CoffmanPhDIs space-time discrete or continuous?

[Cane_Toad]

No that was not my point. All I'm saying that if we take snapshots of the race at those particular points, the process seems endless, because there is always going to be a point when the turtle is ahead of Achilles. The idea is that we never get a chance to see Achilles crossing the turtle, not that we are biasing observation with Achilles always behind the turtle(which in fact happens but leads to no contradictions).

That it seems endless is irrelevant. Are you choosing a set of points which has the possibility of eventually seeing Achilles reach the turtle, or not.

The snapshot doesn't bias the observation, you have already biased it by your choice of how Achilles is allowed to move, and therefore the set of possible points.

 

[arunbg]

If the set of "points" when watching Achilles behind the turtle is never-ending, how can you expect to have a set of "points" where Achilles catches up?

 

[Cane_Toad]

If the set of "points" when watching Achilles behind the turtle is never-ending, how can you expect to have a set of "points" where Achilles catches up?

By this argument, you'll never see Achilles move at all, since any movement will entail going through an infinite set of points.

 

[neurocomp2003]

"By this argument, you'll never see Achilles move at all, since any movement will entail going through an infinite set of points...:"
And that is the point =]

 

[Hurkyl]

If the set of "points" when watching Achilles behind the turtle is never-ending, how can you expect to have a set of "points" where Achilles catches up?

Physical intuition and simple demonstration.

I can simply consider the set of points {start of race, point where Achilles catches the turtle}.

Or, to be fancy, I can consider the set of points:
{your infinite set of points} union {the point where Achilles catches the turtle}

Or, really, I can just consider
{all of the points from the starting point to the point where Achilles catches the turtle, inclusive}

And that's only if I limit myself to points that are involved in the race; at my discretion, I can consider points before Achilles enters the race, and points after Achilles passed the turtle.


More importantly, (and the point I've been trying to drive home this entire time), why would you ever think that

the set of "points" when watching Achilles behind the turtle is never-ending​

suggests

you cannot expect to have a set of "points" where Achilles catches up?​

If it helps, the following statement is false:

Every ordered set X can be "counted" -- that is, there exists a function f from X into some interval of integers such that x < y if and only if f(x) < f(y). (For any x, y in X)​

 

[Cane_Toad]

"By this argument, you'll never see Achilles move at all, since any movement will entail going through an infinite set of points...:"
And that is the point =]

No, it was a statement left unfinished as an exercise for why that approach will lead nowhere. (Other approaches have been posted.)

The idea is, as Hurkyl illustrates above, that you have to pick a useful set of points, and it doesn't matter if that set is big, just that its extent does you some good. But now I'm just repeated myself...

 

[neurocomp2003]

but by your logic don't you create a finite set of points? and thus a finite stepsize?

Because if u consider the path to be Continuous, then at what point does the person go from Ai to Ai+1. or in terms of numbers [0,1]
at what point do they go from 0 to 1 through stepping?

 

[Saladsamurai]

Awesome...

 

[Pythagorean]

but by your logic don't you create a finite set of points? and thus a finite stepsize?

Because if u consider the path to be Continuous, then at what point does the person go from Ai to Ai+1. or in terms of numbers [0,1]
at what point do they go from 0 to 1 through stepping?

You might be confusing the length of a number as it is expressed:

.7529293829829380980329480928344... (and so on)

with it's value.

That 'irrational' number above is not more than .8 just because it has more numbers than .8

it doesn't take infinite time to pass through that point because that's not quite how the math works.

Furthermore, the atoms that make me up (and even the elctrons) can be defined within a finite amount of decimals (i.e. we can round of that irrational number to have a finite number of decimal places without losing any information about a person passing a point).

There's also the convention of units so that it doesn't even have to be expressed as a decimal (i.e. nanometers in place of 10^-9 meters).

 

[arunbg]

{all of the points from the starting point to the point where Achilles catches the turtle, inclusive}

Sure, can you clarify what you mean by considering the set of all points from start to finish?
Note that in the approach discussed, we are merely ruling out the fact that Achilles gets to the set of points where {he catches up with the turtle} union
{he is ahead of the turtle}, and not excluding those points arbitrarily from discussion.
There is nothing physically wrong with watching things from the turtle's perspective.

Physical intuition and simple demonstration.

Of course, we do know that Achilles will cross the turtle(physical knowledge), the "paradox" is that we can't counter Zeno's arguments, by simply saying that the whole scenario was not taken into account.

No, it was a statement left unfinished as an exercise for why that approach will lead nowhere. (Other approaches have been posted.)

The idea is, as Hurkyl illustrates above, that you have to pick a useful set of points, and it doesn't matter if that set is big, just that its extent does you some good. But now I'm just repeated myself...

Note that we are not trying to get a better or the "right" approach here, but rather to find flaws in an argument set.

 

[Cane_Toad]

Sure, can you clarify what you mean by considering the set of all points from start to finish?
Note that in the approach discussed, we are merely ruling out the fact that Achilles gets to the set of points where {he catches up with the turtle} union
{he is ahead of the turtle}, and not excluding those points arbitrarily from discussion.
There is nothing physically wrong with watching things from the turtle's perspective.

Zeno makes it ambiguous as to which set of points we are supposed to be using in order to view his paradox.

