# Approximate truth and dynamical theories.

1 Introduction

Many of our most useful scientific theories are not strictly and completely true. Still, they surely approximate to the truth in some good sense. The problem is to explicate the relevant sense.

Suppose we have achieved a passable theory by ignoring the small effects of (as it might be) friction and air resistance. If we want a more accurate theory, we can try adding the necessary correction terms. Complicating a theory in this way may well make it less mathematically tractable: that's why we will often prefer to stick with an inaccurate theory that idealizes by ignoring small perturbations. But in such a case - we might naively be tempted to say - our preferred inaccurate theory approximates to the truth in the sense that the theory could be developed into an accurate one by adding in enough small correction terms, hitting truth in the limit. In other cases, no doubt, it is better to think of the approximately true theory not as something that can be turned into the truth by adding epicycles, but rather as something that can be derived by starting with the truth and then simplifying. Newtonian dynamics is approximately true, we say, because it is what you get when you start from the relativistic truth and then, inter alia, set a small quantity - the reciprocal of the velocity of light - to zero.

However, there are immediate difficulties with the naive suggestion that we may deem a theory approximately true in virtue of its differing only by small modifications, in one direction or the other, from a true theory.(1) For a start, as soon as we try to think in a principled way about what counts as a 'small' modification of a theory, or try to make good the picture of there being a limit to the needed small adjustment terms, we discover that this story won't really do even in the cases where it looks most promising. But here I want to emphasize a different point - namely, that there are central kinds of case which the story doesn't even begin to fit. Call a theory 'stubbornly unrevisable' if it cannot be turned into the truth by adding small correction terms (and cannot be derived from a true theory by small backwards modifications). Then a theory may still count as approximately true, by any sane standards that respect what we ordinarily say outside the philosophy seminar, even if it stubbornly resists revision. Theories developed within the framework of classical fluid mechanics provide obvious candidates. Such theories cannot be made strictly true by fine-tuning, for the classical framework embodies the essential axiom that fluids are perfect continua - and no piling up of added epicycles is going to cancel that axiom and so deliver a theory which is strictly true of real, atomically granular, fluids.(2) (Further, there is no backward path via easy simplification from, for example, a quantum statistical mechanics of molecular motion to a classical theory.)(3) Hence classical fluid theories are stubbornly false; and yet some such theories - those that guide the design of aircraft wings, for instance - surely count as approximately true in their domain, if any theory does.

Applied chaotic dynamics provides some further, more exotic, examples of stubbornly unrevisable but approximately true theories. Without going into details here, true chaos essentially involves infinitely intricate structures (e.g. 'strange attractors' with a fractal geometry). But we have every reason to suppose that there aren't correspondingly infinitely intricate structures in nature.(4) So applied chaotic theories can't be strictly true of the world: and tinkering round the edges by adding extra complications won't help at all (the trouble with standard chaotic models is that they already have, as it were, too much fine structure to fit the world accurately - which also apparently means that they are not to be derived by simplification from some still more complex truths). Yet some applied chaotic theories seem at least potential candidates for being approximately true.(5)

Grant, then, that a stubbornly unrevisable theory may yet be approximately true. Competent treatments of approximate truth for theories must therefore allow for such cases (and so, by hypothesis, the 'small modifications' account of approximate truth certainly can't apply across the board).

In the rest of the paper, I first (Section 2) briskly outline a framework for thinking about approximate truth. This framework supports the independently plausible thought that the story about approximate truth will need to be filled out rather differently in different domains: so we have to treat various cases piecemeal. I then concentrate on one centrally important case and sketch an account of what constitutes approximate truth for a certain rich class of dynamical theories (Sections 3, 4) where, I will argue, the general framework has a particularly smooth application which nicely accommodates the category of approximately true but stubbornly unrevisable theories. I compare this account with two other well-known treatments of approximate truth as applied to dynamical theories (Sections 5, 6); it is argued that the first of these is evidently unsatisfactory, whilst the second only works if treated as a disguised version of the type of view sketched here. I next (Section 7) fend off a difficulty arising from what we can call 'Miller's problem'. Finally (Section 8), it is briefly indicated how the defended view might be further developed.

The resulting account of what makes for approximate truth, at least in the target class of dynamical theories, is simple and straightforward; but despite its basic simplicity, it apparently isn't widely supported. Hence the interest in telling the story anew.

2 'P' is approximately true if and only if approximately P

I claim that

[A] 'P' is approximately true if and only if approximately P

(here and henceforth, take 'approximately' to mean at least approximately - i.e. not to rule out strict truth). And furthermore,

[Exp] The order of explanation goes from right to left across the biconditional [A].

Hence the task of construing the notion of approximate truth reduces to the task of explaining how the modifier 'approximately' gets applied to propositions of various types.

More generally, with minor explicable exceptions, we always have

[M] 'P' is M-ly true if and only if M-ly P,

with the right-hand side of the biconditional explaining the content of the left. But I will not defend this general claim here, except to note that it should appeal to anyone attracted by a broadly deflationist or minimalist account of the way the truth-predicate works within the class of truth-apt statements. To take the simplest deflationist story: suppose that 'is true' is simply used to endorse assertions, perhaps without explicitly repeating them. The truth-predicate then has a merely formal disquotational role; it doesn't express a property that can come in various modes or degrees. So apparent modifications of the truth-predicate had better be understood as modifications of the proposition being said to be true. Thus just as '"P" is true' says no more than that P, '"P" is allegedly/unfortunately/unexpectedly/probably true' says no more than that allegedly/unfortunately/unexpectedly/probably P. Conversely, robust exceptions to [M] would sink deflationism by revealing truth as a more-than-formal property that can - as it were - come in its own proprietary flavours.

A deflationist about truth, then, should endorse in particular the combination of [A] plus [Exp]. But so should a wide variety of other theorists. Suppose you conceive of truth as a matter of the existence of some corresponding situation or fact (and take a chunkily realistic view of facts). Existence does not admit of degrees; hence, in the last analysis, neither can truth. From this perspective, any account that aims to give content to some notion of approximate truth must respect the constraint that truth, like existence, is fundamentally an all-or-nothing matter. And then if truth is all-or-nothing, what else is there to do but explain apparent modifications of the truth-predicate by appeal once more to [A], read right-to-left?(6)

Take a couple of very simple instances of [A] in operation. (1) 'Jones is six foot tall' is approximately true just in case Jones is approximately six foot. Here, in its second occurrence, 'approximately' serves to fuzzify the main predicate of the sentence (or perhaps further fuzzify the predicate, if it is reasonably supposed that the vernacular 'is six foot' is already vague): the degree of fuzzification indicated will be context-dependent. Of course, there are deep problems about the semantics of vague predicates, and hence serious problems about the semantics of a fuzzifying operator: but the thought is that there is no additional problem about understanding the notion of approximate truth of some proposition when it amounts to the truth of a corresponding proposition with a fuzzified predicate.

