Zeno and Algorithms

I'm wondering if it's possible to demonstrate the paradox of Zeno's race between Achilles and the Tortoise via a software model. In other words if we modelled the race on a computer would the the result be computable?

I suppose that the movement of the contestants would be created algorithmically. Eg. let's say that in 1 unit of time the Tortoise (T) moves forward 1 unit of distance while Achilles moves forward 10 units.

Starting at x the sequence of positions of T are given by x (next) = x+1 and the position of A are given by x (next) = x+10.

(OK so far?)

If we give T a head start of 100 units of length then presumably in this case the programme would show A whizzing past T with no problem.

But what if the units of time and distance are defined as fundamental, ie. each unit of time is an 'instant', and each unit of distance is a 'Planck length' (or some equivalent).

In this case the Tortoise couldn't go any slower, since it is racing at one fundamental quanta of space in one fundamental quanta of time.

Under these conditions it seems that the programme can never specify the time or position at which A and T are equal. As a result A passes T during an instant, which seems to be impossible.

Is this a more modern version of what Zeno was trying to say, (that motion cannot be algorithmic), or have I got it wrong somewhere?

 

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Zeno was trying to say that logical reasoning has flaws, since we all know that Achilles can outrun a tortoise. He postulated a situation where the logical conclusion that we derive does not make any sense... this is why it's called Zeno's Paradox.

Also, you're using a discrete unit to measure time, which leaves Zeno out in the cold, because in the description of Zeno's Paradox, each time increment is 1/n of the last time increment, with n being equal to v1/v2 where v1 is the velocity of Achilles and v2 is the velocity of the tortoise. That's actually why Achilles never passes the tortoise...

If you look into calculus and do this as a related rates problem (velocity of Achilles vs. velocity of the tortoise), then you should be able to get the exact time that Achilles passes the tortoise, assuming that you know their exact velocities. As you observed, the answer will probably not be an integer.

 

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Aha, there is somebody out there. I thought this one had sunk withour trace.

 

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Ah; in principle I don't believe in quantized time or motion; time is continuous and so therefore is motion.

Recently there has been a hubbub in the news because some gent was claiming that classical physics defined time as a series of connected instants. I don't think this is a fair criticism of classical physics; the series of connected instants was created as a means of analyzing systems of forces at a particular time, since it's easier to express a cross section of a set of circumstances than the entire continuum of their being.

Your computer experiment, in my opinion, shows that the idea of quantized time is sort of silly, since Achilles never really passed the turtle because the two were never side by side - Achilles was behind at time t, and then at time t+1 he was ahead. Trying to describe a continuous system with quantized values always has this problem more or less; that's why we can't predict the weather very well.

That reminds me of the story about how a fly can stop a train...

A fly and a train are heading towards each other in opposite directions, the train at 50km/h, the fly at 50km/h. Eventually the two meet.

At this point the fly must stop going 50km/h in the direction it is going, and start going 50km/h in the other direction.

There must be a point in time, no matter how short the moment may be, where the fly is therefore not moving with relation to the ground.

When this happens, the fly is on the train.

So, the train must not be moving either.

Hence, the fly stops the train.

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(edit) Once we get into quantum physics I'm unable to speculate, since my background in that area is kind of weak. Certainly, electrons move, since you can measure their velocity (subject to certain restrictions). However, my idea of time being continuous and thus promoting continuous motion does not readily apply to the orbital jumps of electrons, like say when an electron absorbs and emits photons. I have no idea what to say about this.(edit)

 

In any particular instant of time we are all motionless.

 

Yeah, but instants of time are a philosophical concept and may have no reality, just as seconds and hours do not.

The "connected instants" concept of time may be only as related to the world as mathematics - helpful in modeling, but still only a convention.

 

BBH

I agree with all that, and it seems to be the orthodox view. I'm still trying to figure out why Peter Lynds has made waves in the scientific community by arguing that instants don't exist, I didn't know that anyone ever thought that they did.

I was suggesting in my question that the race between Achilles and the Tortoise was non-computable. Is that a technically correct?

 

In a sense it is correct - a computer will always represent a system of measurements in discrete units, and the time/distance measurements actually described by the Achilles/Tortoise race are part of a continuous motion; there is always the chance that the precise distance or time will fall between the units used by the computer, however small they are.

