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Alex Byrne and Ned Hall

Department of Linguistics and Philosophy, MIT

The textbook presentation of quantum mechanics, in a nutshell, is this. The physical state of any isolated system evolves deterministically in accordance with Schrdinger's equation until a "measurement" of some physical magnitude M (e.g. position, energy, spin) is made. Restricting attention to the case where the values of M are discrete, the system's pre-measurement state-vector f is a linear combination, or "superposition", of vectors f₁, f₂,... that individually represent states that have particular values m₁, m₂,... of M; these vectors are eigenvectors of the (Hermitian) operator corresponding to M. Thus f = a₁f₁+a₂f₂+..., where a₁, a₂,... are (complex) scalars. Typically, more than one of the a_i are non-zero, which means that the pre-measurement system does not have a "definite" value of M. If f and f₁, f₂,... are normalized (i.e. have unit length) then, by the usual statistical algorithm, the probability that the measurement results in value m_i is |a_i|². On measurement, the state instantaneously changes, or "collapses", taking on the measured value of the magnitude; its state-vector thus becomes one of the corresponding eigenvectors.[1]

For familiar reasons, this understanding of quantum mechanics is highly problematic. But for equally familiar reasons some sort of "collapse" postulate would appear necessary, or at least highly desirable. Without it, quantum mechanics will describe some systems (for example, voltmeters) without saying whether they have or lack some properties (for example, the property of pointing to `10'). And plainly voltmeter needles have particular positions. Thus, without a collapse postulate, it seems that quantum mechanics fails (radically) to describe the world completely and correctly.

However, perhaps that is too hasty. Maybe all we can be sure of is that voltmeter needles look to be in particular positions, not that they actually are in them. If we could somehow explain how needles, despite not pointing in any one particular direction, nonetheless look that way, then a simple "no collapse" interpretation of quantum mechanics might be viable. Such an interpretation is suggested by a famous, and famously obscure, paper by Hugh Everett III (1957).

In a recent and deservedly much-discussed book, The Conscious Mind (1996), David Chalmers argues that an "independently motivated theory of consciousness" (349), namely his own, can "lend support" (xv) to the Everett interpretation.[2] Chalmers aims to settle crucial questions left open by Everett's own account, thereby producing an interpretation of quantum mechanics which, he claims, outranks all others in "theoretical virtue" (356).

Chalmers' theory of consciousness "lends support" to the Everett interpretation because, he thinks, it predicts "that a superposed brain state should be associated with a number of distinct subjects of discrete experience" (349)--for example that a brain in a "superposition" of the states perceiving the needle at `10', and perceiving the needle at `5', supports two distinct experiencing subjects, one of whom perceives the pointer as being at the 10 volt mark, while the other perceives it as being at the 5 volt mark.[3] Thus, even though needles almost never point anywhere, our perceptions invariably tell us otherwise.

Chalmers' argument is, then, of considerable interest. Although it fails, it does so instructively, for two reasons. First, Chalmers is one of the few proponents of an Everett-style interpretation who attempts to provide it with a sustained positive argument, derived from an explicit philosophical account of the mind-body relation.[4] Seeing why his ingenious argument doesn't work will lessen the temptation to appeal to the philosophy of mind in interpreting quantum mechanics. Second, examining Chalmers' argument will show the deep and underappreciated flaw in any Everett-style interpretation (to be distinguished from the well-known objection that such interpretations cannot accomodate the quantum mechanical probabilities). And some antidote is certainly desirable, because Everett-style interpretations appear to be getting increasingly popular (see, e.g., Lockwood 1996 and the accompanying commentaries).

Section 1 sketches Chalmers' account of the relationship between consciousness, cognition, and the physical. Then, in section 2, we explain Chalmers' argument that, given his theory of consciousness, the simple no-collapse interpretation can explain why we perceive the world as containing needles at particular positions, etc. We divide this argument into three phases. The first phase is an argument for one major premise. The second phase is an argument for another major premise. And the third phase puts the two major premises together in an attempt to draw the desired conclusion. In sections 3, 4, and 5, we critically discuss, respectively, these three phases. Although Chalmers has failed to establish his Everett-inspired interpretation, there remains the question whether anything resembling it should be taken seriously. The answer is "no": section 6 exposes the deep flaw in such interpretations. Section 7 sums up.

By `consciousness', Chalmers means what Ned Block (1995) has called `phenomenal consciousness': a phenomenally conscious state is a mental state there is something it is like for the subject to be in. Normal cases of pain and visual experience are uncontroversial examples of phenomenally conscious states; a controversial example is conscious belief.

Cognitive states are intentional mental states; that is, mental states with propositional content. Beliefs and other propositional attitudes are uncontroversial examples of cognitive states, and visual experience is relatively uncontroversial. Pain, however, is a controversial example of a cognitive state.

Physical properties are "the fundamental properties that are invoked by a completed theory of physics" (33). Physical facts are "the facts concerning the instantiation and distribution of [physical properties]" (33).

Physicalism is the view that any metaphysically possible world exactly the same as the actual world with respect to the physical facts is exactly the same simpliciter. (Here we ignore some irrelevant complications.[5]) In other words, according to physicalism, everything actual supervenes with metaphysical necessity on the physical.

