What is Metaphysical Equivalence

What is Metaphysical Equivalence?*

Abstract

Theories are metaphysically equivalent just if there is no fact of the matter that could render one theory true and the other false. In this paper I argue that if we are judiciously to resolve disputes about whether theories are equivalent or not, we need to develop testable criteria that will give us epistemic access to the obtaining of the relation of metaphysical equivalence holding between those theories. I develop such “diagnostic” criteria. I argue that these criteria include non-arbitrary inter-translatability, empirical equivalence, the same relative degree of theoretical virtues and the non ad hoc explanation of the existence of unobservable facts. With these criteria in mind, it becomes possible to determine whether theories are metaphysically equivalent or not, and in cases where we do not agree about whether they are equivalent, it allows us to locate the source of the disagreement.

1 Introduction

What does it mean to say that two theories are equivalent; that they amount to the same thing; that we have a case of mere verbal disagreement? How do we know when this is the case? Let us call the strong equivalence relation that holds between such theories metaphysical equivalence. Then intuitively, theories are metaphysically equivalent just if they use different language to describe the same underlying reality. That is, the part of the world described by one theory is identical to the part of the world described by the other theory, thus there would be no fact of the matter that could render one theory true and the other false.

Frequently it is argued, or at least hinted, that certain metaphysical theories are equivalent. Presentists are sometimes accused of espousing a metaphysics that is either trivially false, or equivalent to eternalism. It is suggested that although it appears that presentists and eternalists are making substantially different claims about what exists, specifically about whether any temporal locations other than the present exist, in fact this might be mere verbal disagreement. Perhaps the presentist and the eternalist mean something different by ‘everything’ so that when the presentist says that everything that exists, exists in the present, and the eternalist denies this, they are not making contradictory claims at all.[1]

Equally, one might wonder whether various forms of eliminativism and non-eliminativism are equivalent. You say that there are no composite objects, and I say that there are. It might be that “object” as you use the term is defined by its role in your entire ontological theory and so too with “object” as I use the term. Are we making radically different ontological claims, or do we mean something different by “object”? In order to answer this question we need to know whether or not our ontological theories are equivalent. Only then can we discern whether, given their respective roles in those theories “object” means the same in your mouth as it does in mine.[2]

Though in each of these cases we discover a hint of the idea that two apparently quite radically different metaphysical theories might be equivalent, it is not always clear exactly what this means, nor how we would ascertain whether or not it is true. Recently, however, we find some attempts actually to argue that in certain areas, apparently competing metaphysical theories are not competitors at all. In Platonism and Anti-Platonism in Mathematics, Mark Balaguer controversially argues that there is only one viable version of Platonism: full-blooded Platonism, and one viable version of non-Platonism: fictionalism, and that there is a sense in which these two theories are equivalent.[3]

According to the Platonist, mathematical objects exist. The fictionalist disagrees, arguing that mathematical objects are fictional just like characters such as Sherlock Holmes. Balaguer argues that for all practical purposes there is no difference between Platonism and fictionalism: both offer exactly the same vision of mathematical practice. The only difference between the two theories is with respect to ontology: Platonists maintain that mathematical objects exist, and fictionalists maintain that they do not. Balaguer argues, however, that this is no real difference at all, for there is simply no fact of the matter as to whether mathematical objects exist or not.[4] As Balaguer puts it “the metaphysical question of whether there exist any abstract objects is empty, but the two conclusions [Platonism and fictionalism] cash this out in different ways.”[5]

Though Balaguer does not use the locution of metaphysical equivalence, his conclusion is certainly suggestive of this view. For if Platonism and fictionalism provide the same account of mathematical practice, and differ only in matters of ontology, then if the ontological debate is empty, it would seem that Platonism and fictionalism are equivalent. It turns out that there is no difference between a world in which Platonism is true, and a world in which fictionalism is true.

Balaguer is not alone in arguing that what appear to be substantially different metaphysical theories are in fact equivalent. Storrs McCall and E.J. Lowe argue that two competing accounts of persistence over time, three dimensionalism and four dimensionalism, are equivalent.[6] Three dimensionalism is the view according to which objects have only three spatial dimensions and persist through time by being wholly present whenever they exist. Four dimensionalism is the view that objects have both spatial and temporal dimensions, and persist through time by being the mereological fusion of temporal parts. McCall and Lowe argue that objects can be described in either three dimensional or four dimensional language, and that these descriptions are “equivalent” in the sense that they are inter-translatable and [thus] there is no fact of the matter in the world that makes one of the descriptions true and the other false.[7]So if today I point to a rabbit, described in four dimensional language I point to a temporal part of some four dimensional rabbit. Described in three dimensional language I point to a wholly present rabbit that is strictly identical to some rabbit I pointed to yesterday. In three dimensional language, the claim that the wholly present rabbit is strictly identical yesterday and today, is equivalent to the claim in four dimensional language, that the temporal part of the rabbit yesterday is topologically connected to the temporal part of the rabbit today.[8]