Contrast (2): 'Snoopy is a spaniel' is approximately true just in case Snoopy is a approximately enough a spaniel. Here, there are no relevantly fuzzy predicates: rather approximately being of one kind is naturally construed as exemplifying some other kind placed close enough on a hierarchical tree. Thus, for many purposes, it is close to the truth that Snoopy is a spaniel (while not even approximately true that Snoopy is a bear) because spaniels are close enough to beagles on the zoological family tree, and bears are too far removed?

Now, if the modifier 'approximately' is sensibly to be applied to a given proposition, then typically some focal term(s) in the proposition should be locatable in a domain which is subject to some natural ordering (as e.g. the class of heights, or of mammalian kinds); but the detailed structure of the relevant domain for comparison can of course vary widely between types of proposition. According to [Exp], applications of the idea of approximate truth are to be explained via the application of the approximation operator to whatever proposition is in question. So - as the examples (1) and (2) already indicate - different applications of the notion of approximate truth will require different kinds of detailed elucidation. This gives us a principled reason why we shouldn't necessarily expect a substantially worked out account of approximate truth for one domain to carry over to another. In particular, there is no requirement that an account of approximate truth for, e.g., mathematically framed dynamical theories in the physical sciences should carry over readily to other cases of approximate truth.

3 Approximate truth for geometric modelling theories

How are we to apply the schema [A] to scientific theories? Suppose we present a theory as a conjunction of propositions: then the schema blandly tells us that the theory 'P & Q & R ...' is approximately true just if approximately (P & Q & R ...). But now how do we move the approximation operator across the conjunctions? 'Approximately (P & Q & R ...)' is not equivalent to 'approximately P & approximately Q & approximately R ...', since the latter plainly does not imply the former. We will therefore need, in particular cases, some principled way of distributing the effect of the approximation operator across a theory-conjunction.

There is no general story to be told here. For one class of theories, however, part of the distribution problem is easily resolved. I'll first describe this class in an entirely abstract way; I'll then argue that this class in fact includes a rich group of real theories.

Let's say that a theory is a 'geometric modelling theory' - a GM-theory, for short - if it has the following two components. Component M is to be purely mathematical, specifying a certain geometrical structure (e.g. by giving governing equations, or by specifying transformations for which the structure is the unique invariant, or ...). Component A is to give empirical application to the mathematically characterized structure by claiming that it replicates, subject to arbitrary scaling for units, etc., the geometric structure to be found in some real-world phenomenon. Now, what would it be for a GM-theory comprising M and A to be only approximately true? Given schema [A], to say that the theory is approximately true is to say that, approximately, M & A. But presumably, in the normal case, we won't want to deny that the mathematical kernel of the theory is a perfectly correct characterization of some abstract geometric structure.(8) So the approximation operator is naturally taken to pass by M and apply only to the second empirical component of the theory-conjunction. In other words, the claim that a GM-theory is approximately true is to be taken as the claim that the geometric structure in question approximately replicates the relevant structure to be found in the target real-world phenomenon.

And what is it for one geometric structure to approximate another? A simple but central case is where the structures in question are both curves embedded in the same space: here it is natural to say that one structure approximates to the other if the first curve can be distorted into the second by a transformation which (a) moves points by no more than some [Epsilon], and (b) preserves 'smoothness' (very roughly speaking, first and perhaps higher derivatives at corresponding points are within some [Delta]) - in such a case, one curve will closely 'track' the other. To be sure, different cases may require different treatment, depending on which features of the geometric structures are the prime focus of interest in a given context.(9) However, specifying a geometrical relation that satisfactorily captures the contextually relevant 'closeness' relation between structures will be a technical problem in geometry: there is no deep conceptual problem about a claim that one geometric structure approximates to another (after all, it is precisely in such metric contexts that the notion of approximation is most at home).

In short then, given schema [A], the claim that a certain GM-theory is approximately true should be conceptually unmysterious: elucidating the claim will just require spelling out what it is for an appropriate geometrical 'closeness' relation to hold between structures of the relevant kinds. Further, we have immediately made room for the idea that a given GM-theory, though approximately true, may be 'stubbornly unrevisable' in various ways. For a simple case, suppose that approximate truth for a certain GM-theory is a matter of tracking a real-world structure within some [Epsilon], and suppose that tinkering by adding small terms to the defining equations of its M-component cannot further reduce tracking error below some [Alpha]: then the given GM-theory will be one which is approximately true but cannot be made to approach the truth indefinitely closely by adding epicycles.

So far, so good - but also, so very abstract. All this becomes pointful, however, when we note that classical dynamical theories can readily be regimented as GM-theories.(10) For a simple example, consider the familiar account of the dynamics of a freely swinging pendulum. One standard way of looking at this account is to regard it as first characterizing a pure abstraction, the ideal frictionless pendulum moving in a plane according to Newton's laws. The governing equations determine the allowable patterns for the time-evolution of the ideal pendulum's angular displacement and velocity as a function of the pendulum's fixed length, etc. If we conceive of plotting a three-dimensional graph of time against displacement against velocity, then a certain bundle of three-dimensional curves will trace the allowable behaviours of a pendulum of given length subject to a given force. If we conceive, yet more abstractly, of these three-dimensional bundles being 'plotted' against pendulum length and applied force, we will get a more complex five-dimensional structure that in addition encodes the way that the possible behaviours of the pendulum depend on the length and force. This geometric structure is of course a purely abstract object; it gets put to empirical work, however, when it is claimed - as a first shot - that this structure exactly matches a structure that similarly would encode the physically possible behaviours of, for example, real freely swinging pendulums of varying lengths, etc. Of course, this empirical claim is evidently false (for a start, real physical pendulums are damped by friction, air-resistance, and so on, and eventually stop moving: ideal pendulums swing for ever). Still, considered for shortish periods, real pendulums behave similarly to ideal pendulums; in more geometrical terms, a time-slice of the mathematically defined abstract structure will give, curve by curve in the bundle, a close approximation to a time-slice of the corresponding structure read off the possible behaviour of real pendulums (and approximation here can be given a simple reading along the lines of 'within [Epsilon], preserving smoothness').

So we can say: there is an abstract structure of a certain kind, and in a straightforward sense this approximately replicates a structure in the real pendulum's spectrum of possible behaviours. Whence, by the inference that if (M & approximately A) then approximately (M & A),(11) and an application of schema [A], it follows that the regimented standard account of the pendulum (which indeed says that there is a certain abstract structure and it models a real pendulum) is indeed approximately true.