Chalmers is almost a physicalist. He thinks that nearly everything--including the cognitive--supervenes with metaphysical necessity on the physical. It is impossible, he thinks, for a world physically the same as the actual world to differ from it with respect to who believes what: in particular, such a world contains a physical duplicate of Chalmers himself, who believes that he has written a book on consciousness, and so forth. But consciousness is the residue that does not supervene with metaphysical necessity on the physical[6]: there are (non-nomologically) possible worlds exactly the same as the actual world with respect to physical facts, and so with respect to facts about cognitive states--but from which consciousness is entirely absent. Such "zombie worlds" contain, of course, zombies (in the philosophers' sense): creatures cognitively just like you and us, some of whom are publishing weighty tomes on the problem of consciousness, but who are never in any phenomenally conscious states.[7]

Some terminology: the causal structure, or "functional organization", of a system may be described more or less finely; we shall call whatever is described at the level that is just fine enough for the purposes of functionalist psychology the system's fine-grained functional organization (sometimes the qualifier `fine-grained' will be omitted).[8] Chalmers first argues "for a principle of organizational invariance...given any system that has conscious experiences, then any system that has the same fine-grained functional organization will have qualititatively identical experiences" (248-9). And: "The invariance principle holds that functional organization determines conscious experience by some lawful link in the actual world" (250). Thus the principle of organizational invariance is the following supervenience thesis:

POI
For all nomologically possible worlds w₁, w₂, times t₁ and t₂, and possible systems s₁ and s₂, if s₁ in w₁ at t₁ and s₂ in w₂ at t₂ share the same fine-grained functional organization, then they have qualitatively identical experiences at t₁, t₂, respectively.

Following Chalmers, let "a maximal physical state be a physical state that fully characterizes the intrinsic physical state of a system at a given time" (349). Chalmers then argues that if a system in maximal physical state P has functional organization F, then any system "in a superposition of P with orthogonal physical states" (350) also has functional organization F. The new system might have other functional organizations, but that is not relevant: the claim is simply that it still has F. So functional organization is preserved by superposition (with "orthogonal states"). Let us call this thesis organizational preservation under superposition (OPUS).

Chalmers defines a maximal phenomenal state as "a phenomenal state that characterizes the entire experience of a subject at a given time" (349), and then draws the following conclusion from POI and OPUS:

If...a system in maximal physical state P gives rise to an associated maximal phenomenal state E, then...a system in a superposition of P with some orthogonal physical states will also give rise to E (349).

And he continues:

If [the above is right], then a superposition of orthogonal physical states will give rise to at least the maximal phenomenal states that the physical states would have given rise to separately. This is precisely what the Everett interpretation requires. If a brain is in a superposition of a "perceiving up" state and a "perceiving down" state, then it will give rise to at least two subjects of experience, where one is having an experience of a pointer pointing upward, and the other is experiencing a pointer pointing downward. (Of course, these will be two distinct subjects of experience, as the phenomenal states are each maximal phenomenal states of a subject.) (349).

We now assess this argument, starting with the first phase, where Chalmers tries to establish POI.

Oscar is looking at a ripe tomato in good light. Suppose that his fine-grained functional organization is F, and that POI is false. Then one of the following cases, involving Twoscar, who also has functional organization F, is nomologically possible:

(AQ) Twoscar has no conscious experience at all (so-called "absent qualia").

(IQ) Twoscar has conscious experience, but it differs qualitatively from the experience of Oscar (perhaps, to take the usual example, this is a case of "spectrum inversion" or "inverted qualia" [Shoemaker 1982]).

Chalmers argues (chapter 7) that if either AQ or IQ were nomologically possible, then the world would be extremely bizarre--implausibly bizarre, in fact. He concludes by reductio that POI is true.

Now we are quite unpersuaded by Chalmers' argument here, but it is not necessary to go into the details. For Chalmers' main opponents--orthodox physicalists--hold a stronger version of POI, obtained by replacing `nomologically possible' with `physically possible'. In other words, orthodox physicalists hold that once functional organization and the physical laws are fixed, all facts about consciousness are fixed. Chalmers, on the other hand, only holds that once functional organization and the physical and psychophysical laws are fixed, all facts about consciousness are fixed. So orthodox (i.e. functionalist) theories of consciousness entail POI, and since this is the only premise concerning consciousness in the overall argument this means that Chalmers has both overstated (at least in our opinion) and understated its significance. It is not his own theory of consciousness that supports the Everett-style interpretation, if his argument goes through, but rather almost everyone else's theory of consciousness.[9]

We now need to be a bit more precise about what exactly is supposed to be preserved under superposition. In chapter 9, Chalmers gives an account of how a physical system may implement a certain kind of abstract computing device, a combinatorial-state automaton (CSA). A CSA is simply a set of "state-vectors" {[S¹, ..., Sⁿ],...} "input-vectors" {[I¹, ..., I^k],...} and "output-vectors" {[O¹, ..., O^m],...}, together with state transition rules that map each pair of input- and state-vectors to a pair of output- and state-vectors. The transition rules determine (informally speaking) how the CSA produces an output, and changes its state, given its current state and input. Chalmers' proposal is "that a physical system implements a computation when the causal structure of the system mirrors the formal structure of the computation" (317-8). More exactly:

A physical system P implements a CSA M iff there is a decomposition of internal states of P into components [s¹,..., sⁿ], and a mapping f from the substates s^j into corresponding substates S^j of M, along with similar decompositions and mappings for inputs and outputs, such that for every state transition rule ([I¹, ..., I^k], [S¹, ..., Sⁿ]) -> ([S¹, ..., Sⁿ], [O¹, ..., O^m]) of M: if P is in internal state [s¹,..., sⁿ] and receives input [i¹,..., iⁿ], which map to formal state and input [S¹, ..., Sⁿ] and [I¹, ..., I^k] respectively, this reliably causes it to enter an internal state and produce an output that map to [S¹, ..., Sⁿ] and [O¹, ..., O^m] respectively (318).

