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Old 9th September 2021, 05:29 PM   #121
Robin
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Originally Posted by Myriad View Post
The strange thing is this has turned into the classic Internet argument ".999... can't really be exactly equal to 1" in disguise.
But then again you did make my point for me.

It is a very clear point and not ambiguous at all to say that an iterative process can never in principle get at the sum of an infinite convergent series.
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Old 9th September 2021, 05:30 PM   #122
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Oh and Meadmaker, yes, I did indeed say "series" in the OP instead of "sequence". My bad.
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Old 9th September 2021, 05:34 PM   #123
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Can't go back and fix the original but

Definitions:

P is a continuous path of length l
x is an object that travels through path at a velocity that gets it to the end in a duration of t.

S is a sequence where the first term is 1/2 and the subsequent terms are the sum of the previous term and one half of the previous term.

S={1/2, 1-1/4, 1-1/8, 1-1/16, ...}

Argument:
1. x can travel through p and reach the position 1. (Assumption for reductio ad absurdum)

2. The sequence represents positions that the object must have passed through if it is at 1.

3. S is an infinite sequence and all of the terms are less than one and so the object cannot reach 1.

4. But the definition of motion requires that the object reaches 1.

The definition of motion leads to a contradiction and therefore is incoherent. (contradiction 1,3)

Conclusion: There is no such thing as motion as described in the assumption.

That is the correct one and I think all the premises are true. I haven't detailed the inference but I think it is implied.
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Old 9th September 2021, 05:34 PM   #124
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Originally Posted by Robin View Post
I don't know why you said any of that, don't see how it relates. The point is that the infinite sequence of partial sums doesn't contain the sum of the series by definition. People don't seem to understand that.
We understand it just fine. The problem is that you don't seem to understand that the infinite sequence of partial sums is the sum of the series, by definition. Specifically, by some arbitrary axioms defining an artificial system. Your mistake is assuming some (but not all!) of these axioms properly model movement through spacetime.
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Old 9th September 2021, 05:43 PM   #125
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Originally Posted by Robin View Post
We are talking about a set of numbers that will, by definition not contain the sum that would be calculated. We are talking about a specific case here and not about generalities.

Now here is the scenario again.

An infinite series converges on 3.

The infinite sequence of partial sums for this series can never, by definition, contain 3.

You could, in principle, take the underlying sequence and sum it step by step for an infinite number of times. which would give you the same values as the sequence of partial sums and so also can never by definition, contain the number 3.

Are you with me so far.

The suggestion is that the underlying sequence is summed step by step for an infinite number of times and is able to halt in a finite time.

So the question is, what would be the value of the sum? If it is 3, then where did the 3 come from?
I'm with you, and I see your error. You are conflating doing the sum an infinite number of times with doing the sum once using an infinite number of terms.

The 3 is not contained in the infinitely large set of partial sums. The 3 comes from performing the summation once, using an infinite number of terms.
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Old 9th September 2021, 05:49 PM   #126
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Originally Posted by Meadmaker View Post
I'm with you, and I see your error. You are conflating doing the sum an infinite number of times with doing the sum once using an infinite number of terms.

The 3 is not contained in the infinitely large set of partial sums. The 3 comes from performing the summation once, using an infinite number of terms.
Or as I keep saying, the set is the sum. The set is just a symbol for 3 using sets.

Just like 1+2 is a symbol for 3 using arithmetic.
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Old 9th September 2021, 05:52 PM   #127
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Originally Posted by Robin View Post
Can't go back and fix the original but

Definitions:

P is a continuous path of length l
x is an object that travels through path at a velocity that gets it to the end in a duration of t.

S is a series where the first term is 1/2 and the subsequent terms are the sum of the previous term and one half of the previous term.

S={1/2, 1-1/4, 1-1/8, 1-1/16, ...}

Argument:
1. x can travel through p and reach the position 1. (Assumption for reductio ad absurdum)

2. The series represents positions that the object must have passed through if it is at 1.

3. S is an infinite sequence and all of the terms are less than one and so the object cannot reach 1.

4. But the definition of motion requires that the object reaches 1.

The definition of motion leads to a contradiction and therefore is incoherent. (contradiction 1,3)

Conclusion: There is no such thing as motion as described in the assumption.

That is the correct one and I think all the premises are true. I haven't detailed the inference but I think it is implied.
You are still using sequence and series incorrectly. S, as you have given above, is not a series. The series you are looking for is

1/2+1/4+1/8+1/16+...

