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Talk:Stephenson:Neal:Quicksilver:165:Zeno's Paradox (Matt Zwolinski)

From the Quicksilver Metaweb.

Talk for Xeno or Zeno...

From a Lay Person's Point of View

I can see the argument, that any given space may be, Theoreticaly, divided in half an infinte number of times, however, I believe the Absurdity, that you point out, comes from the intuitive realization that this statement is inherently false.

Would it be a true statement, or a false one, to say that any given piece of wood may only be divided a finite number of times?

I would say that it would be true, for you only need to start with the smallest element you can, of wood, or anything for that matter, and divide it but one more time, after which, it is no longer "that" element, but rather, the components of, that element.
Modern science teaches us that this is true, that all Gross, physical "Things", have components, that something may only be broken down, so far, until it is no longer the "Thing", but the things which compose it.

It would seem that distance, unlike something as solid, and physical, as, say, an Apple, Would "Theorecticaly" be capable of being divided an infinite number of times, since the measuring system I was raised with has no lower limit. An inch isn't composed of anything but fractions of an inch, and you can continue to multiply the denominator of 1/64ths of an inch, by two, an infinite number of times.

To walk 100 Yards might not take forever, but attempting to disprove that the span can be divided an infinte number of times, through Math, would.

But we all know it is false. Because, the conclusion is continually disproved with each and every beating of the Heart, each walk to the corner store, each time a fist is raised in anger, each step of the smallest of babies, each flutter of an eyelash of a young girl, lost in the melancholy Springtime of love.

It seems that the only thing which causes Any confusion, at all, is that Math Proves to us that it Is indeed possible to infinitely, divide, Anything.

If Math shows us, without any hint of Doubt at all, that an Apple can be divided an infinite number of times, and it does, but, in the Doing -- or the Proving -- of it, we find that we start having trouble once we begin dividing the Cell, let alone the atomic structure of the apple, that there comes a point at which, one cannot Prove, that the specimen is, any longer, "Apple", then what?

Is Math wrong, does this prove, instead, that Math is neither relevant nor real?

Somehow my original comment got zapped -- Peek at Einstein's Principle of Equivalence. Xeno or Zeno? This entry has most of the relevant math links - Heinlein:Robert:Have Space Suit Will Travel:40:Almost halfway to the Moon, I'd say (Neal Stephenson).

The mighty Wikipedia says Zeno. - Sparky 16:48, 2004 Jan 22 (PST)

Zeno's Paradoxes

Zeno's paradoxes are a set of paradoxes conceived by Zeno of Elea to support Parmenides's doctrine that all evidence of the senses is misleading, and particularly that there is no motion.

Several of Zeno's eight surviving paradoxes (preserved in Aristotle's Physics and Simplicius's commentary thereon) are essentially equivalent to one another; and most of them were regarded, even in ancient times, as very easy to refute. Three of the strongest and most famous--that of Achilles and the tortoise, that of a rock thrown at a tree, and that of an arrow in flight--are given here.

Zeno's paradoxes may seem trivial today, but they were a major problem for ancient and medieval philosophers, who found no satisfactory solution until the 17th century, with the mathematical results on infinite sequences and calculus. Indeed, it appears very reasonable that their correct resolution wasn't actually acheived until the 21st century, with the work of Peter Lynds on time, motion and position.

Achilles and the tortoise

In the paradox of Achilles and the tortoise, we imagine the Greek hero Achilles in a footrace with the plodding reptile. Because he is so fast a runner, Achilles graciously allows the tortoise a head start of a hundred feet. If we suppose that each racer start running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run a hundred feet, bringing him to the tortoise's starting point; during this time, the tortoise has "run" a (much shorter) distance, say one foot. It will then take Achilles some further period of time to run that distance, during which the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, Zeno says, swift Achilles can never overtake the tortoise.

In the modern analysis, the paradox is resolved with the fundamental insight of calculus that a sum of infinitely many terms can yield a finite result. Adding the (infinitely many) times together that Achilles needs to reach the previous positions of the tortoise results in a finite total time, and that is indeed the time when Achilles overtakes the tortoise.

