THE UNDERDETERMINATION OF METAPHYSICS BY PHYSICS
We have now reviewed a number of different ways in which one might 'flesh out' the metaphysical package of particles-as-individuals, together with some of the more obvious problems that this package must face. If it is agreed that these problems can be overcome, then the conclusion is that the formalism can be taken 12212i820m to support two very different metaphysical positions, one in which the particles are regarded as 'non-individuals' in some sense and another in which they are regarded as (philosophically) classical individuals for which certain sets of states are rendered inaccessible.
There is, then, a kind of underdetermination of the metaphysics by the physics and thus good reason to be sceptical of claims that physics forces us to drop the view of elementary particles as named individuals. 168 It then follows that the claims of certain 'naturalist' philosophers that we can simply read our metaphysics off the relevant physics are highly suspect. Quine, for example, has drawn on the Received View to argue that developments in physics this century support the evaporation of physical objects-at foundation, elementary particles-into nothing more than regions of space-time bearing certain properties. 169 Similarly, and more recently, Resnick has sought support for his 'structuralist' view of mathematics in the same developments (1990). In particular he claims that epistemic differences between mathematical and physical objects are blurred by recent physics. 170 Even Lucas can be set among these odd bedfellows as he argues that the disappearance of transcendental individuality from the 'categories of ultimate reality' 171 has contributed to the opening up of 'new vistas of rationality' and supports, in particular, an anti-reductionist account of theory autonomy. 172
More generally Fine, in his 1988 Presidential Address to the Philosophy of Science Association, urged the assembled cohorts to ". actively engage philosophy with on-going science" and reminded us of ". the potential in science itself for addressing virtually all the sorts of interpretative questions and issues that philosophy traditionally pursues". 173 Someone who has taken this latter naturalistic claim very seriously is Shapere, who has argued that science can in fact resolve traditional philosophical questions concerning, for example, identity and existence. 174 At the same 1988 meeting, he suggested that ". by being internalized into the scientific process, even such concepts as explanation and existence can be subject to alteration in the light of what we learn". 175 The problem is, it is not always clear what it is that physics teaches us!
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Of course, in some cases the lesson does seem pretty straightforward, even blunt: as we have indicated, a strong case can be made that the Principle of Identity of Indiscernibles can be ruled out in the quantum domain. Such conclusions give the lie to the claim that physics cannot yield any metaphysical conclusions at all. And indeed, the fact that the early defenders of the Received View, together with philosophers of the stature of Quine, for example, have attempted to draw such conclusions suggests that the establishment of this metaphysical underdetermination is a significant result. Those readers who are familiar with the realism-antirealism debate in the philosophy of science will, of course, recognize this kind of conclusion. In that context it is theories themselves which are underdetermined by the empirical data and this has been deployed by the anti-realist in order to undermine realist claims that we can believe a particular theory to be true, on the basis of its explanation of such data. (As we will see shortly, this metaphysical underdetermination has also been taken to have implications for realism.) The history of the underdetermination of theories by data can be traced back to the debates of the early twentieth century concerning the nature of space-time. In these debates a 'conventionalist' view was often adopted, whereby the particular view of space-time one held was understood to be based on an initial convention concerning measurement. Similarly, our metaphysical underdetermination in the quantum context has also been linked to a form of conventionalism. Belousek, for example, has insisted that,
Whether or not identical particles are (in)distinguishable [where by this he means (non-) individuality] . is conventional in the (Duhemian-Einsteinian) sense that is justified ultimately by an epistemologically prior choice between two (or more) observationally equivalent sets of initial hypotheses, a choice that is necessitated by neither logic nor experiment, but rather is guided by normative-methodological criteria of theory appraisal. 176
This conclusion has been disputed on the grounds that use of the term 'conventional' in this context suggests that there is something 'arbitrary' in the metaphysical packages above, whereas they are arrived at on the basis of reasoning which is better described as 'fallible'. 177 But this is not the intention at all: again, consider the case of conventionalism in space-time physics. There the choice of an initial measurement standard was not at all taken to be determined arbitrarily or on the basis of an epistemological whim. The point
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was, the physics itself permitted different standards-supported by different lines of fallible reasoning-to be applied and likewise in the quantum case, adopting a different understanding of the Indistinguishability Principle leads to a different metaphysical position. Describing the adoption of that different understanding as 'conventional' should in no way be taken to imply that it is arbitrary.
Having established this last point, let us move on to consider whether there are any reasons for preferring one metaphysical package over another; or, putting it in the form of a question, is there any way of breaking this underdetermination?
One option is to note that the individuals package depends on such metaphysically problematic notions as that of substance and primitive thisness. However, this will cut little ice with defenders of such concepts, who will point a similarly accusing finger at the alternative notion of a 'non-individual'. Indeed, they can argue that at least the former package allows us to deploy our standard logical and set-theoretical resources, whereas it is not clear how the idea of non-individuality is even expressible in formal terms. 178 Subsequent chapters of the present work can be taken as an attempt to respond to this argument.
