On Many-Minds Interpretations Of Quantum Theory HTML version of the revision of November 1997. Matthew J. Donald The Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, Great Britain. e-mail: mjd1014@cam.ac.uk home page: http://people.bss.phy.cam.ac.uk/~mjd1014

Abstract This paper is a response to some recent discussions of many-minds interpretations in the philosophical literature. After an introduction to the many-minds idea, the complexity of quantum states for macroscopic objects is stressed. Then it is proposed that a characterization of the physical structure of observers is a proper goal for physical theory. It is argued that an observer cannot be defined merely by the instantaneous structure of a brain, but that the history of the brain's functioning must also be taken into account. Next the nature of probability in many-minds interpretations is discussed and it is suggested that only discrete probability models are needed. The paper concludes with brief comments on issues of actuality and identity over time.

Introduction

The purpose of this paper is to discuss some philosophical issues which have arisen in “many-minds” interpretations of quantum theory. The paper begins with a brief introduction to the many-minds idea laying particular emphasis on the complexity of quantum states for macroscopic objects. A more thorough introduction at an elementary level is given in Lockwood 1996a. The relationship between mind and brain is of fundamental importance to many-minds interpretations and in the second section, I shall argue that, in the light of quantum theory, we should revise our understanding of what a brain is. In particular, I shall suggest that the history of a brain's functioning is an essential part of its nature as an object on which a mind supervenes. Then I shall turn to the issue of probability in many-minds interpretations and argue against the claim that there should be a continuous infinity of minds at each instant. I shall also make a number of briefer comments on other philosophical questions. In three long and technical papers ( Donald 1990, 1992, 1995) I have presented a version of the many-minds interpretation which I believe to be compatible with special relativity, with quantum field theory, and with the macroscopic and thermal nature of real observers. The ideas discussed here stem from that technical work but also stand independent of it.

Many-minds interpretations are a class of “no collapse” interpretations of quantum mechanics, which is considered to be a universal theory. This means that they assert that all physical entities are governed by some version of quantum theory, and that the physical dynamics of any closed system (in particular, the entire universe) is governed entirely by some version, or generalization, of the Schrödinger equation. From the point of view of a committed quantum theorist, who knows neither of any experimental evidence for any breakdown in quantum theory, nor of any alternative theory which is not both ad hoc and incompatible with special relativity, these assertions may seem plausible (cf.Deutsch 1996). It is certainly the case that, at least over short time intervals, quantum states can be found which will give apparently accurate representations of the physical states of essentially all non-gravitational physical systems, however large or complex they may be. Indeed, states can be found which are not only “apparently accurate” in the sense that they are compatible with all our actual short-term observations, but also in the sense that they represent any particle as being, during the interval considered, sufficiently well-localized to accord with conventional pictures of that type of quantum particle. Thus, in such states, the atoms in our environment are represented as slightly fuzzy balls, while free electrons are represented as moving along slightly fuzzy straight lines. I shall call these states “pragmatic”. For example, a pragmatic state for a gas might use minimum uncertainty wave-packets centred on definite choices of positions and momenta to describe the centre of mass variables of the molecules, together with appropriate choices of molecular wavefunctions for the electronic variables. At a more sophisticated level, using our excellent understanding of quantum states for single molecules, and the possibility of building such quantum states up to describe many molecules, pragmatic quantum states can be ascribed at any time to any chemical system — including the human brain. The fundamental problem — the problem of “Schrödinger's cat” — arises because the time continuation given by the quantum mechanical dynamics does not always lead from a quantum state which provides an apparently accurate description at one time to a quantum state which provides a similarly apparently accurate description at later times. The pragmatism of a state, in other words, is time dependent.

Consider, for example, electrons contributing to the production of an interference pattern by passing, one at a time, through some kind of two-slit device and hitting a position detector. (Pictures from such an experiment are presented in Tonomura et al. 1989.) I have chosen this example, partly because the pictures are such a direct demonstration of the difference between quantum state and observation, partly because the electrons are more likely to be seen to hit some parts of the detector than others, and partly because I think that the two-alternative experiments by which this subject is usually introduced foster a naive view of the complexity of quantum states (cf. Weinstein 1996).

According to quantum mechanics (by which I shall henceforth mean universal, no collapse, quantum mechanics), nothing determines the points on the detector where the electrons hit. Indeed, if a pragmatic state is given for an electron at the beginning of the experiment, then the slits act to amplify the fuzziness of its original trajectory, so that as it approaches the detector, the state associated with it will represent some sort of weighted distribution over all possible hitting points. The “many worlds” idea is that such “superpositions” of possibilities can form part of a correct and complete description of the physical world and that all possible hittings do, in some sense, happen. This is an interesting idea, but it leads immediately to questions as to why each electron is only ever seen to hit at one point and to what the probability of any particular hitting being seen means. That the word “seen” slips naturally into these questions, suggests that here, as in so many other attempts to understand quantum mechanics, “the observer” has some special role to play. Everett himself said (in DeWitt and Graham 1973, p. 117) that, within the context of his theory, “it develops that the probabilistic aspects of [von Neumann's] Process 1” (the collapse of the wavefunction) “reappear at the subjective level, as relative phenomena to observers.”

