The ancient Greek philosopher, Zeno of Elea, conceived of many paradoxes of time and motion. For instance, logically it can be argued that Achilles gives a tortoise a head start in a race, then he could never overtake it, as long as the tortoise keeps moving. This is because in order to first overtake it, he must reach where the tortoise is, but by the time he gets there, the tortoise will have moved on.

So, Achilles must then reach where the tortoise has moved on to, but once he gets there, the tortoise has already gone, and so on, ad infinitum.

Another paradox states that an arrow can never move, since in any given moment of time, the arrow had to completely occupy a certain space. Liek a photograph, at any given moment, the arrow is where it is and not somewhere else. Hence it is stationary. But if time is no more than a series of moments, and if the arrow is stationary at every particular moement, it never moves.

Yet we know that arrows do move, and that Achilles can overtake the tortoise. So what is flawed - our view of reality, or the logic of the paradoxes?

So what do we gather from all this?

Zeno's motion paradoxes has the ancient world running hard just to stay still. But, without the questions posed by the paradoxes, physicists would not have gotten the modern world on the move by explaining the nature of space, time, and matter.

What do you believe is the key to solving paradoxes?

The key is that the ancients should have put down the pipe a little bit sooner.

There is also one where you can never cross a room if you first stop halfway, then go half the next distance and so forth. The idea is a convergent series, but I believe that the LOGIC is flawed. Of course you can cross a room, overtake a tortise and so on.

I say let them die in the past with Zeno

Sorry, but these always annoyed the s*** out of me, nice topic smiless!

The paradox of the arrow is simply semantic sophistry. It tries to equate a single instant with multiple instants, which is an error in categorization.

Also, regarding Achilles andthe Tortise, it is just as much a product of semantic sophistry as the arrow paradox.

According to the way it is stated, at some point, the distance the tortise has moved will eventually equate to the reciprocal of infinity. The sophistry is in the implication that both the Tortise and Achilles will reach that point at the same time. In other words, the the only way for Achilles to not overtake the Tortise is if time stopped at the point where the distance travelled by the tortise was the reciprocal of infinity.

I'm sure a mathematician could say that better though.

Well, I think the first key would be to properly understand them.

Without a proper understanding of what it being claimed, it would be impossible to many any sense of it at all.

First off, what sense would it make for Zeno to run around arguing that motion is impossible in any absolute sense of the idea?

Cleary even Zeno was running around and passing people by. Why would he argue that motion is impossible?

Well, the answer to this requires an understanding of what the Greek philosophers were considering at the time. At that time in ancient Greece, there were debates going on as to whether or not the unviverse is a continuum, or whether it is discrete.

Can things be divided up infintely? Or is there an ultimately limit to how much something can be divided up?

That was the question at the time.

Zeno thought about this question and came to a brilliant epiphany.

He reasoned that if the universe is a continuum (i.e. there is no end to how small things can be divided up), then motion would be impossible, therefore, he concluded that the universe must be discrete.

So Zeno's actual arugment should be stated as follows:

"If the universe is a continuum, then motion would be impossible"

However, "If the world is discrete, there motion would be possible".

He then began to describe his logical arguments that show that in a universe that is continuous motion would be impossible.

Zeno's arguments where brilliant, and they were perfectly true and correct. Zeno of Elea was the first human to discover the quantum nature of the universe.

It took the rest of mankind about 2000 years to catch up. And even then they only caught up because they had no choice.

In 1900 a man by the name of Max Planck accidentally discovered that the world is indeed quantized. Quantum Mechanics was born and the value of the smallest possible increment of action is called "Plank's Constant"

Although in truth it should be called Zeno's Constant.

Zeno had figured out that we live in a quantum universe using nothing more than pure thought.

No one believed him then, and no one still recognizes the man's genius even today.

He'll probably forever be unrecognized for his brillance.

He was my childhood hero.

Here's to Zeno!

The real father of Quantum Mechanics.

Hey Sky, you're beating me to the punch here!

Only because my post was shorter so it took less time to write.

But I think yours was more interesting.

Thank you for explaining. It gives a clearer perspective of what Zeno discovered.

I think it is a little more simple actually...

Speed was not being taken into account. Invoke the speed of the tortoise compared to the speed of Achilles and the paradox disappears.

Well there you go that is simple.

With a solution, would it ever have been a paradox?

The achilles paradox is easy to solve. Achilles reaches the tortoise's spot; and it moved. Then A reaches T's spot again. then a third time.

The time that elapses between each time Achilles reaches the Tortoise's spot is alway getting shorter. There are, yes, an infinite number of reach-gone spots; but the time is decreasing between each subsequent pair, so that their sum, though it's an infinite number of additives, reach no infinite, but a finite amount of total time and a specific location.

The arrow? It is a tougher one. Another way of asking it is: Given the location of the point of a moving arrow, what's the next location after that? There is no next location as there are no two points in a line that are adjacent. If the points are infinitesimally small. Which the actual geometrical axiom is for a point. This problem could only be solved, with our present knowledge, two ways: 1. By calculus (which assumes that differences can be subdivided ad infinitum) or by quantum mechanics (which assumes that things progress without intervening smooth transitions).

This latter, the arrow one, is a better paradox, for it gives rise to the fundamental inadequacy of our "slaigh of hand" tactic in thinking. We, humans, assume those things for granted which we can't see other than as a paradox, and that only when pointed out to us. If someone, like Zeno, points out that everyday occurrances can't exist with our everyday or sophisticated explanations on how things work, we panic, and call in our prowess of rationalization.

I believe it is possible that the arrow moves neither on calculean nor on quantum mechanical principles. It is our perception that is hugely inadequate, and seeing that we, humans, have a miryad of these inadequately equipped senses, we have developed a hugely efficient brain to do the rationalization of our cognitive dissonance for us.