Let's go.........RANDOM!
 bert_the_turtle
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That would be a potential spoiler, so:
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Not a bad game, mind you. They just seem to be making an effort to hit as many cliches as possible. I guess it's an homage?
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Not a bad game, mind you. They just seem to be making an effort to hit as many cliches as possible. I guess it's an homage?
 Jackdapantyrip
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 Joined: Sat Mar 19, 2011 6:33 am
Endless wrote:NeatNit wrote:oh god whyEndless wrote:I have been reading this thread from beginningToo bad books can't be updated in real time.Would make a great book.
As a state trooper you get a lot of down time.
Depends on the state, I'd imagine. A friend of mine does accident investigations, he keeps fairly busy.
 Jackdapantyrip
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 Jackdapantyrip
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 Posts: 250
 Joined: Sat Mar 19, 2011 6:33 am
bert_the_turtle wrote:NeatNit, since one of your troubles seems to be that the 1/2, 1/4... subdivision of the process is arbitrary, try this different problem:
Consider an idealized bouncing ball in a vacuum (no air friction). Assume the bounce is instantaneous, it skips the compression/expansion phase. But it is not lossless: each jump reaches only sqrt(0.5) times the height of the previous jump (which means the air time will be half of that of the previous jump). The ball is also allowed to rest still on the ground.
Drop the ball from 1.25 m height. After 1/2 a second, it will hit the floor (let's arbitrarily define that as our zero point in time, t=0s) and bounce back up, reaching only, umm, about 0.8 m height, and hit the floor 1/2 s later. The next jump will reach 0.625 m and last only 1/4 s.
So, summing up, the first jump finishes at t=1/2 s. The second jump at t=3/4 s. The third jump at t=7/8 s. And so on, the nth jump concludes at 10.5^n. What happens at and after t=1 s?
Well, obviously, the ball is not jumping any more at t > 1s. Each jump demonstrably concludes before t=1 s. So it must be resting on the ground. (*) When did the transition between jumping and resting happen? Again, obviously, at t=1.
And now, the really important question is: How did the transition happen at t=1?
SubQuestion: Was there a last jump that ended at t=1?
Clearly, there must have been, because at t<1, the ball is jumping and at t>1 it is resting. So it stopped jumping at t=1. So there was a jump that ended then.
But equally clearly, there can't have been. All jumps end before t=1, none ends precisely at t=1.
So.... now what?
And before you claim that the model is incomplete and insufficient: Yes, it is. However, a full model, when you afterwards take the limit to infinitely fast bouncing without compression, yields precisely the same motion. The full model gives one result you would need to wrestle from the simplified model: the speed of the ball at t=1 s is zero. You are allowed to use that.
*: You may be tempted to evade by saying the ball was blown up by the increasing frequency of impact shocks. You would be wrong.Yes, their frequency approaches infinity, but their magnitude approaches zero in such a way that any average containing more than one shock has a finite average force bound by about twice the ball's weight. If you want to blow up ideal bouncing balls in thought experiments, put one between two mercilessly converging plates. Or trains.
Oh, I forgot to answer this one!
First of all congrats on finally making a proper analogy... You really had me stumped there. I see the point now!
However, a few points:
1. There isn't anything that doesn't compress. When you add compression into that experiment, it doesn't work.
2. Many balls were harmed in the thinking of this film.
Most importantly:
3. The whole point of Zeno's Paradoxes is that motion is a lie and "all is one" and a whole bunch of other obvious nonsense that no selfrespecting mathematician or physicist would be caught dead saying. THAT is my problem with them. They do this by taking real examples from real life that actually happen every day, and then "explain" why they're impossible. If anyone starts doubting their own eyes because they're wrestling with his paradoxes, then I would lose all my respect for them. I find the paradoxes simple to overcome.
By the way, Zeno's Paradox is mentioned briefly here, around 6:30:

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NeatNit wrote:1. There isn't anything that doesn't compress. When you add compression into that experiment, it doesn't work.
It is a thought experiment. As soon as you start dismissing it because nothing in nature is the Platonic ideal of the object described by the thought experiment, you have failed to understand the point. There are no incompressible objects in nature. There are no perfect circles or spheres. There may not be any continuous actions. That isn't the point of the paradox. The point is to highlight the counterintuitive fact that, in a pure mathematical abstraction, infinite processes can have finite results.
NeatNit wrote:3. The whole point of Zeno's Paradoxes is that motion is a lie and "all is one" and a whole bunch of other obvious nonsense...
No. This statement is incorrect. It is possible that Zeno's intention was to make that kind of statement (thought I doubt it), but in the modern understanding, Zeno's Paradoxes are meant to give intuition about infinite series. In fact, the analogy is perfectwe know that motion is possible, and we know that we can model motion by this halving process, which means that the halving process (which is infinite) must terminate (because either (a) motion is possible (intuition) or (b) sequences of partial sums can converge (mathematics)).
xander
And of course, xander, you've missed the important bits.
and:
And most importantly:
Have I made my point clear? Yes, the thought experiment is a paradox (the ball one, not the arrow one). I can see that.
NeatNit wrote:finally making a proper analogy... You really had me stumped there. I see the point now!
and:
NeatNit wrote:The whole point of Zeno's Paradoxes is that motion is a lie and "all is one" and a whole bunch of other obvious nonsense that no selfrespecting mathematician or physicist would be caught dead saying. THAT is my problem with them. They do this by taking real examples from real life that actually happen every day, and then "explain" why they're impossible. If anyone starts doubting their own eyes because they're wrestling with his paradoxes, then I would lose all my respect for them. I find the paradoxes simple to overcome.
And most importantly:
Of course I said this, don't be silly. wrote:I understand the paradoxical nature of this thought experiment, and yours bert was really the only one that did it right (I wish you'd put that here sooner), but Zeno's original paradoxes were not as paradoxical. The reason they weren't, as I said, is that they take reallife examples that actually happen without any simplifications, and since the real world is not paradoxical, neither are the "paradoxes". You've posted probably the best example of a version of the paradox which actually is paradoxical, with the only problem being that it's not taken directly from the real world without any simplifications.
Have I made my point clear? Yes, the thought experiment is a paradox (the ball one, not the arrow one). I can see that.
NeatNit wrote:Have I made my point clear?
No. You haven't.
NeatNit wrote:Yes, the thought experiment is a paradox (the ball one, not the arrow one). I can see that.
Both are paradoxical. The underlying mathematics are identical. Clearly, the arrow moves, and clearly the ball stops bouncing eventually. The magic computer eventually returns the result "2". The paradox has nothing to do with whatever "real world" analogy we choose to explain it with, but with the underlying mathematics, and the fact that an infinite series can evaluate to a finite result.
xander
 Jackdapantyrip
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