Consider, therefore, someone who is looking an image produced directly by the detector, or even someone who is looking at one of the photographs reproduced from Tonomura et al. 1989 on page 3 of Silverman's 1995 book. They will see a picture of a definite pattern of specks. Nevertheless, if we consider the quantum states that we should associate to the physical structures of these human beings, and, if we start by looking back in time as far as the beginning of the experiment and take as our initial condition a “pragmatic” state for the entire situation at that time, then, solving the Schrödinger equation will lead to quantum states which by the time of the observations considered will also describe some sort of weighted distribution; in this case over brains seeing possible patterns of specks. A many-minds theory aims to accept such “unpragmatic” states, and to interpret them.

The first step in a many-minds interpretation is to make the hypothesis that our nature is such that we cannot see such a picture except as a definite pattern. The quantum state is correct in its description of a weighted distribution of possible patterns. When a single individual comes into contact with the picture for the first time, all those different possibilities do occur, but each different pattern is seen by a different mind; minds which share the same past and the same name, but which experience different presents and different futures and which have no means of communicating to each other. The probability of seeing a given pattern is determined, at least to a first approximation, by the corresponding weight in the quantum state.

In any no collapse interpretation, including the modal interpretation and the Bohm interpretation as well as all versions of the many-worlds interpretation, we start off with only one quantum state: the uncollapsing universal quantum state, which I shall denote by ω. ω can be identified by going backwards in time. Each time we pass back through the appearance of a collapse we get a better approximation to ω. Eventually, we arrive back at the big bang. For the moment, we may ignore the question of whether the big bang itself was merely the appearance of a collapse. The quantum state of the universe coming out of the big bang looks — at least in its non-gravitational aspects — very like a thermal equilibrium state. In the Hamiltonian (uncollapsed) time propagation of that state, the stars and planets which we see now do not exist as definite objects, and certainly neither does any particular measuring device now being used by us on one of those planets. ω seems to be a complete mess. However, it does have a great deal of hidden structure, and it is the job of a no collapse interpretation to explain how that hidden structure comes to be seen.

Most workers in no collapse interpretations have produced no more than elementary models based on the definite existence of specific measuring devices. They have assumed, for example, that the Hilbert space of the universe splits naturally into a tensor product structure compatible with the measurement under consideration. They have also assumed, even when describing the behaviour of macroscopic objects, that it is appropriate to employ models in which only a few dimensions of Hilbert space are used to describe all the relevant behaviour. In my opinion, these assumptions are untenable (cf. Bacciagaluppi, Donald, and Vermaas 1995). The first assumption begs the question of what is natural, and depends on assumptions about the nature of particles which are known to be false according to relativistic quantum field theory (Haag 1992). As far as the second is concerned, a measure of the number of dimensions relevant to a macroscopic object can be given by eS/k where S is the absolute thermodynamic entropy of the object and k is Boltzmann's constant. This number is so large that it must call into question any argument which refers to “the wavefunction (ψ observer ) of the observer”. Possible wavefunctions do exist, but there are something like eS/k orthogonal choices available, and any one of these choices will, by entanglement with the environment, rapidly move into a mixture with the other choices. This vast complexity of available observer states suggests, for example, that Weinstein 1996 is quite right to criticize Albert and Loewer's 1988 invocation of a projection measuring a belief held by a observer.

The analysis of ω is not however a hopeless task. The secret lies in the idea of “correlation”. Quantum states, particularly many-particle quantum states, are patterns of correlations. For example, consider the quantum state describing a volume of hydrogen and oxygen atoms in a two-to-one mixture which has come to chemical and thermal equilibrium at room temperature and pressure. The state is a canonical ensemble equilibrium density matrix. It is not a pure state, but that is only to be expected, because states of non-isolated macroscopic systems are virtually never pure. (As I discuss elsewhere, this lack of purity does not solve the conceptual problems we are considering (Donald 1990, 1992).) The important point is that it is not a pragmatic state because it does not describe a collection of water molecules with well-defined positions. Indeed, the position distribution of each individual atom considered separately will be spread uniformly over the entire container. Nevertheless, if the position of one of the oxygen atoms is taken as given, then the same equilibrium quantum state determines a distribution for the other atoms which will be such that precisely two hydrogen atoms will be closest to the given oxygen position, and those two will be around 0.096nm distant. Similarly in the Schrödinger cat state — if there has been a radioactive decay, then the cat is dead; if there is broken glass, then the flask which contained the poison has been broken, and the cat is dead; if, as he richly deserves, the experimenter's face is scratched when he opens the steel box, then the radioactive atoms have not yet decayed. An adequate analysis of the correlations in ω is the first step towards an interpretation of quantum mechanics.