Which is equal to 1.

Eta:. There are two related sequences. One is

1/2, 1/4, 1/8...
And the sequence of partial sums,

1/2, 3/4, 7/8...
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Old 9th September 2021, 05:55 PM   #128
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Here is the problem. The problem is that the points are chosen as approaching the finish asymptotically so that the object going through them cannot go through then in finitely many steps. They are simply a marker for each successive half way.

The algebraic sum of that implied sequence is one.

But that does not change the fact that there is no logically possible way, given the definition of move, for the object to cross all those places in a finite number of steps and yet it must pass through them.

The convergent series solution seems to say that they will go past all of those infinitely many points in a finite time. Finishing those steps in a finite time would do no good, because none those positions is at 1.

In fact this creates another problem because there is now no fact of the matter about where the object is and so the path no longer meets the definition of 'continuous' and so the argument succeeds. The convergent series solution only makes it worse.

In the real case, of course, the distinction between the "where" of those points would just stop making sense and so there is no problem. But in the real case there is no continuum.
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Old 9th September 2021, 05:58 PM   #129
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Originally Posted by Robin View Post
Actually there could be no possible fact of the matter. It couldn't be any of those because, well where would they come from? It wouldn't be random or chaotic or anything at all.

There just simply could be no fact of the matter.

However I am pleased that someone could finally admit that the iterative summing of an infinite sequence is an impossible occurrence and not something that could result in the sum of the series.

Quote:
The sum of the series, as can be proven based on the relevant axioms of mathematics, is 3.
Yes, again as I said. But it could never even in principle be got at by an iterative process. Not even if it were possible to run infinitely many iterations in a finite time, which you can't.

That is just the point I have been making. But some have been claiming that you could do that.

Okay, the rest of the understanding you need is that summing a series is not what Achilles, the tortoise, or the arrow is doing, any more than a circle must square its radius and multiply by pi to determine how much paint is needed to fill it in. Zeno chooses to evaluate a common situation (to use the Achilles-tortoise example) in a manner that requires an endless number of steps to converge upon either the location or the time that Achilles actually would pass the tortoise in the scenario given. That's Zeno's choice, but it imposes no obligation on Achilles or the tortoise or nature or us to evaluate it Zeno's way. We can determine when and where Achilles passes the tortoise with simple algebra, or just carry out a comparable race and watch "Achilles" win.

We only need to use the mathematics of sums of infinite convergent series to reconcile Zeno's wacky way of evaluating the events with what algebra or observation or common sense tells us will happen. Nature is under no onus to interpret Zeno's ideas.
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Old 9th September 2021, 06:03 PM   #130
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Originally Posted by Meadmaker View Post
You are still using sequence and series incorrectly. S, as you have given above, is not a series. The series you are looking for is

1/2+1/4+1/8+1/16+...

Which is equal to 1.

Eta:. There are two related sequences. One is

1/2, 1/4, 1/8...
And the sequence of partial sums,

1/2, 3/4, 7/8...
No actually, I need a sequence in this case which is why I said it was a mistake. What you are suggesting wouldn't work as what I need is a set of numbers which are a map to all the half ways in between. You notice that the sequence here is a set of numbers getting closer to one, which is just what I need. A sum is not a sequence of numbers and so obviously would not work in this case.

The sequence does nothing but point to some locations. It only needs to be true that the object has to pass though those locations in order to get to 1.

Note I have {1/2, 1-1/4, 1-1/8, 1-1/16, ...} which is what you have in the second row and is just what I need.

In effect any sequence of numbers that is infinite gets nearer to the end without ever reaching it will do.
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Old 9th September 2021, 06:18 PM   #131
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Originally Posted by Myriad View Post
Okay, the rest of the understanding you need is that summing a series is not what Achilles, the tortoise, or the arrow is doing, any more than a circle must square its radius and multiply by pi to determine how much paint is needed to fill it in. Zeno chooses to evaluate a common situation (to use the Achilles-tortoise example) in a manner that requires an endless number of steps to converge upon either the location or the time that Achilles actually would pass the tortoise in the scenario given. That's Zeno's choice, but it imposes no obligation on Achilles or the tortoise or nature or us to evaluate it Zeno's way. We can determine when and where Achilles passes the tortoise with simple algebra, or just carry out a comparable race and watch "Achilles" win.

We only need to use the mathematics of sums of infinite convergent series to reconcile Zeno's wacky way of evaluating the events with what algebra or observation or common sense tells us will happen. Nature is under no onus to interpret Zeno's ideas.
It is actually a much simpler argument that you seem to appreciate and quite sound.

