates for theory acceptance. And we see an important and theoretically manageable relation that cuts across scientific revolutions. Indeed, we can now identify a body of knowledge which does evolve by accumulation rather than replacement: the empirical knowledge that was tested and is retained, still accepted afterward as triumphs of past science. So we can reply to Worrall: YES, there is an accumulation of knowledge through science, but it is knowledge about the observable phenomena. You did make a good guess: there is an accumulation going on throughout all those deep theoretical changes. Moreover this requirement upon theory succession should satisfy the 'No Miracle' intuition! The success of science is not a miracle, because in any theoretical change both the past empirical success retained and new empirical successes were needed as credentials for acceptance. It used to be argued that the older successful theory must be shown to be reducible to the new. A good deal of theoretical work was done to perform such reductions, for example of phenomenological thermodynamics to statistical mechanics. But there is no precise, strict sense in which theories are in general reducible to parts of their successors.

The relationship expressed in the above Requirement upon succession is not a requirement of reducibility. The relation is much looser, but nevertheless very demanding. The difference is that it grants conceptual autonomy to the new theory, which is allowed to re-describe nature entirely in its own terms. Although looser, this relationship admits of rigorous proof, and every advocate of a new theory is at pains to demonstrate it. The loveliest, neatest examples belong to the more mathematical sciences. So, by letting the speed of light in the Special Theory of Relativity go to infinity, you can deduce the relevant Newtonian equations. Literally, that means that Newton's equations are false throughout -- for no finite speed is less than a finite fraction of the speed of light. But it explains, on the basis of the Special Theory of Relativity, why those Newtonian equations performed so well, to such a good approximation, in domains of slow transport over comparatively short time intervals. Similarly if you let the Planck constant go toward zero in quantum mechanical deductions. There is no sense in which models of Newtonian physics can in general be embedded in the models of those newer theories. The transformation groups are too different. But the new theory can explain the empirical rewards earned by the now rejected theory. There is however also an intermediate candidate for the requirement upon candidates for the successor theory in a scientific revolution. That is Reichenbach's requirement of "continuous augmentation": previously employed principles are assumed to hold approximately, in the limit. This requirement is precisely modeled on the first especially nice example (Relativity Theory) I just gave. One finds it expressed elsewhere as part of a certain remaining triumphalism in how we might wish to see the past:  To use a comparison, we could say that creating a new theory is not like destroying an old barn and erecting a skyscraper in its place. It is rather like climbing a mountain, gaining new and wider views, discovering unexpected connections between our starting point and its rich environment. But the point from which we started out still exists, and can be seen, although it appears smaller and forms a tiny part of our broad view gainded by the mastery of the obstacles on our adventurous way up. (Einstein and Infeld, pp. 158-159) Not at all what Galileo, Bacon, or Descartes wished to say about their forerunners! But more recently Reichenbach's requirement has been embraced by Michael Friedman as inspiration for a renewed Kantian perspective on science. The difference between  Reichenbach's  requirement and the above Requirement upon Succession is that the former extends to the entire theory, or parts of the theory still looked upon as having enjoyed empirical success -- as opposed to the empirical successes themselves. It does happen of course that a new theory will resurrect old theoretical principles in some fashion, and not just explain why they managed to reap empirical support. But I see no rationale for this as a stronger requirement. Let's take an illustrative example: isn't the stronger requirement violated in the transition from Aristotelian to Cartesian or Newtonian physics? Coincidentally this is discussed by Friedman (forthcoming), and he denies the violation, in two ways. First, as a general point, he emphasizes the continuity of theory change, so we have to look at small steps. Relevantly to this he mentions Galileo's postulate of circular inertia as intermediate between Aristotle's laws of motion and the modern. Second, more specifically, Friedman describes how we can see Aristotelian physics as approximately correct in a limited domain described in one particular way: near the earth, in a reference frame with earth at the center. But it seems to me that there is no such continuous augmentation at the theoretical level -- or rather that, depending on how we construe it, the principle becomes either trivial or clearly false. Appearances to the contrary come from extremely selective attention to certain features of the old theory, whose relevance is only identifiable retrospectively, so as to function in retrospective rationalization. There is a solid core of continuity (perhaps somewhat overstated by Reichenbach and Friedman), namely that the new theory is (ought to be) demonstratively capable of duplicating the empirical successes which we can still see as support  for the old theory -- at that later time; we need not accept what the old theory advocates saw as support -- and in this way, and this way only, explain why the old theory was as empirically successful as in fact it was. Clearly the principle of continuous augmentation can be trivially satisfied if we allow any kind of gerrymandered attention to any implications we like, of the old theory, applied to selected features of a roughly characterized suitably small domain of application. If that is allowed there was a process of continuous augmentation leading from medieval astrology to modern medicine, and from Paracelsus to Heisenberg. At least -- this is important -- such a description can then always be cooked up retrospectively. The more important point, as I see it, is this: in the examples of continuity we can typically distinguish an empirical element which serves as the operative core, from a theoretical aspect which is in fact not retained at all, not even in highly restricted form or in the limit. (There are of course atypical cases, and these are precisely what inspired Reichenbach's requirement upon royal succession in the sciences.)