The object of supervenience

If we are to make sense of the many worlds idea and demonstrate how the world that we see is extracted from ω, it would appear that we need to be able to say what a world is. Saunders 1996b has pointed out that there are three ways of approaching this problem. One is direct attack, by mathematical definition. In my opinion, “consistent histories” is an example of this approach. So far, however, I do not believe that this direct approach has been successful. Dowker and Kent 1996, in particular, have found many problems with the consistent histories program. It is not clear that the abstract definition of a consistent history does solve the problem of defining a world, because there are many possible sets of consistent histories, and no way of choosing one particular set is available. The second approach is the “many minds” idea. Here we give observers a central role. We do not need a general definition of “worlds”, but we do need to define “observers”. The only worlds which are considered important are identified by correlations to those observers and, as Brown 1996 has emphasized, these external worlds are secondary, derived, concepts. The final approach is the idea, championed by Saunders 1995, 1996a,b, of “relativism”. The claim here is that we do not need to define “worlds”; the total pattern of correlations established by the universal quantum state are sufficient in themselves as a foundation for physics. Saunders claims consistent histories as an example of relativism. As it stands at present, I would accept that it is possible to interpret consistent histories in either way.

Saunders' approach might seem to avoid the problem of definition, but, in my opinion, this is not a problem that we should be trying to avoid. Although the correlations established by ω are enough to provide a description of objective reality, they do not give us any understanding of subjective reality. We are here and we look out at the universe entirely through those particular correlations which ω establishes to our brains. These correlations are the ones that count. We do not need another “view from nowhere” in physics.

I also do not believe that we exist merely “for all practical purposes”. In a state of water at equilibrium, I can localize the oxygen atoms, to a certain extent, by giving the positions of the hydrogen atoms, or I can localize the hydrogen atoms, to a certain extent, by giving the positions of the oxygen atoms. Similarly, in Saunders' program, the state of our brains determines for all practical purposes the state of the world that we experience, and the state of the world that we experience determines for all practical purposes the state of our brains. But this raises the as-yet-unsolved technical problem of producing a theory, at the level of ω, of the “quasi-classical domains” in terms of which, for all practical purposes, our experienced world is to be described (Saunders 1993, Gell-Mann and Hartle 1993). Not only does it seem to me that it is more straightforward to solve the technical problem of finding an unambiguous characterization of an observer (Donald 1995), but also I believe that consciousness is a fundamental pillar of existence and must therefore be something definite.

In my opinion, I am what I am, and I want to discover what that what might be. In particular, I want to discover exactly what the physical manifestations of that what might be. Which physical properties are fixed by the existence of a given state of my mind, and which properties are only probabilistically constrained? Here the ontological issue that “I am what I am” must be distinguished from the epistemological question of whether I can know what I am. No conscious being can be aware of his exact state, but this does not mean that he does not need to be in some exact state. I think that it is the proper task of an analysis of quantum theory to try to give an exact definition of the possible physical manifestations of an observer.

The doctrine of psychophysical supervenience claims that two people cannot differ mentally without also differing physically (Lewis 1986, p. 14). In a classical picture, brains are just there, and the investigation of exactly on what aspects of those brains the corresponding minds supervene hardly seems to be an essential task. In quantum theory, however, the universal quantum state is a grand superposition of possibilities. Some of those possibilities do seem to contain structures which look like brains or which function like brains. Now the doctrine of supervenience no longer merely provides a convenient name behind which questions about the nature of mind can be hidden. Instead, it raises the difficult but intriguing technical problem of analysing such a superposition (cf. Albert and Loewer 1988, Barrett 1995, Lehner 1997). This requires discovering what the relevant physical constitution of a person might be. In other words, the problem is to discover a physical “object of supervenience” such that the doctrine of supervenience holds in the form that two people with identical “objects of supervenience” are mentally identical. Putting this another way, the idea of psychophysical parallelism encounters the problem of identifying just what it is to which the psyche is supposed to be parallel.