McCall and Lowe’s position then, appears to be a straightforward example of a claim of metaphysical equivalence. Both Balaguer and McCall and Lowe’s contentions are interesting ones indeed. The sense in which they use the term “equivalence” is clearly the intuitive sense which I used earlier. In this sense, theories are metaphysically equivalent because they describe the same underlying reality: the sets of worlds in which each theory is true are identical, and the sets of worlds in which each theory is false are identical. Indeed, it might seem that we can simply define metaphysical equivalence in terms of sets of worlds being identical. We might then say that any two theories x and y are metaphysically equivalent iff the set of the worlds in which x is true, is identical to the set of worlds in which y is true, and the set of worlds in which x is false, is identical to the set of worlds in which y is false. On further reflection, however, this clearly will not do. For suppose, pace Balaguer, that in fact Platonism is necessarily true, and thus fictionalism necessarily false. Now suppose also that pace McCall, four dimensionalism is necessarily true, and three dimensionalism necessarily false. Then Platonism will be true in all worlds, as will four dimensionalism. So the set of worlds in which Platonism is true, is identical to the set of worlds in which four dimensionalism is true, and the set of worlds in which Platonism is false (an empty set) is identical to the set of worlds in which four dimensionalism is false. Then Platonism is metaphysically equivalent to four dimensionalism. We do not want it to be the case, of course, that any two necessarily true or necessarily false theories are metaphysically equivalent. Clearly two theories are metaphysically equivalent only if the part of the world in virtue of which one theory is true, is identical to the part of the world in virtue of which the other theory is true. This suggests that we say the following:

Two theories x and y are metaphysically equivalent iff:

(i) the set of worlds in which x is false is identical to the set of worlds in which y is false and

(ii) the set of worlds in which x is true is identical to the set of worlds in which y is true and

(iii) in every world w in which x is true, the part of w in virtue of which x is true, is identical to the part of w in virtue of which y is true and

(iv) in every world w1 in which x is false, the part of w1 in virtue of which x is false, is identical to the part of w1 in virtue of which y is false.

This gives us a reasonably straightforward definition of metaphysical equivalence. Although we might know what it is for two theories to be metaphysically equivalent, however, we do not know how to determine whether or not theories are equivalent in this manner. We have no apparatus with which either to argue that certain theories are metaphysically equivalent, or to dispute such a claim. In this paper I develop what I will call diagnostic criteria of metaphysical equivalence. These are criteria against which putatively equivalent theories can be measured, and it can be determined whether they are equivalent or not. They are also criteria that allow points of dispute between parties who disagree as to whether certain theories are metaphysically equivalent or not, to be clearly located and their significance to be noted. The methodology I employ in arriving at these diagnostic criteria is twofold. I consider the reasoning by which Balaguer, Lowe and McCall reach the conclusions they do about the pairs of theories they consider. If their arguments are sound (and perhaps even if they are not sound with respect to the theories in question) they provide at least the beginnings of some diagnostic criteria. I also consider a number of theories where our intuitions agree that the theories are not metaphysically equivalent. The idea is that the primary data we have with respect to when metaphysical equivalence holds and when it does not, are cases where we are confident theories are equivalent, and cases where we are confident that they are not. We can extract diagnostic criteria by considering these cases, in particular cases where we are confident that there is no metaphysical equivalence, and asking ourselves what it is about those theories in virtue of which they are not equivalent. By following this procedure I develop diagnostic criteria that track our confident beliefs about when equivalence does not hold, and I argue that these criteria are ones that can generally be applied to all cases to determine whether we have metaphysical equivalence or not.

2 Diagnostic Criteria of Metaphysical Equivalence

2.1 Empirical Equivalence

As we have already supposed, two theories are metaphysically equivalent only if the set of worlds in which one theory is true, is identical to the set of worlds in which the other theory is true. But how are we to ascertain whether theories are metaphysically equivalent? How are we even to ascertain whether there are any facts in the actual world that could render one theory true and the other false? For there are facts to which we have no direct observational access. If mathematical objects exist, then they are not the sorts of things you and I can go out and observe. Of course, there might be other sorts of evidence for their existence or not: there might be a priori arguments that show that such objects must exist, or cannot exist. Balaguer contends, however, that this is not so. Indeed, he maintains that there is good reason to suppose that no argument could settle the dispute about whether mathematical objects exist or not.[9]

Still, there are clearly facts to which we do have access. Mathematical objects aside, Balaguer’s contention that Platonism and fictionalism are equally good theories with no reason to choose between them, is based on the claim that they provide the same picture of mathematical practice. The observations that we would make in a Platonist world are the same observations we would make in a fictionalist world. Given that we are attempting to find a way of determining whether two theories are metaphysically equivalent, these sorts of observations provide a first test.

For if two theories are metaphysically equivalent, they must make the same observational predictions. Indeed, they must not only make the same observations in the actual world, but they must make the same observations in all worlds. We will say that two theories are empirically equivalent just if any possible observational prediction of one theory is also a prediction of the other theory.[10] Thus there is no actual or possible piece of evidence that could render one theory true and the other false. Empirical equivalence, then, is a necessary criterion of metaphysical equivalence.

2.2 Inter-translatability

While empirical equivalence is a necessary criterion for metaphysical equivalence, it is clearly not a sufficient one. Recalling McCall and Lowe’s argument for the equivalence of three and four dimensionalism, another criterion suggests itself: inter-translatability. Given the assumption that three and four dimensionalism are empirically equivalent, McCall and Lowe place the weight of their argument on the claim that the two theories are translatable (though they do not show exactly how this translation is to work).[11] We can see the same notion of translatability arising in Balaguer’s arguments.

Zalta and Colyvan suggest that one way to understand Balaguer’s claim that Platonism and fictionalism are equivalent with respect to all matters but ontology, is to understand Platonism and fictionalism as two interpretations of a single formalism: “$xAx”.[12] Then the disagreement between the two resides in the fact that the Platonist reads the formalism such that the quantifier has existential import, and reads the predicate “A” as “abstract”, while the fictionalist reads the quantifier as lacking existential import, and reads the predicate “A” as “fictional”. Hence “$xAx” under one interpretation reads, “there exist abstract objects”, and under the other interpretation reads, “there are fictions”. Since the debate about whether there exist abstract objects or not is, according to Balaguer, an empty one, we can see these two interpretations as two ways of explicating this ontological emptiness, thus explaining how Platonism and fictionalism provide the same understanding of mathematical practice and yet appear to differ so radically on ontological matters.

