backwards or twice as fast

Recently I came across an argument about 'reversal of time' and our conscious experience (I am sure
this type of argument must be at least hundred years old) and I thought I should mix it with an old
idea of mine. I am curious what others think about it; So here it goes:

Imagine that we can describe the world as a Newtonian universe of classical particles so that
xi(t) , where x is the position(vector) of the i-th particle and t is the classical time
parameter, determines the configuration of our world for each moment of time. I am pretty sure that
the following argument can be generalized to a quantum mechanical description, but it is much easier to
stick to Newton for now.

We assume that the world evolves according to the laws of Newtonian physics up until the time t0.
At this moment an omnipotent demon reverses all velocities: vi(t0) = x'i(t0) -> - x'i(t0),
where ' is the time derivative, and the Newtonian evolution continues afterwards.

Obviously, for t > t0 everything moves 'backwards'; If a glass fell on the floor and shattered into many pieces for t < t0,
it will now assemble and bounce back up from the floor etc.; If the entropy S(t) increased with t for t < t0, it now decreases for
t > t0.

One can also check that xi(t0+T) = xi(t0-T) and x'i(t0+T) = -x'i(t0-T) for every T (as long
as we rule out non-conservative forces).

The interesting question in this thought experiment is "what would an observer experience for t > t0 ?".

If we assume that the conscious experience E(t) of an observer is a function of xb(t), where b enumerates
the particles which constitute her brain, then we would have to conclude that the observer does not recognize anything
strange for t > t0, since xb(t0+T) = xb(t0-T) and it follows immediately that E(t0+T) = E(t0-T). So if
all the experiences E(t0-T) contained only 'normal' impressions then the same is true for E(t0+T). In other words, while the sequence of
experiences is 'backwards' no single experience contains the thought "everything is backwards" and nobody feels anything strange.

But this would mean that no observer is able to recognize 'backward evolution' with entropy decreasing and distinguish
it from normal evolution!

One way to avoid this strange conclusion is to assume that E(t) is a function of xb(t) and vb(t).
Of course, we do not have a physical description of conscious experiences and how they follow from the configurations of our brain (yet).
It is reasonable that our conscious experience depends not only on the position of all molecules in our brain but also
their velocities.

Unfortunately, this leads us into another problem. If we rescale the time parameter t as t* = s*t, this would rescale all velocities
so that v(t*) = s*v(t) and thus E(t) = E[x(t),v(t)] -> E(t*) = E[x(t*),s*v(t*)]; But if the function E is sensitive to vb then
it would be sensitive to the scale s too. I find this to be quite absurd, our experiences should not depend on an unphysical parameter.

The summary of my argument is the following:

i) If the world evolves 'twice as fast' we should not notice a difference (the molecules
in our brains would move twice as fast as well).

ii) However, if the world suddenly evolves 'backwards' we would like to be able to recognize this (otherwise how would we know if the 2nd law is correct).

iii) But it seems that one cannot have both i) and ii) if one assumes that our conscious experience is a 'natural' function of the material configuration
of our brain, e.g. if we follow Daniel Dennett and assume that consciousness simply is the material configuration of our brain: E(t) = [xb(t)]
or E(t) = [xb(t),vb(t)] (*).

Perhaps one can solve this puzzle by assuming E depends on higher derivatives x'' and/or perhaps one can find some
clever non-linear function. But I think this would introduce other problems (at least for the few I tried ) and I don't find this very convincing [x].

Of course one can challenge other assumptions too. I already mentioned quantum mechanics instead of Newton or perhaps
we have to assume that our conscious experience is not a function of the particle positions in our brain. But still, none of these
solutions are very convincing in my opinion.

What do you think?

(*) Dennett is never that explicit about his explanation of consciousness.

In general, one could imagine that E is some sort of vector in the 'space of all possible conscious experience' - whatever that means.

[x] e.g. E could depend on vb/N with N = sqrt(sumb v2b) instead of vb. But where would the non-local N come from and also there would be a singularity at N=0, i.e. when all velocities are zero. One would not expect a singularity of E for a dead brain (with all molecules at rest) but rather zero experience.