One way of approaching these problems is to consider the preliminary claim that two brains which are physically sufficiently similar, necessarily give rise to the same mental phenomena. Even without any analysis of “necessarily”, “the same”, or “mental phenomena”, this claim seems plausible. For example, it seems to me to be plausible that mental phenomena would not change if the temperature of the brain was changed by less than one thousandth of a Celsius degree, nor if the pH changed by less than 0.01. So what exactly does “physically sufficiently similar” mean? What physical structures could be used to define such equivalence classes of brains? Can we describe the minimal amount of structure required or discover a simple characterization of the equivalence classes? Initially, I shall refer to anything which is a brain by virtue of having such a structure, as being “an object of supervenience”, although I am working towards a position in which, strictly speaking, the term should be reserved for the equivalence classes rather than for the members of those classes.

It is important that the words “give rise” in the claim also be taken seriously. There are two different techniques by which it is possible to think about what it means for a physical situation to imply the existence of mental phenomena. The more conventional technique is to imagine strange ways in which such a situation could arise and to contemplate whether the proposed mental phenomena remain plausible. For example, someone who claims that it is only the instantaneous structure of a brain which has any importance, could be required to consider the possibility that a functioning piece of flesh could be constructed merely by bringing the requisite atoms together, for a moment, in a vat. A more powerful technique, however, is to imagine the mind as a “ghost in a machine” and to ask what properties of the machine the ghost would use to find its meaning. This is not to suggest that the ultimate aim is not to exorcize all such ghosts. The object of supervenience exists for itself. However, if the physical structure is sufficient to imply the meaning, then the meaning must be interpretable from the physical structure. Imagining a ghost is merely a way of thinking about what might be required in the interpretation of physical structure. For example, if the doctrine of supervenience is to have any force, then the meaning should be interpretable without requiring that the ghost be an educated human being with a training in twentieth century neurophysiology and nineteenth century physics who already knows that the machine is a functioning human brain.

For many of those who have written on the many-minds interpretation, the object of supervenience is an element of a basis of brain states. For example, Lockwood 1996a refers to a “consciousness basis” and Albert and Loewer 1988 to an incomplete basis of “brain (or brain + environment) states”. In neither case, is the particular basis identified. In my opinion, the attachment to the idea of a basis is a mistake based on a false analogy with elementary models of measurement theory. A human brain is not like an atom in a Stern-Gerlach device, nor is it like a Copenhagen-interpretation measuring device which bears on its cover the name of the self-adjoint operator which it is designed to measure. A brain is warm and wet. The number eS/k of available wavefunctions is something like 10 raised to the power 1026. The identification of a suitable basis might be expected to require identification of exactly which molecules are to be included in any given basis state. Is this set of molecules to include the blood which is pumping through the brain, and the molecules which it is breathing? Where is the outside surface of the brain (or in Albert and Loewer's case of the brain + environment) to be drawn? Is Albert part of Loewer's environment, and if not, where is the line of separation? Why should any, arbitrarily small, change in pH — which changes the number of hydrogen ions in the brain, and therefore changes the basis state — be sufficient to change the object of supervenience?

Many-minds authors have also tended to assume that the object of supervenience is a brain at an instant. This too is an assumption with which I disagree, and I continue to disagree, even if “the instant” is modified, as Butterfield 1996 proposes, to cover the duration of a psychological moment. The doctrine of supervenience has become dominant in modern philosophy, at least in part because of the success of neural computation models of the brain (e.g. Dennett 1991, Churchland and Sejnowski 1992). In the framework of classical physics, these models are part of an increasingly convincing demonstration that everything that a mind appears to understand or feel is reflected in detail in neural functioning. However, although behaviour may be governed entirely by instantaneous neural functioning, this does not imply that instantaneous functioning is sufficient for mental understanding. Indeed, in my opinion, even short-term functioning carries, in itself, very little meaning.

Imagine that you were given a perfect snapshot of a brain. How would you begin to understand the information that was being processed? How would you understand the dispositions of the individual under study? Suppose that you find excitement in one part of the brain, for example, the occipital lobe. The name merely tells you that this is at the back of the head. Tracing nerve cells will connect this part to a part at the front of the head which we call the retina, and which you and I know to be involved in vision. How would a ghost in the brain faced with such a snapshot know what the retina is for? How would he know in which direction nerve cells should be traced? How would he even know that nerve cells — rather than glial cells, or blood vessels, or the positions of individual atoms, or electrons — are what he should be studying?

Supervenience suggests that it be possible to make explicit all the information that a ghost would require for understanding. Suppose then, that we tell the ghost roughly what a brain is and how it functions; suppose, for example, that we give him a textbook of neurophysiology. With a perfect snapshot of the brain, he could then perhaps make a model which he could try to use to find out how the actual brain under consideration was functioning. He would have to run that model under all sorts of different, but physiologically “normal”, conditions in order to find out the details of how it worked and exactly what it was doing and remembering at the instant of the snapshot. The required knowledge of the conditions of operation of the brain and the results of his simulations would already seem to go far beyond the physical instant. Exactly the same types of simulation would also be required to explicate what a brain was doing given a “psychological moment”, or even to understand objectively how long such a moment should last.

