Xeno's Parodoxes and Peter Lynd

I'm sure many of you are familiar with Xeno, the greek philosopher and mathemeticion from circa 480 B.C. There are three main paradoxes he discussed, all involving time and motion.

Taken from Aristotle's Treatise on Physics:

Achilles and the Tortoise

"In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead."

In other words, how can one runner overtake the other runner, when he is constantly in a state of reaching a point where the first runner has passed.

The Dichotomy Paradox

"That which is in locomotion must arrive at the half-way stage before it arrives at the goal."

In Layman's terms, this paradox arrises when you consider any motion from a start point to an end point. For example, hold your hand in the air, and then touch your computer. If you take that distance, and move half way, then take the remaining distance, and move half way, you can do this an infinite amount of times. As such, you should never be able to reach the end point of your motion.

The Arrow Paradox

"If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless."

This paradox is about the understanding of motion. If we stop time, and observe an arrow in the air, it has zero motion, and therefore, it is not possible that it is moving.

These three paradoxes are incredibly interesting, and as far as I can understand it, have yet to be solved by today's mathemiticians and philosophers.

But there is a young man named Peter Lynds, from new Zealand seems to think he has solved them. You can read hs papers here.

But in essence, his theory is that time is static. That you can never observe an object at 0 seconds. You can observe the location of an object between .0000001 and .0000002 seconds, and so narrow down it's location to a distance, depending upon the speed of motion, but to exactly define it's location is an impossability. In addition to this, it also implies that everyting else observable about an object, (i.e. time, velocity, mass, momentum, energy, acceleration, etc.) cannot be precisely determined either. They can only be narrowed down to a small static range.

This, of course, would have a huge impact on the world of physics if it became generally thought of as truth. Steven Hawkings work on imaginary time would be utterly meaningless, as this theory would mean it is the relative order of events that have any sort of meaning, not the direction of time as Hawking believes (which is a confusing concept), and it would completely redifine our understanding of the Big Bang (that time and space congealed from the physical and temporal singularity, and the fundamental idea that chronons exist to begin with)

This new way of looking at time solves Xeno's paradoxes. Lynds has been hailed as both a genius and a fraud, so I'm interested to hear what others on this board think about his solution. You can read the paper address the paradoxes here. I'm interested to hear what you think.

I heard about Mr Lynds' work some time ago, even refered to some of in a few of my prior posts. He my Hee-ro!

But as I understood it, the primary thesis of his work was that Time, by its very definition, Could NOT be thought of as Static; and therfore it would be impossible to isolate what one could call a "moment" or an "instant" of time. The implication of thesis is that it renders our traditional understanding of physical processes virtually nonsensical.

If Time cannot be sub-divided into smaller units such as hours, minutes and seconds (without consenually accepting the fiction of our presumption), then measurement of any event within the context of n-space is an expression of shared fantasy; with NO correlation to Reality!

For example: If you cannot divide Time into , say hours (without stopping the flow of Time , and thus nullifying your existence!), then the expression "Miles per Hour" becomes meaningles in the context of the reality in which it is/was to be measured.

Lynds' Revelation seems to be that Time, the concept we thought we were intimately familiar with, is a thing of a piece unto itself, and Linearity, the "passage of time as we seem to experience it, is but a convenience of perception imposed by the mind.

Boggling!

Or perhaps I've got it all wrong?

Edit for Spelling

[edit on 1-6-2006 by Bhadhidar]

That's the "movie-roll" theory of the universe - that there are untold numbers of universes, each universe static in its own time, and that since we're travelling through the universes it appears to us like a movie - a bunch of still images that merged together to produce an apparently moving image.

This theory works well towards multi-verse theories, since many universes could exist simultaneously, each following a different direction of quantum chaotic events that differentiates each from the other.

I believe this to exist myself for spiritual reasons.

I actually have understood this explanation of time for a while now...

In my personal opinion i think the current theories of time and space leave a lot to be desired. They look at time as something that is tangable and can be manipulated...

Time is not some long string or loop or movie we can rewind and fast forward with mathmatics or any other form of funny numbers.

Time is only the here and now, the.00000000000000001 of a second (Just a random number, you could put any number of zeros past that . ) is already gone... It no longer exists... you can't go back, no matter how hard you tried.

I think of time more like age. Things change as they get older, therfore your never the same; as you were a second ago; or even .000000001 seconds ago.

