# Quantum Mechanics and Many Minds

In this post I will explain how measurement gives rise to the many minds interpretation (MMI) of quantum mechanics (QM), an interpretation which entails that there are no ‘matters of fact’ about the outcomes of measurements and so no genuine measurement is possible.  Subsequently, I will discuss how this interpretation affects our understanding of QM and the world.

The measurable properties of electrons are called ‘observables.’  When a property of an electron is measured, the observable assumes one of two values.  We will discuss the properties ‘color’ (whose value is either white or black) and ‘hardness’ (whose value is either hard or soft).  We measure an observable by shooting an electron through a property-box which detects and represents the value of the relevant property.  By the uncertainty principle, the properties of color and hardness are incompatible with each other (because they are orthogonal; absolutely no correlation can be made between color and hardness) — the measurement of color disrupts the value of hardness and vice versa.  A way we can put this: an electron that we know is black is in a superposition of being both hard and soft. Superposition requires some explanation.

A superposition is a mode of being or movement for electrons.  A Total of Nothing box (ToN) is a box with two apertures; an electron (E) passes through it with none of its measurable properties changed or affected, and the time it takes for E to clear the box is the same it takes for E to traverse empty space of the same size.  Shoot a white electron through a color-box, and it will exit white.  If we insert a ToN into the middle of the box, the white electron will emerge black instead of white.  But by definition a ToN changes none of the properties of that which passes through, so it follows that the electrons could not have traveled through the ToN and that they could not somehow have circumvented the ToN (for then they still should have exited the box white).  So we say that an electron is in superposition in the sense that it could not have been black and it could not have been white and so it must somehow be in the state of both black and white until it emerges from the box and is determinately either one or the other.

We also need to understand something about the standard view of QM, namely the roles of the dynamical equations of motion (DEM) and the collapse postulate.  DEM is based on the Schrodinger equation, $\Psi(t_1) \rightarrow \Psi(t_2)$, and says that the states of all physical systems invariably evolve in accordance with DEM which are strictly deterministic principles — there is no room for probability (all motions are theoretically calculable).  The collapse postulate, in contrast, is strictly probabilistic.  It says that a system in a superposition (of color, let’s say) collapses to an observable state (to either black or white, with the probability of either outcome being exactly $1/2$), when disrupted by some measurement.1 So, prima facie, it seems that physical systems are governed by one set of rules when not being measured (the strictly deterministic ones) and are governed by different set of rules when they are the process of being measured (the strictly probabilistic ones).

But there is something strange about saying that physical systems are governed by two different kinds of rules depending whether or not they are being measured.  That physical systems are probabilistic only when measured (and are otherwise deterministic) seems rather ad hoc, especially in light of the measurement problem.  This is to say that, on this standard view, the probabilistic axiom is postulated solely for the sake of preserving our ability to make accurate predictions.  Moreover, we cannot even determine when the collapse of a physical system occurs — we cannot identify the precise physical interaction which causes the wave function to collapse.2 Suppose we wanted to maintain the (more parsimonious) thesis that all physical systems are governed by the same set of rules at all times.  If this is so, then (we will show) no genuine measurement will be possible.

Let’s make more explicit our assumptions.  Suppose that (1) there is just one world, (2) the world has one true description, (3) QM provides a complete description of any physical system, (4) DEM always makes true predictions, (5) normal people can correctly report whether they have some determinate belief, and (6) the evolution of a person’s mental state during the course of measurement is probabilistic.

(1), (2), (3), and (4) should all strike you as intuitively plausible.  Here’s the idea behind (5).  Though not all of your determinate beliefs (like that there is blog post before you) are immediately present to us in our experience, if we are to reflect on any believable proposition $\phi$, we would be able to accurately specify or report our (dis)belief in $\phi$, which has some determinate value.3 We will return to (6).  But notice that all physical systems adhere to DEM; mental states, however, always have a determinate value, so they cannot be governed by DEM and so, by definition, they are not physical systems.4 Let’s see where this takes us.

Sam is about to measure a hard electron E (so E is in $(1/\sqrt{2})(|E_b\rangle + |E_w\rangle)$).  If both QM and Sam’s ability to measure color are true, then after the measurement the physical state of E and Sam’s brain is $(1/\sqrt{2})(|believesE_b\rangle|E_b\rangle + |believesE_b\rangle|E_b\rangle)$ (by DEM).  But if you ask Sam about the outcome, he would report that he has a definite belief (e.g., Sam reports his state as: $|believesE_b\rangle$).  So Sam’s brain state cannot have any definite belief about the outcome.  But we can report our mental states (so they must be determinate) (via [5]).  If this is so, then our mental states are not identical to physical states.  We demarcate mental states and brain states by asserting that the evolution of Sam’s mental state is probabilistic.  Sam’s mind (which is not his brain or brain state), begins the measurement with no beliefs about the state of the outcome, and ends up with a definite belief (a mental state), e.g. $|believes E_b\rangle$.  Sam’s beliefs (mental), are never in superposition.5   The probability of Sam being in a mental state (e.g. $|believesE_b\rangle$) after measurement is exactly $1/2$.  The mind associated with a brain ends up in a mental state (e.g. $|believes E_b\rangle$) associated  with one of his brain states e.g. the brain state $1/\sqrt{2}(|believes E_b\rangle + |believes E_w\rangle)$ (that is — the mental state is one of the two terms of the [superpositional] brain state).