Of course, the elementary textbooks do not talk e.g. of five-dimensional bundles of curves etc.; my characterization of the dynamical theory of the pendulum does indeed involve a radical reconceptualization - however, this kind of geometrical reconceptualization is now absolutely standard in the more advanced study of dynamical systems.(12) Faced with any of a very wide variety of theories in mathematical physics whose concern is the deterministic dynamics of some set of quantities, we can (just as with the ideal pendulum) usefully imagine an assignment of values to these quantities being represented by a point in a multi-dimensional 'phase space', then take the time-evolution of the quantities to be given by the trajectory of one of these representing points, and then reconceptualize the theory as giving the geometry of the bundles of possible trajectories. If treated in this way, empirically applied dynamical theories then can very easily be regimented as a GM-theory - and once so reformulated, we have argued, there is a simple and natural way of construing the claim that the dynamical theory in question is approximately true. It is just a matter of one geometric structure approximating another.

4 Ancestor views: Suppes, van Fraassen, Sneer, Stegmuller

Our emerging account of approximate truth for those dynamical theories that invite the GM-theory treatment is still schematic: but the outlines are clear enough to enable us to make comparisons with some alternative accounts of approximate truth as they would apply to the same cases. But before turning to comparisons, a few comments on ancestors and cousins of the present account.

The basic idea originates with Patrick Suppes in the 1960s:(13) the objects treated by (for example) textbook classical mechanics - the familiar cast of point masses subject only to gravitation, of massless springs, frictionless joints, ideal pendulums and the rest - are really fictional ideals, best thought of abstract objects, which purport to structurally mirror in their behaviour some class of real phenomena. So, on this view, various mathematically articulated theories can be best regarded as having a kernel specifying a certain abstract structure or structures, with the rest of the theory indicating application rules associating these structures with certain worldly phenomena which supposedly exemplify them. What I have done, in fact, is to note that in the case of many dynamical theories, we can finesse general problems about the talk of structure here by giving a very literal-minded geometric reading of the notion: indeed, for certain purposes I would recommend identifying the 'ideal pendulum', etc. with particular geometric structures in appropriate abstract spaces. This geometric identification is ontologically neat, as well as smoothing the way (as we have seen) to an unproblematic account of approximate truth for dynamical theories.

The Suppes view of theories is, however, probably now best known in one or other of two rather special versions, which I should pause to comment on. First, recall the logical empiricists' picture of theories as comprising a formal calculus and 'correspondence rules' relating some formal terms with observables. Now take the formal calculus not as initially uninterpreted but rather (following Suppes) as specifying some formal structure(s); and take the correspondence rules to link not terms with observables but formal structures with observables. This yields an updated empiricist view of theories which is, of course, exactly Bas van Fraassen's. 'To present a theory,' he writes ([1980], p. 64), 'is to specify a family of structures ... and to specify certain parts of these models as candidates for the direct representation of observable phenomena.' It is not clear exactly what van Fraassen means by 'structures', but waive this worry: for there are other familiar reasons which I won't rehearse here for disenchantment with van Fraassen's overall empiricist philosophy of science. Let me emphatically insist, then, that it is one thing to hold that some mathematically framed theories can be regarded as having two components, specifying formal structures (e.g. geometric ones) and correlating these to the world: it is quite another thing to give an empiricist gloss to this picture and hold that the correlation between the textbook abstractions and the real world are to be confined to observable phenomena. I certainly don't want to endorse this empiricism, which is no essential part of the Suppes program.(14) It is entirely consistent with the basic idea to hold that theory-to-world correlations can link abstract geometric structures with (say) structures encoding the time-evolution of a range of physical quantities including unobservables.

Second, let's briefly consider the formalized structuralism of Joseph Sneed and Wolfgang Stegmuller.(15) Three quick comments:

(a) Sneed and Stegmuller are thoroughly wedded to the view that all talk of mathematical structures has to be cashed out in set-theoretic terms - so that predicates like 'is an ideal pendulum' or 'is as a classical system of point masses moving under mutual gravitation' get regimented as predicates characterizing certain sets. In other words, for them, a dynamical system like an ideal pendulum is just an ordered n-tuple, whose elements are themselves characterized set-theoretically, and which satisfy certain formal constraints. However, I don't here want to take a particular stance on set-theoretic reductionism. Of course, if you are happy generally to reduce (more or less) all and any mathematical objects to sets, then you needn't balk at regarding classical point-particle systems as sets either; conversely, if you reject set-theoretic reductionism elsewhere, you won't be happy about it here. I am not officially going to take sides: and my immediate point is that one needn't take sides. The potentially distracting issue about set-theoretic reductionism is independent of the basic issue that concerns me - though I suppose that if the question about the ontology of abstract geometric structures is really pressed, then I can fairly happily join Sneed and Stegmuller in going set-theoretic - thus buying into a familiar sort of ontological story, which should satisfy even the most hard-nosed.

(b) There is a question about which abstract structure best to identify as the 'ideal pendulum', as a 'classical n-point-particle system', and so forth. I have suggested we can in fact - for certain purposes - identify these abstract dynamical systems with certain geometrical structures in abstract spaces. The Sneed-Stegmuller story looks rather different.(16) But we are in the game of regimentation here, and there need be no single best solution. As one would expect, the two accounts are in any case closely related; in particular a deterministic Sneed-Stegmuller dynamical system will fix a geometrical structure in an appropriate abstract space.(17) What is not so clear is whether the reverse connection will hold in general - simply because it isn't clear what the intended constraints are on redescribing dynamical systems in the Sneed-Stegmuller way. It would take us too far afield to pursue this: I will simply advertise my approach as conforming more closely to the geometrical thinking of modern dynamical theorists.

(c) In one respect, however, I do more clearly depart from the official Sneed-Stegmuller line. For Sneed, for example, to say that the standard theory of the pendulum is true of a real world system would be to say of a certain set (comprising the really exemplified time-to-displacement function, real length, etc.) that it belongs to the set of ideal pendulums. But this leaves it difficult to construe talk of a theory being approximately true of the world: the problem of making sense of the idea of one set being approximately a member of another is, if anything, worse than the original problem about approximate truth. Sneed himself notes the worry ([1971], p. 25), but leaves it hanging. However, by saying from the outset that applying an abstract theory to the world is - in the best case - to claim exact structural matching, we make room from the start for the possibility of lesser degrees of similarity and hence for the approximate truth of theories.