Thats why there is no unifyng theme explaining it all. Sure there are theories on physics, space time and such, but even those have been proven to be faulty at some point; even by the authors in some cases...

I don't claim to know how all things came into being and how all things work togethor. But this is just my thoughts on the matter... I don't claim to put in near as much thought as others have, and forgive if my explanations sound a bit elementary... I have never posted before... Lonf time reader of ATS, first time poster...

Those paradoxes are not paradoxes at all; they are erroneous conclusions.

The first "paradox"'s description is not even correct: the correct description is about the tortoise running half the speed of Achilles: by the time it takes Achilles to cover the distance between the start point and the tortoise, the tortoise will have moved half of that distance. Thus it seems that Achilles can never reach the tortoise.

The first "paradox" is wrong because it implicitely says that Achilles halves his speed after he reaches the point that the tortoise was. Indeed, if the speed is halved each time the previous position of the tortoise is reached, Achilles will never reach the tortoise. But in reality Achilles will not cut down his speed, so eventually he will overtake the tortoise.

The second "paradox" is the same as the first one, actually: it concerns halving the distance my hand has to travel after a certain point. If initially I aim for half the distance, and then the half of the half, etc, I will never reach the final point.

The third "paradox" is not a paradox at all. It is the theory of relativity in its simplest form: motion is relative to a frame of reference.

Materp,

I think you're missing the point. Mathematically, they are still paradoxes (although you are right in your statement that the first two are essentially the same). Even taking the motion of your hand as a whole movement, it can be mathematically broken down to 1/2 spacial movements, which means you should never reach your destination, and yet you do. Hence the paradox. What happens in real life, does not match up with what happens mathematically, and somehow, both are correct. Unless of course we have a fundamental misunderstanding of the nature of time, as Lynds explains.

The assumption that causes this paradox, is that space is infinitely divisible, whereas time is atomic (chronons), and therefore, not. I hop ethis makes sense.

These paradoxes have been around for thousands of years, and many mathemiticians have tried, unsuccessfully, to solve them. Which is why Peter Lynds work is so interesting, because it seems to solve the issue.

That aside, the other interesting, and more metaphysical question that arises from this theory is, how does human conciousness work in this context. In other words, if there really is no specific present as it were, how is it that we percieve one? If time cannot be narrowed down to a specific zero point, how is it our minds are capable of percieving the present, and differentiating it from that past and future?

Originally posted by Athenion
I think you're missing the point. Mathematically, they are still paradoxes (although you are right in your statement that the first two are essentially the same).

Athenion,
I think that you must mean that physically "..they are all still paradoxes." Mathematics solved each of these centuries ago with the concept of the limit, the basis of differential calculus.

Harte

The Xeno paradox lies in the fact that the process of the movent is incorrectly serialized. Achilles does not wait till tortoise "makes her move". He just keeps going. So in real life, the two movements are happening concurrently, whereas in this thought paradox, they are serialized, which does not reflect the nature of the actual phenomenon.

By same argument, I can never go broke I spend a thousand dollars I have, but at the same time I earned a dollar. While I'm busy spending the dollar, I earn a penny. No matter how poor I am, I'm never broke. How's that.


[edit on 2-6-2006 by Aelita]

Achilles and the Tortoise:

Sorry, but Aristotle got this wrong, plain and simple. If the runner is going fast enough, they can always overtake the leader.

If the tortoise has a ten meter lead over Achilles, and walks at 10cm/s, whereas Achilles starts at position zero and runs at 10m/s (olympic sprinting speed) he will overtake the turtle in just over 1 second. He will cross the ten meters in one second, after which time the turtle will move 10cm. After two seconds, the turtle will be at position 10.2 meters and Achilles will be at position 20 meters. He has obviously passed the turtle.

The Dichotomy Paradox:

This immediately reminded me of the mean value theorem in mathematics. This idea makes sense to me. To use the example given, if I want to touch my computer monitor, my hand is going to have to cross the intermediate space between them. Outside of quantum mechanics, I can't think of a situation in which this is violated.

The Arrow Paradox:

I'm not completely sure I understand this one, but if I'm interpreting it right, it's basically talking about the physics concept of 'relative velocity'. For example, if I am driving a car at 100km/h, and you are also driving a car in the next lane at 100km/h, if we look out the window, we will see the other persons car, and it will appear to be stationary, because it is moving at the same speed and in the same direction. I have no idea how this can be considered a paradox, but like I said, I might be reading into it wrong.