Let us say that two theories are inter-translatable just in case there is a function that correctly maps each of the sentences of one theory onto the sentences of the other theory. That is, just in case there is a correct translation manual.[13] And let us say that a translation manual is correct just in case it preserves the truth values of the sentences in each of the theoretical languages. This is a fairly weak notion of inter-translatability which will later need to be strengthened. Why not adopt a stronger notion of translatability according to which theories are inter-translatable just in case the translation preserves meaning? That would be problematic. If McCall and Lowe are right, then three and four dimensionalism are inter-translatable. Certainly most three and four dimensionalists would agree that there is some mapping between the sentences of the languages that preserves the truth values in each of the theories. They would presumably not agree, however, that they mean the same thing by the mapped sentences. They would not agree that “wholly present” just means “temporal stage”. This is presumably true in any case where proponents of competing theories do not already agree that they are equivalent. Best then to start with a notion of inter-translatability that side-steps this issue.

So let us consider a test case where we have strong intuitions that the theories in question are not metaphysically equivalent. Let us consider a case where two theories are empirically equivalent, and then ask ourselves whether they might also be inter-translatable. If they are both empirically equivalent and inter-translatable, then we should conclude that these two diagnostic criteria provide insufficient grounds to conclude that theories meeting these criteria are metaphysically equivalent. These two theories are limited pantheism and meteorology.

Let us suppose that limited pantheists hold that thunderous noises in the sky are the manifestation of Thor’s anger, and that water falling from the sky is the manifestation of Athena’s sorrow. Limited pantheists, however, do not hold that wind is the manifestation of some great god’s flatulence, or that snow is the manifestation of an bleak aesthetic temperament. Indeed, limited pantheists hold that most weather phenomena are correctly described in meteorological terms. So the limited pantheist will utter the sentence “Thor is angry” when and only when you or I would say “it is thundering”. Similarly, when you and I say “it is raining” the limited pantheist says “Athena is sad”. None of this means that the limited pantheist disagrees about the proximal cause of thunder or rain. Asked what causes thunder, the limited pantheist will tell a story about how electron streams heat up surrounding air which when it cools produces a partial vacuum that causes the column of air to vibrate and produce a thunderous sound. This is not the full story though according to the limited pantheist, for this process is the manifestation of Thor’s anger, and there would be no such process if Thor were not angry.

We can see how it is possible to map the sentences of the limited pantheist, onto the sentences of meteorology. Most sentences will be straightforward. “It is snowing” will straightforwardly map onto “it is snowing”. “Thor is angry” will map onto “it is thundering” and so forth. Since this mapping preserves the truth values of the sentences in each of the theories, it is a correct mapping. So we have a correct translation manual between limited pantheism and meteorology.

Given that our test case is one where we strongly believe the theories are not metaphysically equivalent, this provides good reason to suppose that these two diagnostic criteria are not sufficient for diagnosing the obtaining of metaphysical equivalence. What additional criteria might we adopt? One possibility might be that metaphysically equivalent theories are not merely inter-translatable, but rather are inter-translatable in a specific way. Consider limited pantheism and meteorology. Meteorology is an overarching theory that explains a range of diverse weather phenomena. Indeed, according to meteorology, weather phenomena are deeply causally interrelated. Not only does lightening cause thunder (by heating air volumes) but the weather system in one part of the world directly effects the weather system in other parts of the world. The meteorological theory then, is a highly integrated theory that explains a wide range of states within the apparatus of a single theory.

Now of course, limited pantheism also explains all of these weather phenomena in terms of the same causal mechanisms as the meteorologist. Ultimately though, some weather phenomena turn out to have some additional feature: being the manifestation of some god’s emotional state, a feature which other weather phenomena lack. But is there any relevant distinction to be made between thunder and rain on the one hand, and snow, lightening, storms and so forth on the other? Well of course, thunder is different to snow, and rain is different to lightening. But this does not seem to be a relevant difference: it does not show that there is some feature that both thunder and rain share and which other weather phenomena do not, that would lead us to believe that only the former phenomena are the manifestation of some psychological state. Indeed, if lightening is the proximal cause of thunder, and lightening is not the manifestation of some psychological state, then it seems odd that thunder should be such a manifestation.

What is noteworthy about the translation between limited pantheism and meteorology then, is that it seems arbitrary. Where one theory, meteorology, views a range of weather phenomena as being of the same broad kind, the other theory, limited pantheism, sees those same phenomena as belonging to two distinct kinds: meteorological phenomena and manifestational phenomena. Thus we can see the sense in which the mapping of the sentences of the two theories is arbitrary despite the fact that it is truth preserving. So let us say that there is a non-arbitrary mapping of the sentences of one theory onto the sentences of the other just in case if one theory recognises some difference in kind between certain phenomena, then so does the other theory. Then a translation manual is a non-arbitrary one just if the correct mapping of the manual is a non-arbitrary mapping.

We want our diagnostic criteria to diagnose limited pantheism and meteorology as failing to be metaphysically equivalent. The criteria of empirical equivalence and inter-translatability do not accomplish this task: we need the further criteria of non-arbitrary translation. So let us adopt this as a further criteria of metaphysical equivalence.