5 Approximate truth and verisimilitude: Oddie, Niiniluoto

To summarize: I am advocating a geometrized version of the Suppes view of mathematically framed theories, at least for a wide class of dynamical theories, while eschewing any empiricist restriction on theory-world correlations. We can canonically regiment such a theory into GM form, with two components - one specifying a certain abstract geometrical structure; the other giving its application. But we will standardly want to recognize the integrity of the mathematical kernel as a precise description of some intended abstract structure. So when it comes to attributing approximate truth to the theory and applying schema [A], we will let the approximation operator applied to the theory-conjunction pass transparently by the mathematical kernel and attach it only to (some of) the application claims - yielding the plausible claim that a theory is approximately true if the world exhibits a relevant structure sufficiently similar to the abstract structure specified by the theory (though which similarities to weight will, no doubt, be interest-relative).

What alternative accounts are available? I suggested at the outset that the idea that a theory is approximately true if 'small' modifications will make it strictly true is scuppered by the possibility of stubbornly strictly false yet approximately true theories. So, having set aside accounts of that type, I'll turn next to the most technically developed line on approximate truth, found in the work of Graham Oddie and Ilkka Niiniluoto, among others, who are responding to the well-known failure of Popper's own original definitions of verisimilitude.(18)

We need not pause to consider how far this line was intended to cope, for example, with an application of the notion of approximate truth to dynamical theories, but I think that it is evident that in this domain at least, the neo-Popperian approach points us in entirely the wrong direction. Popper, recall, tried to define verisimilitude in terms of plain truth and plain falsehood (no approximation involved); roughly speaking, one theory has greater verisimilitude than another if it accurately hits the truth more often. Likewise for Oddie and Niiniluoto; when the wraps are off they are revealed as still wedded to the idea that in the end it is the number of bull's-eyes that matters. Their construction of a metric for distance from the truth begins from ideas like this: one basic conjunction of atomic propositions or their negations is nearer the truth than another if more of its conjuncts hit the truth.(19) So for them, getting near the truth is - at bottom - like getting a good score on a test asking a lot of yes/no questions (and getting nearer the truth is - at bottom - switching answers by inserting or deleting some negation signs).

But this looks to be a hopeless initial model for approximate truth of theories apt for regimentation in GM form. Surely, most approximately true dynamical theories aren't theories that get a suitable proportion of basic claims dead right (other than the non-empirical, purely mathematical claims). Rather, they overall get enough things near enough right. And certainly most approximately true mathematically framed theories can't be repaired by the simple expedient of twiddling a few negation signs in the basics in some canonical formulation. This seems particularly clear in the case of the unrevisably false dynamical theories: you are not going to turn those approximately true classical fluid theories into strictly true theories just by a judicious sprinkling of negations. Or that, at any rate, is the intuitive challenge to the Oddie-Niiniluoto approach.(20)

Of course, it might yet be urged that the intuitive notion of approximate truth behind this challenge - the very notion I am trying for an account of - is fatally flawed. If so, then (as a fall-back position) we could consider replacing the intuitive notion e.g. by some construct based on the very different idea of scoring a sufficiency of direct hits on the truth. But such a replacement would be radically revisionary of prephilosophical ideas. It would be good if we could do better for the intuitive notion - and I am arguing that we can.

6 Approximate truth and truth in nearby worlds: Lewis

Here's another approach to approximate truth found in the literature - still off target, in its first crude version at any rate, but certainly a move in the right direction.

Start with the truism that a theory is approximately true if the way things actually are approximates to the way that the theory says they are (approximates, that is, to the way things would have been, had the theory been strictly true). Now make the standard move from talk about 'the way things are' or 'the way things would have been' to talk about possible worlds. And we get the following: a theory is approximately true if the actual world is close to the possible world where the theory is strictly true. Except, of course, that there will be many worlds at which a given theory is true: so we need to say, rather, that a theory is approximately true if the actual world is close to the set of possible worlds where the theory is true.

This sort of account has been endorsed by David Lewis.(21) Obviously the account needs fine tuning, by elucidating the correct closeness relation: is it simple 'separation at closest approach', or some more complex function of the distances between the actual world and the worlds where the relevant theory is true? But, relatively independently of the tuning, the account has evident virtues. For a start, the basic framework is familiar - possible worlds ordered by similarity. Of course, we may differ about how best to interpret the framework (how realist should we be about possible worlds?), but just take the interpretation of the framework which you favour when explicating counterfactuals, for example, and you've now got the framework of an account of approximate truth at no extra cost.(22)

Most importantly, the possible-world account correctly locates the approximation at the level of the world rather than at the level of representation. To see what I mean, consider again the momentarily tempting view according to which a theory is approximately true if it could be turned into an accurate one by adding small correction terms (a small adjustment of the representation gets things right). The problem with this is that a theory can easily be structurally unstable, which is to say (crudely speaking) that the behaviour described by the theory radically changes if you make even vanishingly small changes to the theory by marginally altering values of parameters or adding tiny correction terms. Now, take a theory which descriptively gets things very badly wrong about the behaviour of its target system: we won't normally deem this theory to be approximately true - even if, because of some local structural instability, it just so happens that the theory can be adjusted to fit the facts by making changes of small order. Hence it can make a crucial difference which way you put the point: to assert that a theory is approximately true is primarily to say that the world is roughly as the theory says it is, not to say that another theory, related by small adjustments, is right. And that, of course, is why a theory that stubbornly resists modification can still be approximately true: a theory might be true of a nearby world, even if no 'nearby' theory is true.

The possible-world account has virtues, then. But, at least in a simple form, it won't do. Consider classical fluid theories again. Any possible world of which such a theory is strictly true is one where fluids are perfect continua. So, none of the fluid stuffs in our world is exemplified in that world and vice versa; and very many of the laws - for a start, those governing phase transitions from the fluid to the gaseous state - must be unimaginably different. Overall, the physics of such a world will be radically unlike that of the actual world. But this means that any world of which classical fluid mechanics is strictly true will be homologically a very remote world. So, assuming that we assign nomological considerations anything like their usual weighting in ranking similarities, the classical theory is not true at any nearby world - and hence (according to the Mark I possible-world theory, however fine-tuned) isn't approximately true. That's a reductio, given that what we are after is precisely an explication of the pre-theoretic sense in which such theories are approximately true.

Now, it might well be protested that the objection is rather a cheap shot and is very easily met. It is, after all, familiar that different applications of the possible-world framework may require different standards of similarity. So in the present case we just need an appropriately qualified account of what makes for relevant similarity between words - an account which allows words to count as relevantly similar for the purposes at hand even though differing widely in many other nomological respects.(23) But then how is the Mark II story to go?

