3.1. Emergence

The first example of fundamental rethinking is a reconception of the relationship between fundamental ontology and the manifest structure of the world. The relationship between the ontology of physics and the everyday world seemed relatively simple in the classical context. Big things were made up of little things, themselves localized in space and time. The cracks in this simple view were showing already in the phenomena that originally prompted talk of emergence in biology and complex systems, but it is much more obviously unworkable in the Everettian context, where the gap between the way things seem to the inhabitants of the universe and the way that the theory says they are is very large. Wallace was among the first to recognize the need for a reconception of this relationship and to look to emergence as the proper account (see also Ladyman and Ross Reference Ladyman, Ross, Spurrett and Collier2007). He makes a considerable contribution toward turning it into a workable explicit account. So conceived, emergence has general import quite outside the context of Everett. This is a nuanced, explicit account of the relationship between fundamental and nonfundamental ontology, as well as a radical improvement on old and outmoded ideas of reduction, explicit definition, and mereological composition.

According to the method for doing naturalistic metaphysics that Wallace advocates, we take our ontology from our best physical theories, interpreting them at face value. How do we find in the world described by the theory structures that correspond to things like tables, chairs, tigers, and chipmunks? We first figure out what role they play in our epistemic and practical lives (how we learn about and interact with them), and then we find something in the base ontology that satisfies that role. This makes Wallace a functionalist about nonfundamental structures of all kinds. We see the method applied in part 1 to identify structures that correspond (well enough) to worlds to warrant the description of EQM as a many-worlds theory. The emerging quasi-classical decoherent histories count as worlds because they satisfy the ‘world’ role. We see it applied in part 2 of the book to the concept of probability.

This method, quite generally, is the strategy for ‘interpreting’ a fundamental theory. It turns the Ramsey/Lewis/Horwich method on its head. Instead of trying to implicitly define theoretical primitives in everyday or observational vocabulary, it treats the theory’s basic concepts as ontological primitives and interprets everyday concepts in that ontological setting by identifying something that plays that role.

3.2. Probability

The second example of this rethinking is the discussion of probability. Common wisdom in the nonquantum setting is that we have probability only when we have indeterminism. The Everettian has to reject this link. Looking in the philosophical literature for an explicit account of what probability is even in the classical setting, Wallace finds it wanting. So he starts from scratch. What goes for tigers and worlds goes also for probability. If chances are whatever justifiably plays the role of probabilities in practical and epistemic reasoning, then with the decision-theoretic proof, which intends to show that the numbers derived from the Born rule do just that, Wallace has shown that the Born probabilities are chances.

We noted in section 2 that Wallace is a rationalist about probability. He wants to establish that the use of Born probabilities in decision is not just a strategically good policy but a policy that is rationally compelled, so that one would not just be making a pragmatic mistake if one failed to set one’s credences to the Born probabilities; one would be making a logical mistake. These rationalist leanings, however, are not a necessary part of EQM, and we want to suggest here that one could adopt a purely pragmatist view of probabilities. According to such a view, the ontological content of the theory makes no use of probabilities. There is a story that relates the ontology to the evolution of observables along a family of decoherent histories, and probability is something that plays a role in the cognitive life of an agent whose experience is confined to sampling observables along such a history. In so doing, one would still be doing EQM, and this interpretation may have quite strong advantages. Here is how it might be developed.

Let us begin by looking at the kind of emergent probabilities in a classical setting that casinos, lotteries, and insurance companies rely on, the probability that a roll of an ordinary pair of dice will come up sixes, that a quarter drawn from the general population will land heads if tossed, or that a person with the BRCA1 gene will contract cancer. These probabilities are emergent in the sense of the previous subsection in that the fundamental ontological story is deterministic and that probabilities are a theoretical construct useful in describing and predicting higher-level behavior. These kinds of probabilities are general in that they are defined for types rather than tokens, and their basic form is conditional. Hence, for example, we have the probability of heads in the toss of a coin under specified conditions. They are not defined for any old reference class (the best-known system for which there is a rigorous analysis is the coin toss), but only in those in which there is the right sort of ‘randomness’ so that any not too carefully chosen subset has a relative frequency that more or less matches the others. The dynamical underpinning of these probabilities, where they exist, varies from case to case (see Diaconis Reference Diaconis1998; Hoefer Reference Hoefer2007; Ismael Reference Ismael2009). They are, of course, related to frequencies, insofar as they may be inductively derived from stabilized relative frequencies, and insofar as they will be empirically successful in predicting them; but some of the complaints about frequentist accounts are inapplicable since this account does not aim for a content-preserving reduction of probability to frequency. There is no direct route from observed frequencies to probabilities. Theory is needed to prescribe reference classes. Whether I can apply probabilities derived from stabilized relative frequencies over the class of fair dice to generate expectations for the next roll of this particular pair depends on whether this pair is fair. And ‘fair’ does not mean any pair that happens to exhibit certain regularities. It is a theoretical concept defined in terms of certain (fundamental and hence intrinsically nonprobabilistic) structures, in particular, dynamical symmetries. The theory contains a rule with certain physical structures as inputs and certain numbers (the probabilities) as outputs. The inputs are solidly objective,Footnote ³ but the outputs need not be reified (see Ismael, Reference Ismaelforthcoming).