But if the mind is associated with just a single brain state in the superposition of brain states, then all but one of the brain states (which we represent: $|B\rangle =c_1|\acute{B_1}\rangle +c_2|B_2\rangle +c_3|\acute{B_3}\rangle + ... + c_n|B_n\rangle$, where $|B\rangle$ is the overall brain state6) represent mindless hulks in the sense that there is a potential infinity of brain states with no corresponding mental state.7 We correct this by supposing that every sentient physical system is associated with a continuous infinity of minds8 as opposed to a single mind and that probability of some infinite subset of those minds in a particular time, is given by $|\langle b_i | \Psi \rangle|^2$, meaning when the outcome of Sam’s measurement is a superposition, $1/2$ Sam’s continuous infinity of minds will be $|believesE_b\rangle$, and remainder is $|believesE_w\rangle$.  The reason for supposing an infinite collection of minds is so that each mind will end up associated with some element of the brain state in the superposition resulting from measurement; a measurement or a sequence of measurements may have an infinite number of outcomes, so an infinite number of minds is required (to guarantee that every brain state will have a corresponding mental state).  When Sam’s brain is $c_k|B_k\rangle$, his mind is in a determinate mental state $|M_k\rangle$.  Each individual mind in the continuum still evolves probabilistically and will never be in superposition (in the same manner as previously discussed).  But the totality of the infinite set of minds (thglobal’ mental state) associated with some $c_k|B_k\rangle$ evolves deterministically since the evolution of the measurement process is deterministic.9 The evolution of the global mental states will be determined, but the course of each individual mind will be subject to QM probabilities.  DEM determines the evolution all physical systems, including brain states, and the varieties of brain states determine the possible mental states (what beliefs Sam can collapse to and so have), so mental states will depend on what brain states are present and thus the determined physical state of the world.

Let’s revisit Sam’s measurement of E.  He is $|B\rangle$ and no mind has |color belief$E\rangle$.  After the measurement, Sam’s brain state will be $(1/\sqrt{2})(|believes E_b\rangle + |believes E_w\rangle)$, by DEM. $1/2$ Sam’s minds will be $|believes E_b\rangle$ and the other $1/2$, $|believes E_w\rangle$.  Each mind can be thought of as a unique ‘POV,’ representing some state of affairs, and each mind will have a seemingly continuous experience.10 So the $1/2$ minds associated with $|believes E_b\rangle$ are in a different ‘mental world,’ so to speak, than the other $1/2$.

This has a curious consequence: our beliefs about physical reality will turn out to be false.  This would be problematic, except for that the future evolution of Sam’s mental states will proceed exactly as if his beliefs were true.  Prima facie, this seems like a bitter pill to swallow.11 But it needn’t be too bitter, for our beliefs have effective validity — they will always evolve as though they were true.  Suppose that after Sam’s measurement (which happened at $t$), Sam$_1$ is associated with $|believes E_b\rangle$.  Sam$_1$ repeats the measurement with same E and the physical outcome is12 $1/\sqrt{2}(|believes E_b\rangle|D_b\rangle|E_b\rangle + |believes E_w\rangle|D_w\rangle|E_w\rangle$.  Sam’s mental state must be $|believes E_{b_1}\rangle|believes D_{b_1}\rangle + |believes E_{b_2}\rangle|believes D_{b_2}\rangle$, because all his minds$_1$ after $t$ know that the value of measurement$_2$ must be $|E_b\rangle$, for empirically the outcome of $E_b$ going through a color box (without any other tampering) must be $E_b$.  But this is the same mental state that Sam would have ended up in had in the event that his belief about the first measurement (at $t$, which was false) were true, for suppose measurement$_1$ resulted in $|D_b\rangle|E_b\rangle$ (corresponding to his belief) instead of the superposition — then measurement$_2$ must result in $|believesE_b\rangle|E_b\rangle$ for the minds know that if $E_b$ goes through a color box the outcome is $E_b$ and this is entailed by DEM.  The fact that our beliefs about reality are false has no effect on our lives or what we take to be our experience of the world.  We shouldn’t be too resistant to accepting that most of our beliefs our false, because if it happened to be the case that our beliefs were true, there would still be no detectable difference in our perceptual experience.  We won’t end up questioning our beliefs, even if we know them to be false about veridical reality, because we will still believe that the fire we (falsely) believe to be before us is hot — for if we do not, then we will likely get burned; so our beliefs must be valid even if false.

