History

Re: Achilles and the Tortoise

Smiling Dave — Sat, 22 Dec 2012 11:58:37 GMT

I guess you just cant imagine that someone would decline to make that assumption in the first place, it has to be you, after struggling with it for a while.

Glad to see you are one step ahead of me, in your world. Good for you.

Re: Achilles and the Tortoise

Malachi — Sat, 22 Dec 2012 05:41:27 GMT

Smiling Dave:
Malachi, thank you for opening my eyes to how you think. .

I see that we are too far apart on the very basics. So good luck to you.

But what about this?

So what's the flaw? The flaw is in assuming his model makes sense. Slicing something up into a finite number of pieces makes sense to us, and we justified to feel we know the laws that apply to finite slicing, because we see countless examples of it every day of our lives. But we have never seen anything divided into an infinity of pieces. Therefore we have no basis for assuming we know the laws that apply to infinite slicing in the real world [even granting that such a thing is possible]. So we have no basis for the assertion that nature is not a glue that melds infinitely many objects together. On the contrary, we see that it is.

I guess you just cant imagine that someone would decline to make that assumption in the first place, it has to be you, after struggling with it for a while.

Re: Achilles and the Tortoise

Smiling Dave — Sat, 22 Dec 2012 04:11:25 GMT

Malachi, thank you for opening my eyes to how you think. .

I see that we are too far apart on the very basics. So good luck to you.

Re: Achilles and the Tortoise

Smiling Dave — Sat, 22 Dec 2012 03:49:27 GMT

Glad to see you get it, Blagg.

Here's a summary of the various resolutions of Zeno's paradox I have seen so far.

1. Since in reality, Achilles catches the hare, Zeno must be wrong and/or not worth bothering about. [Just dealt with that one].

2. Now that we know how to sum an infinite series, Zeno's paradox just melts away of itself. [It doesn't, since the definition of the sum of an infinite series is of a least upper bound. The problems inherent in that are in the thread].

3. Now that we have a rigorous definition of continuity, Zeno is no problem anymore. [Same kind of problem as with 2.]

4. Zeno did not get that 1+ 1/2+ 1/4+... actually adds up to one. [It doesn't actuallly add up to one, except in the sense of least upper bound. See 2.]

5. Now that we know about non standard analysis, the paradox is solved. [Red herring. Non standard analysis adds nothing to the discussion].

6. Space, time, and/or numbers do not really exist, and so are not worth bothering about. [Same rebuttal as for 1.]

All of these are flawed, to put it politely. This long thread is pretty much devoted to showing what's wrong with each of them.

The only answer I like is Smiling Dave's, posted here: http://mises.org/community/forums/p/29288/470667.aspx#470667

The gist of that answer is that Zeno is making an assumption with no basis, the assumption that we understand how infinity works. See the post for more details.

Re: Achilles and the Tortoise

Blargg — Sat, 22 Dec 2012 02:25:12 GMT

I concur that the point of these paradoxes is to show that taking a simple situation and applying some logic to it, we arrive at something that seems like it can't happen. If our tools can seem to innocently get us into situations where it's obvious that it's absurd, what of all the situations where we can't see the actual situation but can only approach it with our tools, and when we run into absurdities like in this paradox? Will we recognize that it's not the situation, but our tools (or use of them, or something in our field) that are the source of an illusion?

Re: Achilles and the Tortoise

Malachi — Sat, 22 Dec 2012 02:11:10 GMT

So Zeno must have been a real dummy, or else never watched a foot race. He never saw someone winning a race after trailing initially. Same with Bertrand Russell, who wrote that it requires advanced calculus to solve Zeno's paradox. Those two fools never got out of their ivory towers to actually observe events, I guess. One big dunce cap for them, hey?

if thats what youre into. I dont go in for arguments from authority. "how dare you say this dead guy solved a silly problem in a ridiculously complicated way and some guy on the internet did it simpler and faster!"

The paradox is not germane to reality, but it is germane to the only tool we have in our possesion to make sense of reality, aka logical thinking. It is challenging the validity of that tool, showing it produces results that do not fit reality. That's what bothered Zeno and Russell and everyone else.

it shows that youre not thinking logically. Blargg showed you how that is exactly the case.

No. He asserts that although reality shows us Achilles catches the tortoise, and we all see it with our own eyes, logical thinking forces us to conclude it impossible.

The cognitive process by which you conclude it to be impossible is not properly described as logical.

He is challenging the validity of the accepted rules of logic and math. This is a huge problem, for it says we cannot rely on the many formulas we constantly use to predict reality. And yet, they have worked so well. There must be some escape hatch, but what is it?

easy, youre misapplying the formula by focusing on repeated division of units, when you dont need to.

Re: Achilles and the Tortoise

Smiling Dave — Sat, 22 Dec 2012 01:58:59 GMT

We both know that real observers actually do observe events.

So Zeno must have been a real dummy, or else never watched a foot race. He never saw someone winning a race after trailing initially. Same with Bertrand Russell, who wrote that it requires advanced calculus to solve Zeno's paradox. Those two fools never got out of their ivory towers to actually observe events, I guess. One big dunce cap for them, hey?

The paradox is not germane to reality.

The paradox is not germane to reality, but it is germane to the only tool we have in our possesion to make sense of reality, aka logical thinking. It is challenging the validity of that tool, showing it produces results that do not fit reality. That's what bothered Zeno and Russell and everyone else.

...the person who poses the paradox asserts that the oberserver never reaches that instant.

