No Present: Debate on Time's Definition

[hypnagogue]

Originally posted by Canute
I didn't know that, I've never looked into the orthodox quantum mechanical view on this. This is our very best scientific theory of the motion of subatomic waves? Wow.

Yes. Assuming energy is transmitted continuously actually leads to some much more grave paradoxes in physics than Zeno had in mind-- for instance, the paradox of why we aren't burned to a crisp by looking at a fireplace.

I can't agree with any of your analysis of the divisibility of time or space. You've 'renormalised' the infinities by sleight of hand.

What have I renormalized, and what is my sleight of hand? It's math, pure and simple.

Well, I can only say I don't agree. I think Zeno's paradoxes have lasted so long because he had a point.

Zeno's paradoxes lasted so long because it took a while for calculus to be invented. Using calculus, we see that Zeno can cross a finite distance in a finite time, even if it can be theoretically broken down into infinitely many subdivisions. Specifically, we can show that in the limit as the number of steps approaches infinity, the size of the steps trails off quickly enough that their sum approaches a finite number rather than infinity (likewise for the time needed to complete the tasks).

Zeno clearly did not understand that an infinite sum can have a finite value, as evidenced by his argument against plurality (see http://faculty.washington.edu/smcohen/320/zeno2.htm). His argument here basically was that "All objects are composed of infinitely many parts; all parts must have some finite size; therefore, all objects are infinitely large." But calculus shows us concretely that infinite sums can in fact have finite values. This simple fact alleviates both Zeno's paradox and his argument against plurality.

 

[hypnagogue]

Originally posted by Jeebus
Zeno notes that if physical objects exist discretely at a sequence of discrete instants of time, and if no motion occurs in an instant, then we must conclude that there is no motion in any given instant. (As Bertrand Russell commented, this is simply "a plain statement of an elementary fact".) But if there is literally no physical difference between a moving and a non-moving arrow in any given discrete instant, then how does the arrow know from one instant to the next if it is moving? In other words, how is causality transmitted forward in time through a sequence of instants, in each of which motion does not exist?

This is quickly becoming a thread about Zeno. I propose we re-name this thread, "there is no paradox."

In any case, you are referring to Zeno's arrow paradox. Zeno's arrow paradox is solved at http://faculty.washington.edu/smcohen/320/ZenoArrow.html .

 

[Canute]

Originally posted by hypnagogue
Yes. Assuming energy is transmitted continuously actually leads to some much more grave paradoxes in physics than Zeno had in mind-- for instance, the paradox of why we aren't burned to a crisp by looking at a fireplace.

Well, you've got a choice. You can have the paradox in physics or you can have it in real life. Take your pick. I tend to think motion is possible and that it is mathematical physics that can't handle the infinities involved in dealing with a continuum.

What have I renormalized, and what is my sleight of hand? It's math, pure and simple.

To eternally approach a finite number is not the same as being a finite number.

Zeno's paradoxes lasted so long because it took a while for calculus to be invented. Using calculus, we see that Zeno can cross a finite distance in a finite time, even if it can be theoretically broken down into infinitely many subdivisions.

Calculus just hides the infinties. That's what it's for.

Specifically, we can show that in the limit as the number of steps approaches infinity, the size of the steps trails off quickly enough that their sum approaches a finite number rather than infinity (likewise for the time needed to complete the tasks).

It approaches a finite number for all eternity. At what particular arbitrary point do you assume that it arrives?

Zeno clearly did not understand that an infinite sum can have a finite value, as evidenced by his argument against plurality (see http://faculty.washington.edu/smcohen/320/zeno2.htm). His argument here basically was that "All objects are composed of infinitely many parts; all parts must have some finite size; therefore, all objects are infinitely large." But calculus shows us concretely that infinite sums can in fact have finite values. This simple fact alleviates both Zeno's paradox and his argument against plurality. [/B]

No offense, but I feel that Zeno (and Russell) gave this more thought than you think they did.

Thanks for the link, but it does not appear to be a solution, more a missing of the point. Jeebus summed up the problem very well. How would you answer his post?

 

[hypnagogue]

Originally posted by Canute
Well, you've got a choice. You can have the paradox in physics or you can have it in real life. Take your pick. I tend to think motion is possible and that it is mathematical physics that can't handle the infinities involved in dealing with a continuum.

Physics is real life.

Again, Zeno's paradox relies on space being a continuum to draw its paradoxicalness. Discretely quantized space is not the central figure to Zeno's paradox, and further there is nothing inherently paradoxical about motion through quantized space. Counter-intuitive != paradoxical.

To eternally approach a finite number is not the same as being a finite number.

Calculus just hides the infinties. That's what it's for.

The point is that an infinite numbers of steps in the race does not imply infinite distance or an infinite amount of time to complete the steps. This is not a matter of hiding infinity, but solidly reasoning about it.

It approaches a finite number for all eternity. At what particular arbitrary point do you assume that it arrives?

So you are arguing that \sum^{\infty}_{i=1} 1/2^i does not equal 1? If you are not, then my argument stands. If you are, I would like to see your proof to the contrary.

No offense, but I feel that Zeno (and Russell) gave this more thought than you think they did.

As I have explained, Zeno clearly did not understand that an infinite sum could be finite. This is no fault against him, since he couldn't have known without inventing calculus himself. But his paradox depends on this incorrect notion.

Thanks for the link, but it does not appear to be a solution, more a missing of the point. Jeebus summed up the problem very well. How would you answer his post?

Jeebus brought up Zeno's arrow paradox, which is un-paradoxed in the link I replied with. The central idea is that motion is only defined on an interval of time. If an object changes its physical location during a given interval of time, it has been in motion; if it does not, it is at rest. For an arrow at a given instant of time, we cannot infer whether it is in motion or at rest, since by necessity we need to define these terms with respect to an interval of time.

 

[Canute]

Oh well. No point in going through it again. We'll have to disagree.

 

[mikelus]

If there's a fixated point in time doesn't that mean that time has been separated from itself. The present is where a fixated point disappears and comes one with it all.