There seems to be no prospect of an account that makes the kinds of similarity relevant to approximate truth independent of the theory whose approximate truth is up for consideration. Rather, I think the story has to run (very roughly): theory T is approximately true if the actual world and some world where T is hue sufficiently resemble each other in key respects which T is concerned to describe. Which respects to give weight to, and what to count as sufficiency in these respects, will no doubt be interest-relative. Thus classical fluid mechanics is approximately true because the actual world and some world where this theory is true resemble each other as far as the medium and large-scale behaviour of fluid flow is concerned (and nomological divergences in micro-respects don't matter). Classical genetics is approximately true because the actual world and some world where that theory is true resemble each other as far as the gross patterns in inherited characteristics are concerned. And so on.(24)

However, while this revised possible world account of approximate truth may meet the earlier objection, it does so at the price of a certain built-in redundancy. For on anybody's view - whether you are officially a Lewis-style realist, a Stalnaker-style ersatzer, a Rosen-style fictionalist, or whatever(25) - full-blown possible worlds are richly structured. To talk of some world of which a given theory is true is in the general case to talk of something with vastly more content than is required just for satisfying the theory. So if we move from a theory to a world of which it is strictly true, but then compare that world with the actual world only in the limited respects which the theory targets, we have in effect first added a great deal of unspecified structure and then carefully ignored it. And this manoeuvre evidently involves redundancy. Compare possible-world treatments of counterfactuals, where the invocation of full-blown worlds seems to be doing some real work. The fundamental insight is, roughly, that 'if P were the case, then Q' is true if the state of affairs P along with lots of relevant background facts suffices for Q. So giving truth conditions for the counterfactual in terms of what obtains at near P-worlds, with all their detail, is precisely a way of ensuring that all the required background gets into the story (with the nearness requirement keeping us as far as possible to background facts). To be sure, there is overkill here: the worlds will contain vast numbers of background facts irrelevant to evaluating the counterfactual; but overkill at least guarantees that we get all the relevant background facts into the frame. By contrast, when comparing worlds just in respect of fitting a given theory in targeted respects, all the background is effectively to be ignored, so we don't need to get it into the frame at all.

To see this, return to my particular concern, dynamical theories in GM form. As a first shot, such a theory would be strictly true of some physical system if that system precisely replicates just the geometrical structure required by the theory.(26) So the revised Lewisian account of approximate truth, applied to such a theory, comes to this: the theory is approximately true when the actual world closely resembles some other possible worlds wherein the target phenomena exactly exemplify the geometric structure required by the theory - where the resemblance is in the relevant respects which the theory addresses. But when will the actual world suitably resemble a possible world which exemplifies a given geometric structure? When the target phenomena in the actual world exemplify a structure which - in some appropriate sense - is close to the target structure. So the Mark II Lewisian theory entails that a theory is approximately true if the actual world exhibits some geometric structure sufficiently close in respects the theory cares about to the structure required by the theory. But that consequence is itself already a direct account of approximate truth, the very one we have already given; and the detour via other possible worlds has indeed proved redundant.(27)

7 Miller's problem resolved

There is a familiar argument due to David Miller (see Miller [1994], Ch. 11 for a recent version) which must now be faced, for if sound, it would sink any account of approximate truth for GM-theories that invokes ideas like 'close-tracking'.

To take a simple illustration, suppose we have rival dynamical theories, H and K, which give values for the state variables x and y at time t. Then, Miller notes, we can have the following situation. (1) the trajectory predicted by K manifestly tracks the true values of x and y better than H. But (2) there is a transformation of coordinates, from x, y to [x.sup.*], [y.sup.*], and H now gives values for [x.sup.*] and [y.sup.*] which manifestly track the true values of the new variables better than K's values. So, on the close-tracking account of approximate truth, we have: H is further from the truth than K in one coordinate frame, but nearer the truth in a second frame. But any plausible account of distance from truth should make comparative approximation to the truth coordinate invariant. And that's bad news for the close-tracking proposal.

Miller's own example is this: take

H: x(t) = t, y(t) = 5t

K: x(t) = t + a y(t) = 2t

and suppose that the true values of x and y are given by

x(t) = t + 2a y(t) = t

Uncontroversially, K tracks the correct values closer than H. Now take the coordinate transformation

[x.sup.*] = x + (5a/12t)y [y.sup.*] = (5t/2a)x + y

The theories H and K become

H: [x.sup.*](t) = t + 25a/12, [y.sup.*](t) = 5t2/2a + 5t

K: [x.sup.*](t) = t + 22a/12, [y.sup.*](t) = 5t2/2a + 9t/2

while the true values of [x.sup.*] and [y.sup.*] are given by

[x.sup.*](t) = t + 29a/12, [y.sup.*](t) = 5t2/2a + 6t.

And equally H, it seems, now tracks the correct values more closely than K. But H cannot be both nearer and further from the truth than K; hence close-tracking cannot constitute nearness to truth.

How seriously should we take this kind of example? The coordinate transformation required to reverse the fortunes of H and K is time-dependent in a pretty odd way. Consider: a constant unit line along the original x-axis becomes (applying any reasonable non-time-dependent metric in the new coordinate system) a line whose length grows with time; while a constant unit length along the original y-axis becomes a line whose length shrinks with time. With this kind of gerrymandered time-dilation, no wonder we can get strange results ('lengths' will vary without cause or effect, laws will cease to be time-invariant, and so forth). So the obvious initial response to Miller's actual example is to complain that the unnatural type of time-dependent coordinate transformation he invokes is illegitimate.

Still, while this complaint has intuitive force, it isn't immediately obvious how best to make the complaint stick. And in any case, banning weirdly time-dependent coordinate transformations won't resolve the matter. For consider a second example. Again there are two hypotheses

H[prime]: x(t) = 5t y(t) = 3t

K[prime]: x(t) = 4t y(t) = 2t

whereas the truth is that

x(t) = t y(t) = t

Neither H[prime] nor K[prime] is a very good shot at tracking the actual values, but K[prime] is surely the better. However, take the coordinate transformation

[x.sup.*] = x - 3y/2

[y.sup.*] = y - 3x/4

Then we have

H[prime]: [x.sup.*](t) = t/2 [y.sup.*](t) = -3t/4

K[prime]: [x.sup.*](t) = t [y.sup.*](t) = -t

and the true time-evolution is

[x.sup.*](t) = -t/2 [y.sup.*](t) = t/4

And now H[prime] appears to track better than K[prime]. So we have a coordinate transformation which this time is nicely time-independent but which again reverses the fortunes of two hypotheses.

It might for a moment be wondered whether this phenomenon can only arise because we are considering cases where there are alternative hypotheses for single phase-space trajectories, not hypotheses about whole families of trajectories from different initial states. So is there safety in numbers or can there still be Miller-type reversal between generalized hypotheses?