I am focusing on one of them only, the "approach" dichotomy argument. I have set it out and the only important thing is to ask if the premises are true, which I believe they are

It is true that the object must pass by all of those points listed in the sequence.

It is true that the object can never pass all of those places in a finite number of steps.

As I said, the convergent series only makes things worse because it predicts that the object is at the end of the path when it logically cannot, therefore making the argument succeed.

And if we were to attempt to iteratively sum the underlying sequence of the convergent series then it would exactly agree with wacky old Zeno!
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Old 9th September 2021, 06:23 PM   #132
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Focus on this.

Is it true that the object must pass through all of those positions in order to get to 1.

Yes.

Is it true that the object cannot pass all of those positions in finite number of steps.

Yes.
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Old 9th September 2021, 06:27 PM   #133
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Originally Posted by Robin View Post

Why are convergent series inconsistent if you try to use them as a description of actual motion?
Don't try to use them as something they are not, and your confusion disappears.
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Old 9th September 2021, 06:32 PM   #134
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Originally Posted by Robin View Post
Focus on this.

Is it true that the object must pass through all of those positions in order to get to 1.

Yes.

Is it true that the object cannot pass all of those positions in finite number of steps.

Yes.

"Steps" are features of certain ways of modeling events in nature. Not all models include them. If you model an object's movement with differential equations, that model has no "steps," just relationships.

What is a "step" in nature? How do you count "steps" in nature?
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Old 9th September 2021, 06:33 PM   #135
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Originally Posted by Robin View Post
It is actually a much simpler argument that you seem to appreciate and quite sound.

I am focusing on one of them only, the "approach" dichotomy argument. I have set it out and the only important thing is to ask if the premises are true, which I believe they are

It is true that the object must pass by all of those points listed in the sequence.

It is true that the object can never pass all of those places in a finite number of steps.

As I said, the convergent series only makes things worse because it predicts that the object is at the end of the path when it logically cannot, therefore making the argument succeed.

And if we were to attempt to iteratively sum the underlying sequence of the convergent series then it would exactly agree with wacky old Zeno!
I still think you're confusing things. Sure, the object must pass by all those points, and the state or position of that object cannot be expressed in a finite number of steps, but the object is not traveling to its destination by division. It is subtracting, and whether or not you or Zeno finds the description of the trip sufficiently informative or complete, the object is happy simply to move in the way things actually move, which is not in convergent series; which is why when if I shoot an arrow it flies past the target and gets lost in the long grass.
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Old 9th September 2021, 06:36 PM   #136
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Originally Posted by Steve View Post
Don't try to use them as something they are not, and your confusion disappears.
Exactly! My point precisely.

Convergent series are perfectly consistent things when you regard them as static objects.

Once you attempt to impose any concept of change and time on them, they go all Zeno on you,
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Old 9th September 2021, 06:38 PM   #137
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Originally Posted by Myriad View Post
"Steps" are features of certain ways of modeling events in nature. Not all models include them. If you model an object's movement with differential equations, that model has no "steps," just relationships.

What is a "step" in nature? How do you count "steps" in nature?
Things that the object did.

Is moving though a particular interval of space not something the object did?
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Old 9th September 2021, 06:39 PM   #138
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Originally Posted by Robin View Post
Exactly! My point precisely.

Convergent series are perfectly consistent things when you regard them as static objects.

Once you attempt to impose any concept of change and time on them, they go all Zeno on you,
Your point seems rather pointless then.
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Old 9th September 2021, 06:40 PM   #139
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Originally Posted by bruto View Post
I still think you're confusing things. Sure, the object must pass by all those points, and the state or position of that object cannot be expressed in a finite number of steps, but the object is not traveling to its destination by division. It is subtracting, and whether or not you or Zeno finds the description of the trip sufficiently informative or complete, the object is happy simply to move in the way things actually move, which is not in convergent series; which is why when if I shoot an arrow it flies past the target and gets lost in the long grass.
The argument does not suggest at all that the object is travelling by division. That is nowhere in the premises.
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Old 9th September 2021, 06:41 PM   #140
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Originally Posted by Robin View Post
No actually, I need a sequence in this case which is why I said it was a mistake.
It looks like you corrected it in step 3, but not in the definitions.

So, S is the sequence of partial sums. With that change, your terms are consistent.

I'll get to the rest later
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Old 9th September 2021, 06:43 PM   #141
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Originally Posted by Robin View Post
Focus on this.