The rock thrown towards a tree

The next paradox, that of the rock thrown towards a tree, is a variant of the previous one. Now Zeno stands eight feet from a tree, holding a rock. He throws his rock at the tree. Before the rock can reach the tree, it must traverse half the eight feet. It will take some finite time for the rock to fly four feet. After that time, it will still have four feet to go, and to traverse that distance must first cover half of it: two feet, and more time. After it travels two feet, it must travel one foot, then half a foot, then a quarter foot, and so on ad infinitum. Therefore, Zeno concludes, the rock can never hit the tree.

The Arrow Paradox

Finally, in the arrow paradox, we imagine an arrow in flight. At every moment in time, the arrow is located at a specific position. If the moment is just a single instant, then the arrow does not have time to move and is at rest during that instant. Now, during the following instances, it then must also be at rest for the same reason. The arrow is always at rest and cannot move: motion is impossible.

This paradox is resolved by calculus as follows: in the limit, as the length of a moment approaches zero, the instantaneous rate of change or velocity (which is the quotient of distance over length of the moment) does not have to approach zero. This nonzero limit is the velocity of the arrow at the instant.

  • Zeno's Arrow
  • Zeno's Arrow applied to Atomism -- “Zeno’s argument that an (apparently) moving arrow is really at rest throughout its flight seems easy to evade if one insists that space is continuous (and hence infinitely divisible).  But an atomist who insists on theoretically indivisible atoms seems bound to deny that space is infinitely divisible. And Zeno’s Arrow Paradox poses an especially troubling problem for such an atomist....”

This Frank & Ernest cartoon may help in understanding when Zeno is practical for humans.

I know I just read this, perhaps this example of an arrow in flight is used within one of Neal's books?

But I'll admit to being confused. We know that the arrow is in motion, and thus not at rest, ever. If we were to take a high speed photo, then yes, we would be able to freeze it's image, and, for all apparent purposes, it would seem to be at rest. Showing that picture to another, say, someone from a zero gravity environment, there likely would be no prejudice of motion conveyed to them, within the picture itself.

I would wonder, though, whether the arrow can be said to be in any "specific" point, at any given moment in time. For one thing, it's following a trajectory, and, as far as time is concerned, is the arrow really a single arrow, or, more of a Wave form. Does it exist as an object, or does it embody the path between orgin and destination.

In just an aside, it seems as though the author is making very clear that the concept of time, is, purely a human one. Thinking about this earlier, while out snowshoeing, I wondered whether the example where motion was impossible, was, not also saying that a Mathmatian would never be able to arrive at the point of orgin, from any given destination, by contintually halving the distance known to be between the two. The converse being true as well.

In the example I read, the difficulty was in the first step, never being able to define, and thus make, that very first step -- A paradox of decision. However, if the Paradox is turned on it's head, it becomes a problem of never being able to Complete one's journey, rather than start it, as long as that journey remains defined.

Perhaps, what Zeno was Really trying to say, was that one is "Never Able to return home"

And(!)

that "Life is a journey, not a destination".

as soon as the distance becomes an unknown, the point of orgin, or destination lost, balance is restored. But one is still unable to go home. timberbee 17:22, 22 Jan 2004 (PST)

Damn, it's fun to think about these things.

Zeno may be saying you can't cross the river in the same place twice. - Sparky 01:03, 2004 Jan 23 (PST)

Actually the Greeks had some bits of what we would today call the calculus, although not as a unified system. Eudoxus developed the method of exhaustion, that permits finding areas and volumes, integral-calculus-style. Archimedes did a lot of "practical" work on this line to apply the method to specific cases. Check out for example http://www.nationmaster.com/encyclopedia/History-of-calculus I can't find evidence of differential calculus in ancient Greece, but I believe I've read something of the sort somwhere, just can't find it now. At any rate, the use of integrative methods shows some insight into the ideas of infinity and the problem of the continuum sufficient to deal with the paradoxes in question. 213.60.77.115 20:36, 23 Feb 2006 (PST)