Ponder Stibbons said...

But this would mean that no observer is able to recognize 'backward evolution' with entropy decreasing and distinguish it from normal evolution!

I'm not sure there's anything strange about this conclusion. If you think the second law holds strictly, then no observers would ever see the universe's entropy decrease, because assuming a universal arrow of time, the 'arrow of time of their consciousness' would always track the universe's arrow of time, which points towards increasing entropy. It seems to me that it would be a problem only if you think it's possible that the universe's entropy sometimes decreases. Boltzmann did think that, but then he made a weird argument that the direction of increasing time just is the direction of increasing entropy.

wolfgang said...

Ponder Stibbons,

>> If you think the second law holds strictly

it does not hold in this thought experiment ( dS/dt > 0 for t < t0 and dS/dt < 0 for t > t0) thanks to the 'omnipotent demon'.

>> I'm not sure there's anything strange about this conclusion. [..]
> the 'arrow of time of their consciousness' would always track the universe's arrow of time

Well, at least the 2nd part of my argument suggests that it could be different (e.g. if E depends on particle velocities and not only positions) - but of course we dont know anything about this strange function E.

>> if you think it's possible that the universe's entropy sometimes decreases

if the Newtonian universe would be closed and finite then the entropy would decrease at some point if we just wait long enough even without an omnipotent demon.

Cosma said...

I am pretty sure Norbert Wiener discusses the "backwards" version of the thought experiment in either Cybernetics or The Human Use of Human Beings.

As for the twice-as-fast argument, would it be cheating to invoke special relativity?

wolfgang said...


I am not familiar with Wiener's argument or conclusion but like I said, this type of thought experiment must be hundred years old.

>> would it be cheating to invoke special relativity?

This would be strange imho.
But on the other hand the particles in our brains (electrons and photons) move to some extent at relativistic speeds...

Ponder Stibbons said...


your comment that 'it doesn't hold in this thought experiment that...' alerted me to a more general flaw in your argument. Let's accept that consciousness is indeed determined the 'Dennettian' way (but with no dependence on velocities). I argue that the following two statements are still compatible:

1. It is physically possible that an observer [correctly] observes the entropy of the universe to decrease. (I insert 'correctly' to rule out erroneous measurements of entropy.)

2. The observer in your Newtonian time-reversal thought experiment would not observe the entropy of the universe to decrease after t0.

To see this, suppose that for some time T<t0 entropy actually decreases throughout the interval (T,t0). The observer sees this. (I am assuming here that it's possible for the observer to see entropy decreasing, as you think.) Then time reverses at t=t0. By the same reasoning you've used, the observer would continue seeing entropy decrease for t>t0. In other words, your thought experiment allows for the possibility that for certain dynamic sequences of events, the observer will observe entropy to decrease whether the arrow of time is flipped or not.

So, yes, in the scenario in your thought experiment, the observer will never see the sequence of world-states delineated in that thought experiment as decreasing in entropy. But it doesn't generalise to every sequence of world-states --- by the same reasoning there could be sequences of world-states for which the observer would always see entropy decreasing. I think your reasoning for that thought experiment proves only the following statement:

3. Given a sequence of world-states in which the observer had seen entropy increasing, in the time-reversed sequence, he would still see entropy increasing.

The antecedent of 3. is important, since not every sequence of world-states is one for which entropy is observed to increase! This also answers your point about the recurrence of states in a closed Newtonian system. In general, a system may return to a low-entropy starting point in phase space without having exactly traced its path in phase space in the reverse direction (it may go round phase space in a 'circle', for example). So it can return to a low entropy state via a different path from the one it took, and not through an *exact* reversal of events. 3. applies only when the reversal is exact, so the observer in the closed Newtonian system, so long as the world doesn't retrace its path through phase space in the opposite direction, might still observe entropy to decrease as the system evolves back to its initial state.

wolfgang said...

Ponder Stibbons,

I agree with you that one cannot directly draw a conclusion from my thought experiment (with an omnipotent demon!) for the general case.

But notice that your argument also relies on an (unproven) assumption. You assume (in part 3 of your argument) that there is an essential difference between 'exact reversal' and the general case.
But I think it is quite strange that for the exact reversal one would not notice anything and yet for a partial reversal you suggest we might.