But now consider a different test case. Consider Russell’s sceptical scenario according to which a demon created the world five minutes ago to appear as though it has existed for billions of years.[14] Now suppose that this sceptical scenario is a competing theory to the naturalistic theory as it is presented by cosmologists, evolutionary theorists, physicists and so forth. Call these theories the demon theory and the naturalistic theory. Do our strongly held beliefs reveal that demonism and naturalism are metaphysically equivalent or that they fail to be so equivalent? Surely they do: surely they reveal the latter to be so. So we want our diagnostic criteria to track this belief.

Now suppose further that after the demon creates the universe it then ceases to interfere in any way in the universe’s causal processes. Again, these two theories are empirically equivalent. They are also inter-translatable. Whenever the naturalistic theorist says “a” the demon theorist will say “looks like a” or “the demon made the universe such that it appears that a”. We can formalise this a little by considering an example. Consider the sentence “this fossil is 3000 years old.” We can formalise this as $x (FxŸOx) where F is the predicate “is a fossil”, and O is the predicate “is 3000 years old.” This sentence will be true in the naturalistic theory, and false in the demon theory. We can, however, map the sentences of the naturalistic theory onto the sentences of the demon theory by defining a new operator “A”. The operator “A” will function a little like a fictional operator, but instead of reading “in the fiction” “A” will read “the demon constructed the world such that it has the intrinsic properties it would have had, if”. Then A $x (FxŸOx) will read: the demon constructed the world such that it has the intrinsic properties it would have had if there existed some fossils that were 3000 years old.[15] Then each sentence of the naturalistic theory will straightforwardly map onto the same sentence in the demon language, but with the addition of the operator A. It follows that any difference in kind recognised by the naturalistic theory will be recognised by the demon theory. So the mapping of the sentences of naturalism onto the sentences of demonism would appear to be non-arbitrary.

If it were a sufficient criteria of metaphysical equivalence that theories be empirically equivalent and non-arbitrarily inter-translatable, then various sceptical scenarios such as the demon theory will turn out to be equivalent to their naturalistic counterparts. Given that we do not want this to be the case, we need some further diagnostic criterion, and the case of the demon and naturalistic theories provides the hint of an insight into what that criterion might be.

3 Theoretical Virtues

3.1 Simplicity

There is something that demonism and limited pantheism have in common. Prima facie they appear to be less simple than their rival theories in that they both posit some entity, a demon or a host of gods, that their rival theories do not. This suggests that the notion of theoretical simplicity might play a role in helping us determine whether theories are metaphysically equivalent or not. A further diagnostic criteria might be that theories are metaphysically equivalent just if they are equally simple. But what sort of simplicity?

It is notoriously difficult, if not impossible, to provide any formal account of simplicity.[16] Instead we have a family of related notions, ranging from the parsimony of ontology, the descriptive length of the theory, the number of adjustable parameters in the theory, and so forth. There appear to be two obvious notions of simplicity that we could adopt as diagnostic criteria: ontological parsimony and descriptive length and complexity of the theory.

Let us consider the latter variety of simplicity first. The demon theory appears to be less simple in this respect. For it requires all of the explanatory and predictive mechanisms of the naturalistic theory. It requires that we are able to say that (the universe is as if) the dinosaurs became extinct in virtue of massive climate change due most probably to the impact of a large meteor. It requires that we are able to say that (the universe is as if) the human ancestor Australopithecus stood on two feet while its brain was still relatively small. The demon language, however, requires an extra operator that the naturalistic language does not: the A operator. So the demon theory is less simple than the naturalistic theory.

We might think that this sets the case of demonism and naturalism apart from that of Platonism and fictionalism, where the very same formalism can be translated in terms of either Platonism or fictionalism. Recall that Zalta and Colyvan’s model for understanding Balaguer involves holding that the very same formalism “$xAx” is interpreted in two different ways by the Platonist and fictionalist. The Platonist reads the quantifier as having existential import, and reads the predicate “A” as “abstract”. The fictionalist reads the quantifier as lacking existential import, and reads the predicate “A” as “fictional”.

In the case of demonism versus naturalism though, we do not simply interpret either the quantifier or the predicates differently, rather, we introduce a new operator. The cases are therefore significantly different. Or are they? If the predicate “A” in “$xAx” can be translated either as “abstract” or “fictional”, then so too the predicate “F” in “$x (FxŸOx)” can be translated as either “is a fossil” or “is an ersatz fossil”. So too just as “$” can be translated as either “there exists something” it can also be translated as “A demon created something.” As De Vito notes, it is always possible to make a theory appear simpler by burying the complexity in the atomic predicates of the language.[17]

So let us for a moment put this variety of simplicity aside, and consider ontological parsimony as a kind of simplicity. Again, prima facie we might think that the demon theory is less simple than the naturalistic theory insofar as it posits the existence of an entity, the demon, that naturalism does not. And so too for the pantheistic theory that posits the existence of “weather” gods. The case of pantheism is a good deal more straightforward in terms of its relative simplicity than is demonism. For pantheism posits the existence of gods, but clearly makes no other “ontological savings” elsewhere in its theory. It is straightforwardly less ontologically parsimonious. That falls out of the fact that the causal aspects of the pantheistic theory are the same as those of meteorology, with the gods being merely epiphenomenal. Any theory that has this structure will be less simple than its naturalistic counterpart.