An important consequence of this view is that it preserves locality.  On the many minds picture, it is an illusion that measurements have outcomes (for measurements like Sam’s will result in a superposition), while the underlying assumption of nonlocality is that measurements do have outcomes.  At the conclusion of Sam’s measurement, the physical state of E is just $1/\sqrt{2}(|E_b\rangle + |E_w\rangle)$, and so there is no determinate outcome.  No matter what outcome is measured by whatever half of Sam’s continuous infinity of minds, one half of the minds will be $|believes E_{b_1}\rangle$ and the other half, $|believes E_{w_1}\rangle$.  This is not, however, a local realist theory.  Our beliefs will always be determinate and so we will always be in a belief state representing locality, but the actual state of the physical world is in superposition so veridical locality can never in fact be established (again because measurements do not have outcomes).

While it may seem problematic that mental states (and their probabilities) are inserted into this view by fiat, it is not necessarily so.  For the physicalist, the aforementioned seems ad hoc, inserted for the sole purpose of accounting for QM probability distributions.  This inevitably weakens the theory (as argued by Hempel).  This worry gets traction from its physicalist assumptions.  But prima facie, we are not compelled to physicalists over dualists; our inclination toward one over the other is contingent upon whatever background hypotheses we entertain and which concepts we choose to take as primitive (as argued by Lakatos).  If we take perceptual experience to be primitive in the same sense as matter or energy,13 then physicalism does not have any prima facie advantage over dualism.  And so, if we start with these equally plausible dualist intuitions, then MMV actually accounts for why our perceptual experience is such as it is, in a way that is both parsimonious and not-ad hoc.14  But moreover, because on MMV mental states still depend on brain states, we can preserve our intuitions about the brain somehow determining the state of our minds — and so stay consistent with contemporary psychology and neuroscience.

We’ll conclude by summarizing what we have shown MMV means for QM and the world.  All physical systems turn out to be governed by DEM, so there is no need to square DEM with the collapse postulate.  This also entails realism, that there is some one correct physical description of the world at any $t$, and that description will not be changed or altered by consciousness or other non-physical phenomena.  Because the evolution of the physical world is determined by DEM and is quantum mechanical, the ‘global’ mental state of Sam is fixed by his physical brain state — that is, mental states importantly depend on physical brain states.  Locality is preserved.  Probabilities (which is where the collapse postulate came in) turn out to be objective but can only refer to sequences of mental events in individual minds.  And measurement will not be possible because there is simply no matter of fact about the outcome of the measurement (because the measurement will result in superposition).

1. That is, the states of systems undergoing measurement evolve in accordance with the collapse postulate, not DEM.
2. Does it collapse when the microscopic particle interacts with the macroscopic measuring device?  Or does it collapse when the information from the position of a measuring dial passes through our retina into our visual cortex?  There is no good means of determining this.  A theory invoking collapse ought to what, where, when, and why the collapse occurs.
3. It does not matter if there are degrees of belief, for even if there were, we should be able to assess the exact degree of our belief in $\phi$
4. Here’s the main thrust: if an observer can measure some property of an electron and can correctly report his mental states, then it cannot be the case that his brain state is identical to his mental state, for brain states are physical systems conforming to DEM (and so will be superpositional) while mental states are never in superposition and so will be better described by the probabilities of the collapse postulate.
5. This explains why we never see a superposition.  Your perceptual experience, like your seeing a tree, amounts to a representational belief state. Because this is a mental state, there is no experience of superposition, and all beliefs are definite and of definite things.
6. I.e. the totality of all occurrent brain states.  Note: each $|\acute{B}\rangle$ term is associated with an infinite set of minds in the corresponding mental state $|M_k\rangle$.)
7. As thus far we have made it sound like there is some one mind corresponding to the mental state associated with a some, one, brain state.  Consequently, the remaining brain states have no associated mind.
8. A continuous infinity of minds can be thought of as the mental (or dualist) analogue of Everett’s continuous infinity of worlds.
9. This will become clear in the next example, but the idea is that the physical superpositions are determined, and each mind collapses to a mental state associated with one term of the superposition, so the totality of the evolution of the mental states will be determines, but the course of each individual mind will be subject to QM probabilities.
10. Similar to how, on the many-worlds view, each world represents some outcome of physical affairs/measurement, and no person notices their continuous instantiation of possible worlds — they will have a continuous experience.
11. For, on reflection, it seems as though our beliefs turn out to be true, and that we organize our lives around these beliefs that do correspond to reality in some determinate way.
12. The term ‘$D$‘ represents the measuring device.
13. As has been done here: ‘Facing Up to the Hard Problem of Consciousness,’ David Chalmers.
14. It is parsimonious because it simply explains the evolution of both physical systems and mental systems; it is not ad hoc because there is no reason to think that ontologically distinct systems should be governed by the same set of rules.