There can. Just generalize the last example. Suppose our rival hypotheses say that, for any initial values x(0) = a, y(0) = b the resultant trajectories will be

H[double prime]: x(t) = 5t + a y(t) = 3t + b

K[double prime] : x(t) = 4t + a y(t) = 2t + b

whereas the truth is that

x(t) = t + a y(t) = t + b

K[double prime] is better than H[double prime] on the time-evolution of the values of x and y from any initial state; but with the same transformation, it is immediate that H[double prime] is better than K[double prime] on the time evolution of [x.sup.*] and [y.sup.*] from any initial state.

Recall, however, a point already stressed - that what makes for approximate truth is interest-relative. Suppose, then, that in our last example x and y respectively represent, as it might be, temperature and circulation velocity in a steadily heated convected liquid (and so H[double prime] and K[double prime] are hypotheses about how these quantities increase over time). Then [x.sup.*] represents temperature minus 150% of the circulation velocity. And what - we may well ask - is the real physical significance of that? So, given that we are not interested in the values of [x.sup.*] and [y.sup.*], why care whether a theory close-tracks them? Surely dynamical theories should aim to track the time evolution of physically significant quantities, and a theory will count as approximately true just so long as it gets the values of those quantities near-enough right for long enough (so, for example, K[double prime] counts as approximating the truth better than H[double prime] since it is more accurate on the values of the physically significant quantities).

Does concentrating on certain properties as 'physically significant' require appeal to heavy-duty metaphysical reasons for privileging certain 'elite' properties (in the manner of David Lewis)? Not so. It is enough for our purposes to note that, in real cases, the quantities represented by variables in a particular dynamical theory - temperature and circulation velocity, in the case we just imagined - will (according to other theories) feature in a wide range of functional relationships to other quantities (pressure, volume, viscosity, etc.). And while the {x, y} [approaches] {[x.sup.*], [y.sup.*]} transformation is tailor-made in order to transform the H and K pairs of hypotheses so as to preserve functional simplicity while reversing intuitive accuracy, there is no reason at all to suppose that the same transformation (or any extension thereof to include variables over other quantities) will produce anything but mess when it operates over a wider domain of hypotheses. In the wider, many-theory setting, standard simplicity considerations can privilege a certain choice of quantities as the significant ones.

So the state of play is this. In the general case, to get the Miller reversal phenomenon we will have to use coordinate transformations between phase spaces which are time-dependent in a quite anomalous way. Only in very special cases can well-behaved transformations lead to reversal. And in most of those special cases, we can still reasonably prefer working in one phase space rather than another on the grounds that the state variables in the one case but not the other represent physically significant quantities (as evidenced by other theories). So the only case we would really have to worry about is the case where transformations which were not time-dependent in an anomalous way took us from one space to another, reversing the fortunes of some hypotheses, but where the state variables of both spaces had equal claim to physical interest.

I know of no such case; but if there were one, we could just live with it - the relevant hypotheses H and K would, as it were, score one goal each, so the match would be a score draw, and neither preferable. An account of approximate truth doesn't have to rule out such cases.

8 Conclusion

I have argued, then, that combining a certain framework approach to approximate truth ([A] plus [Exp]) with a certain attractive reconceptualization of dynamical theories delivers a very natural account of their approximate truth. It would be surprising if no version of an account that can be arrived at so readily had ever been suggested in the literature; and a close approach is to be found in Ronald Giere's [1985] and [1988]. He too takes dynamical theories of the kind we are interested in to delineate abstract mathematical models. But he handles the issue of approximate truth rather differently. He briskly proposes that 'theoretical hypotheses ... have the following general form: The designated real system is similar to the proposed model in specified respects and to specified degrees' ([1985], p. 80]). Or, as he rephrases it later, 'To claim a hypothesis is true is to claim no more or less than that an indicated type and degree of similarity exists between a model and a real system. We can therefore forget about truth and focus on the details of the similarity' ([1988], p. 81]). We might put it this way: for Giere, claims for theoretic truth are already claims about approximate truth (so explicitly talking about approximate truth will simply serve to emphasize the looseness of the model-to-world similarity being asserted). By contrast, I have preferred not to 'forget about truth', but to take an unqualified claim to truth to be what it seems to be, i.e. the assertion of an exact model-to-world match, and to take (mere) similarity of match to be the mark of (mere) approximate 'truth. But perhaps this difference can be set down as largely one of rhetoric.

A much more significant difference between Giere's discussion and mine here is that Giere sets his sights higher: although he uses examples of simple, classical dynamical theories, he seems to be aiming for a rather general treatment of the content of theoretical hypotheses. Since theoretical hypotheses are all approximative, by his lights, that in effect means saying something general about approximate truth. But I doubt, given the discussion earlier, that we can say anything both general and substantive. Certainly, the price Giere has to pay for his generality is reliance on the notoriously slippery notion of similarity - about which, unfriendly critics might say, he is suspiciously quiet. (Those who are unhappy about a general appeal to the idea of approximate truth to underpin a version of scientific realism are unlikely to be soothed by an equally general appeal to similarity.) Giere thus doesn't pause to note or exploit the very real gain in the geometric reconceptualization of classical dynamical models that I have wanted to stress. To repeat, it allows us to use the metric structure of the space in which a model lives to talk in conceptually unproblematic ways of its geometric structure approximating a structure representing possible real-world time-evolutions. In the domain of classical dynamical models, then, we can avoid relying on any inchoate notion of similarity, and we can ground an attractively clear proposal about what would constitute at least the simplest cases of approximate truth - namely, close tracking of phase-space trajectories.

But we are not out of the woods yet. Even in some relatively straightforward instances, the close-tracking story will need development. Consider the following obvious case.(28) We have a pair of dynamical theories of planetary motion, where [T.sub.1] is Ptolemaic and [T.sub.2] Newtonian. It could well be that the parameters in [T.sub.1] and [T.sub.2] are so chosen that in fact [T.sub.1] tracks the actual behaviour of the planets better than [T.sub.2]. But must we say that the Ptolemaic [T.sub.1] will therefore be nearer the truth than [T.sub.2]? Of course, we've allowed that judgements of approximate truth will be interest-relative, and if our concern is (say) merely navigational - if we just want to predict the position of the planets against the fixed stars - then [T.sub.1] will by hypothesis indeed be nearer the truth in the respects we care about. But our interests could be more explanatory (even while concentrating on pure dynamics - cf. fn. 10); and from such a perspective, surely the Newtonian [T.sub.2] can rate as nearer the truth. The objection is that a simple tracking proposal cannot allow us to say this.