Matters are not so simple when we consider theories such as the demon theory. The demon theory posits the existence of a demon, which the naturalistic theory does not. But the naturalistic theory posits the existence of numbers of animals that the demon theory does not. The demon theory holds that a certain number and kind of ersatz fossils exist. The naturalist theory holds that a far greater number of living animals once existed: for not all of the animals that once existed have left fossil traces. So there is a perspective from which we might argue that demonism is actually more ontologically parsimonious than naturalism. Of course, this is controversial, and it is so because frequently theoretical simplicity is employed as a method of choosing between competing theories: the most simple theory is considered to be the preferable. Naturalists therefore want it to be the case that naturalistic theories are more simple than various sceptical theories such as demonism, and than theories such as creationism. We should note though, that for our purposes all that matters is that it be possible to distinguish the relative degree of ontological parsimony of competing theories. If it turns out that naturalism is less parsimonious than demonism, this does not necessarily imply that it is less simple overall. It does imply, however, that the two theories are not metaphysically equivalent. So we can certainly use ontological parsimony as a diagnostic criterion of equivalence.

Given that theory complexity as a variety of simplicity is subject to the problems it is, should we employ only ontological parsimony as a measure of relative simplicity? I think not. The purpose in creating diagnostic criteria is in part to clarify where issues of disagreement lie between parties who dispute whether certain theories are equivalent or not. The more that the notion of metaphysical equivalence can be “broken down” into different components, the greater the possibility of determining precisely where disagreement lies. If it turns out that you and I disagree about whether two theories are metaphysically equivalent in virtue of disagreeing about whether they are equally theoretically complex, then this is an important discovery. It may not be a simple matter to resolve the disagreement, given that perhaps you maintain that there is complexity buried in the predicates of one theory, and I disagree. Nevertheless, that the matter is difficult to resolve does not make it any the less important to note its existence.

We should, therefore, hold that two theories have the same relative degree of simplicity just if they are both equally ontologically parsimonious, and equally theoretically complex. If we adopt the idea of relative simplicity as a further diagnostic criteria of metaphysical equivalence, this allows us to reject as metaphysically equivalent a range of empirically equivalent non-arbitrarily inter-translatable theories such as demonism and naturalism.

It might seem then, that if demonism and naturalism are ruled out as being metaphysically equivalent because they are not equally ontologically parsimonious, then on these same grounds we should rule out the equivalence of Platonism and fictionalism. For Platonism posits the existence of unobservable, causally inert abstract objects not posited by fictionalism. I think this would certainly be the case if Balaguer did not argue that there is no fact of the matter as to whether mathematical objects exist or not. If Platonism and fictionalism are equivalent, then this is so because Balaguer is right when he maintains that there is a crucial difference between abstract objects and objects such as demons and gods.

How so? Balaguer contends that since abstract objects do not exist in spacetime, you and I have no idea what a world would be like in which such objects exist. Whereas we might have a reasonable notion of what it would be for some god or demon to exist, we really have no conception of what it would be for a spatially unextended, unobservable, causally inert object to exist outside of spacetime. Sentences such as “there exist mathematical objects” fail to pick out any particular set of worlds, for we do not know what those worlds would need to look like in order for the sentence to be true. But since our usage of sentences containing terms such as “mathematical object” is all that could determine their truth conditions, it must be the case that such sentences have no truth conditions. Therefore there is no fact of the matter as to whether there exist mathematical objects.[18]

Whether or not this argument is successful is not something to be discussed here. The point is that if Balaguer’s broad contention that Platonism and fictionalism are metaphysically equivalent succeeds, then it must be because the two theories are equally simple in a way that demonism and naturalism, for instance, are not. And this equal simplicity would issue from the claim that the notion of “abstract object” is so under-conceived that the debate about whether there are any such objects is simply empty.

3.2 Explanatory Power

This brings us to the issue of whether the existence of a non-arbitrary translation of equally simple empirically equivalent theories is sufficient to conclude that the theories in question are metaphysically equivalent. In part this will depend on how one understands the relations between the various theoretical virtues. To the extent that, for instance, explanatory power is seen as a kind of simplicity, equal explanatory power will be purchased for free with equal simplicity. Generally though, it is held that explanatory power is more than just a brand of simplicity, so it remains possible that equally simple empirically equivalent theories not be equally explanatory.

Indeed, although there is a good case to be made for the equal simplicity of three and four dimensionalism since neither theory posits entities that the other does not, there is an equally good, though I think ultimately unsuccessful case to be made that they are not equally explanatory.[19] Yuri Balashov argues, for instance, that in the context of truths about special relativity and the structure of Minkowski spacetime, four dimensionalism has explanatory resources that three dimensionalism lacks.[20] Only four dimensionalism, he argues, can explain why it is that the various three dimensional objects that exist, fit together to form nice unified four dimensional volumes.

If Balashov were correct, then this would be good reason to suppose that three and four dimensionalism are not metaphysically equivalent. Why so? Why should we think that a difference in the explanatory resources of theories means that they are describing different metaphysical realities? Well clearly in some sense it does not. An incomplete or sketchy theory might in some sense be “describing the same reality” as a more complete theory. Or at least, it might be trying to describe the same reality. Still, an incomplete theory and a complete theory are almost certainly not metaphysically equivalent, since they will not pick out the same set of worlds as being true and false. If theories have different explanatory resources, then this is good reason to suppose that they will not pick out the same set of worlds as being true and false.

Lowe and McCall’s claim then, that three and four dimensionalism are equivalent in virtue of being inter-translatable and empirically equivalent, is clearly flawed. If we are to show that these theories are indeed equivalent, we will need to show that Balashov is mistaken. For theories are metaphysically equivalent iff they are empirically equivalent theories for which there is a correct, non-arbitrary translation manual, and where both theories share the same relative degree of theoretical virtues.