No. He asserts that although reality shows us Achilles catches the tortoise, and we all see it with our own eyes, logical thinking forces us to conclude it impossible. He is challenging the validity of the accepted rules of logic and math. This is a huge problem, for it says we cannot rely on the many formulas we constantly use to predict reality. And yet, they have worked so well. There must be some escape hatch, but what is it?

Re: Achilles and the Tortoise

Malachi — Sat, 22 Dec 2012 01:57:23 GMT

1. Zeno also assumes that time passes at a specific rate. What makes you think otherwise?

the part where the arrow never hits the target because we keep looking at it whwn it isnt hitting the target.

2. What does "passing in an infinite regression" even mean? It has nothing to do with what Zeno or Blagg or I were talking about.

ok, the arrow never hits the target, because it has to be infnitely small distances away from the target first. Thats your infinite regression. Thats wha youve been talking about, its right here in the thread.

To elaborate, you make it sound like the study of infinite series is invalid when it comes to time, because time passes at a specific rate. But that is wildly wrong, like a batter swinging for the umpire's head instead of the ball. Both math and physics, at all levels, admit that the infinite series point of view is valid with respect to time.

study infinite series all you want, just dont get butthurt when your paradox is easily solved by pointing out how confused you are.

So for someone else other than observer the rate is different? Can you explain to whom it is different, and what the rate is for them?

the person discussing the paradox proposes to discuss the passage of time in a manner that exponentially decellerates.

So Zeno thought that time does not pass at the rate of one second per second? What rate do you think he was assuming? Two seconds per second?

apparently he was discussing a universe where time passes at a rate that changes, sometimes decelerating exponentially. Otherwise there is no paradox.

Re: Achilles and the Tortoise

Smiling Dave — Sat, 22 Dec 2012 01:47:39 GMT

...time doesnt pass in an infinite regression, but at a specific rate..

1. Zeno also assumes that time passes at a specific rate. What makes you think otherwise?

2. What does "passing in an infinite regression" even mean? It has nothing to do with what Zeno or Blagg or I were talking about.

To elaborate, you make it sound like the study of infinite series is invalid when it comes to time, because time passes at a specific rate. But that is wildly wrong, like a batter swinging for the umpire's head instead of the ball. Both math and physics, at all levels, admit that the infinite series point of view is valid with respect to time.

...from the perspective of the observer...

So for someone else other than observer the rate is different? Can you explain to whom it is different, and what the rate is for them?

...one second per second.

So Zeno thought that time does not pass at the rate of one second per second? What rate do you think he was assuming? Two seconds per second?

Re: Achilles and the Tortoise

Blargg — Sat, 22 Dec 2012 00:22:21 GMT

My point was that what makes it seem like Achilles never reaches the tortoise is that we choose to observe the race not in normal flowing time, but in a way that causes us to keep slowing down as we get close to the point where he does catch up. Mentally we associate the iterations with a clock of sorts, and see that we'll never see Achilles catch up. It's an explanation for it coming across as a paradox. Clearly, he catches up, so we're not asking whether this proves that he doesn't, just to resolve the seeming inconsistency.

Re: Achilles and the Tortoise

Malachi — Fri, 21 Dec 2012 23:29:54 GMT

What I'm saying is that by shifting the problem to the observer and saying he will never get to the time when Achilles catches up, you haven't solved the paradox, but merely shifted it into a different version. That the observer will never observe the moment of catching up, because before he sees that etc.

the person who poses the paradox asserts that the oberserver never reaches that instant. We both know that real observers actually do observe events. The paradox is not germane to reality.

Re: Achilles and the Tortoise

Malachi — Fri, 21 Dec 2012 23:28:04 GMT

Smiling Dave:

What do you mean? I've shown that it's not that Achilles never catches up to the tortoise, it's that the observer never reaches the time where that occurs. The observer keeps going some fraction of the whole way, which always leaves some other fraction of time remaining until they meet.

So Zeno's Paradox proves we never advance in time.

No, the passage of time disproves Zeno's paradox because time doesnt pass in an infinite regression, but at a specific rate: from the perspective of the observer, one second per second.

Re: Achilles and the Tortoise

Smiling Dave — Fri, 21 Dec 2012 23:21:38 GMT

Can you elaborate on what you mean? I'm detecting some subtext but it'd help if you were explicit.

Zeno stated several versions of his paradox. One was the tortoise and Achilles. He also had an arrow paradox, that an arrow will never hit its target, because before it can get to the target , it has to get halfway there. And before it does that, it has to get a quarter of the way there, etc. There is a third that I don't remember.

What I'm saying is that by shifting the problem to the observer and saying he will never get to the time when Achilles catches up, you haven't solved the paradox, but merely shifted it into a different version. That the observer will never observe the moment of catching up, because before he sees that etc.

Re: Achilles and the Tortoise

Blargg — Fri, 21 Dec 2012 23:03:32 GMT

Can you elaborate on what you mean? I'm detecting some subtext but it'd help if you were explicit.

Re: Achilles and the Tortoise

thetabularasa — Fri, 21 Dec 2012 22:57:12 GMT

Smiling Dave:

What do you mean? I've shown that it's not that Achilles never catches up to the tortoise, it's that the observer never reaches the time where that occurs. The observer keeps going some fraction of the whole way, which always leaves some other fraction of time remaining until they meet.

So Zeno's Paradox proves we never advance in time.

Exactly what I was thinking.