Is it true that the object must pass through all of those positions in order to get to 1.
Yes.

Quote:
Is it true that the object cannot pass all of those positions in finite number of steps.
No.

This is trivially easy to prove. You're proving it yourself right now just by breathing. Let alone typing and posting.
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Old 9th September 2021, 06:45 PM   #142
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Originally Posted by Steve View Post
Your point seems rather pointless then.
Imposing time and change concepts onto convergent series is just exactly what the convergent series solution to the Zeno Paradox does. That is exactly why it fails to "solve" the paradox and actually ends up agreeing with Zeno.

Note that in 2,500 years the convergent series solution is the best solution that anyone has come up with for the paradox.
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Old 9th September 2021, 06:46 PM   #143
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Originally Posted by theprestige View Post
This is trivially easy to prove. You're proving it yourself right now just by breathing. Let alone typing and posting.
Again, you appear to have failed to notice about a century of science right there.
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Old 9th September 2021, 06:48 PM   #144
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Originally Posted by theprestige View Post
We understand it just fine. The problem is that you don't seem to understand that the infinite sequence of partial sums is the sum of the series, by definition. Specifically, by some arbitrary axioms defining an artificial system. Your mistake is assuming some (but not all!) of these axioms properly model movement through spacetime.
Actually I am not the one who suggested that convergent series could be applied in this case.

Specifically I said that it couldn't.
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Old 9th September 2021, 06:51 PM   #145
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Originally Posted by theprestige View Post
No.

This is trivially easy to prove. You're proving it yourself right now just by breathing. Let alone typing and posting.
Also you should note that the argument refers to the definitions and premises of the argument.

The statement is true of the definition of travelling a continuous path given in the argument because that is specifically what it is about.
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Old 9th September 2021, 06:55 PM   #146
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Originally Posted by Robin View Post
Things that the object did.

Is moving though a particular interval of space not something the object did?

"Things that the object did" is a feature of narrative descriptions of reality, not of reality itself.

Here, I'll prove it. Unbeknownst to you, the archer and the target we've been talking about all along are on a moving train facing the rear, and the arrow in flight actually did not move at all; the target moved toward it instead.

By relativity (Newtonian or Einsteinean, take your pick) this makes no difference in the physics. So the same event with the same physics can have different narratives, one in which the arrow moved through intervals of space and one in which it did not move at all. Therefore moving-through-space is a feature of the narrative you choose, not "what the arrow did" (or didn't do).
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Old 9th September 2021, 06:56 PM   #147
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Originally Posted by Robin View Post
Imposing time and change concepts onto convergent series is just exactly what the convergent series solution to the Zeno Paradox does. That is exactly why it fails to "solve" the paradox and actually ends up agreeing with Zeno.

Note that in 2,500 years the convergent series solution is the best solution that anyone has come up with for the paradox.
So Zeno was wrong. We can move. The paradox is imaginary. A "solution" involving numbers has no bearing on reality.
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Old 9th September 2021, 07:00 PM   #148
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Originally Posted by Robin View Post
Exactly! My point precisely.

Convergent series are perfectly consistent things when you regard them as static objects.

Once you attempt to impose any concept of change and time on them, they go all Zeno on you,
We already know that our tools have limitations. You've discovered a limitation of convergent series, not a limitation of motion in spacetime. Meanwhile motion in space time observably does not have this limitation.
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Old 9th September 2021, 07:04 PM   #149
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Originally Posted by Robin View Post
But that does not change the fact that there is no logically possible way, given the definition of move, for the object to cross all those places in a finite number of steps and yet it must pass through them.
Well, yes, if you try to force it to be a 'problem' as fundamentally nonsense as 'how to traverse an infinite graph in a finite number of steps', then that's your problem right there.

But ultimately, as I keep telling you, the only thing that's confusing you is your insistence to use the wrong model. Starting with thinking that the arrow moves in steps, or that the number of steps must be finite.

You don't even need Zeno for that. Just think of the it in terms of the real numbers in that interval to see why it's nonsense.

That object goes through all real values between 0 and 1. The whole infinite number of them. Whatever real number x you pick between 0 and 1, yep, that object was in point x too. I can pick weird numbers like the square root of 1/pi, and yep, it hit that point too.

Can you go through all those points in a finite number of 'steps'? No, because it's reals. There's an infinite number of them.