>> Let's accept that consciousness is indeed determined the 'Dennettian' way (but with no dependence on velocities)

Actually I think that it would be much more natural if E depends on x and v for the simple reason that a 'dead brain' static configuration should not have conscious experience associated with it.

Kelly said...

> a 'dead brain' static configuration

A static configuration is not necessarily the same as a "dead brain". A static configuration is just the configuration at a given point in time. I wouldn't necessarily rule out the idea that a particular static configuration is directly equivalent to a particular "Experience". What would be the problem with this being true? What law would be broken?

Also, what about Einstein's "Block Time", where there is no future, present, or past, but instead all points in spacetime exist simultaneously?

Isn't there an open question about what Time even is? Whether it exists at all (a la Julian Barbour)?

I actually think that the "flow" of time is an artifact of how human's perceive change. Things don't have to change with respect to only the Time dimension. There can be change with respect to some other point of reference.

For example, a non-horizontal line's Y value "changes" with respect to the X axis. There is change, but no Time.

With the human perception of time I think we are experiencing change in the informational content of our brain state realtive to it's previous state. That's what time is.

Kelly said...

To clarify: In the above scenario, the human perception of the flow of time comes from the Experience of a series of static brain configurations, analagous to our perception of a "flowing" movie being due to the projection of a series of static images.

Cosma said...

Any sort of neural activity depends on current flow ("action potentials" or "spikes") in neurons, plus processes by which the arrival of spikes at synapses triggers the release of neurotransmitters at one side of the synapse and their reception and transduction at the other. These are all exemplary irreversible processes. If the helpful demon reversed all the velocities, what you'd see wouldn't even be any recognizable kind of neural activity, not even a pathological condition.

Since Dennett believes that mental states are functions of neural activity (and environmental states), he'd say they have to be functions of microscopic velocities as well. But I think he'd be within his rights to say that the microstate-to-mental-state mapping simply isn't well-defined for the weird, velocity-reversed microstates under consideration. It'd be a bit like asking what the nematic order parameter is for the Ising model.

Kelly said...

> These are all exemplary irreversible processes.

As far as I've ever heard, except possibly in some instances for kaon decay, all physical processes are symmetric with respect to time. If they can occur one way, then they can occur the other way as well. Just watching a recording of the process it would be impossible to tell whether the recording was being played forwards or backwards, except by looking at the before-and-after entropy.

Reversing these processes is not impossible...just very very improbable, correct? Which is enough, given that this isn't the point of the thought experiment.

The usual example I've seen is of a broken egg. It CAN unbreak. It's just not very likely to do so.

> what you'd see wouldn't even be any recognizable kind of neural activity

It would definitely be a recognizeable kind of neural activity. It's normal neural activity, reversed. And recognizeable as such.

> he'd say they have to be functions of microscopic velocities as well

I'd say that a the brain is basically an evolved form of computer. If you double the speed of a computer, and other things remain the same, does that mean you'll no longer get the correct results?

Going back to the reverse-world thought experiment, would computers running backwards in this world give incorrect results (in terms of working backwards towards the input from the output)?

I'd say that consciousness is a function of the information represented by the brain at any given time, not the physical processes of the brain per se.

I don't agree with Dennett on a lot of things, so I'm not going to hitch my wagon to his star anymore than is absolutely necessary.

Kelly said...

>> I'd say that consciousness is a function of the information represented by the brain at any given time

For instance, let's say that we set up a computer simulation of the molecules of your brain. Assuming that we have accurate molecular simulation software, a powerful enough computer to run it, and accurate starting data, we should be able to calculate what your response to a given series of inputs will be.

In my opinion, in addition to simulating your responses we will also have produced your conscious experience of having had those responses to the input data.

So in this case, what does the velocity of the particles mean? Given that there are no real particles, it's just a simulation of particles...?

Does it matter if you run the computer at a higher clock rate? No. Does it even matter if you run it on a different type of computer? Or print out the simulator source code and data and execute the program by hand with pen and paper? No.