I don't believe that Peter Lynds is rocking the scientific community... I am more inclined to believe that he is rocking the popular science community, which is usually a little bit behind. Most physicists are willing to allow that their attempts to model the universe do not amount to a perfect representation; this is part of what the field of chaos theory was developed to study so many years ago, to see if patterns could be derived to make predictions more precise.

(In any iterated system, small differences will expand after a predictable number of iterations to become as large as the system. This, as I said, is why we can't predict the future very well.)

 

Just to be completely clear - Is it the orthodox scientific view that spacetime is a continuum?

 

There isn't an orthodox scientific view in the sense that you're asking... most fields are split on almost every issue you could think of. That's why no one ever really "rocks the scientific community".

General relativity postulates that space and time are a continuum. If you believe in general relativity, then yes... but I don't.

 

Thanks. Good point about rocking the popular science community rather than the real one by the way.

 

No problem. Sorry I can't help you with a metaphysical description of space and time... I haven't yet heard one that I like, and I'm not really an expert anyway.

 

That's ok. I'm happy with my metaphysics.

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I case you haven't heard, Zeno's paradox was solved a long time ago. Zeno erroneously assumed that it must take in infinite amount of time to pass through an infinite number of intervals. We know that this isn't the case. With calculus it is easy to show that an infinite number of intervals can be passed through in a discrete amount of time. Look up taking the limit of an infinite summation series if you want to see the specifics of solving this sort of problem.

 

But calculus cheats. It gets around Zeno's problems but doesn't actually solve them. They can only be solved by getting rid of finite quanta.

 

Not all problems are soluble without changing the system - Russell's Paradox (or the Barber of Baghdad) is an example of a flaw in set theory, which set theory can't readily be changed to fix.

It might be a problem with our number system... maybe the "successor of" system that we've been using all our lives isn't the best after all.

 

In what way is it 'cheating'? The paradox centers on the time it takes to pass through an infinite number of points. Calculus allows us to calculate that. Xeno, who wasn't able to perform such a calculation, naively assumed that it would take an infinite amount of time to pass through an infinite series of discrete intervals. We now know that this isn't the case.

 

Xeno was trying to demonstrate a problem with logic, not with physics; he probably knew that it was possible to traverse a distance in the real world, and so his effort was to postulate a valid argument with evidently true premises that nonetheless did not agree with the observable world.

The problem with Xeno's paradox, as I said earlier, is that if Achilles never passes the tortoise, then neither do we ever pass that moment in time where he would have - the whole world grinds to a halt waiting for Achilles to pass the tortoise.

Xeno's paradox does not suggest that it takes an infinite amount of time for Achilles to pass the tortoise... if you look at the time interval of each step it is 1/n of the last one for n=(speed of Achilles/speed of tortoise). If we assume that n=2 (that is, Achilles is twice as fast as the tortoise), then we have the following set of iterations:

t=-2 -> t=0 (tortoise head start)
t=0 Achilles has gone 0 units, tortoise has gone 2 units.
t=1 Achilles 2 units, tortoise 3 units.
t=1.5 Achilles 3 units, tortoise 3.5 units.
t=1.75 Achilles 3.5 units, tortoise 3.75 units.
t=1.875 Achilles 3.75 units, tortoise 3.875 units.
t=1.9375 Achilles 3.875 units, tortoise 3.9375 units.

And so on. If you pay attention to the time, it will keep increasing by 1/2 the amount it increased last time. This means that in Xeno's paradox we never actually reach t=2.

That's why it is a logical paradox, not a physical one. In the real world we reach t=2 without any problems most of the time. Xeno was not naive... he was trying to demonstrate that reason doesn't always give us a sensible answer.

 

I agree with all that until right at the end, where you say that he was trying to demonstrate that reason does not always give us a sensible answer.

I would say that his view was that poor reasoning was giving us the wrong answer. Ideas of discrete quanta of time or space, and algorithmic methods of representing motion based on those discrete quanta, don't work logically. His paradoxical race was a reductio ad absurdam designed to prove it.

(I think his proof is most easily seen by reducing the speed of the tortoise to one quanta of distance travelled per one quanta of time passing and then calculating the race, rather than by making the calculations simply proportional as above. By my calcs. the race becomes immediately and obviously absurd as soon as you do that. If I'm not right about this I'd like to know).

 

Yes... what I mean is that it showed up the (sometimes invisible) differences between the real world and a logical representation of the real world, much as Russell's paradox shows up the problem of self-referential definitions in set theory.