One of the most puzzling features of the decoherent multiverse of Everettian quantum mechanics is that the number of branches it contains is typically indeterminate. David Wallace describes the situation as follows:There is no sense in which [chaotic] phenomena lead to a naturally discrete branching process: … while a branching structure can be discerned in such systems it has no natural “grain”. To be sure, by choosing a certain discretisation of (configuration-)space and time, a discrete branching structure will emerge, but a finer or coarser choice would also give branching. And there is no “finest” choice of branching structure: as we fine-grain our decoherent history space, we will eventually reach a point where interference between branches ceases to be negligible, but there is no precise point where this occurs. As such, the question “how many branches are there?” does not, ultimately, make sense. (Wallace, in Saunders et. al, 2010, Many Worlds?, p.67-68)
Advocates of EQM use a variety of terminology to express this feature of the theory: ascriptions of branch number are ‘interest-relative’ (Saunders), are ‘arbitrary conventions’ (Saunders), are subject to ‘some indeterminacy’ (Wallace), are ‘not well-defined’ (Greaves) or ‘presuppose a piece of structure which is not present in the theory’ (Greaves). The question ‘how many branches?’ is said to be ‘a non-question’ (Wallace), with ‘no good answer’ (Saunders) or simply with ‘no answer’ at all (Wallace). Branch number ‘has no categorical physical significance; it is not part of what is really there.’ (Saunders)
This treatment of an apparently meaningful question as lacking any good answer is reminiscent of Carnapian, Wittgensteinian and neo-verificationist rejections of metaphysics. One of the best responses to these critics of metaphysics is the appeal to classical logic. If mountains have no boundaries, but it is not the case that everything overlaps with every mountain, it follows that there are no mountains. And if there is no positive number N such that N is the number of branches, it follows that there are no branches. Of course, critics of metaphysics do not deny that there are mountains; and Wallace, Greaves and Saunders do not deny that there is branching. Our puzzle, then, is how to make sense of how it can be that there is branching despite there being no positive N such that N is the number of branches.
The key to resolving the puzzle, I think, is to grant that there is something defective about the question ‘how many branches?’ but to deny that this requires it to be nonsensical or to lack a true answer. We can instead say that the number of branches is indeterminate: that there is some N such that N is the number of branches, but there is no N such that N is determinately the number of branches. The question ‘how many branches?’ is defective because it has no determinate answer, even though (determinately) it is meaningful and thus (determinately) does have an answer. This sort of response is hinted at by Wallace’s talk of ‘indeterminacy’ and by Greaves’ talk of ‘vagueness’, but it has not to my knowledge anywhere been made explicit; it also appears to be in tension with phrases like ‘non-question’ that Wallace and Greaves use elsewhere. I therefore propose it not as an interpretation of Greaves, Wallace and Saunders but as a friendly amendment to their strategy, designed to make decoherence-based EQM more palatable to the mainstream metaphysician.
The bivalent indeterminacy explanation of the defectiveness of questions about branch number is only satisfying if we can give an appropriate account of the determinacy operator. The contemporary literature on vagueness provides various frameworks for thinking about determinacy and indeterminacy that allow us to retain classical logic. I will discuss only the most prominent approaches, each of which makes use of the notion of admissible precisifications. A precisification is, as the name suggests, a way of making precise a vague expression. The characterization of admissibility varies from theory to theory.
Epistemicist approaches to vagueness say that some precisification is picked out by patterns in our community’s global linguistic usage as objectively the correct precisification, but that it is typically unknowable (for a distinctive sort of reason) which is the correct one. According to epistemicism, admissibility is epistemic in nature, and indeterminacy is a kind of ineliminable ignorance, the ineliminability of which derives from our lack of knowledge of the semantic values of the words we use.