Importantly, the substates and inputs and outputs of P have to be the sorts of properties that can enter into causal relations--"causally efficacious properties", for short. For dropping this restriction trivializes the notion of a physical system implementing a CSA: any system would implement any CSA, because, by various "grue-like" devices, we can always find properties of the system that map in the right way to the elements of the CSA. Indeed, one of Chalmers' main purposes in proposing this account is to rebut Searle's charge (1990) that to say that a system implements a computation places no interesting physical constraints on it.

With the account of implementation in hand, OPUS can be expressed more exactly as follows:

OPUS
"If a computation [i.e. a CSA] is implemented by a system in maximal physical state P, it is also implemented by a system in a superposition of P with orthogonal physical states" (350).

Recall that Chalmers' argument for POI was an argument only for the supervenience of phenomenology on "fine-grained functional organization", in other words on causal structure (at the appropriate fineness of grain).[10] Nothing in that argument appealed to the idea that mental processes are computations. So talk of "computation" in the statement of OPUS is a minor distraction. For the purposes of Chalmers' overall argument we do not have to regard an implementation of a CSA as actually computing anything: the crucial point is that Chalmers has provided a way of taking a CSA to be a description of the causal structure of a physical system.

Before we turn to the argument for OPUS, three observations are in order.

First, Chalmers follows common practice in speaking of a physical state as being a "superposition" of other physical states, and of two physical states as being "orthogonal". But although widespread, this way of talking can encourage serious confusion. In quantum mechanics, remember, vectors represent physical states; more precisely, to each physical system there corresponds a Hilbert space (a particular kind of vector space), and certain of the vectors in that Hilbert space represent different possible physical states of the system. The terms `orthogonal' and `superposition' refer in the first instance to relationships among vectors, not physical states. When a state is said to be a "superposition" of other states, this means, or should mean, that the vector representing the state is a superposition of the vectors representing the other states. Similarly when two states are said to be "orthogonal": this means that the vectors representing the states are orthogonal.

Unfortunately, talk of "superposed" or "orthogonal" states encourages the view that these terms are physically significant. In particular, it can suggest that "superposed" states are somehow physically composite. But without further argument, this is a mistake. It is the same kind of mistake as thinking that the property of being a monosyllabic primate (Ned is a monosyllabic primate because his name has one syllable, but Alex isn't) or the property of having prime mass (for some unit of mass, a mass of prime-numbered units), are physically or biologically significant properties. In fact, the latter is a better analogy: everything with mass has prime mass, and every state is "superposed"--for every state-vector can be expressed as a superposition of other state-vectors (indeed, in countlessly many ways). The importance of these cautionary remarks will become clear shortly.

The second observation is that OPUS can be more perspicuously rewritten as follows:

OPUS*
If a CSA is implemented by a system in maximal physical state P (represented by f), it is also implemented by a system whose state-vector is not orthogonal to f.

Since the state-vectors not orthogonal to f are exactly the state-vectors that are non-trivial superpositions of f with orthogonal vectors, OPUS* and OPUS are equivalent.

The third observation is that OPUS has the following remarkable consequence: any two systems simultaneously implement exactly the same CSA's, provided only that their state-vectors belong to the same Hilbert space. For suppose a system in state P implements a certain CSA, and that its state-vector f is not orthogonal to the state-vector y of a system in P*. Then OPUS (in the guise of OPUS*) immediately delivers the result that the P*-system also implements that CSA. Suppose, alternatively, that f is orthogonal to y. Consider the superposition f + y. It is not orthogonal to f, so by OPUS* the system whose state it represents implements the CSA.[11] But f + y is not orthogonal to y either, and so applying OPUS* once more yields that a system in state P* also implements that CSA.

For short: all systems of the same type (i.e., whose states are representable on the same Hilbert space) implement the same CSA's.

We now turn to Chalmers' argument for OPUS, expressed concisely as follows:

Assume that the original system (in maximal physical state P) implements a computation [i.e. a CSA] C. That is, there is a mapping between physical substates of the system and formal substates of C such that causal relations between the physical substates correspond to formal relations between the formal substates. Then a version of the same mapping will also support an implementation of C in the superposed system. For a given substate S of the original system, we can find a corresponding substate S of the superposed system by the obvious projection relation: the superposed system is in S if the system obtained by projecting it onto the hyperplane of P is in S. Because the superposed system is a superposition of P with orthogonal states, it follows that if the original system is in S, the superposed system is in S. Because the Schrdinger equation is linear, it also follows that the state-transition relations between the substates S precisely mirror the relations between the original substates S. We know that these relations in turn precisely mirror the formal relations between the substates of C. It follows that the superposed system also implements C, establishing the required result (350).

Drawing on the third observation in the previous section, if this argument is right then any system A simultaneously has the causal structure of any other system B whose states are representable on the same Hilbert space. But all that requires is for A and B to comprise the same number of each type of fundamental particle (same number of electrons, protons, etc.). Thus, if the universe has causal structure at all, this structure exhibits an astonishing amount of duplication. And this conclusion is apparently supposed to follow without assuming any controversial interpretation of quantum mechanics (let alone controversial theories of consciousness): we just observe that quantum mechanics applies to physically possible systems with causal structure, and then appeal to the "projection relation" and the linearity of the Schrdinger equation. There's got to be a catch in that!

We will split Chalmers' argument into two parts. The first part sets up a mapping between substates of the original system and substates of the new system. The second part argues from the linearity of the Schrdinger equation to the conclusion that "the state-transition relations between the substates S precisely mirror the relations between the original substates S," and thence to the conclusion that, if the original system implements C, so does the new system.

