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
x_i(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: v_i(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 x_i(t0+T) = x_i(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 x_b(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 x_b(t0+T) = x_b(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 x_b(t) and v_b(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 v_b 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) = [x_b(t)]
or E(t) = [x_b(t),v_b(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 v_b/N with N = sqrt(sum_b v²_b) instead of v_b. 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.