But it doesn't matter, because that object doesn't move in steps. Its movement in space time isn't a listable sequence of discrete {x, x/v} pairs. It's not discrete points on a graph. It's a continuous line. It's EVERY {x, x/v} pair, where x is between 0 and 1. The whole infinite number of them.

Picking your own arbitrary subset from those, which is what your sequence is, doesn't change anything. (As I was saying, choosing a different model doesn't change reality. It's a model, not a voodoo doll.) You just picked an arbitrarily defined subset of those pairs. Sure, it's infinite, but so was the original set you picked from. If the object doesn't have a problem hitting every real number in that interval, it won't have a problem hitting the subset too.
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Old 9th September 2021, 07:12 PM   #150
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More generally speaking, you should consider again the example of a circle's area I mentioned before. Just because a given circle has an area, doesn't mean the circle calculates an area. That's clear enough, right?

Just because an arrow has movement doesn't mean an arrow does movement. The arrow has no volition. Its movement in flight is a property imparted to it by the bow, not some series of operations the arrow is carrying out.

The "steps" you're trying to invoke have no intrinsic meaning. Assuming nature has no intrinsic frame rate, steps are a feature of a narrative description of events, not of the events themselves. I can describe the arrow's flight as one step described by a parameterized equation over a time interval of a second or two, or as many steps (each step necessarily spanning an infinite number of positions and of different intervals of position), or (as Zeno does) an infinite number of them. The arrow does not care about my narratives. (And if nature does have a frame rate, all infinite divisions of time become invalid anyhow.)
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Old 9th September 2021, 07:19 PM   #151
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Originally Posted by Steve View Post
So Zeno was wrong. We can move. The paradox is imaginary. A "solution" involving numbers has no bearing on reality.
So the best solution anyone has found for it in 2,500 years and it is, as you say, using a convergent series for what it is not for. And it failed to solve the paradox.

And therefore Zeno was wrong?

Something is wrong with that.
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Old 9th September 2021, 07:22 PM   #152
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Actually, now that I think more about it, I think I can pinpoint even more exactly the underlying cause of why Zeno got confused.

And it's this: the ancients didn't use time as a coordinate. The notion of describing movement as a set of {x, x/v} pairs, which don't change if you pick some arbitrary subset, wouldn't even be understood by anyone until the 17'th century.

So basically Zeno kinda has to think of it in terms of the time it take to perform all those sets, instead of seeing that he'd just picking from the same {x, t} pairs no matter how he slices it. It doesn't change the t for a given x.

Anyone still getting confused in the modern day, though...
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Old 9th September 2021, 07:23 PM   #153
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Originally Posted by Robin View Post
Here is the problem. The problem is that the points are chosen as approaching the finish asymptotically so that the object going through them cannot go through then in finitely many steps. They are simply a marker for each successive half way.

The algebraic sum of that implied sequence is one.

But that does not change the fact that there is no logically possible way, given the definition of move, for the object to cross all those places in a finite number of steps and yet it must pass through them.
So far, so good, although I notice the undefined terms "move" and "step", but we'll try to use the colloquial definitions and see if it works out.

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The convergent series solution seems to say that they will go past all of those infinitely many points in a finite time. Finishing those steps in a finite time would do no good, because none those positions is at 1.
Tow problems here. You have introduced time. But...again, we might be ok using "common sense". The bigger problem, though, is the assertion that none of those steps is 1. Well, that's only sort of true. None of the "steps" is 1, but if you take an infinite number of steps, you end up at 1.

Keep in mind that we can't talk about the "last step", because there is not last step. There are an infinite number of steps, and the sum of the distances of each step is 1. That gets a bit counterintuitive, and strains our "common sense" definition, because we seem to think of "steps" as finite things, but each step can be infinitessimally small, and require in an infinitessimally small amount of time to take. So, it works.

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In fact this creates another problem because there is now no fact of the matter about where the object is and so the path no longer meets the definition of 'continuous' and so the argument succeeds.
As soon as you start talking about "steps", you have left "continuous" behind. If you go from 0 to 1/2, without passing through 1/3, and all of the infinite number of points between, you aren't continuous. We could talk about traversing the interval [0,1/2], then [1/2,3/4], etc. And we can call each interval a "step", in which case the size of each step has a limit of 0, and the size of the sum of the steps is 1, and the time to traverse each step has a limit of 0, and the time to traverse the total is is equal to 1/r.

Not that I said the size of each step has a limit of 0, but I said that the size of the sum equals 1. It doesn't have a limit of 1. It's equal to 1. The terms of the sequence of partial sums has a limit of 1. That difference between "is equal" and "has a limit" matters.