A more robust line would be to claim that the non-individuals package meshes better with the framework of quantum field theory and is thus to be preferred on these grounds alone. 179 There are two aspects to this 'meshing' which need to be separated. The first emphasizes that on standard accounts of quantum field theory particle indices are not assigned from the word go and thus particle permutations are undefined. 180 However, state spaces for QFT can be constructed which do involve such indices. Thus van Fraassen has insisted that appealing to QFT cannot resolve the underdetermination, since this theory is,
. equivalent to a somewhat enriched and elegantly stated theory of [individual] particles. That we can take it as a description of a world that is particle-less only masquerades as an incompatible alternative. 181
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The basis for such a claim is the relationship of representation which is purported to hold between models of QFT and 'concrete' constructions of Fock space carried out within de Muynck's 'labelled particle' approach. 182 This suggestion has been criticized by Butterfield, who has cast doubt on the empirical equivalence that is supposed to exist between QFT and many-particle quantum theory. 183 More relevantly in this context, he has suggested that the existence, within QFT, of states that are superpositions of particle number render van Fraassen's claim false. The idea here is that on any reasonable account of individuality, the number of individuals must be definite. To deny the latter leaves one open to the objection that the entities concerned are not individuals at all. 184
We shall return to these issues in Chapter 9, where we shall see that difficulties emerge in the quantum field theoretic context as well.
The second aspect of this 'meshing' between the particles-as-non-individuals package and QFT is more methodological in nature. If the particles are conceived of as individuals subject to restrictions on the sets of states they may occupy, then those states that are inaccessible to the particles of a particular kind correspond to just so much 'surplus structure' in our theoretical description. In particular, if the particles are regarded as individuals and particle indices introduced, then it is entirely mysterious as to why, in addition to the standard symmetrized, anti-symmetrized and mixed-symmetric (corresponding to parastatistics) combinations, a particular subset of these inaccessible, surplus states, namely those that are non-symmetric, are not only not realized in nature but appear to serve no useful theoretical purpose. They might be thought of as both empirically and theoretically 'surplus'. Applying the general methodological principle that a theory which does not contain such surplus structure is to be preferred over one that does, Redhead and Teller conclude that we have grounds for preferring the non-individuals approach and this mystery simply does not arise. 185
Now, this methodological principle of 'avoid surplus structure' is an interesting one. The phrase itself derives from a paper by Redhead in which he used this notion to explore the idea that when a theory of physics is embedded
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in some mathematical framework, we often get more mathematics than we need to express that particular theory. 186 Giving a physical interpretation to certain elements of this surplus mathematical structure can then lead to fruitful extensions of the theory, as in the case of the positive and negative energy solutions of the Dirac equation, for example. Thus in its original incarnation, the notion of surplus structure was introduced as a positive aspect of the applicability of mathematics to physics, which embodied the heuristic advantages of introducing mathematical structures. Somehow this was transformed into a negative methodological principle whereby a package which is taken to contain less surplus structure than another is deemed to be more acceptable. Given the original heuristic understanding, however, the dangers of adopting such a methodological principle are obvious.
Consider the case of parastatistics, for example. If physicists had adopted Redhead and Teller's principle, then paraparticle states would have been ruled out as just so much surplus structure. Yet, as we indicated in Chapter 3, the exploration of the possible application of parastatistics to quarks was absolutely crucial for the early development of quantum chromodynamics. Teller does appear to acknowledge this point 187 but insists that, leaving the parastatistics case to one side, other, non-standard statistics have not found such a heuristic role and count as just so much surplus baggage. Even more robustly, in their reply to Belousek's 'conventionalist' understanding of the under- determination, Redhead and Teller claim that the general quantum mechanical formalism, 'together with an extraordinarily large array of experimental facts', support the conclusion that only Bose-Einstein and Fermi-Dirac statistics exist in nature. 188 The kinds of 'facts' they have in mind here are the following: if photons, for example, did not obey Bose-Einstein statistics, then the absorption and emission of light would be very different; if electrons were to exist in superpositions of symmetric and anti-symmetric states, then all the elements in the periodic table from lithium on up would possess states in which the 1s shell would have more than two electrons and hence the chemical properties of these elements would be very different.