It would only be possible not to go beyond the physical instant, if human brains were “off-the-shelf” devices for which one could, in theory, provide a handbook mapping instantaneous molecular structure into function. However, human brains seem to be more like “neural net” machines, whose function is best discovered by simulation, than they are like progammable computers, whose function can be explained more compactly by provision of the manual and the program. This applies in particular to the details of the functioning. Moreover, even considered classically, human brains do not work deterministically at the level of neural processing. In the details of its functioning, a brain is metastable and information is processed by the accumulation of small and uncertain effects. Thus we cannot tell by looking at a brain, how it has arrived at its current state. Different, physiologically normal, prior histories could lead to physically identical brains. It seems to me to be a bold assumption that these different histories would necessarily result in identical awareness of the same present.

My preferred alternative is to take the object of supervenience to be the entire lifetime history of the brain in question up to the moment of the snapshot. Michael Lockwood has commented that this gives a picture of us as dragging our histories behind us like “Marley with his chain” (Dickens 1843). The relevant history of a brain is a history of patterns of neural firings. An explicit description of a history of patterns would be a much simpler description of the information required by a ghost than would be a textbook of neurophysiology and a full record of simulations. It would be simpler in two senses. It would be shorter, because a full record of simulations would need to analyse all the possible behaviours of the brain, and not just the actual past behaviour. It would also be more abstract and less contingent: a history of patterns, for example, would not even require a carbon-based biochemistry for the object of supervenience. Such a history may be thought of as a minimal structure sufficient to define the causal relations required by functionalism.

An appeal to the idea of functionalism seems to me to be an important part of the hypothesis that the brain is, in some sense, an adequate model of the mind. (For more about this hypothesis in the present context, see the non-technical second section of Donald 1995.) Most many-minds authors, although paying lip-service to materialism, seem to me to lose sight of this hypothesis by referring to instananeous mental states distinguished by unexplicated labels (e.g. Albert and Loewer 1988 “belief states”, Lockwood 1989 “phenomenal perspectives”, Page 1996 “perceptions”). Barrett has advised me that “belief states” should be interpreted as “dispositions” (Albert 1992, p. 129), but once again I think that our “dispositions” can be more parsimoniously represented by an account of what we have done, or said, or thought, than by an account of what we might do, or say, or think. The theory I am proposing is not functionalism because I require that the causal relations be incarnated according to specific quantum theoretical rules — rules explained in the technical sections of Donald 1990 and 1995. It is also more explicit than conventional functionalism in that the mental is taken to be constructed from a specific type of finite pattern of causal relations between elementary events. My dream is that such an explicit and finite formulation could ultimately be used to reduce functionalism to a kind of linguistics concerned with the study of possible meanings of finite stuctures built up from elementary ideas like “this is the same as that”, “this is not the same as that”, “this is that continued indefinitely”, “this is pleasant”, “this is not pleasant”. Using this linguistics, I would hope that it would possible to argue that any “instantaneous” mental state could be interpreted as the culmination of something like a (very long) book written in a language of such elementary ideas. However while the books we read have only a one-dimensional causal structure — one word comes after another — the structures I have proposed would also allow for spacelike separated “words”.

In as far as we are ghosts stuck in our brains, I think that we do make sense of our present pattern of neural firings as a development of earlier patterns. It seems to me to be wrong to suggest that one particular pattern is a tasting of a rather too strong cup of coffee, merely because of the present arrangement of the atoms in my brain. I have become aware of the meaning of arrangements of atoms in my brain, not by a process of analysis, but by existing as that functioning brain, throughout a lifetime of many cups of coffee. It will perhaps be objected that I cannot, in fact, remember anything which is not somehow contained in the present arrangement of atoms. I accept that. However, the question is not what I can remember, but how I give meaning to my physical structure. I do not activate memories of previous cups of coffee in order to discover what I am now tasting. Instead, by the activation of patterns in my brain which are correlated with earlier patterns (and hence with earlier drinks), I become a tasting of coffee. Consciousness develops. It is not born anew at each instant.