I'll make this point specific for Friedman's own example. In Aristotelian physics each sublunary body has its natural place, toward which it will naturally move if unconstrained. Any other motion is not 'natural' but 'violent' motion, subject to the principle that nothing moves unless it be moved by something. This natural motion is in a straight line toward the natural place. For the element earth the natural place is the center of the Universe, which is accordingly the center of the Earth. Stones, which are predominantly earth, if released in air or water (whose natural places are above that of earth) above but not too far from the surface of the Earth, will accordingly fall straigt down. That they will do so (to a great degree of approximation, which ignores the gravitational pull of anything other than the Earth) is also implied by Newton's theory. So we see an agreement in a special case, in the limit (of zero contribution to the resultant force on the body by anything other than the gravitational pull of the Earth). But what is this agreement? It is certainly an agreement on the important empirical prediction of a quite commonly observed phenomenon. This phenomenon did fit very well into Aristotle's theory and provided empirical support for that theory. But if we look at Aristotle's theory, even the very small part that divides motion into natural and violent components, even for a stone near the surface of the Earth, we can immediately see a theoretical conflict with Newtonian physics. The disparity is of course evident only if we pay attention as well to slight modifications in the circumstances. Imagine the stone to be released at the edge of a bowl. It will slide towards the bottom in its natural motion toward the center of the Earth --  deflected 'violently' by the resistance of the bowl to penetration. But then it will refuse to continue to display its natural tendency, by leaving the bottom and continuing its slide in an upward direction. This violates the theory, for there is nothing in the situation to which we can attribute violent action upon the stone to account for this. There is no upward push by the bowl. If there were, it would tend to retard the downward slide equally, and -- judging by the diminishing speed upward -- more so nearer the bottom than near the edge. In other words, the natural motion downward would diminish as the stone neared the bottom of the bowl (the valley floor, in a larger case); the larger the bowl, the more noticeably. It may be objected that I am unfairly switching attention to another situation, in which inertia plays a role as well as gravity. A small part of the Aristotelian theory coincides with a relevant small part of Newton's theory, a larger part fails to do so. But what is this coincidence besides agreement on an observable phenomenon? My example of slightly altered circumstances brings out the fact that the theoretical element (that is, the concept of natural motion and its distinction from violent motion) is entirely foreign to Newton's theory. It is not as if Newton's theory says that in certain restricted cases stones will move rectilinearly toward their natural place. Newton's theory says only that in this restricted case stones behave as if they were moving rectilinearly toward what Aristotle identifies as their natural place. Even that Newton's theory can say only in an extension of its own proper language, in which both theories can be expressed. Both my example and the objection to it highlight the extreme selectivity that marks such theory comparison. Now I add to this: what guides the selectivity is an interest not in the theoretical relations at all, but rather in the empirical successes which credential the theories. The new theory needs to establish its credentials as serious contender, and this requires that it can also claim the empirical success which supported the old theory, and which we would not gladly do without. It may achieve this by tailoring itself in the limit and/or to a certain degree of approximation, to certain implications for known sorts of phenomena. But it can also achieve it in a quite different, theoretically novel way. Either is perfectly fine, but only the former illustrates Reichenbach's principle of continuous augmentation. Finally, perhaps the most important consideration is the following. While the weaker principle/requirement upon succession has a clear rationale, the stronger one can only be proposed on vague general grounds. One can at best cite theoretical conservativeness or calculational advantage, neither of  which can be more than an inclining consideration. That the new theory must establish its empirical credentials with respect to what we still consider the empirical phenomena successfully handled in the past, that is obvious. But beyond this, what authority does the old theory have, that would entitle it to be respected and even duplicated in any part of its world picture? Reichenbach, as far as I have noticed, speaks only of conservativeness. Friedman, I think, is swayed by the conviction that theoretical transitions, including the most revolutionary, must be guided by a priori principles at a meta-theoretic level.

