I have read a paradox by Zeno, which defies logic, and seems to defy a logical solution... personally, I agree with the term seems, because I believe it can be logically solved, if it already hasn't been. The paradox is interesting to me, however, and I thought it would get people thinking, so here it is... paraphrased of course...


Achilles and the tortoise...


Achilles the athlete is going to run a race with the tortoise. Now obviously we know by experience and common reasoning that Achilles will win such a race. Achilles thinks so also, therefore, he gives the tortoise a generous head start to point A. After the tortoise has reached point A, Achilles begins to run until he also reaches point A. At this time, the tortoise has also continued to run and has reached point B. As Achilles reaches point B, then the tortoise has reached point C, etc.,etc.

Because of the headstart, following the construct, as Achilles reaches each of the last points that the tortoise had been, the tortoise has reached yet another. Achilles never catches the tortoise within this structure, yet there is no apparent flaw in the construct.

The issue lies within the notion of the measurement of space and time.

I would like to hear others thoughts on this paradox.

In the case of Achilles and the tortoise, suppose that the tortoise runs at a constant speed of v metres per second (m/s) and gets a head start of distance d metres (m), and that Achilles runs at constant speed xv m/s with x > 1. It takes Achilles time d/xv seconds (s) to travel distance d and reach the point where the tortoise started, at which time the tortoise has travelled d/x m. It then takes further time d/x²v sec for Achilles to travel this new distance d/x m, at which time the tortoise has travelled another d/x², and so on.

maybe because like the Rabbit in the other story...Achilles took a nap and never woke up too late...

Marley:

I am attempting to follow your construct here...and failing miserably...

Not sure where the value placement came from and why the need for the complexity...

If Achilles runs twice as fast as the tortoise is a simple construct which accurately exemplifies the paradox.

As Achilles reaches point A, then the tortoise will have traveled half as far down the path to point B. As Achilles reaches point B, then the tortoise will have traveled half as far... and so on.

Sorry for my inept thinking, but I could not follow your construct... wheres Abra?

I suspect that we are making the same claim, no?

Because both Achilles and the tortoise are traveling simultaneously, as Achilles reaches the last point at which the tortoise had been, the tortoise is at the next point, always staying ahead. Achilles gets closer and closer without ever catching the turtle.

It is quite intriguing...

Kinda like half of a hole...

It makes no sense to me. Eventually, the tortoise will win the race or Achillies will pass the tortoise. If two objects are moving in the same direction at different speeds on the same course, they will pass one another eventually.

Of course they will spider... that is the problem...

The logical construct is irrefutable, in and of itself, especially when one considers the speed at half/twice...

BUT...

The conclusion is a known falsehood...

That is the problem with it.

Let's put it another way...

If Achilles runs twice as fast as the tortoise then the tortoise will travel half as far as Achilles in the same amount of time.

That is irrefutable.

If the tortoise begins the race 100 feet ahead of Achilles, then by the time Achilles reaches the 100 foot mark, the tortoise will have reached the 150 foot mark.

That is irrefutable.

Following the irrefutable premise above, the tortoise always stays 1/2 the distance traveled by Achilles ahead of Achilles...


So then, what is wrong with the premise?

Can it be denied?

Why then, does it obviously lead to a false conclusion?

i don't know much, but i think there is a point missing here. what was the speed of each. i mean at least i can't lay out this problem without numbers which tell me the distances and the speeds.
I may not be a very logical person though.

Whatever speed you choose Miguel...

Half is half... and twice is twice... the distances traveled will vary accordingly.

because the acceleration is constant.
therefore, Achilles never speeds up in order to reach the tortoise.
that is why the tortoise is always going to be ahead.
the sames happens with gravity
no matter how heavy an object is compare with another, they always are going to reach the ground at the same time.

I get what your saying, but I am not sure if that applies here Miguel...

The speeds are different and gravity is not the only force at work in this example.

Achilles will pass up the tortoise, just not within the construct, which seems to be irrefutably sound.



The division of any given measurement into infinitely smaller pieces is only possible if each has a value of zero. An infinite number of pieces all of which consisting with a length of zero.

All of the measurements in this example of the division of infinity have a value of more than zero, which is impossible.

It is impossible to dig half of a hole.

Gosh, I hope I have this right...

James? I am waiting on what you have to say about this, I suspect it is quite familiar to you already, along with Zeno's other 3 paradoxes...

the speeds are different, but constant at any time.
therefore, it will keep hapenning. that is why the tortoise is always going to be ahead half of the distance that achilles traveled to reach the point the tortoise reached previously.

since numbers are infinite they can be divided infinitely. that is why theoretically it seems that the achilles won't ever reach the tortoise. however, in real life it seems to me that eventually achilles will reach the tortoise.

I'm working on typing it up.

It can't be explained in a few words, plus the modern day mathematical solution is actually invalid. But I'll address that issue as well, just hang on a few more minutes, I'm typing as fast as I can.

Oh, by the way, all of Zeno's paradoxes are solved in one fell swoop. So just hang in there.

Owl be back.

Oh, of course, Miguel...

I am with you there. Achilles will indeed catch and pass the tortoise, but the way the problem is defined makes it logically impossible, although everyone knows that it will happen.