Unpacking the shorthand that allows Chalmers to speak loosely of systems being projected onto hyperplanes, and so forth, his suggested mapping is this:

the superposed system is in S' if the state represented by the vector that results from projecting the state-vector of the superposed system onto the hyperplane of the state-vector representing P, represents a system in S

Now some jargon needs to be explained. The "hyperplane" of a vector is the one-dimensional space of vectors that are parallel to it. For any vector f, there is a unique vector y lying in this space such that f is a superposition of y with some orthogonal vector; the "projection" of f onto the space is simply y.

Since parallel vectors represent the same state, the only state that a vector lying in the hyperplane of the state-vector of the original system could represent is P itself. Of course, such a vector might not represent any state: the zero vector, for example, does not. Might Chalmers intend that only normalized vectors represent states? No: For it would immediately follow that the superposed system is not guaranteed to be in a "corresponding substate S'"--since the projection of its state-vector onto the hyperplane of P will necessarily have length less than one. So we must interpret Chalmers as supposing that non-zero vectors of length less than or equal to 1 represent states.

Putting all this together, the suggested mapping simplifies rather dramatically:

the superposed system is in S' if its state-vector is not orthogonal to the state-vector that represents P

(For if the state-vector of the superposed system is not orthogonal, then its projection onto the hyperplane of P will yield a non-zero vector, which represents P, which is a state which places the original system in S.)

Is this adequate for Chalmers' purposes? No. He needs to show that there is a mapping from the relevant substates of a CSA-implementing system to substates of a "superposed" system which preserves causal organization. And to do that, he must--minimally--show (i) that the mapping takes a given substate S of the original system to a causally efficacious property, and (ii) that the mapping is one-one.

Chalmers offers no argument whatsoever that the "corresponding substates" of the superposed system are causally efficacious. And since he gives only a sufficient condition--and not necessary and sufficient conditions--for the superposed system to be in such a substate, it is clear he has failed to define a one-one mapping.

In fact, matters are considerably worse, for any mapping meeting Chalmers' sufficient condition actually makes (i) and (ii) false. Suppose f represents a system in (inter alia) substate S. Then every state-vector not orthogonal to f (and perhaps others) represents a system in corresponding substate S. Consider an arbitrary such state-vector y. By a second application of the sufficient condition, every state-vector not orthogonal to y represents a system in a substate S corresponding to S. In other words, every state-vector not orthogonal to some state-vector that is not orthogonal to f represents a system in substate S.

But that is every state-vector, period. And it is hardly believable that the property represented--a property the system must have, no matter what its state--is causally efficacious.

Furthermore, suppose our original system is in both substates S₁ and S₂. Then any system whose state-vector is not orthogonal to that of the original system will be in corresponding substates S₁ and S₂. But the substates that, by a second application of Chalmers' sufficient condition, correspond to these will be exactly the same: the trivial substate just introduced. Thus no mapping meeting Chalmers' condition is one-one.

Suppose these difficulties can somehow be fixed, and a mapping produced that meets conditions (i) and (ii) above. Does it follow from the linearity of the Schrdinger equation that the causal relations between successive substates of the superposed system exactly mirror the causal relations between successive substates of the original system?

Take a simple example. Suppose S₁ is a substate of the P-system (represented by f), mapped onto substate S₁ of the superposed system (represented by a vector that we can write as `af + by'). And suppose that (the instantiation of) S₁ causes, t seconds later, the P-system to be in a new substate S₂. Let Schrdinger evolution over this time period take f to f* and y to y*. Then the P-system state-vector at time t is f*, and by the linearity of the Schrdinger equation the state-vector of the superposed system at t is af* + by*. Then the mapping guarantees that the superposed system is in a substate S₂ corresponding to S₂.[12]

Does it follow that S₁ causes, t seconds later, the superposed system to instantiate S₂? Well, either the original mapping guarantees that S₁ causes S₂, or it doesn't. If it does (and so goes beyond the minimal desiderata of (i) and (ii)), then the linearity of the Schrdinger equation is irrelevant. So suppose it doesn't. By linearity, the superposed system at the later time has state-vector af* + by*, a component of which is af*, which (since parallel vectors represent the same state), represents the P-system at the later time. But, if we heed the cautionary remarks about "superpositions" made earlier in section 4.1, there should be no temptation to think that this fact about the representation of states implies anything interesting at all about the states represented. From the two premises (a) that af* + by* has af* as a component, and (b) that the system represented by f* has a substate caused in such-and-such ways, nothing follows at all about the causal history of any substates of the system represented by af* + by*. Analogously, from the two premises (a) that `Alex' has `Al' as a component, and (b) that the person named `Al' has tenure, nothing follows at all (regrettably) about the tenured status of the person named `Alex'.

Recapitulate the cautionary remarks about "superpositions" made in section 4.1 with the example of the spin states of spin-1/2 particles. Spin is not spin simpliciter, of course, but spin in some direction: spin in the x-direction, spin in the y-direction, etc. Spin "up" (equally, "down") in any one direction is always represented by a non-trivial superposition of the "up"- and "down"-vectors for any other direction. So unless we pick out some direction--hence, by implication, some pair of "up" and "down" eigenstates--as being somehow physically privileged, there is just no physically interesting sense in which some spin states are "superposed" and some are not.[13] Of course, this observation applies not just to spin states, but to any kind of quantum mechanical state.