His paradox is based on this ambiguity, IMHO. He sets up the story focusing on the points behind the turtle (infinite bifurcation), but he then states the problem in a context that supposed includes all the points up to and beyond the turtle. He leaves it ambiguous, and thus creates a confusion, and not a true paradox. This is why I say it is a trick question.

Note that we are not trying to get a better or the "right" approach here, but rather to find flaws in an argument set.

Finding a "right" approach is just a possible part of the deconstruction to understand the problem.

 

[Hurkyl]

Note that in the approach discussed, we are merely ruling out the fact that Achilles gets to the set of points where {he catches up with the turtle} union
{he is ahead of the turtle}, and not excluding those points arbitrarily from discussion.

HOW? By what means do you arrive at that conclusion?

If you were simply trying to argue that Achilles doesn't pass the turtle during the time when he's catching up, but I do not see how you are arriving at the stronger conclusion that Achilles doesn't pass the turtle at any other time. How are you ruling that out?

 

[-Job-]

There are an infinite number of (infinitely small) steps between A and B. But each of those steps require infinitely small time to cross.

The time required to go from A to B is the addition of an infinity of infinitely small time units.

Zeno's paradox is proportional to saying, since there are an infinite number of steps between A and B, and because it takes us an infinitely small amount of time (i.e. 0) to cross an infinitely small step (moving at some speed c), then the time required to go from A to B is 0.

 

[neurocomp2003]

Pythagorean: no one said anything about infinite time. The conversation is about the concept of infinite actions in a finite space [0,1] & time bounds [0,1]. Or as hurkyl put it perhaps we cannot assume infinite actions(or to my understanding of his points).

Put it another way how does motion/time evolution occur when we assumse space to be continuous?

 

[arunbg]

If you were simply trying to argue that Achilles doesn't pass the turtle during the time when he's catching up, but I do not see how you are arriving at the stronger conclusion that Achilles doesn't pass the turtle at any other time.

Why not, if Achilles doesn't get to the turtle ever when he's catching up, how can he pass the turtle at any other time? Of course all this is in context of Zeno's arguments, not actual physical experience.
Infinitesimally small distances and times are approached, however absolute 0 is never reached.
Again I state the problem for clarity(sorry I couldn't exclude the terminologies)

The turtle has a head start on Achilles, who is actually faster. At a certain time during the race, Achilles reaches the initial position of the turtle. However this process has taken some time, during which the turtle has moved a little foreward. The process continues(like a series of mini-races).

How can Achilles catch up with the turtle, if it always takes him some finite time to get to the turtle's last position(however small), during which the turtle would have moved a small distance foreward ?

Finding a "right" approach is just a possible part of the deconstruction to understand the problem.

I partly agree, the "right" approach is essential, but it should also be able to explain why the other approach was wrong. I don't think anyone so far has done that.

Zeno's paradox is proportional to saying, since there are an infinite number of steps between A and B, and because it takes us an infinitely small amount of time (i.e. 0) to cross an infinitely small step (moving at some speed c), then the time required to go from A to B is 0.

Can you clarify how you arrived at this conclusion from Zeno's arguments?

 

[Universe_Man]

I think zeno's paradox tells us something about the nature of what we think reality is. Reality, or the construct of time and space that we presently see around us is only a part of something much larger. I personally believe that the fact that there exists a paradox indicates that there is a larger category of reality which encompasses or is somehow associated with our universe. I like to think of this "realm" if you will, that is infinite; it's a place of pure thought. Of course this will most likely be regarded as speculative, but I once read that Plato or some other greek philosopher held a similar idea.

 

[DeaconJohn]

If I understand correctly, the question discussed here is related to the question of whether or not space and/or time are continuous or discrete in their underlying nature.

I don't think modern physics has a definite answer to that question. Mainline physics uses continuous models, but, discrete models have been seriously investigated off and on for at least the last 50 years (I heard Pere Lax say he had spent a lot of time looking at them at a lecture he gave in 1968).

The place in physics that I have personally seen discrete models discussed the most is in various attempts attempts to unify the gravitational force with the other forces in a single coherent theory. (String theory is one of many attempts to unify gravity with the other fundamental forces of nature.)

So far, the discrete models have not produced any interesting descriptions or predictions or whatever, as far as I know. People make that complaint about string theory, but, string theory is far better in that area than the discrete models are. At least string theory has a place for the graviton (that must be a spin two particle (a boson?) if the Einstein field equations for general relativity are correct) without the divergent integrals of "the standard model" (quantum field theory, QFT) to compute the possible values of its observables.

However, string theory (according to vol 2 of Joe Polinski's book "String Theory," still the baisc graduate level text in the field) implies that the Heisenberg uncertainity principle applies to the underlying space-time continum. If this suggestion matches reality, it amounts to a perturbation of our normal concepts of space and time that is far greater than the question of whether or not space-time is discrete or continuous. It is saying that the underlying structure of space time is properly described by Noncommutative Geometry - a field dominated by the name of Allain Connes.

Not to worry, it is likely we will never know for sure. Polchinski points out that if the basic space-time vaiables of reality in fact do not commute, the lack of commutativity is only observable (i.e., measureable by experient) if you have access to enough energy to get you within two orders of magnitude of the Plank length (10**(-33) meters). That means 1% of all the energy in the visible universe. That much energy never existed in one place in the entire universe except for a very short time after the big bang.

Yours,

DJ