That is why it is perplexing.

Thanks James, I suspect that they all would be. They all actually define the same thing anyway... the division of infinity...

How are you Miguel?

Are you going to Texas? I read that somewhere in a forum...

I'm good my friend. I was on Texas a couple of weeks ago.
How r u?
Thanks God, we have James to help us with this math problems.

That’s precisely where the solution lies. The premise is indeed incorrect. The universe is not a continuum. At least not for this problem.

Although, you’ll find that even modern day mathematicians have not solved this problem correctly even though they believe they have solved it with calculus. They have not.

The solution actually comes from physics, and the work of Max Planck (i.e. Quantum Physics)

Before I being (this will probably be a lengthy post), let me say that Zeno of Elea and Leucippus of Miletus were two of my most revered childhood idols (at least from ancient Greece).

Let me begin by stating the real problem.

The real problem was the idea of the continuum. The idea that everything can be divided up continuously forever and ever with out end. Leucippus was the first to propose that this is not the case. That the world is actually made up of atoms (individual discrete parts). Of course he wasn’t taking about the atoms we know of today, he was speaking on a purely philosophical level.

Well, the mainstream Greek philosophers of the time didn’t believe like Leucippus, they believed that everything is continuous.

Zeno knew about the work of Leucippus, and Zeno had a wonderful epiphany. He realize that if the world was truly continuous then motion would be impossible (for the very argument you gave in the OP). Although I he actually gave a dozen or so arguments for this. They all were based on the same idea. The idea that it would be impossible to complete an infinite number of tasks. That’s his basic premise – it would be impossible to complete an infinite number of tasks, therefore if the world is truly continuous then motion would be impossible.

He was actually giving support to Leucippus’ idea that realty must be discrete. Obviously he wasn’t trying to argue that things can’t move, he clearly knew that things can move!!! What he was arguing is that if the world it a continuum motion would be impossible.

Well, the mainstream philosophers weren’t prepared to accept that the world is discrete so they saw this as a ‘paradox’ that must be solved within the framework of the continuum. But they could not resolve the paradox so it become known as Zeno’s Paradox of Motion.

In fact, no one could solve this so-called paradox until Isaac Newton and Gottfrid Leibniz invented Calculus. Then it was proclaimed that the paradox had been solved. But they really didn’t solve it! However it is still believed today that Calculus solves the paradox whilst preserving a continuum. This is false, so I won’t spend a lot of time on this,….

First off Marley is correct. He has given you the modern calculus solution. And that solution has to do with derivatives. For example dx/dt is the velocity dx²/dt is acceleration and so on where x is position, t is time, and d means to take the derivative. (just as you learned in high school calculus)

The idea here is that it can be shown mathematically that rates of change between time and distance are sufficient to complete motions required. And they give (feeble) arguments that even though it appears that you need to keep cutting things in half, you don’t really need to because the time it takes to move is also heading toward zero. They claim that the time it takes to move is heading to zero faster than the distance is heading to zero, thus solving the apparent paradox.

Actually their full of baloney.

This doesn’t solve the problem. The problem that Zeno actually proposed it that it would required an infinite number of tasks to complete the move. In other words, you’d have to close that halfway mark an infinite number of times if the world is a continuum!!!

Zeno is right and the modern day calculus solution is wrong. Yet this is still accepted as the correct solution by modern day mathematicians.

However, there is nothing in calculus that states than an infinite number of tasks can actually be completed. On the contrary if you look at the formal definition for the calculus limit (I can type it in here without symbols but it probably wouldn’t mean anything to you anyway, even most mathematician truly don’t understand it!)

You will see, that it does not even imply that an infinite number of task can be completed. In fact any mathematician worthy his salt will tell you that to prove that a limit exists all one must do is prove that it is bounded, and has a certain trend toward a given number. Then you can say the limit exists. However, even when you can prove a limit exists that doesn’t even mean that the actual quantity exists! In other words you can prove that some equation that is approaching zero in it’s denominator will have a limit at that zero. However, that doesn’t mean that the a actual limit can be reached at that zero mark. It only means that this is the quantity it is heading toward as the denominator approaches zero!

In other words, the calculus limit (formally defined by Karl Weierstrass) doesn’t prove anything. It’s just means of calculating trends with great precision. So the calculus limit can’t even be used to disprove Zeno anyway.

Ok, so what’s the real solution?

The real solution is that Zeno was right all along!

If the world truly is a continuum then motion would be impossible! But the world isn’t a continuum! We know this know from Quantum Theory. On the quantum level you can either be here, or there but not in-between.

There is a smallest distance that can be traveled. A smallest amount of energy that can be expended, and even a smallest tick of time that can tick. At some point as you are cutting those distances in half you’re eventually going to make a quantum leap and pass the tortoise.

Quantum Physics has confirmed that Zeno was indeed correct. Our universe is not a continuum. At least not with respect to the motion of the physical things within it. It still has a oneness about it, that the Greeks refused to give up. What they didn’t realize is that they can have both, a universe that is physically discrete and still has a nature of oneness. In fact, this is the modern day paradox of Quantum Mechanics. We solved Zeno’s paradox, only to discover a newer and deeper paradox.

So Zeno was right. Can we send a Nobel Prize back in time????