Some of your comments afterwards also bring to mind the concept of

Heisenberg's Uncertainty Principle. (the ones about not knowing speed and time precisely)

Actually, the "paradox" was answered shortly after it was proposed. There are also a number of mathematical solutions and proofs of those (I learned about them briefly in calculus.) Wikipedia gives an overview:
en.wikipedia.org...'s_paradox

The solutions folks are proposing here are those discussed well over 2300 years ago. Haven't read Lynd's paper yet, but it looks interesting. He does review the history of the proofs and various applications of them in his paper.

Again to quote from the wipikedia articel that Byrd kindly provided for us:

A problem with using calculus to try and solve Zeno's paradoxes is that this only addresses the geometry of the situation, and not its dynamics. What is at the core of Zeno's paradoxes is the idea that one cannot finish the act of sequentially going through an infinite sequence, and while calculus shows that the sum of an infinite number of terms can be finite, calculus does not explain how one is able to finish going through an infinite number of points, if one has to go through these points one by one. Indeed, saying that there are an infinite number of points or intervals within some finite interval is of course the very assumption in the Achilles and Dichotomy Paradoxes, and it is this assumption regarding the geometry of the situation that leads to a paradox regarding its dynamics.

It should also be noted that calculus-based solutions that are offered often object to the claim that "it must take an infinite amount of time to traverse an infinite sequence of distances". However, Zeno's paradox doesn't contemplate the time it would take for Achilles to catch the Tortoise; it simply points out that in order for Achilles to catch up with the Tortoise, Achilles must first perform an infinite number of acts, which seems to be impossible in and of itself: time has nothing to do with it. Thus, calculus-based solutions to Zeno's paradoxes often make the paradox into a straw man.

...

In short, trying to use calculus to resolve the paradox simply reaffirms the idea that space and time are infinitely divisible, and thus still suffers from the basic question as to how one can possibly reach the end of an endless series.

I'm aware that we have found a way, through using a geomotric series, we can make the math work, but this doesn't "solve" the paradox. As Harte kindly pointed out, we have mathematically figured out how to deal with these paradoxes, but we physically still do not understand it. That's why Lynds' paper is so intriguing. It makes it work both mathematically and physically.

I hope I'm making sense.

Originally posted by Athenion
Materp,

I think you're missing the point. Mathematically, they are still paradoxes (although you are right in your statement that the first two are essentially the same). Even taking the motion of your hand as a whole movement, it can be mathematically broken down to 1/2 spacial movements, which means you should never reach your destination, and yet you do. Hence the paradox. What happens in real life, does not match up with what happens mathematically, and somehow, both are correct. Unless of course we have a fundamental misunderstanding of the nature of time, as Lynds explains.

No, you are simply wrong (sorry to say this, but you are).

When you constantly divide the remaining distance in 2 parts, you never travel the upper part. Let's see a mathematical example:

Let's say our distance is from 0 to 20 m.

step 1: travel to 10 m.
step 2 : travel to 15 m.
step 3: travel to 17.5 m.
step 4: travel to 18.75 m.
step 5: travel to 19.375 m.

etc

So the formula for computing the travelled distance D at step N is:

distance(D, N) = D - distance(D / 2, N + 1)

Which is a recursive formula, and the end result is never the starting distance.

And that is because we always cut the remaining distance in two, and we never travel the upper part.

There is no paradox, no mystery, no metaphysical questions.

Mind you, mathematically, though you also always have to travel to the next half-way point, that next half-way point is reached in HALF the amount of time as the last point!

So as the number of points approaches infinite, the amount of time it takes to get to that point approaches 0.

In the end, when points DOES equal infinite, the time to reach that point is Zero, and the destination is reached instantaneously.

The paradox is resolved!

Originally posted by Yarium
Mind you, mathematically, though you also always have to travel to the next half-way point, that next half-way point is reached in HALF the amount of time as the last point!

So as the number of points approaches infinite, the amount of time it takes to get to that point approaches 0.

In the end, when points DOES equal infinite, the time to reach that point is Zero, and the destination is reached instantaneously.

The paradox is resolved!

A very elegant presentation, Yarium.