Supervaluationist approaches in contrast hold that admissibility is semantic in nature: admissible interpretations are those whose correctness is not ruled out by the various conventions we have (explicitly or implicitly) laid down to govern our language. For supervaluationists, indeterminacy derives from semantic indecision. Fine, Lewis, and McGee & McLaughlin give influential (though differing) versions of supervaluationism. I prefer McGee & McLaughlin’s version, since (like epistemicism) it allows for the preservation of bivalence.
Let’s see how these two views apply to the indeterminacy of branch number present in decoherence-only versions of EQM. We treat ‘the number of branches’ as a vague expression, and consider various ways of making it precise. Each of these ways corresponds to some particular coarse-graining of the decoherence-based decomposition of the quantum state into components. The coarse-graining we choose is subject to constraints: if it is too fine-grained, then interactions between the components will become non-trivial and decoherence will be lost. If we make it too coarse-grained, there will cease to be any branching at all. And no coarse-graining produces non-integral values for branch number. We can therefore place two extremely minimal constraints on admissible precisifications for ‘the number of branches’ in any episode of branching: every admissible precisification must correspond to some natural number of branches, and this natural number must be greater than 1.
The constraints on precisifications described in the previous paragraph ensure the right results in the case of questions about world number. According to both epistemicism and supervaluationism, it makes perfect sense to say that there is some N such that N is the number of branches, but it is indeterminate which N is the number of branches. Despite the indeterminacy of branch number, it remains determinately true that there is branching.
As it happens, I expect that most Everettians, if they accept my account of branch number indeterminacy in terms of a precisification-based theory of vagueness, would prefer a supervaluationist reading of the indeterminacy operator. Epistemicism is widely considered implausible even in classical cases of vagueness – it is simply hard to believe in all those unknowable facts of the matter – and I see no particular reason why epistemicism should be any more or less plausible in the case of branch number than it is in classical cases of vagueness.
The claim that branch number is vague could be resisted in a number of ways. For example, van Fraassen suggests as a constraint on the usefulness of vague predicates that there be clear cases and clear counter-cases. While there are clear counter-cases for ‘is the number of branches’ (for example zero) there are no clear cases – no numbers N such that determinately N is the number of branches. This seems to me an overly restrictive conception of vagueness, however. For example, consider the predicate ‘is the smallest large number’. We might think that zero is a clear counter-case: it is as far from large as anything could be. But there are no clear cases – no number is such that it is clear that it is the smallest large number. Van Fraassen’s suggested characterization of vagueness accordingly looks inadequate. Vagueness should instead be thought of as the phenomenon of borderline cases; that characterization (which is favoured by Williamson) allows branch number to count as indeterminate.
Wallace prefers to say that it is not vague but arbitrary how many worlds there are. However, this seems to have highly implausible consequences. Arbitrariness is a property of choices, or of decisions: if a choice is arbitrary, then it is up to us how we make it. But it is not, in any coherent sense, up to us how many worlds there are. (It may in some sense be up to us what we mean by ‘world’: but given any choice of what is to count as a world, it is not up to us how many such objects there are.) Of course, it makes sense to say that for practical purposes we must pick some coarse-graining to analyse a particular physical interaction, and to say that the choice we actually make in any given situation is arbitrary; but this is perfectly compatible with (and indeed can potentially be explained by) the thesis that world number is indeterminate. Knowing that something is indeterminate but – for pragmatic reasons – picking a single precisification to work with invariably involves making an arbitrary choice.
The claim that world number is indeterminate an unsettling one. Quineans might well worry how we give adequate identity conditions to entities indeterminate in number; and prima facie such entities might give rise to a version of Evans’ argument against vague identity. There are many reasons of this general sort to doubt that vagueness is possible in fundamental reality; the Everettian, though, has a multi-purpose response. It is that, in EQM, worlds (like people and minds) are not fundamental; rather, they are emergent phenomena. Saunders and Zurek have emphasized that there need be no vagueness at the more fundamental level, the level of the quantum state. And Wallace argues forcefully that approximation and indeterminacy are characteristic features of all emergence in the non-fundamental sciences. According to this line of thought, EQM is altogether unexceptional in using vague terms in its explanations.