It would not matter greatly to our understanding of classical physics whether the object of supervenience was a brain at an instant or the history of a brain. However, in the many-minds interpretation, the distinction is absolutely fundamental. A history of a brain cannot be recorded in a single wavefunction. As has already been discussed, although it is always possible to find wavefunctions which form apparently accurate representations of a brain as it can be observed, or experienced, at one instant, there are no single wavefunctions which are simultaneously apparently accurate representations of the brain as it is observed over periods long enough to include, for example, performance of the electron interference experiment mentioned above. Indeed, in Donald 1990, I argue that “pragmatic” states in the brain would have to be replaced on a very short (e.g. millisecond) time scale, whether or not the brain was involved in the observation of quantum experiments. Thus if the object of supervenience is to be a history, then it has to be represented either by some sort of sequence of quantum states, or, as in consistent histories, by a sequence of projections. (In fact, these alternatives are, to some extent equivalent because of the duality between states and operators. For example, in my own technical work, in which an explicit construction of a quantum model of the history of a brain is given, the sequences of quantum states invoked are themselves defined by corresponding sequences of projections.)

I suggested above that the proper task of an analysis of quantum theory is to try to give an exact definition of the possible physical manifestations of an observer. Ultimately, I think that any many-minds program aims to characterize an observer abstractly as an information-processing structure and to explain how that structure manifests itself physically as some sort of objectively real quantum mechanical structure probabilistically constrained by some sort of universal quantum state ω. For example, an observer might be manifested by a physical system with a wavefunction which was an element of a “consciousness basis”, and abstractly characterized by a definition of such a basis. In my own work, in order to deal with the wide range of imperceptible variations in possible descriptions of the physical structure of a given observer, I have found myself moving ever further from the “pragmatic wavefunction” picture of elementary quantum mechanics. For example, I have moved from sequences of quantum states, corresponding to a brain history, to abstractly characterized sets of sequences of quantum states. This is a progression away from naive physical realism towards a position in which the the physical world external to the observer exists only as something which provides (observer-independent) probabilities for his (objectively real) present and future existence. The progression is driven by the aim of finding an exact “object of supervenience”. The elements of my sets of sequences of quantum states are sequences which cannot be distinguished by the observer. Yet each of these indistinguishable sequences has the same relation to the universal quantum state, and should have the same ontological status. I am, incorrigibly, what my mind is. That subjective incorrigibility corresponds objectively to something definite which is governed by physical law. I am not an approximation. If I am a set of possible brain states, then I am that set, not some element of that set. This position may be contrasted with that of Lockwood 1996b, p. 458, who refers to calculating transition probabilities by considering sets of minds, which he refers to as sets of “identical maximal experiences”. In as far as there are such sets of identical experiences, I would associate them with the same mind.

At the abstract level — the level of functionalism — I associate any observer with a finite structure. Moreover, within a given bound on complexity, only finitely many observers are possible. I shall argue below that this finiteness is crucial in understanding probability in a many-minds interpretation.

It is possible to take the progression away from naive physical realism further than I have previously suggested, to a point at which the definiteness of the universal quantum state itself is called into question. I calculate probabilities by maximizing likelihood over indistinguishable possibilities. This maximization can be extended to allow variation in ω. In this way, I believe that it may be possible to arrive at a physics with no a priori physical constants at all. According to such a theory, our observations fix the value of physical constants for us, in exactly the same way that our observations fix for us the particular position at which an electron has hit a position detector. If this goal can be achieved, then it may indeed be claimed that we have a theory in which “arbitrariness” or “contingency” has been reduced. Lewis (1986, paragraph 2.7) argues that modal realism does not reduce arbitrariness on the grounds that the existence in our world of a specific value for a physical constant would be as arbitrary as the unique existence of that value. However, in the theory which I am sketching, physical constants would not have precise values in “our worlds”. There might, for example, depending on what information was available, be no fact of the matter about whether the reciprocal of the fine structure constant was for a given observer closer to 137.03601 or to 137.03602, let alone about whether it was a rational or an irrational number. Only the finite amount of information which determines one's structure as observer would determine the “world” in which one lives, while only a finite set of axioms, containing no arbitrary constants, would determine the set of possible worlds.

Probability

There has been much discussion of the meaning of probability in the many-minds interpretation. Here I agree with Butterfield 1996 that specific definitions of probability measures can be justified both by formal and by dynamical considerations. In Donald 1992, I present formal justifications for probabilities in a many-minds theory. Papineau 1996 has stated that correct probabilities “(1) have their values evidenced by frequencies, and (2) provide a decision-theoretic basis for rational decisions.” He goes on to say that these stipulations have no good justification. As far as the justification of (1) is concerned, many authors have pointed out that the laws of large numbers are circular. These laws show only that there is negligible probability of a long-run relative frequency diverging significantly from the probability which is to be justified (cf. Kent 1990, who gives a critical survey of quantum mechanical versions of these laws). Here I would merely comment that, at least in quantum theory, negligible probability is the same as small in the topology of the space of quantum states. This topology is a cornerstone of the entire theory of quantum physics. Thus, any justification of quantum physics as a whole, including, for example, from over-all consistency or beauty, may help to justify specific numerical probabilities. Although DeWitt, (in DeWitt and Graham 1973, p. 168) was certainly wrong to claim that the “mathematical formalism of the quantum theory is capable of yielding its own interpretation”, nevertheless, it seems to me that, because the topology is so natural and fundamental a part of the theory, arguments based on the topology can be at least as intuitively satisfying as counting arguments in a theory of equiprobable events. We may not be able to explain why the world we happen to have experienced was typical, but, if we can give meaning to the idea of a “world”, then at least it is possible for “typical” to be well-founded and consistent.

