The reply to our rebuttal is likely to be that such phenomenological meanderings are unreliable, and that we seem to be flirting with dualism. People are physical things; consciousness is hence ultimately a physical brain process; ergo, since the Beethoven experience is a stretch of conscious experience, it corresponds to some brain process stretching from to -- and surely (so the reply here continues) such a process can be reversed.

Unfortunately, this reply fails.
Recall that we explicitly ruled out dualism from the outset by
affirming agent materialism. In keeping with this affirmation
[an affirmation, specifically, of the proposition we labeled (ii)], and with
the spirit of the present objection in mind,
let's view the brain in overtly materialistic terms: Suppose that it's
composed, not of neurons and dendrites and the like, but of tiny colliding
billiard balls. A brain state is thus a ``snapshot" of these billiard balls
which
``freezes" them in a certain configuration. Suppose, further, that we have
on hand a microscopic but prodigious helper named Max, a (distant)
relative of Maxwell's demon.
Max can manipulate neurological
billiard balls *par excellence*. It should be possible for Max to
reverse billiard
ball mentation, if genuine mentation, bound as it is to be infinitely more
complex than its homey pool-hall analogue, is reversible. But Max, no matter
how clever, no matter how fast, no matter how conscientious, can't pull it off.

In order to see this let's return to Bob's mentation from to . In keeping with our billiards setup, we are now free to view Bob's mentation as the temporally extended interaction of tiny billiard balls inside his cranium. Suppose that Max has kept his eyes on this - progression, and that he remembers, impeccably, the passage of these balls through time -- their velocities, spatial and temporal positions, and so on. And suppose that we give Max the opportunity to reverse the mentation. Can he manage? No. And the reason he can't is quite straightforward: Suppose he begins by noting that the 8 ball, which moved from place to , must be reversed; so he moves the 8 ball from to (making sure that the temporal factors match up perfectly with the ``forward" recording of the 8 ball). In order to perform his task, the demon must remember what he has done. However, at some point Max's memory will be filled and he will have to erase some stored information. Since erasure is a setting process (i.e., an operation which brings all members of an ensemble to the same observational state) and this operation increases the entropy of the system (billiard balls plus demon) by an amount equal to or greater than the entropy decrease made possible by the newly available memory capacity, the demon cannot execute the reversed mentation.

``Slow down!" exclaims our critic. ``If Max's memory is large enough to store the entire sequence of states, then it's large enough to do the reversing; and the obvious way to reverse a sequence of billiard ball collisions is to simply reverse the velocities of the balls in the final state and let the whole system cycle backwards. There is no theorem of statistical mechanics which rules out reversing physical processes."

The central claim here is that since the dynamics of the billiard ball system is completely described by classical statistical mechanics, all we need do to reverse the system is to change the algebraic signs of the motion variables. Any state of the system, so the story goes, is sufficient to allow us to compute any previous and subsequent state.

Unfortunately, this objection conflates the formal system called
`classical' or `rational mechanics' with a particular instantiation
of it by point particles colliding elastically. Reversing
is achieved for our critic by manipulating the
*equations* of classical statistical mechanics,
that is, it is a formal operation, not a physical one.
We introduced Max to
see what it would take to *actually* reverse a physical process,
not simply reverse the formalism used to represent a physical process.
Max is supposed to bring into focus the conditions and consequences of
executing such a reversal. So it doesn't suffice to say
``reverse the velocities of the balls in the final state and let the
whole system cycle backwards." We want to know *how* this is done --
and this leads us to consider fanciful creatures like Max.

And what Max reminds us of is that while the idealized (point particles, elastic collisions) billiard ball model is logically reversible, the non-ideal (non-point particles, inelastic collisions) is not. The former is logically equivalent to the formalism it re-represents, and there is a one-to-one correspondence between the discrete states of the idealized model and the continuous state of the mathematical description. The non-ideal model is non-ideal precisely because it lacks those characteristics: there is no simple one-to-one correspondence between an ``inelastic collision" and the formalism which tries to capture it. And this is why Max cannot win when working in the real world of friction, dissipation, inelastic collisions, uncertain initial conditions, etc. His memory is soon exhausted, he needs to erase, but the erasure only causes him further difficulties and increases his task.