Let’s call these general probabilities. They are quite familiar philosophically. Their existence is entirely compatible with determinism. Indeed, the deterministic dynamics will explain how they arise in many cases (although only for ‘typical’ initial conditions). They play a readily explicable role guiding belief where we are otherwise ignorant. The general probability of (x/y) gets transformed into the single-case probability that some particular y will be an x, and adopted as credence if all that one knows about the system or event in question is that it is a y. On this view, general probabilities are statistical probabilities grounded by a theory that has the right holistic fit with relative frequencies. Chances of the kind implicitly defined by the Principal Principle might be thought of as single-case probabilities extracted from general probabilities by conditionalizing on everything we generally think knowable about a system and deployed to set credence and to guide decision.Footnote ⁴ The recommended credences are, of course, only as good as our theories, but they function (more or less) as hedged predictions. This view makes general probabilities more basic than chances, and so it is available even where the underlying dynamics is deterministic. It does not attempt to identify probability with frequency, and so it does not fall afoul of the familiar objections to frequentist accounts (or other reductive accounts like Best Systems analysis) but rather treats chances as encoding inductive content of our theories in a form suited to guide credence in the face of ignorance.

Most of this goes over rather smoothly to the Everettian case. If there are decoherent histories, there are relative frequencies across macroscopic ensembles, so there are general probabilities of macroscopic events. These are the probabilities that feature in Wallace’s account of statistical inference. They are well-enough defined to support the familiar probabilities that casinos and insurance agents rely on, as well as the probabilities of measurement results. We can say that the Born probability of x is a measure of the probability that a postmeasurement branch will have an x-result. In the newer setting probability is not a measure of our ignorance of what the result will be, since all results will be realized. But it is none the worse for that. Ignorance about which of a set of possibilities gets actualized turns into distribution of belief over a set of downstream alternatives, all of which are actualized. And it still plays the same role in decision, but for pragmatic reasons. The Everettian agent walking into a casino is not betting on the singular outcome of a chancy event, but choosing a betting policy in a universe in which all outcomes will be realized.

It is not hard to see in pragmatic terms how to justify a policy that tells him to bet evenly on the outcomes where the roulette wheel is fair, and in general according to the branch weight of the outcome. Indeed, we can reproduce the epistemic and practical situation of an Everettian agent as follows. As customary, we will suppose that the decisions are betting scenarios and that the dollar is the measure of value and the same for everyone at all times. The Everettian agent knows that roulette wheels are classically chaotic, so he knows that the quantum description is branching. We will suppose, moreover, that in the agent’s branch we have stabilized relative frequencies for roulette wheels, where this is given precise content in terms of relative frequencies across randomly sampled subclasses. We now use those frequencies to introduce general probabilities of the form pr(x/y), where x is a possible result and y is a generic description of the spin of a wheel. These probabilities are not treated as representing any kind of frequency, but they are understood through their inferential role (the conditions and consequences of assignment). You are placing bets on a roulette wheel. You theorize that there is an equal general probability that an arbitrary spin will produce any particular result. You have no specific information about the outcome of a particular spin. You will be playing for an indefinitely long period of time, you are certain that over the course of play all results will be realized, and you must bind yourself now in a single decision to a strategy that you must use throughout.

Is there a pragmatic justification for a strategy that assigns equal expectations to each result in the classical setting? If there is, then it equally supplies a pragmatic justification for the adoption of Born probabilities in the Everett setting. We can add relative frequencies that have any distribution whatsoever: one has the same pragmatic justification for adopting them as credences. We can add ignorance of what the stabilized relative frequencies are: now bets should be a mixture. We can allow that you might get lucky or unlucky. And we can tell any story that we need to tell (e.g., a Best Systems account) of how we form beliefs about the general probabilities, since the inference from observed frequencies to general probabilities is by no means direct.Footnote ⁵ What we have said here only links beliefs about general probabilities to credence.

