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LobowolfXXX
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Being a mathematical, philosophical kind of guy, maybe you can explain this to my satisfaction. I think it's a lesser paradox of Zeno's, or perhaps just based on Zeno's Paradox.

X and Y are racing a 100-yard dash. X is much faster, so Y gets a 10-yard head start. The race begins. It takes X some amount of time to get to where Y started, so in that time, Y, who is in motion, advances to a place beyond that point (call it Y1). It takes X some amount of time to advance to Y1, during which Y has advanced to Y2; thus Y is still in the lead when X reaches Y2. And so on and so on. We know that it's possible for X to pass Y, but the incremental analysis seems to suggest otherwise. It almost feels like the only explanation is that at some point(s), Y is not in motion, e.g. Midstride. However, that's obviously not the explanation, because it works for cars, too. So what's up?
"Torture doesn't work" lol
Guess they forgot to tell Bill Buckley.

"...as we reason and love, we are able to hope. And hope enables us to resist those things that would enslave us."
balducci
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This is my quick take on it ... The time intervals In the incremental analysis are becoming increasing small, and tend towards the length of an infinitesimally lengthed moment. During that moment car Y is for all intensive porpoises (*) motionless compared to car X.

You could see this in real life with a properly timed stop motion photo.

(*) Smile
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S2000magician
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The underlying assumption is that the sum of an infinite number of positive values must be infinity. This assumption is false: there are convergent infinite series.

For example, suppose that the faster runner can cover that 10-yard head start in one second (about 20 mph), and that the slower runner is 90% as fast as the faster. One second after the start, the faster runner has reached the slower runner's starting point, and the slower runner has covered 9 yards. To reach the slower runner's next point will take 0.9 seconds, during which the slower runner will cover 8.1 yards. The sequence of time intervals (in seconds) is: 1, 0.9, 0.81, 0.729, 0.6561, and so on; it's a geometric sequence with common ratio of 0.9. The corresponding infinite series (1 + 0.9 + 0.81 + . . . .) is convergent (because |r |< 1), and the (infinite) sum is 1 / (1 - 0.9) = 10 seconds; thus, after 10 seconds, the faster runner will overtake the slower runner (coincidentally, at the finish line).
LobowolfXXX
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I knew there would be a brief, not-overly-complex, mathematical explanation.
"Torture doesn't work" lol
Guess they forgot to tell Bill Buckley.

"...as we reason and love, we are able to hope. And hope enables us to resist those things that would enslave us."
Pakar Ilusi
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Quote:
On 2013-07-20 14:51, LobowolfXXX wrote:
I knew there would be a brief, not-overly-complex, mathematical explanation.


That was the brief version?! Smile

Smile
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Dannydoyle
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I used to have to explain this to street racers all the time. Over a standing quarter mile it is easier to illustrate.

In physics and math often people assume "all other things being equal" and even say it. But in math and physics that is rarely the case.
Danny Doyle
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mastermindreader
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Even briefer and without the math:

The seeming paradox is false because it wrongfully assumes that motion is a series of static points in time. It's not.
Pakar Ilusi
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On 2013-07-20 15:50, mastermindreader wrote:
Even briefer and without the math:

The seeming paradox is false because it wrongfully assumes that motion is a series of static points in time. It's not.


Okayyy... Thanks Bob! Smile
"Dreams aren't a matter of Chance but a matter of Choice." -DC-
Zombie Magic
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Quote:
On 2013-07-20 15:00, Dannydoyle wrote:
I used to have to explain this to street racers all the time. Over a standing quarter mile it is easier to illustrate.

In physics and math often people assume "all other things being equal" and even say it. But in math and physics that is rarely the case.


You'd have been a great teacher!
Magnus Eisengrim
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BTW Lobo this is one of three paradoxes attributed to Zeno. It's usually called "Achilles and the Tortoise". You can guess who Zeno gave the head start to Smile
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S2000magician
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Quote:
On 2013-07-20 15:50, mastermindreader wrote:
Even briefer and without the math:

The seeming paradox is false because it wrongfully assumes that motion is a series of static points in time. It's not.

Not exactly, but a good way to visualize it.
mastermindreader
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I know it's not exact, but I was trying to explain it without the math.
The runner is never statically AT a given point- he is continuously passing through them.
S2000magician
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On 2013-07-20 17:25, mastermindreader wrote:
I know it's not exact, but I was trying to explain it without the math.
The runner is never statically AT a given point- he is continuously passing through them.