If, however, some basis is physically privileged, in that its elements represent states that are somehow of a physically distinctive kind, then there is a physical distinction between non-superposed and superposed states: the former are precisely the states represented by some element of the privileged basis.

So assume that there is a privileged basis, and in addition that any (physically possible) CSA is implemented by some system whose state-vector is in the privileged basis.

Second, assume that when a system is in the state represented by a superposition of some privileged basis vectors f_A, f_B,..., it has physically distinct parts that are in states represented by f_A, f_B,... Thus any system in a state represented by the superposition will contain a physical duplicate of a system whose state-vector is f_A, etc.[14]

A version of OPUS can be derived from these two assumptions, as follows.

Suppose a system in state P implements a certain CSA. By the first assumption, P is represented by a vector f in the privileged basis. Let y be a superposition of f with some privileged basis vectors: y = af + .... By the second assumption, one physically distinct part of a system in the state represented by y will implement the same CSA as the P-system, for it is a physical duplicate of that system.

Therefore, given this understanding of the formalism, causal organization is indeed preserved under superposition. Unfortunately, this is hardly an argument that Chalmers can endorse, because he explicitly rejects the appeal to a privileged basis. That Everett's original interpretation seems to require one is his main complaint against it (348).

Let us temporarily waive these problems with Chalmers' premises, and ask if we can get anything out of putting them together.

Return to the conclusion Chalmers draws from POI and OPUS, mentioned earlier in section 2:

If...a system in maximal physical state P gives rise to an associated maximal phenomenal state E, then...a system in a superposition of P with some orthogonal physical states will also give rise to E (349).

This is not yet the desired result--that even if "the Schrdinger equation is all" (346), there will still be minds who experience the world as discrete. To get that, Chalmers must discharge the antecedent, by showing that on his picture of what the world is like, there are maximal physical states that give rise to subjects who experience the world as discrete.

But now there is an obvious difficulty. For Chalmers' interpretation implies that perceptual experience is (more or less) entirely illusory.[15] When you seem to see a voltmeter needle pointing to `10', your perceptual experience is probably not veridical: the needle (if, indeed, we can sensibly speak of such a thing) is not pointing to `10' or anywhere else. Likewise for any other sort of perceptual experience, whether of voltmeter needles, string, sealing wax, cabbages or kings. Russell's remarks could have been written with Chalmers in mind: "Naive realism leads to physics, and physics, if true, shows that naive realism is false. Therefore naive realism, if true, is false; therefore it is false." (1950, 15).

Although a whiff of paradox surrounds the suggestion that empirical enquiry might terminate in a theory that implies that almost all the observational "data" with which it began are false, we are not making this complaint. Rather, we are complaining that Chalmers has not told us what the world is like if "the Schrdinger equation is all"; in particular, he has given us no reason to suppose that there could be systems with the appropriate fine-grained functional organizations. Of course, assuming that our experience is by and large veridical, there is no problem: in that case, we have an abundance of information about the causal organization of various systems, including human brains. But on Chalmers' view, our experience is no such guide.

This complaint can be strengthened. Chalmers can't answer the question of what the world is like if "the Schrdinger equation is all" because no one can. The constraints on an acceptable answer are too few for the question to be an occasion for genuine inquiry, rather than mere stipulation. We shall argue for this in the following section.

The "bare theory" is simply the result of subtracting the "collapse" postulate from orthodox quantum mechanics. The idea that there is something significant to be learned from studying this theory has a great deal of currency: Deutsch (1996) claims, in effect, that the bare theory straightforwardly entails that there are many worlds; Lockwood (1989, 1996) holds a view similar to Chalmers'; Albert and Loewer, who propose to augment the bare theory with certain psychophysical postulates, still turn to it for a non-arbitrary specification of the "eigenstates of mentality" (Albert 1992, 129, fn. 16); both Albert (1992, ch. 6) and Barrett (1994) find merit in pursuing the more modest aim of uncovering the "suggestive properties" of the bare theory.

These discussions apparently assume that the bare theory, while perhaps obviously false, at least possesses powers of representation and conditions of application that are approximately as rich as those of orthodox quantum mechanics. And sometimes a stronger assumption is made: if orthodox quantum mechanics can represent that a system has a certain property, so can the bare theory. That is, if there is a state vector f₁ (interpreted as part of orthodox quantum mechanics) such that necessarily any system represented by f₁ has property P, then there is a state vector f₂ (interpreted as part of the bare theory) such that necessarily any system represented by f₂ has P (typically f₁ and f₂ are taken to be identical). If the stronger assumption is right, then the problem raised for Chalmers in the previous section is straightforwardly solved: the bare theory can represent that systems have such-and-such functional organizations after all--at least if orthodox quantum mechanics can.

Both these assumptions are unfounded. There is an instructive way to dispose of the stronger assumption while granting the weaker; that is what we will do first.

The point is an abstract one that applies not just to quantum mechanics. Suppose T₁ is a physical theory formalized as a family of state spaces together with a set of trajectories through these state spaces--the "dynamics" of T₁. And suppose T₂ is obtained from T₁ merely by changing the dynamics. Unless this change implies otherwise, the interpretation of the formal apparatus of T₂ is taken to be the same as that of T₁. The two pertinent theories related in this way are of course orthodox quantum mechanics and the bare theory: the state spaces are the same, but only the dynamics of the latter is deterministic. Another example is the elementary textbook theory of the pendulum and a theory that adds to it a correction for air resistance.