Since Wallace has established in the discussion of statistical probability that branch weight and statistical probability go together (i.e., the branch weight of an x-result is [more or less] the statistical probability that a typical downstream descendant of the measurement will be observing x), someone who holds this view of probability has little problem making sense of its role in EQM: both decision and confirmation work fine, and it all seems rather innocuous.

Why then does Wallace work so hard to make sense of probabilities, offering the decision-theoretic proof? One possibility is pessimism about the prospects of a pragmatist account. He does consider a frequentist view, rejecting it on the grounds that it does not do a good job as an interpretation in the classical context, but he remarks that it does at least as well in an Everettian context as it does in the classical case. This pragmatist alternative is (among other things) a much more adequate relative frequency account of classical general probabilities. An Everettian might offer some such account of Born probabilities and be done with it.

Perhaps more plausibly, Wallace may think that the decision-theoretic proof is necessary whether one adopts a rationalist or a pragmatist account, in order to provide a theoretical underpinning of the Born probabilities. Although the proof of the Born rule is formulated within the decision-theoretic framework, the mathematical core of the proof does not depend on it: as Wallace remarks, it “establishes that if probability basically makes sense, and has the usual qualitative features, in unitary quantum mechanics, then quantitatively it is given by the Born rule” (155). Several critics have suggested that Wallace’s decision-theoretic axioms fall short of being uniquely rationally compelling, but that does not mean they fail to lend the Born measure a significant degree of naturalness.

One may indeed feel that, without some argument such as the Deutsch‐Wallace proof, the Born rule is merely a phenomenological add-on to the theoretical structure of quantum mechanics, so that pragmatists may have an extremely good fit to observed frequencies but little theoretical underpinning for the Born rule. However, while Wallace can point to the advantage of his proof as compared to classical symmetry arguments (no symmetry broken in the Everettian case), it is not true that the Born rule has little theoretical justification. As several commentators have pointed out, Gleason’s theorem provides another natural way of justifying the Born rule (perfectly acceptable as part of a pragmatic justification). And to do justice to Everett, he presents such a theoretical argument himself.

Wallace, in section 4.6, appears to hold the common view that Everett’s justification for the Born measure is based on the quantum law of large numbers (making Everett a species of frequentist). Instead, Everett justifies the Born measure as the unique function of branch weights that is consistent over time, in explicit analogy to the temporal conservation of the Liouville measure in classical mechanics, interpreted as a typicality measure (see Barrett and Byrne Reference Barrett and Byrne2012, 123–30, 190–92, 261–64, 294–95). Only after the Born measure has been thus justified is the law of large numbers applied in order to establish that typical memory sequences of observer systems display quantum statistics. Everett’s derivation of the Born rule is thus mathematically much closer to the Deutsch–Wallace proof than it seems: the crucial ingredient in both proofs is diachronic consistency, and one plugs it into one’s favorite view of how measure-theoretic concepts are used (typicality measures for Everett, credence measures for Deutsch and Wallace).

In sum, the Deutsch–Wallace proof perhaps retains its interest as part of a pragmatic account of the Born probabilities, but rather less may be resting on it for establishing the intelligibility of probabilities in an Everettian world. In either the classical or the Everettian setting, probability emerges uniformly as a secondary or derived notion, and we can tell an intelligible story about the role it plays in setting credences.

3.3. Uncertainty

Yet another example of rethinking occurs in the discussion of uncertainty, where Wallace challenges the common wisdom that accepting EQM would mean a radical overhaul in worldview. He focuses on three topics: uncertainty, possibility, and personal identity. We were puzzled by the status of these analyses. One way of conceiving the task that the Everettian faces is that of reinterpreting familiar language against an unfamiliar metaphysics. In that capacity, he is likely to face choices about meaning that are made on pragmatic grounds, perhaps to preserve as much as possible of the old usage or to answer to new demands.

If this is the spirit in which Wallace is offering these analyses, they seem harmless enough, but we find it less plausible that they capture the content of the beliefs of pre-Everettian agents. Does he intend his semantics to capture the contents of language users’ beliefs, or is he just pointing out that our linguistic practices could be reinterpreted in an Everettian setting in a manner that kept them in place? Here is what he says: “Some advocates [of EQM] have embraced the idea that so radical a theory ought to have comparably radical consequences for our everyday world-view. My purpose in this chapter is to defend a more conservative position: I will argue that the ways in which we use talk of uncertainty, identity, the future and so forth will remain fully justified in the event that we come to accept EQM as true” (259).