Only the information state at any given point in time matters.

wolfgang said...


unfortunately we (at least I) cannot ask Dennett, but I agree that a dependence on the velocities seems more natural.
The proposal that a reversal of the velocities would 'knock us unconscious' is a new idea (for me) and I need to think about it.

However, my problem with E =E[x,v] is then the dependency on the time scale s.
Although, meanwhile the 'solution' proposed in the footnote does not seem so bad after all - the singularity could be mapped to 'unconscious' I assume.

wolfgang said...


>> Also, what about Einstein's "Block Time", where there is no future, present, or past, but instead all points in spacetime exist simultaneously?

I tried to avoid discussing relativity, but let me just make one comment.
In my post the x were basically Newtonian 3-vectors, in relativity we would have to consider 4-vectors x(t) , where t is a time(like) parameter chosen by a particular observer O1.

If we consider E[ x_i(t) ] then the experience at a particular moment t1 would be E[ x_i(t1) ].

The problem is that different observers (moving relative to each other) would choose different frames so you could in general never get all x_i(t1) to be the same as x_i(t2) , where t2 corresponds to the time slicing used by observer 2.

Now you could say that the obvious choice for the time slicing is the reference frame corresponding to the brain under consideration.
Unfortunately it consists of many fast moving parts (including photons!).

So I am afraid if we include relativity in our discussion the definition of E[ x ] is already on shaky grounds from the start.

In other words, the strong experience of "here I am now" is very difficult to reconcile with relativity and thus I tried to avoid this issue in my thought experiment. But I promise that I will come back to it in another post 8-)

wolfgang said...

>> In the above scenario, the human perception of the flow of time comes from the Experience of a series of static brain configurations

would this not require something in between ?

in the case of the movie frames the perception apparatus and the short term memory of the observer is necessary for the experience of change and flow in time.

If consciousness is indeed a sequence of experiences, then I think the impression "time is flowing" must be part of each experience - somehow;
But not generated by the sequence.

Kelly said...

>> would this not require something in between ?

I don't see why it would, but if you think so then I'm interested in hearing your theory as to how something "in between" states provides the necessary magic to produce consciousness, and why the information within a state is not sufficient.

So let's say you take the information represented by your brain at t0. Then after the smallest theoretically possible instant, you take the state again (t1), and then again (t2), and then again (t3).

What is it, exactly, that is so important to the production of consciousness that happened between t0 and t1?

I would say that what ever it was that happened "between states" is an implementation detail of the physical system, and not important to the production of consciousness , which in my opinion is dependent on the information represented by the system, not the system itself. You can represent the same information in multiple physical systems (a.k.a., Multiple Realizability), and the results produced by the various systems should be equivalent, assuming that they are functionally isomorphic. Note my above mention of computer simulations.

BTW this is why I mentioned Einstein's "block time". Let me quote from Brian Greene's "Fabric of the Cosmos":

"Just as we envision all of space as really being out there, as really existing, we should also envision all of time as really being out there, as really existing too. But, as Einstein once said: 'For we convinced physicists, the distinction between past, present, and future is only an illusion, however persistent.' The only thing that's real is the whole of spacetime.

In this way of thinking, events, regardless of when they happen from any particular perspective, just are. They all exist. They eternally occupy their particular point in spacetime. This is no flow. If you were having a great time at the stroke of midnight on New Year's Eve, 1999, you still are, since that is just one immutable location in spacetime.

The flowing sensation from one moment to the next arises from our conscious recognition of change in our thoughts, feelings, and perceptions. Each moment in spacetime - each time slice - is like one of the still frames in a film. It exists whether or not some projector light illuminates it. To the you who is in any such moment, it is the now, it is the moment you experience at that moment. And it always will be. Moreover, within each individual slice, your thoughts and memories are sufficiently rich to yield a sense that time has continuously flowed to that moment. This feeling, this sensation that time is flowing, doesn't require previous moments - previous frames - to be sequentially illuminated."

Kelly said...

>> Does that help? :-)

Errrrrrrrrrrrrrr...I don't think so...BUT, the attempt is appreciated!

>> the night sky is no longer black the day sky no longer blue!; we are rapidly toasted along with everything else on the planet.