We need, then, to complicate the story somewhat. However, the required extra step doesn't take us beyond the resources already available. Explanation (at least in pure dynamics), as Hempel long since argued, is unification.(29) And the Newtonian [T.sub.2] scores in the explanatory stakes because it can be seen as the instance of a more general recipe which unifies other dynamical theories, where enough of these other theories are approximately true in the basic tracking sense. Even if we have wrongly set the parameters so that [T.sub.2] doesn't track the world well, it can still get the reflected credit (as compared with a Ptolemaic theory) of belonging to a unified family with many successful trackers. So true enough, as the objection insists, we may count a particular dynamical theory as approximating the truth even though it doesn't itself track the relevant quantities so well, but at bottom it can still be close tracking (albeit of other quantities by other specific dynamical theories in the family) that underpins this judgement of approximate truth.

Thus, with modest embellishment, the tracking proposal can cope with the sketched objection. Other cases will require more complex treatments. For example, what about chaotic systems with sensitive dependence on initial conditions where our theoretical plots of quantities won't track the world since errors inevitably keep exploding? Here, we need to stress the fact that chaotic theories prioritize other, more abstract, metric and topological similarity-relations between the chaotic models and worldly behaviour, and the theories can count as getting near the truth in virtue of these similarities. And the conjecture is that, all along the line, the ingredients for an account of the application of the notion of truth to a particular dynamical theory will be geometric. In other words, there is no remaining conceptual problem about describing such theories as approximately true.(30)

Acknowledgements

A short version of this paper was first written as a talk for the Moral Sciences Club at Cambridge; Section 7 was given as part of a talk to an HPS seminar at Cambridge. Thanks are due to the Cambridge audiences, to my Sheffield colleagues, to an anonymous referee, to David Papineau, and to Rosanna Keefe.

1 Van Fraassen might seem to have in mind such an account when he writes that the belief that a theory is approximately true is the 'belief that some member of a class centring on the mentioned theory is (exactly) true' ([1980], p. 9); later he writes: 'to say that a proposition is approximately true is to say that some other proposition, related in a certain way to the first, is true.' ([1985], p. 289). But this is - I think - just rather careless, and van Fraassen's preferred view is much nearer the one I'll be defending.

2 To be sure, from the perspective of our interests in the macroscopic character of fluid flow, the difference between a continuous fluid and a finely granular one might seem very small: so it might be asked why tinkering with the continuity axiom itself can't count as a small modification of our theory. But drop the axiom, and the whole mathematical apparatus would have to be abandoned for something very different: some 'small' change!

3 Or at least, I'm encouraged to place a confident bet on this claim given the well-known difficulties of getting the simplest phenomenological laws even out of classical statistical mechanics - see e.g. Sklar [1993].

4 For some introductory elucidation, see e.g. Kellert [1993] or my own [1993]. For discussion of the lack of appropriate intricacy in the world see my [1993]: but the thought is already in Mandelbrot [1983], pp. 7-8.

5 That's what chaos theorists claim: thus Edward Lorenz ([1993], p. 143) notes that while his now famous equations with their strange attractor don't adequately describe their originally intended subject matter, it turns out that they 'might yield a fair description of some [other] real world phenomenon' such as the so-called chaotic water-wheel.

6 A comment about the idea of degrees of truth. You can't disquote in degrees; so the thoroughgoing deflationist cannot countenance an appeal to degrees of truth as doing ground-level, explanatory work, any more than the correspondence theorist. But one needn't on that account reject all talk of degrees of truth. For example, since we can make sense of claims about Jack's height approximating to six foot extremely closely/fairly closely/very roughly/etc. we can (via applications of a modified schema [A]) give sense to corresponding claims about the sentence 'Jack is six foot tall' approximating extremely closely/fairly closely/very roughly/etc. to the truth. And then instead of saying that the sentence approximates more or less closely to the truth, we could instead speak (if we must) of the sentence being true to various degrees. Still, our grasp of any such constructed notion of degrees of truth must be elucidated by independently explaining how adverbs such as 'approximates to a high degree' work and then reading [A] (or its close kin) from right to left rather than vice versa.

7 This special sort of case is given prominence by Aronson [1990]: but it just seems a mistake to suppose, as he apparently does, that the ordering of natural kinds into subtypes and supertypes could provide the basis of some general theory of approximate truth.

I don't hang much on the particular example. Some find it very odd to claim that 'Snoopy is a spaniel' is approximately true; but such objectors seem to find it equally odd to say that Snoopy is approximately a spaniel - and that pair of reactions confirms, rather than subverts, the thesis that [A] holds. For that thesis implies that judgements about the degrees of acceptability of each side of the biconditional should march in step.

8 Compare Weston [1992], who seems to want also to apply the approximation operator to the mathematical kernel of regimented theories, thus - as he sees it - necessitating a conception of how arguments from approximately true axioms to approximately true conclusions might be 'approximately valid'. I will forbear from going into details here: but the ensuing grim tangles seem wildly unpromising.

9 Take a quite trivial example. Start with a hexagon, and consider the result of microscopically smoothing each vertex to a continuous curve: if what we care about is some overall closeness measure, then since the 'smoothed' hexagon differs by very small distortions from its original. the one might be said to approximate the other. On the other hand, if the matter of prime importance is the number of 'sharp' turning points on the figure, then smoothing - i.e. turning a six-vertex curve in to a zero-vertex curve - could make all the difference, and the result not count as even approximately the same for the relevant purposes,

10 We are concerned here with what might be called pure dynamical theories, which tell us what goes where and when. Our construction of a pure dynamical theory may often be guided by substantial assumptions about causal mechanisms, etc. But the pure theory itself just tells us about the 'how' of the time-evolution of various quantities, not the 'why'.

11 The inference '(P & approximately Q), hence approximately (P& Q)' is not universally valid - e.g. it might be true that Harry weighs (exactly) 160.17 pounds and so weighs approximately 160 pounds; it does not follow that it is approximately true that (Harry weighs 160.17 pounds and weighs 160 pounds). However, when P is necessarily true, then the inference is valid - cf. the formal semantic treatment of 'approximately' noted in fn. 22, and close variants. Hence, in the present case, where M is a composite necessary truth about the existence and properties of a certain geometrical structure, the inference does go through.

12 For a survey, see (from dozens of possibilities) Jackson [1990].

13 See e.g. Suppes [1960]; the spirit of his programmatic suggestions is preserved in the more developed treatments by Frederick Suppe - see e.g. Suppe [1972].