Our
metaphorical Max
can be supplanted with a technical discussion making the same point,
but such a discussion would quickly render the paper inaccessible to
most.
This discussion would begin by charting the research devoted to
instantiating purely conceptual, reversible models in terms of physical
models. This research has lead to a family of devices with intriguing
properties; one such device is is the **ballistic computer**, which
shows, *in principle*, how a computation can be performed without
dissipating the kinetic energy of its signals. The device employs a set
of hard spheres and a number of fixed barriers with which the ``balls"
collide and which cause the ``balls" to collide with each other.
The collisions are elastic, and between collisions the balls travel in
straight lines with constant velocity according to Newton's second law.
But what happens when one departs slightly from the idealizations upon
which the ballistic computers is based? We find what Max already
showed us: when we move from the realm of the ideal and ask how
ballistic computers might be physically embodied,
we immediately confront two problems: (i) the sensitivity of the ballistic
trajectories to small perturbations and (ii) how to make the collisions
truly elastic. Initial random errors in position and or velocity of one
part in 10 are successively magnified with each generation of collisions,
so that after a few dozen generations, the trajectories cannot sustain a
computation. Even if we had perfect accuracy and a perfect apparatus,
the ballistic computer would still be subject to fluctuating
gravitational forces in the environment, so that after a few hundred
collisions, the ``perfect" trajectories would be spoiled. This
dynamical instability of the ballistic computer could be countered
by corrections after every few collisions, but then we would no longer
have a thermodynamically reversible device: the computer becomes
dissipative and logically irreversible. (For a detailed survey
of the ``Max phenomenon," in the form not only of
ballistic computers, but also the enzymatic
Turing Machine and the Brownian computer, see
[5].
For a detailed, book-length treatment of
irreversibility in general, see the book one
of us (Zenzen -- with Hollinger) has written
on the subject: [42].)

Some skeptics might persist in objecting that our argument founders in its use of physics -- and the dialectic could go on, at the cost of producing a paper suitable only for consumption by students and practitioners of physics. Someone might say, for example, that with the possible exception of some rare weak interactions, all of fundamental physics is thought to be reversible in the most straightforward sense. One of the central puzzles of statistical thermodynamics, so the objection goes, is exactly to explain why the world seems to contain irreversible processes when at base it doesn't.

The puzzle here alluded to gives rise
to a fundamental rift in physics between those who hold that the world,
despite appearances, can't contain irreversible processes
(because the formalisms designed to capture the microlevel
imply that all physical processes are
in principle reversible), and those
who hold that no one ought to deny what appears to be clear
as day (for reasons
of the sort canvassed above in our discussion above of ballistic computers),
viz., that *real* physical processes are irreversible. And
because there is such a rift, the present objection is
uncompelling. In order
to see this, first consider someone
who, taking P-consciousness seriously, makes a careful case *C* for
property dualism (the best example is probably [22]).
Now consider an
eliminative materialist
who literally denies the existence of
P-consciousness (e.g., Dennett [12]).
And next consider someone who, apprised of this clash,
objects to *C*
as follows.
``Look, one of the great puzzles in philosophy is
that there appears to be this phenomenon called `P-consciousness' which
resists reduction to purely physical terms. But by the tenets of
physicalism, the world contains nothing that cannot be captured
in physical terms." This objection leaves *C* intact for all those
who, unlike the eliminative materialist, haven't unalterably placed
their bets. Similarly, our argument is intact for all those who
haven't permanently affirmed the view that certain formalisms in use
in physics
imply that nothing is irreversible.

Fri Sep 6 11:58:56 EDT 1996