Suppose that state-space element f, taken as part of T₁, represents that a system has property P. And suppose--as might well be the case--that this is not semantically stipulated (as, for example, it is stipulated that `l' will represent the length of the pendulum). It would be a "fallacy of subtraction" to think it follows that f, taken as part of T₂, also represents that the system has P. For, in particular, if P is a dispositional property, then a system's possession of P will be sensitive to how the system evolves, or would evolve, over time--in other words to the dynamics. Thus, if f represents, taken as part of T₁, that a system is soluble in water, it won't follow that f also represents, taken as part of T₂, that the system is soluble in water. What's more, it won't even follow that, according to T₂, there is any state of the system such that necessarily, if it is in that state, it is soluble in water. If the dynamics of T₂ is sufficiently different, the theory might hold solubility in water to be a nomological impossibility.

This point seems to have been overlooked in discussions of the bare theory. For authors who claim to be exploring its consequences routinely assume that, according to the bare theory, there are possible states of an observer's brain that constitute her having a certain belief (e.g., that the measurement outcome was "up"). And, while no one is very explicit about this, it seems also to be assumed that the vectors representing these "eigenstates of belief" are the very same as the vectors which, according to orthodox quantum mechanics, assign to the observer the given belief. Thus, a typical discussion might begin by assuming that, according to orthodoxy, there is some state-vector f for a given observer (or her brain) such that, when she is in the state represented by f, she has the belief that the measurement outcome was "up".[16] This assumption is simply carried over to the discussion of the bare theory, at which point the standard problematic takes over: according to the bare theory observers typically won't be in states represented by "belief eigenvectors" like f.

But this is quite wrong, because the property of having such-and-such a belief is exactly the sort of property one would expect to be sensitive to the prevailing dynamics (this is especially clear on the common view that belief-properties are kinds of functional--therefore, broadly construed, dispositional--properties). Hence, if f is, according to orthodox quantum mechanics, a belief eigenvector, it does not follow that f is a belief eigenvector according to the bare theory. In fact, it's not at all obvious that the bare theory has any belief eigenvectors.

We now turn to the other assumption, that the bare theory has substantive content, comparable to that of orthodox quantum mechanics. Let us begin by briefly reviewing how the representation of a system's state works in the orthodox case.

First, the physical properties that the system retains over time--being composed of such-and-such particles, and so forth--can be thought of as either straightforwardly encoded in the state-vector or else specified separately.

Second, the physical properties of the system that it can gain or lose over time are encoded by the eigenstate-eigenvalue link: a system possesses the value m_i for physical magnitude M, represented by Hermitian operator M, iff the state-vector f for the system is an m_i-eigenstate of M (i.e. iff Mf = m_if).

Third, the state-vector encodes a certain kind of information about the system's dispositions or propensities: it tells us, via the usual statistical algorithm, the probability that "measurement" of M on the system will produce result m_i.

The point just made against the first assumption shows that it cannot be taken for granted that the orthodox eigenstate-eigenvalue link is completely preserved in the bare theory. Suppose magnitude M is represented in orthodox quantum mechanics by operator M, and suppose that this is not semantically stipulated. It doesn't follow that the bare theory represents M by M (or, indeed, that the bare theory represents M at all). But let us waive this problem. In fact, let us suppose that if orthodox quantum mechanics represents M by M (for a certain system), so does the bare theory.

Now the bare theory rejects the last way of encoding a system's properties: according to it, these propensities to produce various outcomes do not exist. That is because, for well-known reasons, the bare theory does not give outcomes probabilities.[17] But there is nothing else left in orthodox quantum mechanics that might be used to assign some information about M to a state-vector that is not an eigenvector of M. Therefore the bare theory has no resources to interpret such state-vectors as containing information about M.

This has an important consequence. Not every operator can be intelligibly taken to represent a property independently of the statistical algorithm. For example, consider the operators E and B, whose eigenvectors are those that are non-trivial superpositions of, respectively, energy and belief eigenvectors. What properties do these operators represent? The standard explanation appeals to the statistical algorithm and to operators that can be understood independently of it. Thus, the properties represented by E and B are certain propensities to produce, respectively, energy and belief "measurement" outcomes. Disallowing appeal to the statistical algorithm, there seems to be no way of understanding which properties are represented by E and B. And since there can hardly be a hidden fact of the matter, this means that operators like E and B will not represent properties at all.

Therefore, if we say that an "ordinary" operator is one that can be interpreted independently of the statistical algorithm, the important consequence can be put thus: the bare theory will be unable to interpret any state-vector which is not the eigenvector of any ordinary operator.

The fact that the bare theory cannot interpret vectors that are not eigenvectors of any ordinary operator seriously degrades its representational capacity for reasons that are entirely familiar, although their significance is perhaps not widely appreciated. Because of the way (almost any) system--macroscopic or microscopic--interacts with its environment, its state will typically not be represented by an eigenvector of any ordinary operator.[18] Taking energy as an illustration, we can think of the energy states of almost any system s as "measuring" the momentum of some system s in its environment. That is, there will be different initial momentum states of s, represented by f₊ and f_-, and an initial state of the "apparatus" (i.e. s), represented by y, such that Schrdinger evolution takes f₊ x y (i.e. the state vector representing the initial state of the composite system s-plus-s[19]) to f₁ x y₊, and f_- x y to f₂ x y_-, where f₁ and f₂ represent different post-measurement states of s and y₊ and y_- represent different post-measurement energy states of s. Now, s at almost any time will have just acted as this sort of measuring device for the momentum of something in its environment, which moreover we can safely presume was initially in a non-trivial superposition of vectors representing momentum states that the system can measure. So, if we represent the initial state of s by the non-trivial superposition af₊ + bf_-, then the current state of s-plus-s will be represented by af₁ x y₊ + bf₂ x y_-. This state is one in which s is not in an energy eigenstate.[20] The example can be generalized to all ordinary operators. Almost every system at almost any time will not be in an eigenstate of any ordinary operator, and so when interrogated on what the world is like, the bare theory will fall almost entirely silent.[21]

