Cable Guy Paradox

[noel emits]

The unarticulated assumption in this 'paradox' is that there is a discrete event, an isolated system, with definable beginning and end points.

Or rather that such a thing can be said to 'exist'.

This is implicit in the notion that there is a point in time when you can say you have a 'full cone', the moment the experiment begins. So you are allowing for a beginning.

In doing this you are then for the purposes of the experiment treating that moment as the 'beginning of time'.

The 'end of time' as then defined as the moment that the conditions that define the end of the experiment are met. In this case that the water has no depth and / or surface area.

So this is the entire scope of the experiment. It must 'happen', otherwise there is no experiment, it never begins!

The event 'happens', it has beginning and end points, this defines its existence.

The 'trouble' comes when you measure its progress through time 'fractally'. In that sense it will apear to be eternal.

This happens because the mathematical stance you are taking to analyse it is that of being at a relative point 'outside' time.

But if the observer is included as part of the system, or relative to it, then the temporal relationship is less than infinite, the ratio of rate of motion through time is less than 0:1.

I suggest then that the apparent 'error' is in the original assumption - i.e. that such a thing as a discrete event or isolated system can be said to to exist apart from an observer, or even to exist at all.

So does this mean that time has no beginning and no end?

And whence the second law of thermodynamics?

 

[Mr. Tea]

I think I see what you mean, though I'm not sure I get what you mean by the hypothetical observer existing 'outside time'. It's a bit like the old 'paradox' about the divisibility of matter; chop an iron bar in half, the chop one of the halves in half and so on, and you'll 'always' have a piece of iron of finite size - except of course you won't, because sooner or later you'll get down to a single iron atom and you can't chop that any smaller without it ceasing to be iron. Same here with the water, in that you can't have an arbitrarily small amount of water.

What's the second law of thermodynamics got to do with this, by the way? Or did you chuck that in in a moment of whimsy?

 

[noel emits]

Same here with the water, in that you can't have an arbitrarily small amount of water.

It's about the granularity of time and space, man. / The Dude



More later...

 

[noel emits]

Gettin' deep on a Friday afternoon

So, is there a paradox? Of what kind? Or if not, why not? Is it a mathematical 'fallacy' of a kind? A failure to include an observer?

Is it to do with comparing unlike quantities?

Is it to do with introducing a meta-temporal element in comparing the rate of decrease of the rate of evaporation against the rate of decrease in height?

Is there a mistake in essentially measuring the progress of one event against itself within arbitrarily defined boundaries, leading to a kind of recursive calculation loop?

If experiment time is measured at smaller and smaller divisions does that mean that the posited (or implied) observer would be approaching temporal stasis relative to the progress of the experiment through time? Time slows to a halt as the quantity of water approaches zero?

...

I suggested that there was an assumption implicit in talking about an event as having a definable starting point. That is this -

that there can be said to be such a thing as a discrete event, or an isolated system if you like, with definable boundaries in time and space

Of course this assumption is made as a matter of course in experiments, thought experiments and just generally most of the 'time'.

But as we see, it seems to lead to a 'paradox'. Where does one thing end and another begin?

Is it possible that the reason the apparent paradox arises is that this assumption is simply not correct? That is to say, for the sake of argument, that it is not possible to truthfully say there can be such a thing as a discrete event with definable boundaries in space and time. There is no such thing as a thing!

The experiment, indeed our experience of the universe, is dependant on the progress of time. In other words the observer is moving through time. If this were not so it could be experienced as taking infinitely long to complete - it would appear eternal.

So is it our progress through time that gives rise to the appearance of 'things'?

 

[Mr. Tea]

OK, looks like I was wrong. Assuming evaporation rate is proportional to area, which is proportional to V^(2/3):

-dV/dt = k*V^(2/3) for some constant k (dependent on, amongst other things, the angle of the cone).

The only function V(t) that satisfies this is a cubic polynomial, so that:

V(t) = V_0*(1-a*t^3 + terms in t^2 and t)

where a is some coefficient and V_0 is the initial volume.

So the water really does evaporate in finite time (with the evaporation rate going like t^2, i.e. accelerating).

Interesting problem, all the same.

EDIT: something just occurred to me - the rate of decrease of *depth* must be constant, whatever the area!
Consider a thin slice of water, at the very top, open to the air. If it has thickness dD and area A, its volume is dV = A*dD. Now if evaporation rate is exactly equal to exposed area, then the time taken for this volume to evaporate, dt, is given by:

|dV/dt| = k*A so dt = |dV/(k*A)| = dD/k since dV/A = dD...

...which is independent of the area. Integrate for total time taken, and you get t = D/k. So the total evaporation time is constant for a body of water of a given maximum depth, regardless of the cross-section.