I wonder if anyone will still cling to the paradox even after reading this, the way the hangers-on have done with Titor, Billy Meier, etc?

Harte

In regards to pausing time, and observing a once travelling object... I often argue that it is not possible to pause, or rewind time.
You can speed it up infinitely, and slow it down, but you can never hit 0, or go lower than 0. You can make time slow down so much that it appears to be on pause. You can make something infintesimally slow, but never actually stop it.

That however is a theory, it has groundings, but only observational of other laws... take it or leave it.

Math is just a language of symbols that represent other symbols in our reality.

It is not the reality itself.

I want to get from point A to point B. First I must get to point A.5 etc. etc. BUT I can reach point B because it really is just halfway to point C.

Simply stating that we can mathematically resolve the paradox using calculas, does not resolve the logical paradox involved.

Once again to quote from the Wipekedia article Byrd provided (which states things much more eloquently and clearly than I ever could):

Mathematicians thought they had done away with Zeno's paradoxes with the invention of the calculus and methods of handling infinite sequences by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, and then again when certain problems with their methods were resolved by the reformulation of the calculus and infinite series methods in the 19th century. Many philosophers, and certainly engineers, generally went along with the mathematical results.

Nevertheless, Zeno's paradoxes are still hotly debated by philosophers in academic circles. Infinite processes have remained theoretically troublesome. L. E. J. Brouwer, a Dutch mathematician of the 19th and 20th century, and founder of the Intuitionist school, was the most prominent of those who rejected arguments, including proofs, involving infinities. In this he followed Leopold Kronecker, an earlier 19th century mathematician. Some claim that a rigorous formulation of the calculus (as the epsilon-delta version of Weierstrass and Cauchy in the 19th century or the equivalent and equally rigorous differential/infinitesimal version by Abraham Robinson in the 20th) has not resolved all problems involving infinities, including Zeno's.

As a practical matter, however, no engineer has been concerned about them since knowledge of the calculus became common at engineering schools. In ordinary life, very few people have ever been much concerned.

As a practical matter, yes, the paradoxes have been solved. Calculas does this. But as a philosophical understanding of the nature of infinity, it has not been resolved. The question still remains, how can something travel through an infinite amount of points in space in a finite amount of time. This is a contradiction, any way you write it. A paradox!!!!

It's really suprising that so many people aren't able to understand why it's still a philosophical paradox.

Actually, I think that the philosophical community misses the significance that the mathematical answer to the paradox gives (being that, at the point of infinite, the time it takes to traverse the distance is 0, and so the time to traverse all of infinite is also 0, and thus can be traversed).

However, perhaps one that would work better in TODAY'S world as a paradox would be;

Sir Isaac Newton and Albert Einstein are placed in a contest to determine which was the greatest physicist. Since Newton came before Einstein, Einstein's reputation first has to catch up with Newton's. However, since Newton's reputation is also growing (though at a slower pace than Einstein's), Einstein must first reach the reputation that Newton had before he existed, before he can continue to catch up with the lead that Newton had developed in the process.

As he does this, however, whenever his reputation catches up, it will always be at a point where Newton was before him. In the time it takes to catch up, Newton's reputation will have the time to get just that little bit further.

So how is it that Einstein can become more popular than Newton?


This is the same kind of paradox, but it too can be solved, with the concept of death. Since Newton and Einstein's reputation only count amoung those people who are alive, then both only have their reputation's grow so long as there are people. Eventually, people for both Newton's and Einstein's theories will die, though since Newton came first, his followers will die of sooner than Einstein's, and so Einstein's will get ahead because Newton's will fall behind when the death of just 1 of Newton's followers tips the scales of who's more popular.

The paradox is answerable, but the important information comes out of TRYING to answer it. The fun of the riddle is not hearing the answer, but trying to figure out the question.

planck length=smallest unit of measure, about 10^-35meters. once thats all the space left, you gotta go past it and your done.or you consider taht since your moving only half the distance it only takes half the time, and an infinitly small distance will take an equally small amount of time, you actaully will make it.

Lonemaverick,

I don't see how this resolves the issue. Just because Planck's Length is the smallest measurement we have, doesn't mean that smaller things exist.

For example, couldn't one use 1/2 of planck's length in an equation?

That would be like saying "An inch is the smallest measurement, therefore the shortest distance something can move is one inch".