Fortunately, to argue that two theories are metaphysically equivalent need not involve showing that every theoretical virtue is exemplified to the same degree by both theories. For it seems plausible that many of the virtues, such as beauty, supervene on the more fundamental virtues of simplicity and explanatory power. For my part, I tend to think that theories that are equally simple and explanatorily powerful will also exemplify the other virtues to the same degree, though nothing rests on this being true. All that matters is that if all of the theoretical virtues are exemplified to the same degree by empirically equivalent inter-translatable theories, then we should consider those theories to be metaphysically equivalent.[21]

4 Why is this Metaphysical Equivalence?

But should we really conclude that theories are metaphysically equivalent if they meet all of these criteria, given that the proponents of such theories often claim to be making radically different metaphysical claims? That is, why should the mere fact of non-arbitrary inter-translatability, empirical equivalence and so forth be sufficient to conclude that there is no real metaphysical difference? Are these diagnostic criteria sufficient for metaphysical equivalence?

Suppose that we accept that two competing theories meet each of these diagnostic criteria. Then there are three possible conclusions we could draw about what to say about relation between these two theories. We could conclude that the two theories are metaphysically equivalent. We could conclude that they are not metaphysically equivalent, or would conclude that there is no fact of the matter as to whether they are metaphysically equivalent. Which response would be the correct one? If we think that the first response would be correct, then we think that our diagnostic criteria are sufficient. If we think that the latter response would be correct then we think that the diagnostic criteria are not sufficient. One reason we might think that we should conclude that the theories in question are equivalent is if we follow Lowe and McCall in accepting the principle of indifference, according to which if it is immaterial which of two theories is true, then the ontological dispute is merely apparent. According to Lowe and McCall, combining the principle of indifference with the claim that three and four dimensionalism are inter-translatable provides sufficient reason to conclude that the theories are metaphysically equivalent.[22]

What we should conclude from this principle, however, depends entirely on how we choose to understand it being “immaterial” which of the theories we choose. On the weakest interpretation of the principle, it is immaterial which of two theories we choose just if for pragmatic purposes it does not matter which of the theories we adopt: if we achieve the same result using either theory. Clearly on this reading, the principle is insufficient reason to conclude that the theories are equivalent, since the principle could be true even if the theories in question were not even empirically equivalent. On the strongest reading of the principle, it is immaterial which of two theories one adopts just if the theories are equally explanatorily powerful, equally simple and empirically equivalent. This seems to amount to little more than the claim that it is immaterial which theory one adopts just if those theories meet the diagnostic criteria. But so what? That fails to address the underlying question of why it is that theories that meet these criteria should count as metaphysically equivalent.

So let us consider the position according to which there is no fact of the matter as to whether theories that meet the diagnostic criteria are metaphysically equivalent. In most cases this would seem to be an implausible view. We might think though, that if Balaguer is right, then the case of Platonism and fictionalism is a case where this is the conclusion that should be drawn. For if Balaguer’s argument succeeds and there is no fact of the matter as to whether abstract objects exist, then perhaps the correct conclusion to draw is that there is also no fact of the matter as to whether Platonism and fictionalism are metaphysically equivalent. Platonism and fictionalism are equivalent only if the set of worlds in which Platonism is true is identical to the set of worlds in which fictionalism is true. But Balaguer’s central contention is precisely that we have no conception of what a world would need to be like with respect to the existence of abstract objects. That is, he maintains that even if we knew all of the facts about some world, we would not know whether or not that was a world in which there exist abstract objects. When we consider more carefully, however, we will see that even in the case of Platonism and fictionalism it is implausible that there is no fact of the matter as to whether the theories are metaphysically equivalent.

Why so? Well suppose Balaguer is right and statements about abstract objects have no truth conditions. If we think that in principle there is no way that sentences such as “there exist mathematical objects” could have truth conditions, then it will always be the case that the set of worlds expressed by the sentence “there exist mathematical objects” and “there do not exist mathematical objects” will be the same. In that case though, we should surely conclude that Platonism and fictionalism are metaphysically equivalent. On the other hand suppose we think that in virtue of some future discovery or conceptual breakthrough “there exist mathematical objects” has truth conditions, but these are conditions of which we are currently unaware. In that case, assuming that the truth conditions are determinate it is now either true or false that there exist mathematical objects, and thus either Platonism or fictionalism are true, but not both: they are not equivalent.

We might think though, that there is no fact of the matter as to whether Platonism and fictionalism are metaphysically equivalent, if we hold that “abstract object” is so semantically under-described, or so vague, that there are any number of ways of specifying or precisifying it. On some specifications or precisifications it comes out as true that there exist such objects, and on other specifications or precisifications it comes out as false. So it is indeterminate whether or not there exist mathematical objects. So we might say that it is indeterminate whether or not Platonism and fictionalism are equivalent.

But of course this is no special feature of metaphysical equivalence, it is simply a case of semantic indeterminacy. The semantic indeterminacy might be mere semantic indeterminacy, that is, it might issue only from an indeterminacy of semantics. Or, it might, according to some, at least in some cases issue from a deeper ontological indeterminacy.[23] Even if this latter is true, however, as long as each precisification of the sentence in question has truth conditions, the sentence will be true or false relative to a given precisification. So, for instance, even if “there exist abstract objects” is indeterminate in virtue of some ontological vagueness, on each of the precisifications it will either be true that abstract objects exist or that they don’t. Relative to a given precisification then, either Platonism will be true, or fictionalism will be true, and thus relative to any precisification they will not be equivalent. So for every precisification it will be the case that Platonism and fictionalism are not metaphysically equivalent: it will be supertrue that they are not equivalent.

4.1 The explanation of unobservable

What all this tells us is that except in cases of semantic indeterminacy, we should expect there to be some fact of the matter as to whether theories are metaphysically equivalent or not. But on what basis should we favour the view that two theories are metaphysically equivalent over the claim that they are not? What licenses moving from the claim that two theories meet our diagnostic criteria, to the conclusion that they are metaphysically equivalent? Well if two theories meet these criteria, then they fail to be metaphysically equivalent just if some unobservable fact obtains in virtue of which one theory is true and the other false.