As far as the basis for rational decisions is concerned, the question is whether or not, if we accept a many-minds theory, it makes any difference to how we make decisions. Here, despite Papineau's 1995 and 1996 arguments that it should make no difference, I do retain a suspicion that, if one does take many-minds theory seriously, then it does make a difference. I think myself that it makes one think more carefully about all the possible outcomes of an action. If all futures happen, then one cannot get away with anything any more. A lucky escape in this world is merely confirmation that many other worlds are unpleasant. Perhaps if we accept many-minds then it ought to lead us to be more “rational”; to take seriously, for example, the very large negative utilities that we might attach to some very unlikely events. Whether this is a good thing or not depends on how we draw the line between being careful and being neurotic; how we manage to accept the reality of risk without being overwhelmed by it.

Papineau 1995 claims that it “is simply a basic truth about rational choice” that “rational agents ought to choose those actions which will maximize the known objective probability of desired results”. Lockwood 1996b responds that “choosing those actions which maximise the expected return, means maximising the total actual return, as integrated over the successors of whatever instantaneous mind is making the decision”. My problem with this response is that I can see no reason to integrate over successors. However much an individual may believe in a many minds theory, all he can experience, and all he is ever going to experience, is one mind. Suppose, for example, that you were given the opportunity to take part in a single trial in which you would win £1000 with probability 0.6 and lose £1000 with probability 0.4. Your expected return would be £200. But, whatever may happen, you are not going to experience receiving £200. Your only possible experiences, whether or not your future contains both, will be that of winning a large sum and that of losing a large sum. Even if the monetary quantities are accurate representations of the personal utilities to you of the individual events of winning or losing, it is these individual possibilities which matter and not their average.

Suppose that we accept Papineau's view of the difficulty of justifying probability. It would not follow that we do not understand what it is like to experience probabilistic events. Even young children enjoy games of chance and can eventually learn that their wishes do not affect the throw of a die. In my opinion, it would be sufficient for a many minds theory to provide a theory of transition probabilities according to which we experience reality as being like watching a particular, identified, stochastic process. My own published work does not at present succeed in achieving this goal. Nevertheless, by modifying some technical details in my approach to identity over time and by an analysis of the finite number of immediate descendants of a given minimal ordered switching structure (Donald 1995, Section 5), I now believe that it is possible to modify it so that it does.

Lockwood also claims that his theory can be modified to achieve such a goal. There is, however, a fundamental difference between the types of stochastic process which we invoke. My work is based on the calculation of a priori probabilities of existence for completely and finitely specified individual observers with finite-step histories which are completely specified up to a given moment. These a priori probabilities are in general non-zero. Thus I am proposing a theory of transition probabilities which says that we experience the world as being like, for example, the development of a random walk on a lattice. We know what it is like to watch such a process. Examples can easily be constructed on a computer, or even just by throwing a die. Lockwood 1989 and Albert and Loewer 1988, on the other hand, require the existence of an uncountable number of minds at each instant. It is impossible to assign non-zero probabilities at any time to more than a countable number of distinct entities. If there were an uncountable number of minds, then experience would be like Brownian motion, but it seems to me that we have no idea of what it would be like fully to experience such a process. Brownian motion is a continuous time process on a continuous state space. We cannot construct explicit physical models of continuous processes of this type, which involve infinitely many states. We can only make finite models. Sample paths of a genuine Brownian motion have infinite complexity. Indeed, almost all of the individual elements of any infinite set must have infinite complexity in some sense. I certainly do not know what it would be like to watch, in all its detail, something with infinite complexity.

This returns us to the question of Lockwood's sets of “identical maximal experiences”. If it is not accepted that those sets themselves are the objects of supervenience, then what is it like to be a mind? Probabilities are assigned to measurable sets. Which measurable sets are meaningful in Lockwood's theory? (cf. Loewer 1996.) Does it, for example, mean anything that I might be in one half of a given set of “identical maximal experiences” rather than another? What does distinguish one individual object of supervenience from another?

