h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Here's the supposed "paradox" as stated:

"It postulated that motion is impossible because the moving object always has to cover half the distance. Since the number of halves is infinite and they become infinitely small, the moving object never really gets itself going."

This statement contains two assumptions, at least one of which is false, probably both.

The first assumption is that the object "always has to cover half the distance". What does that mean? This way of phrasing it converts a simple, mundane observation (ie, before covering 100% of the distance, the object first covers 50% of the distance) into a conditional statement (ie, before covering 100% of the distance, the object MUST FIRST cover 50% of the distance), as if covering 50% of the distance is a discrete seperate step that must be "completed" before the next discrete step. Yet no attempt is made to prove such an extraordinary assumption. It's just not how things work.

Or look at it this way - covering 50% of the distance will, all else being equal, take only 50% of the time! So nothing is delayed or prevented, and with every pointless halving of "what the object has to do first" (a silly way of thinking about it), so too the time taken to do it is halved. So in the end we wind up where we started, with an object moving along a distance at a certain speed.

The second assumption is that "the number of halves is infinte". Maybe in maths it is, but in the real world it may well not be. There just might be limits on how many times you can keep halving. But that's a whole different debate.

I've never been able to "see" the paradox in this paradox. It mistates the obvious in a simple and apparantly logical (but based on false assumptions, so irrational) way, and people call it a paradox. Beats me.

Arthur

Aside from this, how many replys to your post? Indicative of something perhaps????

I agree, a completely nonsensical hypothesis, which is characteristic of anally retentive philosophical debate (not that all philosophy is anally retentive of course).

It strikes me that by creating such a hypothesis, Xeno has actually managed to disprove it immediately in one fell swoop.

By creating such a hypothesis, Xeno obviously had his head stuck well and truly up his own backside, I'd say as far up as his caecum. But how did he manage to do this? Well, by an amazing feat of contortion, he must have managed to bend his spinal column to an extreme degree to allow access of his head through his anus, after having started in an upright position. This of course involved linear as well as rotational acceleration, i.e. a change in velocity of his skull relative to his anus. Therefore, he himself initiated motion and subsequently came into contact with his own motions.

QED

Can you add up an infinite number of non-zero distances and still get a finite number? We know now that you can, but this wasn't clear to the Greeks, who didn't have our advanced concept of limits.

I don't know if anybody is still alive or a member of the group since this thread was started but experience proves this is nonsense.It is like an argument Dr Johnson had with Alexander Pope. The latter was spouting subjective nonsense and the former said I disprove it thus, through objective action.

I'm still alive.

A lot of ancient Greek debates look like nonsense now, because we have a different way of thinking about things.

Isaac Newton worried that the whole foundation of differential calculus (which is an extremely useful branch of mathematics) was dividing a very small number by another very small number. The two numbers had to be "immeasurably close to zero" for the result to be accurate, but not zero because you can't divide by zero. He called them infinitesimals, because they differed from zero by an infinitely small number.

Modern mathematics has a much more boring way of defining differentiation which gets around this problem.

Zeno was trying to get a grasp on the idea of something that it so small that it is effectively zero but that if you have enough copies of it, it adds up to something. Not an easy concept, and not necessarily "just rubbish".

Ironically, the ancient Greek Archimedes figured all this out, but his solution was lost / forgotten about and only rediscovered in the 20th century.