Does Wallace mean that our worldview is not significantly altered, or that our ordinary ways of speaking can be preserved in the new setting? Consider the ancient medical practices that were justified by magical thinking. We might be able to rationalize the practices as justified from a scientific perspective, but surely we would not be vindicating the beliefs of the ancient practitioners. We are happy to agree that we can make sense of the practical and epistemic behavior of agents in ways that are applicable irrespective of the move from a classical world to an Everettian one. We are not happy to say that the move would not involve significant reconfiguration of our worldview.

Part of the trouble is that the relationship between linguistic meaning and cognitive content associated with a term by the user is both complex and contested. Wallace draws on a tradition in philosophy of language holding that when we are interpreting speech, we are engaged in radical interpretation from a third-person point of view (Davidson Reference Davidson2001). The principle guiding the assignment of meanings to terms is charity: assign meanings that make most of the claims of users come out true. In the Everettian context, then, we are to think of ourselves as having the full multiverse in view, looking down at Everettian agents, and assigning meanings to their expressions. A principle of charity will have claims of uncertainty under branching meaning “happens in some branches and not in others.” But even a defender of such a view about meaning should recognize a notion of content fine-grained enough to capture discriminations that we make from a first-person perspective. The way to see that there is a real and substantive difference between the pre-Everettian notion of uncertainty and the post-Everettian notion is to ask what an agent who underwent a conversion to EQM would say. Even if she continues to use the language of uncertainty, would she say that her subjective state of uncertainty has not changed? We think obviously not. We all know the difference between “one thing happens, I know not which” and “happens in one branch but not all.” If I ask you whether Elm Street curves to the right or left at Tanner Park and you tell me you don’t know, and if I find later that you know that it has branches on both sides, I think you have lied to me.

Nothing really hangs on the issue for Wallace, because the task of making room for probability was treated in part 2 with the decision-theoretic proof. The reason that issues about uncertainty became central in discussions of EQM was in part that they were closely tied to issues about probability. If there were something to be uncertain about, that would be the bearer of probability. Indexical uncertainty—or, self-locating uncertainty—presented itself as a natural candidate, but that only works in the “Vaidman window,” in which a measurement has already taken place and an observer has not opened his eyes to view the result (Vaidman Reference Vaidman1998). In that window, the agent can raise a question about a particular result that he can identify indexically, as this result, because at that stage he can distinguish that result by its relationship to his own location. The reason that this kind of self-locating uncertainty cannot be transformed into uncertainty about the future is that it demands that self-directed thought about the future latch onto particular downstream descendants. If our future-directed uses of ‘I’ did have a well-defined subject, there would be uncertainty about what ‘I’ will see—and statements about what I will see could be the bearers of probability. But because (at least on most standard readings of Everett) they do not, we cannot frame a question at the premeasurement stage about the results in a branch in a way that leaves the outcome open. But Wallace does not need such bearers of probability, and he should not be shy about acknowledging that EQM may do violence to some of our ordinary ways of thinking. The Copernican analogy to which Wallace repeatedly draws attention is apt here: Kepler and Tycho mean different things when they say the sun rises in the east at dawn.

3.4. Space-Time and the Quantum State

Wallace reiterates the view that the best way of doing metaphysics (or the metaphysics of our best physical theories) is by (i) taking our theories literally about the world and (ii) looking for perspicuous structures in our theories. He notes that to identify reality as a point in Hilbert space—although in a sense that may be the fundamental metaphysical picture—is not yet to tell us how this reality is structured. What he calls “space-time state realism” is supposed to make some of the structure of the quantum state explicit. At first it strikes one as trivial. If the state is real and if we assume some background space-time structure as in current quantum field theories, then it follows straightforwardly that what is real in a space-time region is the reduced density operator on that region, and that, although this ontology is localized (in the sense that it represents what is real in a region), it is nonseparable, because the states of larger regions in general do not factorize into products of states on subregions. What is meant to be original in the proposal is that this particular way of analyzing the structure of the universal quantum state is more perspicuous and helpful than others. To see this, we might consider the simpler case of nonrelativistic quantum mechanics and compare the result of writing down the quantum state in various representations. If we write it as a wave function on (the ‘correct’) configuration space, suddenly the Hamiltonian that generates the abstract evolution of the state takes on a familiar form, and it is this that gives meaning to the factorization into subsystems, as well as meaning to various operators such as ‘position’ and ‘momentum’. Now do the same with space-time: we have the abstract state, and we find a representation of it as a state on space-time. Technically this might mean a state over a net of algebras of local observables (Haag), or a family of states over arbitrary hypersurfaces (Tomonaga–Schwinger). And suddenly we see dynamical structure manifesting itself, more specifically causal structure: dynamical influences propagating locally along the light cones of a ‘branching space-time’. Not only is this choice of representation one that makes the (dynamical) microstructure explicit, but one might argue that this is how space-time emerges from Hilbert space. Indeed, even though the quantum states in this representation are nonseparable, the structure of how influences propagate between states on subregions is a structure of Minkowski light cones. And Minkowski space-time arguably just is the causal structure of its light cones. Hence, even though the way we have introduced space-time originally was as a mathematical background structure defining a representation of the quantum state, we now recognize the same mathematical structure in the physics.