Many facts are unobservable. There are unobservable facts about the big bang. There are unobservable facts about the existence or not of other worlds that are causally inaccessible to this world. If modal realism is true, then it is so in virtue of facts that are unobservable to us. Although these facts are unobservable, however, we understand what sort of facts they are. Scientific theories about the big bang, or the inside of a black hole, tell us not only that there are certain facts and that some of these facts are unobservable, but they explain why it is that the facts that are unobservable are so. So too although it is impossible for us to observe worlds that are causally isolated from us, we at least have some understanding of what such worlds would be like, and why it is that they are unobservable. In each case proponents of these theories can point out what facts would need to obtain for the theory to be true. Further, broad features of the theory itself explain why it is that some of those facts are unobservable.

There are two senses in which we can talk about unobservable facts. There are unobservable facts that are posited by a theory in an integrated, systematic way that provides additional explanatory power to the theory, and where this is an explanation for the unobservable status of the facts. Positing the existence of viruses, bacteria, or atoms prior to being able to observe such entities would be a case in point. Then there are unobservable facts posited by a theory in a non-integrated way, as an ad hoc measure that provides no additional explanatory power to the theory. An example of this would be the case where those who hold that two theories are not metaphysically equivalent, hold that there must be some fact that determines that one theory is true and the other false, and that since the fact is not observable, it must therefore be unobservable.

Whether theories are metaphysically equivalent or not depends on the particular details of the theory. If there is a plausible non ad hoc explanation for why the relevant truth making fact is unobservable, then there is some reason to think that the theories in question are not metaphysically equivalent. On the other hand, if no such explanation is forthcoming, then we should conclude that the theories are metaphysically equivalent. This will be true in general: if two theories meet all the requirements for being metaphysically equivalent, then we should conclude that they are metaphysically equivalent, unless there is a plausible, non ad hoc explanation for the unobservability of the relevant truthmaker. For unless there is some explanation as to the nature of this special fact, then we must ask ourselves, what could this fact be? Unless we can answer that question, positing its existence can amount to no more than a desperate, unwarranted attempt in the face of all evidence to the contrary, to maintain that two theories are not metaphysically equivalent.

So a further diagnostic criterion for metaphysical equivalence is that there be no principled, non ad hoc explanation for the existence of the relevant unobservable facts.

5 How does this account help?

We now have some diagnostic criteria of metaphysical equivalence. Theories are metaphysically equivalent just if they are:

(i) empirically equivalent

(ii) non-arbitrarily inter-translatable

(iii) have equal degrees of the theoretical virtues

(iv) there is no principled explanation for the existence of the relevant unobservable facts in virtue of which they would fail to be equivalent.

It might seem though, that some of these criteria are unnecessary. We could, it might be argued, first determine whether there are any observable facts in virtue of which one theory is true and the other false. If we determine that the theories are empirically equivalent, then there is a further question about unobservable facts: are there any unobservable facts in virtue of which one theory is true and the other false? To determine whether there are such unobservable facts we turn to the idea of explanatory ad hocness, and ask whether or not the positing of such facts is part of the apparatus of the theory, or merely an ad hoc measure. If it is purely ad hoc, then we can conclude that the two theories are metaphysically equivalent, if it is not ad hoc, then we can conclude that they are not metaphysically equivalent. We then need no recourse to the criteria of translatability or relative degree of theoretical virtues.

We might think that such a proposal is a good one. After all, there is considerable controversy about inter-theoretic translation, and more still about the notion of theoretical simplicity and explanatory power. So using such notions as diagnostic criteria is surely adding little to our ability to determine whether theories are metaphysically equivalent or not. But there are two reasons why rejecting these additional criteria is not a good idea. First, consider how we are to ascertain whether the positing of the relevant unobservable facts is principled or ad hoc. Suppose I am considering the three and four dimensionalism as theories. Everyone agrees that the theories are empirically equivalent. Is there some non observable fact in virtue of which one is true and the other false? Well if four dimensionalism is true, then it is because objects perdure: they are composed of temporal parts. That fact is unobservable, but there is clearly a sense in which it is not ad hoc: after all, it is precisely part of the apparatus of the theory. Does this resolve whether or not the theories are metaphysically equivalent or not? No. For we have assumed that the theoretical terms as they are used by the four dimensionalist, are the same as the terms as they are used by the three dimensionalist. But if these terms are in part defined by their role in the theoretical apparatus in which they are embedded[24], then there is no reason to suppose that when the three dimensionalist denies that there exist temporal parts and thus that objects perdure, that she is denying what the four dimensionalist is affirming. The claim that three and four dimensionalism are metaphysically equivalent and thus inter-translatable, is the claim that they mean something different by terms such as “part” such that although they appear to be making contradictory claims, in fact they are not. Whether or not there is some principled unobservable fact in virtue of which four dimensionalism is true and three dimensionalism is false, is impossible to determine independent of knowing whether the two theories are inter-translatable.

Second, the additional diagnostic criteria are invaluable in determining precisely where parties disagree about whether theories are equivalent or not. This is not to say that the lack of a clear account of translation, simplicity and explanatory power is not troubling, and for a whole host of reasons. At present though, it is often unclear why it is that proponents of the view that certain theories are equivalent think that they are so, and equally unclear why those who disagree do so. Just where does the disagreement lie? By providing an account that sets out a number of criteria that metaphysically equivalent theories must meet, it becomes considerably more straightforward to see where disagreement lies. This is all important.