To an empiricist, this must be ruefully classed as of a piece with other general convictions deriving from the metaphysical instinct -- though an instinct that has been regrettably common also in empiricist ranks. Examples would include the conviction that there must be an inductive logic, or logic of discovery, or other such principles of right reason -- indeed, that rationality must consist in rule-following. This instinct has here retreated to a meta-theoretic level, but is otherwise unchanged.

Structure: an empiricist view Worrall made a second guess: that what is retained through theory change is structure. Such assertions as that there are mechanical atoms or a plenum, caloric or phlogiston, ether or Rutherford atoms -- these are not retained. But something of the structure of the old scientific image goes over to the new, and the key word is structure. As almost every commentator has somewhat sadly remarked, this key word has its own problems. What exactly is the difference between matter and form, content and structure? Aren't such distinctions painfully context-dependent? Is there really an objective difference in nature, as opposed to merely in our representations of nature? Is it not a little embarrassing to start with the thesis that what is preserved through scientific revolutions is the structure attributed to nature, and then to have to identify structure by noticing what has been preserved? Must this philosophy of science ultimately rest on a metaphysics to distinguish intrinsic and extrinsic properties (essences and accidents, substantial form and prime matter, relation and pure quality)? But we can look at the matter quite differently, and prevent ourselves from sinking into this metaphysical morass that swallows all seekers for the true foundations of being. The empirical successes of the older theories were partial successes of a very distinct sort: their representations of nature, the models they  made available for representation of the observed phenomena, were partially accurate. These successes consisted in their success of fitting the data, the deliverances of experimental and observational experience. There was something they got right: the structure, at some level of approximation, of those phenomena. Here the word "structure" is used to point specifically to a certain character, defined by certain measurable parameters both old and new theory use to describe those empirical successes. Just look at those empirical phenomena! They have, in an intuitive sense, both structure and intrinsic qualities, it seems. What the intrinsic qualities really are, each new theory has something to say about. Colors, for example, first accepted as qualities of light rays by Newton, are later described in terms of wavelength. But there are a good many 'low level laws' which take the form of simple equations, describing the structure of those phenomena. These are closely connected with the empirical successes that every succeeding theory will have to duplicate, at least by approximation in a limited domain. These phenomenal structures must fit, in a certain way, into the new theoretical models. The laws of reflection and refraction of geometric optics, the laws of Archimedes, the laws of inertia, of free fall, of the pendulum -- these are all simple mathematical descriptions of certain aspects of the phenomena. They are not retained in their early precise and unrestricted form. But they are retained as the structure phenomena take when observed at a certain level of discernment. We can plausibly think of the empirical description as the stable evolving surface structure of science on the face of a radically, rapidly altering theoretical content within.

There are just two realms of scientific investigation, hand in hand by experimentalists and theorists. On the one hand there are the phenomena which are investigated. On the other hand there are the models, abstract structures studied in mathematics, which the theory advances as representations of those phenomena. The representation is always partial and selective. You and I are mechanical systems, that is, we are correctly represented by certain mechanical models. But however good those models are, they omit quite a lot about us. Since those models considered in their own right are mathematical structures, they are known only in the way things are known mathematically. In mathematics, things are described only up to isomorphism -- it makes no sense there to speak of differences between isomorphic structures -- and that is why it makes perfect sense to say that here we are dealing solely with structure.  True, the distinctions are context-dependent; so is the very word "isomorphism", and even more so is the selection of "relevant" parameters for theoretical description. But by becoming comfortable with this context-dependence we make sense of the intuition that science presents us with the structure, and that it is knowledge of the structure of the empirical phenomena which is accumulated. For what that means now is that there must be mathematically describable relationships between the new and the old models which pertain in the required way to just those measurable parameters in which the older empirical successes were couched. To put it briefly: Science represents the empirical phenomena solely as embeddable in certain abstract structures (theoretical models), and those abstract structures are describable only up to structural isomorphism. There is warrant for the assertion of an accumulation of empirical knowledge through theory change precisely if it can be demonstrated for phenomena counted among the empirical successes of earlier science that, if they are embeddable in the new models then they are 'approximately' embeddable in the old models. This empiricist re-construal is scant comfort to the scientific realist, of course. It also sets aside as unimportant the conceptual puzzles about how to distinguish structure from content or quality, which beset so-called structural realism. But it provides a balanced view of scientific theory change, taking some of the mystery out of scientific revolutions. All it takes, to achieve this more balanced view, is to dispel the lazy illusion that we could do this by means of the simple expedient of either reifying the models or regarding them as delineating the objective structure of a hidden qualitative content.

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