As a way of making explicit structure encoded in an otherwise implicit way in the wave function, space-time state realism is unobjectionable. But it is not particularly novel. We have known of the analysis of locality and nonseparability of section 8.5 since the classic discussions of nonlocality in the 1980s and 1990s. These apply to EQM because it is just standard quantum mechanics without collapse. (The novel question is, rather, whether the branching process should be thought of as local. In this regard, Wallace’s remarks on Everett and Bell’s theorem are indeed very helpful.) The analysis of sections 8.6 and 8.7 points out how the effects of a decoherence interaction (of course) propagate along the forward light cone, as well as how the nonseparable structure of the state on a larger region leads to the subtle way in which the different local branchings intersect. But this analysis of branching was provided by Bacciagaluppi (2002). Nor does it provide a full analysis of the structure of reality according to EQM because it leaves space-time itself unanalyzed.Footnote ⁶

3.5. Plausibility

Perhaps the biggest prima facie challenge that the multiverse faces in winning over adherents is what many see as the intrinsic (im)plausibility of the view. Sometimes the challenge is expressed in terms of the distance from common sense, sometimes Ockham’s razor is invoked, sometimes just incredulity.

In the interludes, where Wallace addresses his fictional (or composite) opponent, it is the philosophical questions that are mostly in play, and his strategy in answering them is repeatedly to point to episodes in the history of physics, revolutions that challenged the common sense of the day. The way that progression has worked is to expand our view of the universe in two ways: many of the structures at the forefront of experience and everyday thought turn out to be emergent from deeper structures, and many of them turn out to be parochial. And one can certainly make the case that the Everett view simply continues that progression along both fronts. It turns out that our view of reality is far more parochial than we imagined, limited not just to a small part of space and time but to a single decoherent history. There is also a deeper way in which it arguably continues a progression seen in space-time theories. The development from Aristotelian through relativistic physics can be seen as successively restoring symmetries at the level of ontology that are broken in experience. Spatial isotropy is restored by relativizing the difference between up and down to a frame of reference defined by the observer’s orientation, and the difference between being at rest and traveling with constant velocity to a frame of reference defined by the observer’s state of motion. In the Everettian case, the symmetry-breaking transition that picks out one result as actual is replaced with branching, which relativizes the distinction between actual and not to a branch defined by the observer’s location. The analogy is not exact, but we agree with Wallace that the complaints that EQM violates Ockham’s razor involve a misinterpretation of the principle. The right way of reading Ockham’s razor as it actually figures validly in physics has less to do with reducing the size of the universe than with restoring symmetries.

A lot of the debate surrounding EQM has revealed very different standards for rendering it intelligible that we live in the multiverse. Wallace’s brand of naturalistic metaphysics gives complete authority to science, paying no heed to everyday ‘intuitions’ or judgments of intrinsic plausibility. This style of metaphysics contrasts sharply with twentieth-century analytic metaphysics, which relies largely on a priori methods and where intuition and common sense play a role in choosing metaphysical frameworks. It also contrasts with the (so-called) primitive ontologists, who typically hold that every theory, regardless of its prima facie ontology, must be about spatially localized building blocks evolving through time. The burden of argument for the authority of intuition and the justification for primitive ontology would seem to fall on the practitioners of these approaches. We think that the most important and urgent question in metaphysics right now is the choice between these methodologies, and that Wallace’s book provides the best and most compelling example of what metaphysics is like performed in this new, scientific key.