It's true that in "reverse mode" we absorb energy from the night sky as invisible infrared radiation...BUT, now we lose energy during the day by sending visible wavelength photons back towards the sun, where their energy is used to help split helium atoms back into hydrogen (remember, we're in reverse). So, the overall result is the same: We don't get toasted.

>> consciousness is simply what it is like to be the updating of the brain's model of self in the world

You might find "Facing Up to the Problem of Consciousness" by David Chalmers an interesting's available on his website.

Anonymous said...

There is an erroneous statement in the original post. Even if the velocities are reverted, the statistical entropy will still continue to increase for t>t0. It is well-known today that entropy increases both 'forward' and 'backward' in time (I don't like this terminology). It may sound strange only if you think of the statistical entropy as a dynamical variable, which it is not.

There is a nice explanation of this, with concrete simple examples, in

Luke, Am. J. Phys. 56 (1988), 734-736.

See also

Hurley, Phys. Rev. A 22 (1980), 1205-1209; Phys. Rev. A 23 (1981), 268-271; Am. J. Phys. 54 (1986), 25-28.

wolfgang said...

I will (try to) read the papers you mentioned, but for now I would argue that at least in the case that we 'know reliably' that the omnipotent demon reversed all velocities, we can assume that the system will run backwards to its initial state (but with velocities reversed) and thus I would conclude that the entropy S(t+T) is the same as S(t-T).

But to borrow a line from the Aeolist (again), "many of the terms used in the debate, beginning with the all-important definition of entropy, and including terms like ‘preparation’ and ‘reversal’ (and its cognates), are still used in so many different ways that many of the participants are speaking at cross purposes".

wolfgang said...

I should be a bit more precise. At t0 + T the number of micro-states compatible with the macroscopic state is the same as for t0-T and thus the Gibbs entropy is necessarily the same.
I would like to know why this conclusion would be wrong.

The irony is that I assumed that we 'know reliably' that the omnipotent demon reversed all velocities - when at least the first patr of my blog post is about the fact that no observer would notice ...

Anonymous said...

I've not so much time now, but let me try to explain the point briefly.

The key here is the term 'macroscopic state'. This will not be the same at t0+T and t0-T, because a macroscopic state (and hence the TD entropy) is not a dynamical variable. This seems paradoxical, but consider that the definition of macroscopic state is tightly connected with macroscopic reproducibility.

In a sort of slogan I could say like this: in a 'time-reversed world', glass pieces that assemble into a drinking glass and jump on a table are the norm, not the exception. Hence the definition of macroscopic states and their interrelations, which is based on what we find reproducible, will also be different.

Reproducibility of a phenomenon means, from a microscopic point of view, that the set of microstates corresponding to a given macrostate must evolve into a subset of the set of microstates corresponding to the evolved macrostate. Otherwise, we would end up in different macrostates and the phenomenon wouldn't be macroscopically reproducible. This implies, with the conservation of phase-space volume, that the phase-space volume corresponding to a macrostate is smaller than the volume corresponding to the evolved macrostate. Hence the (TD) entropy of the latter will be larger than the former's.

Saying it in yet another way: even if all microdynamical variables are time-reversal invariant, the notion and definition of 'macroscopic state' is not (since it is related on reproducibility), nor is the entropy, as a result.

It is also helpful to realise that in phase-space a time-reversed motion (p -> -p) is not a simple backtracking of the phase-space trajectory, but a new trajectory which is the specular image of the original one in respect of the coordinate hyperplanes. For a harmonic oscillator, eg, time-reversed trajectories still flow anti-clockwise, like the original ones.

PM said...

Correction: two posts of mine above I wrote 'the statistical entropy will still continue to increase for t>t0'. That's wrong of course: I meant the thermodynamic entropy, not the statistical one, ie the Kullback-Leibler entropy of the Liouville function. The latter is always constant for a non-dissipative system of course.

And Wolfgang above writes 'we can assume that the system will run backwards to its initial state'. Of course not. You mean that it will run backwards to its initial configuration, ie the values of the coordinates; because the momenta will have the signs reversed in respect of the initial state.

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