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The convergent series solution only makes it worse.


In the real case, of course, the distinction between the "where" of those points would just stop making sense and so there is no problem. But in the real case there is no continuum.
I'm not sure why you say "in the real case there is no continuum". We usually think of motion as continuous. Maybe some day we will discover that it's all quantized, but then there won't be an infinite number of steps involved.
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Old 9th September 2021, 07:24 PM   #154
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Originally Posted by Myriad View Post
Just because an arrow has movement doesn't mean an arrow does movement. The arrow has no volition. Its movement in flight is a property imparted to it by the bow, not some series of operations the arrow is carrying out.
Um yes, pretty much what Zeno said. That there is no difference between an arrow in flight and a stationary arrow. Movement is the consequence of comparing two positions.

It took centuries before the rest of humanity woke up to the fact that movement was not some property of the thing moving.

But, yeah, silly old Zeno pointing out that there was no such thing as movement and being right about it.
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Old 9th September 2021, 07:26 PM   #155
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Originally Posted by Robin View Post
It is true that the object must pass by all of those points listed in the sequence.

It is true that the object can never pass all of those places in a finite number of steps.
Well, then, he had better use an infinite number of steps.

What's the problem?


(I suspect the problem is that, like Zeno, you aren't contemplating steps that are infinitessimally small, requiring an infinitessimally small amount of time to traverse.)
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Old 9th September 2021, 07:28 PM   #156
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Originally Posted by Meadmaker View Post
I'm not sure why you say "in the real case there is no continuum". We usually think of motion as continuous. Maybe some day we will discover that it's all quantized, but then there won't be an infinite number of steps involved.
A continuum is some infinitely divisible path. We probably don't have infinitely divisible space. I thought that was pretty much implied by current science. Bad news for physics if we can actually find we can measure the both particle's momentum and precision to an arbitrarily high precision.
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Old 9th September 2021, 07:31 PM   #157
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Originally Posted by Robin View Post
A continuum is some infinitely divisible path. We probably don't have infinitely divisible space. I thought that was pretty much implied by current science. Bad news for physics if we can actually find we can measure the both particle's momentum and precision to an arbitrarily high precision.
If that's true, then we don't have an infinite series, and the problem disappears. We can take a large, but finite, number of steps, and as soon as we get to the smallest possible distance, we don't go halfway to the goal. We go all the way.

So whether or not space is continuous, Zeno is still wrong, but for different reasons.
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Old 9th September 2021, 07:33 PM   #158
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[quote=Meadmaker;13595267]Well, then, he had better use an infinite number of steps.

What's the problem.
Well the problem is that infinity doesn't finish.

If you take an infinite number of steps then finish that infinity, what was your last step? Basically there was none because infinitely many steps don't have a last step.

So you have then contradicted the definition of "continuous" and the argument succeeds.

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(I suspect the problem is that, like Zeno, you aren't contemplating steps that are infinitessimally small, requiring an infinitessimally small amount of time to traverse.)
I seem to think that this is just exactly what he is considering. In an infinitely divisible space infinitessimally small steps behave just like any other and can be treated the same.

Infinitessimally small has no meaning in a continuum.
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Old 9th September 2021, 07:34 PM   #159
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Originally Posted by Robin View Post
A continuum is some infinitely divisible path. We probably don't have infinitely divisible space. I thought that was pretty much implied by current science. Bad news for physics if we can actually find we can measure the both particle's momentum and precision to an arbitrarily high precision.
That was already told to you in the other thread by both me and Ziggurat: that's not what QM says. ALL that the Planck unit was supposed to be is a limit on the accuracy you can measure with, NOT any kind of proof that space is quantized.

In fact, as I told you in the other thread too, there's a fundamental problem with thinking that space is neatly on integer coordinates, because you can't square that even with SR.

But basically that's your even more fundamental problem: you've turned into Pixie Of Key. You just keep repeating the same nonsense again and again -- quite literally, in fact, before Darat's intervention -- and ignoring everything that would contradict your nonsense ideas.
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Old 9th September 2021, 07:34 PM   #160
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Originally Posted by Meadmaker View Post
If that's true, then we don't have an infinite series, and the problem disappears. We can take a large, but finite, number of steps, and as soon as we get to the smallest possible distance, we don't go halfway to the goal. We go all the way.

So whether or not space is continuous, Zeno is still wrong, but for different reasons.
Sure, he is wrong, but in 2,500 years nobody has been able to show why.
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