This is a strong claim, but it can be contested. It turns out there is a large body of experimental and theoretical work exploring the possibility that 'standard' quantum statistics might be violated, albeit if only at the margins, as it were. 189 Greenberg and Messiah pointed out as early as 1964 that then possibility that known particles, such as electrons, might obey parastatistics could be easily ruled out by consideration of collision experiments, for example, or simply examining the helium spectrum. Nevertheless, the possibility remains that such particles might exhibit small deviations from standard quantum statistics under certain circumstances. Exploration of this possibility has proceeded in both experimental and theoretical directions. 190 With regard to the theoretical, Greenberg and Mohapatra have proposed a form of 'paronic' statistics, according to which two indistinguishable fermions, say, can be in the normal antisymmetric state with a certain probability and in the anomalous symmetric state with the related probability. 191 Unfortunately, as they acknowledge, there are difficulties in constructing a quantum field theoretic version of such a theory, since the representation of number operators and creation and annihilation operators in a positive metric Hilbert space rules out all but standard and parastatistics. Nevertheless, the relationship between paronic theory and experimental violations of the standard statistics has been pursued at the 'phenomenological' level in the context of quantum mechanics. 192
Greenberg himself has attempted to introduce an alternative operator formalism which accommodates small violations of standard statistics through the introduction of a parameter, q, which can occupy intermediate values between −1 and +1. 193 According to the formalism of these q-mutators, there is a probability of (1 + q)/2 for two particles to show bosonic behaviour and a probability of (1 − q)/2 that they would show fermionic behaviour. For electrons, say, small violations of Fermi-Dirac statistics would be represented by q being close to −1.
The objective of much of this work is to test the experimental and theoretical validity of the spin-statistics theorem which reduces the statistical behaviour of an assembly of particles to one of their intrinsic properties, namely spin; thus, particles with integral spin should obey Bose-Einstein statistics and those with half-integral spin should obey Fermi-Dirac. Obviously, if the theorem could be proven then non-standard statistics would be ruled out; 194 unfortunately, no satisfactory proof has been forthcoming. 195 Hence the situation
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is rather more complex than Redhead and Teller let on. It would certainly be a bad move to adopt a methodological principle that ruled out all non-standard wave functions, including those relevant to parastatistics, fractional statistics, ambistatistics and so on. 196 We have already noted the possible application of parastatistics to quarks and ambistatistics to geons and if violations of the spin-statistics theorem were to be discovered experimentally, then Green's q-mutator formalism might well be the appropriate one to adopt. If we understand surplus structure as it was originally intended, then such structure in these cases might prove rather fruitful. Of course, there may be mathematical structures-corresponding to completely non-symmetric functions for example-which are unlikely to prove at all fruitful in this way, but it is not clear whether, as a methodological principle, 'avoid surplus structure' is honed enough to cut these out whilst leaving the more interesting combinations. At the very least, we can conclude that there exists a tension between the heuristic role of surplus structure and its incorporation into methodology.
From the perspective offered by the former understanding, it should come as no surprise that when we embed quantum mechanics in group theory, we get 'surplus' representations that we don't apparently need to represent the particles that we know of. 197 One could then simply adopt the attitude that the apparent mystery is a mere fabrication: the inaccessible non-symmetric states can be ruled out as simply not physically possible. 198 The surplus structure, then, is a consequence of the representation chosen and has no further metaphysical significance. 199
Still, the point can be pressed that this leaves a mystery as to why this structure is not realized and that one of the jobs of a theory is precisely to explain away such mysteries. One can draw an analogy with statistical mechanics: this theory can explain why we never observe a cold cappuccino starting to boil, whereas the quantum-particles-as-individuals view cannot explain why we never observe states of non-standard symmetry. 200 Even without taking into account the points raised above, the analogy itself is problematic. Let us reflect on our cold cup of cappuccino. Statistical mechanics does not tell us that it will never boil, but only that the probability of this occurring is extremely low. The question then reduces to that of 'why is this probability so low?' This in turn is answered by appealing to the very low number of states which correspond to the coffee boiling compared to the vast number
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of states for which it remains cold. If we were to press on and demand to know why there should be this disparity in the number of states accessible to the cappuccino-or, equivalently, why we should find ourselves in a region of entropy increase-then an answer might be given in terms of the initial conditions pertaining to our region of the universe. Why do we never observe entropy decreases in our cups of coffee? Because that's the way our region of the universe is, as expressed in the initial conditions. A similar line can be taken in the case of quantum statistics. Why do we never observe non-symmetric states, given this metaphysics of individual particles? Because that is the way the universe is and we should not expect quantum mechanics alone to have to explain why certain initial conditions obtain and not others. Here we recall that the symmetry of the Hamiltonian ensures that if a particle is in a state of a particular symmetry to begin with, it will remain in states of that symmetry. Hence, if non-symmetric states do not feature in the initial conditions which held at the beginning of the universe, they will remain forever inaccessible to the particles.
At some point the explanatory buck has to stop and the theory itself can't do all the work. Of course, you might prefer the alternative explanation in which non-symmetric states are ruled out from the word go, but then you might legitimately be urged to flesh out the metaphysics and logic of 'non-individuality'. No explanation of this sort comes without a price attached. In subsequent chapters we shall indicate how this cost may be off-set by setting out just such a logic. Before we do that, however, we need to address a further issue associated with the view of particles as individuals-namely the nature of the indices used to label the particles. As we shall see, regarding such labels as forms of names generates certain problems which will lead us nicely into our consideration of an appropriate formal framework.
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