Lockwood 1996a proposes a continuous infinity of minds becauses he argues that if there were only a finite set of minds then the probability distribution “should” be uniform. He maintains this argument in the face of objections by Saunders 1996b, by Papineau 1996, and by Butterfield 1996 (the last, in particular, remarking that the principle of indifference is “a notorious dead horse”). Lockwood 1996b appears to insist that probability on a finite number of possibilities all of which are actual must necessarily be a simple matter of counting, on the grounds that “where it is stipulated that the history of a mind, beyond a certain point, has just n discrete continuations, all of which are actual . . . there is no freedom . . . to partition this n-fold continuation in any other way than that stipulated”. In my opinion, however, there is no reason why with a finite set of actual minds, the probability of one mind should not be greater than that of another. The a priori probability of a given mind is part of the objective structure of the universe. It is a number. A central concern of physical theory is to define such numbers. The numbers which arise will depend on the details of the theory, and their justification will be a fundamental part of the justification of the theory. In a theory which uses a consciousness basis (φ i ) i > 0 and a universal wavefunction ψ, the numbers would be defined as |< φ i | ψ >|2, and would be justified by arguments and evidence for square amplitudes as probabilities. The axiomatic definition which I give in Donald 1992 and 1995 is more sophisticated, and depends on the past physical structure of the observer in question and on the universal state ω. Nevertheless, ultimately, the justification of my definitions once again stems from, or is parasitic on, the evidence for the standard probabilistic interpretation of quantum states.

Actuality

I am very grateful to Jeremy Butterfield for the interest he has shown in my work, and for his kind comments about it (Butterfield 1995, 1996). In Butterfield 1996, however, he express a dissatisfaction with my position on the issue of actuality. Let me then address that issue here. Butterfield claims that this is an issue “irrespective of mind” and which “applies equally well to ‘many worlds’ versions of Everettian interpretations”. I disagree with this claim. In a many worlds framework, the question is whether one should accept all possible worlds as being actual. I do not think that anything hangs on how one answers this question. In a many minds framework, however, the question is whether one should accept all possible minds as being actual. In this case, there is a fundamental issue at stake. I can see no plausibility in solipsism. (Why me?) Any of your possible minds and any one of my possible present-time minds which shares part of my past but which is not what I am now experiencing, have the same type of abstract characterization and the same kind of physical structure and, because of the “no collapse” hypothesis, all those physical structures are “real” parts of the universal state. On the basis of these similarities alone, I would be inclined to accept the actuality both of all of your minds and of all of mine.

As Lockwood 1996a points out, a theory in which only one mind is actual for each individual would be a hidden variable theory, with actuality as the hidden variable. In such a theory, if, in the usual way, I measure an up-spin in one electron from a singlet state in a EPR-Bohm situation, then, when you, at a spacelike separation, measure the spin of the other electron, either all the well-known non-locality problems arise in forcing the “actual” you to see a down-spin, or I can encounter a “non-actual” you — a “mindless hulk” (Albert 1992). This is bad enough, but such a theory would also require the identification of the set of “individuals”. At a technical level, I do not know how this could be done. In a theory with a “consciousness” basis, how am I to decide to whom a given element of the basis belongs? In an appropriate set of 10 raised to the power 1026 orthogonal wavefunctions available to a warm wet brain, there is an imperceptible passage from wavefunctions which represent me to those which represent you. Where is this line to be drawn? Even in my theory, in which individuals have pasts, all those pasts are ultimately undifferentiated. And finally, we cannot identify the set of all individuals present at a given moment, unless we are prepared to define “a given moment”. To do that would be to attempt a “many-worlds” theory not a “many-minds” theory. At the very least, incompatibility with special relativity would surely follow.

Identity over time

There has also been much discussion in the literature of the problem of identity over time (e.g. Butterfield 1996, paragraph 4). In this connection, I note that at the heart of my technical construction there is a definition (Donald 1990, Hypothesis V and 1995, Definition E), which attempts to capture, within the mathematics of quantum field theory, the idea of an object existing through time by looking for the paths in spacetime along which the local quantum state changes least. This is a very direct approach to the problem, and one which shows that identity over time can be defined without even using the idea of an “object” as being something composed of particles.

Unfortunately, there is a technical flaw in the application of this definition in my 1995 paper. The problem is that I have not allowed for the effect, on the path under consideration, of information gained elsewhere in the brain. I believe, however, that I can solve this problem fairly straightforwardly by disconnecting the times of quantum state change from the times of object “switching”. Work is in progress on the details of this revision and I hope in due course to publish it, together with the details of the two other suggestions I have described above; one, to allow the experience of an individual observer to be modelled as the experience of observing a particular, identified, discrete, stochastic process, and the other, to provide a physics without physical constants.

Acknowledgements

I am grateful to Jeff Barrett, Katherine Brading, Jeremy Butterfield, Michael Esfeld, Michael Lockwood, Simon Saunders, and Steve Weinstein for useful conversations and comments.

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