Chalmers makes it abundantly clear that he considers it an extremely important desideratum for any interpretation that it neither modify the Schrdinger equation (by admitting exceptions to it in the case of "measurements"), nor supplement it (by introducing "hidden variables"). Apparently such a conservative approach is warranted because, as he puts it, "the heart of quantum mechanics is the Schrdinger equation. The measurement postulate, and all the other principles that have been proposed, feel like add-on extras". We thus are supposed to have excellent reason for exploring the consequences of "extend[ing] superposition all the way to the mind" (346).

However, as we have argued, this "bare theory" is no real theory at all: it is quantum mechanics eviscerated, and consists largely of dry bones of uninterpreted mathematics.

Further, although we have examined just one proposal, we hope our discussion has poured some cold water on the idea that a theory of consciousness might help rescue quantum mechanics from the difficulties with the standard interpretation. After all, Chalmers' attempt strikes us as about the best that can be offered, at least if we forego exotic psychophysical postulates, as in the "many minds" theory of Albert and Loewer (1988).

Why was consciousness thought to be relevant in the first place? Because, supposedly, although we can deny that needles are hardly ever at particular positions, and so forth, what we cannot possibly deny are facts about our conscious mental states, in particular that we seem to see needles at particular positions. Of course, the claim that the mental enjoys a peculiar epistemic security is hardly new: it has pervaded Western philosophy since Descartes. But it is a controversial assumption, and one that much epistemology in this century has been devoted to overturning. Any interpretation of quantum mechanics motivated by it deserves to be viewed with suspicion.[22]

Albert, D. 1992. Quantum Mechanics and Experience. Harvard University Press.

Albert, D, and B. Loewer. 1988. Interpreting the many-worlds interpretation. Synthese 77, 195-213.

Barrett, J. 1994. The suggestive properties of quantum mechanics without the collapse postulate. Erkenntnis 42, 89-105.

Block, N. 1995. On a confusion about a function of consciousness. Behavioral and Brain Sciences 18, 227-87.

Byrne, A., and N. Hall. 1988. Chalmers, Papineau, and Saunders on probability and many minds interpretations of quantum mechanics. MS.

Chalmers, D. J. 1996. The Conscious Mind. Oxford University Press.

Dennett, D. C. 1991. Conciousness Explained. Little, Brown.

Deutsch, D. 1996. Comment on Lockwood. British Journal for the Philosophy of Science 47, 222-8.

Dretske, F. 1995. Naturalizing the Mind. MIT Press.

Everett, H. 1957. `Relative-state' formulation of quantum mechanics. Reprinted in J. Wheeler and W. H. Zurek, eds., Quantum Theory and Measurement, Princeton University Press, 1983.

Hall, N. 1996. Composition in the Quantum World. Ph.D. diss., Princeton University.

Harman, G. 1988. Wide functionalism. In S. Schiffer and S. Steele, eds., Cognition and Representation, Westview Press.

Lewis, D. K. 1980. A subjectivist's guide to objective chance. Reprinted in his Philosophical Papers, vol. 2., Oxford University Press, 1986.

Lewis, D. K. 1983. New work for a theory of universals. Australasian Journal of Philosophy 61, 343-77.

Lewis, D. K. 1986. On the Plurality of Worlds. Basil Blackwell.

Lockwood, M. 1989. Mind, Brain and the Quantum. Basil Blackwell.

Lockwood, M. 1996. `Many minds' interpretations of quantum mechanics. British Journal for the Philosophy of Science 47, 159-87.

Loewer, B. 1996. Comment on Lockwood. British Journal for the Philosophy of Science 47, 229-32.

Lycan, W. G. 1996. Consciousness and Experience. MIT Press.

Russell, B. 1950. An Inquiry into Meaning and Truth. Allen & Unwin.

Searle, J. 1990. Is the brain a digital computer? Proceedings and Addresses of the American Philosophical Association 64, 21-37.

Shoemaker, S. 1982. The inverted spectrum. Journal of Philosophy 79, 357-81.

Tye, M. 1995. Ten Problems of Consciousness. MIT Press.

[1]For magnitudes like position and momentum, which admit of a continuum of possible values, the mathematical representation is a little more complicated, although the basic idea is the same.

[2]All references are to this book unless otherwise noted.

[3]Here `perceives' is not supposed to be factive: one may perceive that p when it is false that p.

[4]`Proponent' is perhaps a bit too strong: "We may never be able to accept the view emotionally, but we should at least take seriously the possibility that it is true" (357).

[5]Problems arise because a physicalist might well agree that there could have been "epiphenomenal spirits" that do not interact with anything physical (Lewis 1983, 362); in which case physicalism as stated in the text is false. For discussion and different suggested repairs, see Lewis 1983, 361-4, and Chalmers 1996, 38-41.

[6]So, according to Chalmers, those facts with "a dependence on conscious experience" (72)--e.g., on some views, facts about color and other "secondary qualities"--do not supervene with metaphysical necessity on the physical either.

[7]Chalmers makes a slight qualification to the claim that our zombie twins share absolutely all our intentional states (203-9).

[8]Chalmers use of `fine-grained' is different: "at a level fine enough to determine behavioural capacities" (248). But we think our characterization is more faithful to his overall intentions.