14 Suppe [1972] - for example - quite explicitly rejects the observation/theory distinction as inessential to an adequate analysis of the structure of scientific theories

15 See Sneed [1971], Stegmuller [1976, 1979].

16 For example, the story about an ideal pendulum could go like this (compare Sneed [1971], Ch. 6 on classical particle systems): x is an ideal pendulum iff there exists a T, s, l, and f such that

(1) x = (T,s,l,f)

(2) T is an interval of reals (a 'time interval')

(3) s is a function from T to the real numbers (yielding the 'angular displacement' at t)

(4) l,f are reals greater than 0, giving the non-zero 'length' of the pendulum and the constant downward force (in effect in units chosen to give the pendulum unit mass).

(5) For all t [element of] T. we have d2s(t)/dt2 = -fsin(s(t))/l - i.e. Newton's second law.

17 Thus, the set of Sneed-Stegmuller ideal pendulums determines a structure in a corresponding five-dimensional space (coordinates: time, displacement, velocity, length, force).

18 See Oddie [1986] and Niiniluoto [1987]. The original Popperian definitions of verisimilitude are to be found in his [1963] and [1972]. For a review of early discussions, see Newton-Smith [1981]; and for very useful review discussion of Oddie and Niiniluoto, see Brink [1989].

19 We can then set the distance between a proposition in disjunctive normal form and the truth to be some function of the distances between each of its disjoined basic conjunctions and the one true basic conjunction. The fun really starts, however, when we try to generalize again to give an account of distance from the truth not just for propositional languages but (at the least) for languages with first-order quantification. The fundamental Oddie-Niiniluoto claim remains that 'the simple counting procedure that works so admirably in the case of finite propositional languages can be carried over to languages of any desired degree of complexity' (Oddie [1986], p. 176). There is some technical interest in seeing how they propose to do this. And there are familiar problems: most basically, there is David Miller's classic [1974] objection that this whole approach to measuring distance from truth cannot yield a measure that is invariant between languages interrelated so that atoms in one translate into non-atoms in the other.

20 Here is as good a place as any to suggest the possibility of a related treatment of approximate truth (one that, so far as I know, hasn't been explored). The idea would be to measure the distance of theory T from the truth by the informativeness of the least informative P such that T + P is true, where the dotted addition function '+' is a belief revision function of the kind familiar from such discussions as Gardenfors [1992]. In the toy case of a finite propositional language with n atoms, recast P as a disjunction of basic conjunctions and informativeness will vary inversely with the required number of disjuncts: and we might propose plausible measures of informativeness for more complex cases. There could be some technical interest in considering how this approach could relate to the Oddie-Niiniluoto treatment. But I also conjecture that the philosophical pay-off, at least for my particular concerns, would be small.

21 Lewis [1986], pp. 24-7. Lewis credits the idea to Hilpinen [1976] who in turn acknowledges Lewis [1973] as the inspiration.

22 On some tunings, you also get for free a quite plausible modal logic for the monadic operator 'It is (at least) approximately true that'. Take the standard Kripke semantics and make one world accessible from another if it is close enough; this accessibility relation will evidently be reflexive and symmetric but not transitive. So if 'Approximately P' is true at a world just if P is true at some close enough world, then 'approximately' has the logic of 'possibly' in the familiar so-called Brouwerian system. Cf. Hughes and Cresswell [1968], pp. 57-8, 74-5, on the system they label 'B', or Chellas [1980], pp. 131ff., 163-4, on the system he labels 'KTB'. Hilpinen proposes this formal logic for 'approximately' in his [1976].

23 Lewis ([1986], p. 27) writes: 'An idealized theory is a theory known to be false at our world, but true at worlds thought to be close to ours', and then cheerily mentions 'ideal gases' and 'massless test particles'. He doesn't, however, address the crucial question: by what standards are the nomologically exceedingly remote worlds with ideal gases and massless test particles 'close' to ours?

24 After writing this, I discovered quite a long discussion of approximate truth in Richard W. Miller [1987]. He sums up by saying that 'The judgement that a false scientific theory is approximately true is the judgement that certain aspects of the theory are true, and that its superiority in these respects to definite alternatives is more important than the failures of other aspects, defeated by other alternatives. The rating of the success as more important than the failure is valid if it reflects the actual role of the theory's original triumph in the development of scientific theories' (p. 411). An approximately true theory need not be partially (strictly) true; and talk of 'triumph in the development of scientific theories' is overblown. But the underlying thought that judgements of approximate truth will be guided by, among other things, considerations of what role the theory actually serves seems right.

Weston ([1992], p. 53) also makes my point explicitly: 'an assertion of the approximate truth of a particular theory can only be formulated correctly with the aid of information about the specific subject matter of that theory.'

25 For some options, compare Lewis [1986], Stalnaker [1976], Rosen [1990].

26 'The structure'? What about non-standard models, etc.? Well, our geometric structures are built in the reals (more exactly, in manifolds that are piecewise [R.sup.n]), and given well-behaved equations, etc., a target structure will be unique relative to fixing the reals. So the question is: what about deviant models of the reals? A good question, perhaps. But not one I need to tackle here; for if there is a real problem here that can't be avoided by going second-order, it is equally a problem for everybody, Lewis included.

27 It might be said that the possible-world story at least was frank about its ontological commitments. We still need a story about the ontology of 'structures'. But the firm intuition is that however the story goes in detail, its commitments can be relatively limited, and certainly less than the panoply of possible worlds - for example, as suggested before, a modest amount of set-theory should give us (a la Sneed and Stegmuller) all the structures we need.

28 Obvious, that is, once it was pressed on me by George Botterill and David Owens.

29 'What scientific explanation, especially theoretical explanation, aims at is ... an objective kind of insight that is achieved by systematic unification' (Hempel [1966], p. 83). For elaboration see Friedman [1974], Kitcher [1981], and Redhead [1990].

30 But where's the metaphysical pay-off? Isn't the real interest in questions about approximate truth supposed to be due to the fact that the concept is essential in defending realism about science? - for as Boyd ([1990], p. 355) characteristically notes: 'No realist conception that does not treat theoretical knowledge and theoretical progress as involving approximations to the truth is even prima facie compatible with the actual history of science.'

True, in this paper I have no wider metaphysical ambitions: I just want to understand how the notion can be applied in a limited class of cases - and I take it the notion has application ('unless,' as David Lewis remarks, 'with absurd disdain for what we understand outside the philosophy room, we junk the very idea of closeness to the truth' (Lewis [1986], p. 24)). As far as I can see, the basic shape of my account of what makes for approximate truth in a dynamical theory of a certain sort should be equally congenial to the realist and to various brands of non-realists alike. Enthusiastic realists like Boyd, I would add, are oddly reticent about giving detailed treatments of the concept of approximate truth.

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