It's not the static vs. dynamic distinction that's the explanation, however. It's the convergent vs. divergent infinite series distinction that is. That distinction has nothing to do with static vs. dynamic.
mastermindreader
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The problem is that Zeno's paradoxes can be viewed as either philosophical/metaphysical problems, as well as mathematical ones:

Quote:
Aristotle's objection to the arrow paradox was that "Time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles."[21] Saint Thomas Aquinas, commenting on Aristotle's objection, wrote "Instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time."[22] Bertrand Russell offered what is known as the "at-at theory of motion". It agrees that there can be no motion "during" a durationless instant, and contends that all that is required for motion is that the arrow be at one point at one time, at another point another time, and at appropriate points between those two points for intervening times. In this view motion is a function of position with respect to time.[23][24] Nick Huggett argues that Zeno is begging the question when he says that objects that occupy the same space as they do at rest must be at rest.


http://en.wikipedia.org/wiki/Zeno%27s_paradoxes
balducci
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Comment 1: Taken to its limit, I think the calculus explanation assumes infinite divisibility of space and time. I think physicists are still debating issues related to this. So I do not believe that the calculus explanation is entirely satisfactory / complete.

Comment 2: "[T]here's a tradition among some high school calculus teachers to present ... "Zeno's Paradox" ... then "resolve the paradox" by pointing out that an infinite series can have a finite sum. This may be a useful pedagogical device for beginning calculus students, but it misses an interesting and important philosophical point implied by Zeno's arguments. To see this, we can re-formulate the essence of these two arguments in more modern terms, and show that, far from being vitiated [i.e. invalidated / explained] by the convergence of infinite series, they actually depend on the convergence of the geometric series [i.e. to create the paradox in the first place]."

- Reflections on Relativity By Kevin Brown ( http://www.amazon.com/Reflections-Relati......57033026 )

He goes on to say that the actual explanation has more to do with special relativity and the concept that space and time are not independent (whereas Zeno implicitly believed that they were).

http://www.mathpages.com/rr/s3-07/3-07.htm

Obviously, I am dumbing down the argument at the link above. And it is not something I really care to spend time debating (not on here, anyway), so anyone interested in reading the essay can find someone else to discuss it with. Smile
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MobilityBundle
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Quote:
On 2013-07-20 18:14, balducci wrote:
Comment 1: Taken to its limit, I think the calculus explanation assumes infinite divisibility of space and time. I think physicists are still debating issues related to this. So I do not believe that the calculus explanation is entirely satisfactory / complete.


But if space is quantized (that is, indivisible at some small scale), then the paradox itself doesn't exist. The paradox requires that time and/or space is infinitely divisible. (How? By the assumption that that by the time Achilles gets to where the tortoise used to be, the tortoise actually moved. Or even that such a time exists in all cases. As S2000 mentioned, both of these measures form a geometric sequence converging to zero. So if space or time is quantized at some dimension h, then eventually either or both of these numbers falls below h.)

Quote:
Comment 2: [stuff about relativity]

Obviously, I am dumbing down the argument at the link above. And it is not something I really care to spend time debating (not on here, anyway), so anyone interested in reading the essay can find someone else to discuss it with. Smile

Fair enough. FWIW, I think neither relativity nor quantum mechanics is required to analyze any of Zeno's paradoxes. That's not to say versions of Zeno's paradoxes don't exist in any of those frameworks, but it's just a different issue. Happy to provide details to anyone interested.

Okay, can't resist providing a few details: there's something called the "Quantum Zeno Paradox." Skipping a lot of QM, take as an assumption that it's possible to prepare a system that is in a "mixed state," say a mixture of states X and Y. If observed, suppose the system is 99% likely to be found in state X and 1% likely to be found in state Y. (According to the rules of the game, one only OBSERVES "pure" states, not mixed states.) Finally, it's well known how the system evolves with time. Suppose, for example, that after an observation, the system gradually mixes back to the original 99% / 1 % mixed state in, say, 1 millisecond.

Great. So suppose you take an observation and find it in the unlikely state Y. If you do another observation RIGHT AWAY, the you don't give the system time to mix back with state X. And if you can continually machine-gun observation after observation, you can keep a system in an unlikely state for a really long time. Colloquially, this is akin to being able to keeping a car floating in the air, just by looking at it quickly enough over and over.

And best of all, the quantum Zeno paradox isn't actually a paradox... it's been observed in the lab.
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