If parties disagree about whether or not theories are equivalent in virtue of a disagreement about whether they are equally simple, this tells us that we need to be clear about whether the parties are using the same notion of theoretical simplicity. If a disagreement hangs on a controversial understanding of simplicity, it is imperative for both parties to be aware of this difference. So too if a disagreement arises from an odd construal of explanatory power it will be important first to sort out the underlying dispute before attempting to determine whether or not there is genuine disagreement about equivalence. Similarly, if you and I disagree about whether or not there is some correct translation of two theories, are we disagreeing about which sentences should come out as true in each theory?; are we disagreeing about how the mapping of sentences occurs?; are we disagreeing about how the process of translation itself should be undertaken? All of these complexities need to be sorted out if we are to prevent parties from simply talking past one another instead of genuinely engaging with the issues at hand. My criteria of metaphysical equivalence allows us to begin to see what other issues need to be resolved in order to come to grips with claims of equivalence.

So although this account is by no means a panacea, it does the apparatus with which to clarify the nature of disputes, and to provide a framework within which to argue that theories that we might have thought were radically different, are in fact equivalent.

References

Balaguer, M. (1998). Platonism and Anti-Platonism in Mathematics. New York: Oxford University Press.

Balashov, Y. (2000a). “Relativity and Persistence.” Philosophy of Science. 67(3): 549-562.

Balashov, Y. (2000b). “Enduring and Perduring Objects in Minkowski Space-Time.” Philosophical Studies 99: 129-166.

Balashov, Y. (2000c). “Persistence and Space-time: Philosophical Lessons of the Pole and Barn.” The Monist. 83(3): 321-240.

Carnap, R. (1936) “The elimination of metaphysics through logical analysis of language.” in A.J. Ayer Language, truth and logic 1946 Harmondsworth: Penguin.

DeVito, S. (1997). “A gruesome problem for the curve-fitting solution.” British Journal for the Philosophy of Science 48(3): 391-396.

Hoefer, C. and A. Rosenberg. (1994). “Empirical Equivalence, Under-determination and Systems of the World.” Philosophy of Science. 61: 592-607.

Howard-Snyder, F. (1991). "De Re Modality entails De Re Vagueness. Philosophical Quarterly 72: 101-112.

Hyde, D. (1998). “Vagueness, Ontology and Supervenience.” The Monist. 81: 297-312.

McCall, S. (1994). A Model of the Universe: Space-Time, Probability, and Decision. Oxford: Clarendon Press.

McCall, S. and E.J. Lowe (2003). “3D/4D equivalence, the twins paradox and absolute time.” Analysis 63(2) 114-123.

Merricks, T. (2000). “No Statues.” Australasian Journal of Philosophy, 78(8): 47-52

Quine, W.V.O. (1975). “On Empirically Equivalent Systems in the World.” Erkenntnis 9: 313-328.

Russell, B. (1921) The Analysis of Mind London: Allen and Unwin

Sider, T (1999). “Presentism and Ontological Commitment” Journal of Philosophy 94(7) 325-347.

Sober, E. (1975). Simplicity. Oxford: Clarendon Press.

Tye, M. (1990). “Vague Objects.” Mind 99: 535-557.

Unger, P. (1979). "I Do Not Exist". in Perception and Identity ed G. F. Macdonald. London Macmillan Press.

Zalta, E.and Colyvan, M. (1999). “Mathematics: Truth and Fiction?” Philosophia Mathematica 7(3): 336-349.

* With thanks to David Braddon-Mitchell, Mark Colyvan and Dominic Hyde for helpful discussion of these issues.

[1]Ted Sider raises this possibility in his (1999).

[2]For a discussion of eliminativist issues regarding composition, see Merricks (2000) and Unger (1979).

[10]Empirical equivalence as I define the term is sometimes referred to as strong empirical equivalence. This is opposed to weak empirical equivalence where any observational prediction about the actual world made by one theory is also a prediction of the other theory. For a discussion of this distinction see Hoefer and Rosenberg. (1994). Notice also that we might be tempted to say that theories are empirically equivalent just if any nomologically possible prediction of one is a prediction of the other. For we might think that what happens in some radically different logically possible but nomologically impossible world is irrelevant. But notice that there is no nomologically possible piece of evidence that could render two different laws of nature differentially true or false, yet we probably do not want to say that different laws of nature are empirically equivalent, and certainly do not want to say that they are equivalent. So we need to countenance the nomologically impossible worlds. For what is relevant is that there are nomologically impossible states of affairs which render one but not the other theory false.

[11] For a defence of the claim of inter-translatability and an account of this translation see my “The metaphysical equivalence of three and four dimensionalism” forthcoming.

[15]The details of the A operator would need to be more fully spelled out, but the idea is clear enough.

[19]For a defence of this claim see my “The metaphysical equivalence of three and four dimensionalism”forthcoming

[20] Balashov (2000a); Balashov (2000b); Balashov (2000c).

[21] Once we conclude that metaphysically equivalent theories must share the same degree of theoretical virtues, it seems that the non-arbitrariness constaint on translation is superfluous. For if one theory recognises a difference in kind that the other theory does not, then it will surely be the case that these theories have different degrees of simplicity and explanatory power. Still, since theoretical simplicity, and even explanatory power, are such controversial notions, it will be as well to include this feature in an account of metaphysical equivalence as a useful additional marker.

[23]For a defence of the view that there is ontological vagueness see Howard-Snyder (1991); Hyde (1998) and Tye (1990).

[24]Where their role in the theory can include causal facts about various theorists and so forth.