[9]E.g. Dennett 1991, Tye 1995. There is a complication because Tye, at least, would only endorse POI if a "functional organization" were taken to be individuated in part by actual environmental causes and effects (cf. Harman 1988), and in Chalmers' own presentation the other--"narrow"--kind of functionalism is tacitly assumed. Still, if Chalmers' argument works, it can be adapted to cover Tye's position. Dretske (1995) and Lycan (1996) both claim that what phenomenal state a subject is in depends on his evolutionary history, and so would deny POI (at any rate on the usual readings of `functional organization'). But it is possible to adapt Chalmers' argument even here.

[10]Not exactly. The notion of causal structure that a CSA specifies is highly abstract, and in fact more abstract than the functional supervenience base delivered by the argument for POI. The latter kind, as far as Chalmers' argument for POI goes, might be individuated by the specific nature of the (possible) inputs and outputs (e.g. whether the input is a red tomato before the eyes); analogously, the functional analysis of a mousetrap will mention what it does to mousey input. By contrast, the former (CSA) kind might count a mousetrap and an elephant trap as having the same causal structure. This is a problem with the overall argument, because we want the latter kind of causal structure to be "preserved under superposition", and at best all we get is the preservation of the former. Since there are more serious objections, we can afford to set this difficulty aside.

[11] Of course, f + y may be replaced by its normalization if desired, without affecting the argument. (On whether non-normalized vectors represent states, see section 4.21 below.)

[12]Does the linearity of the Schrdinger equation come into play here? After all, it does guarantee that y* will not be orthogonal to f*. But that is quite irrelevant: for if there is a mapping between states with non-orthogonal state-vectors, then by the two-step maneuver of the previous section there is a mapping between any two states whatsoever.

[13]Although Chalmers makes precisely this point at 335, on the previous page he seems to have temporarily forgotten it, writing that spin "has only two basic values...[which] can be labeled `up' and `down'", and that "In quantum mechanics the spin of a particle is not always up or down" (334).

[14]Cf. Chalmers on a "superposed pointer state": "the theory predicts that the pointer is pointing to many different locations simultaneously!" (339-40). This is presumably a slip.

[15]Chalmers at one point writes that on his view "we are experiencing only the smallest substate of the world" (356), which suggests that he does not think that perceptual experience is illusory. But without a privileged basis (see 4.23 above) there is no avoiding it.

[16]Of course, orthodox quantum mechanics doesn't stipulate that certain state-vectors represent belief states (at any rate as such)--the vocabulary of the theory doesn't contain any psychological expressions.

[17] Assuming we can make sense of the bare theory attaching numbers to outcomes, they cannot be "objective chances" (Lewis 1980), because the theory is deterministic. Neither can they be probabilities resulting from our ignorance of relevant aspects of the system in question: they are outcomes that the theory tells us will happen, and hence there is nothing relevant of which we are ignorant. But these are the only two options, and so the numbers cannot be probabilities. (See, e.g., Albert and Loewer 1988, 201; Loewer 1996.) Chalmers himself holds out some hope that this argument can be defused (355-6); for discussion of his (and others') suggestions see Byrne and Hall 1998.

[18] Of course, irrespective of this point, its state won't be represented by eigenvectors of (e.g.) the position and momentum operators because they don't have eigenvectors.

[19] f₊ x y is a vector in the tensor product of the two Hilbert spaces representing, respectively, states of the object and the system.

[20] Of course the illustration itself shows that it is an idealization to suppose, as we have, that the "apparatus" and the measured system are in pure states before the measurement. Strictly speaking, the two systems before the measurement will be "coupled" with other systems. It is also worth noticing that the sort of coupling exhibited by s-plus-s just after the measurement is unlikely to go away: given realistic assumptions, Schrdinger evolution won't take s-plus-s to an "uncoupled" state in which s is in an energy eigenstate. (Cf. Albert 1992, 88-92.)

[21] It might be objected--for two quite different but equally misguided reasons--that the bare theory can say quite a lot about what a system is like when the composite state-vector is af₁ x y₊ + bf₂ x y_-, at any rate when y₊ and y_- are eigenvectors of an ordinary operator A, with eigenvalues a₊ and a_-.

First, although the system is not in an eigenstate of A, it is in an eigenstate of a "coarser" magnitude A*, represented by an operator A* which is just like the operator A, save that it assigns to y₊ and y_- the very same eigenvalue a*. And isn't the fact that the system has value a* for the magnitude A* perfectly intelligible, even by the lights of the bare theory? It just means that it has either value a₊ or value a_- for the magnitude A. Well, no, it doesn't: given the assumption about A, the "disjunctive" property of having A-value either a₊ or a_- cannot be represented by an operator in accordance with the eigenstate-eigenvalue link. (The erroneous reasoning just sketched involves the "disjunction fallacy"--see Hall 1996, 123-35.)

Second, it might be objected that the state represented by the superposition af₁ x y₊ + bf₂ x y_- is indistinguishable from the corresponding mixture (represented by weighting f₁ x y₊ by a² and f₂ x y _- by b²), which can be given a straightforward "ignorance" interpretation (i.e. the system is either in the state represented by f₁ x y₊ or by f₂ x y_-, but we don't know which). So the states can be identified. This is doubly wrong. The two states are not the same, and anyway `indistinguishable' means: the same statistics for all feasible measurements--which brings in the prohibited statistical algorithm (cf. Albert 1992, 84-92).

[22] Many thanks to David Chalmers, Tim Maudlin, and two anonymous referees for Philosophy of Science.