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Old 11th September 2021, 06:50 AM   #201
HansMustermann
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Originally Posted by Robin View Post
And the sum of a convergent sum really isn't something that can be calculated in time, at least I got one person to agree with that. But ask a maths teacher.
And as was pointed out to you repeatedly, it's irrelevant: all you're doing there is confusing movement and computation. The arrow doesn't compute a series. It just moves.

Originally Posted by Robin View Post
Everybody seems to be trying to say why it is wrong instead of asking if it is right or wrong.
Because it IS wrong.
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Old 11th September 2021, 07:44 AM   #202
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Originally Posted by bruto View Post
I know I'm just being simple and all but I really don't understand why this argument has any merit.
The reason you don't understand why Robin's argument doesn't have any merit is that it doesn't have any merit.

Although Robin's arguments have changed over time (!), all of Robin's arguments beg the question (in the original sense of that phrase: assuming the conclusion) by assuming motion through a continuum is impossible. In Robin's most recent formulation of the argument, that implicit assumption is the only way to get from assertions 1 and 2 to assertion 3.

It should be noted, however, that the modern understanding of motion also begs the question: by assuming motion through a continuum is possible. That illustrates something important about the philosophy of mathematics, which we can more appropriately discuss in this subforum than in the "Science, Mathematics, Medicine, and Technology" subforum where Robin first raised the subject.

In that earlier thread, Ziggurat wrote:
Originally Posted by Ziggurat
Originally Posted by W.D.Clinger View Post
It seems therefore to me that Zeno's paradoxes of motion (the Stadium, the Achilles) were probably designed not to show that motion doesn't exist, but that his contemporaries' understanding of mathematics was inadequate to explain how motion can exist. If that was Zeno's intent, then Zeno succeeded wonderfully well.
Even more broadly, it's not at all clear why mathematics should describe reality to begin with. And yet, it does. But yes, how mathematics and reality connect is, even today, still something of a mystery.
To explain how mathematics and reality connect, let me first explain what mathematics is about.

Modern mathematics is about the invention of axiomatic systems and exploring the logical consequences of those axioms. Mathematicians have their own ideas about which systems are more interesting or beautiful than others, but while the relevance of those systems to reality can indeed be a consideration, it is not the only consideration, and it is often not the primary consideration. G H Hardy, for example, famously sought comfort in his belief that his number theory research would never have any practical applications.

Although G H Hardy turned out to be quite wrong about that, mathematicians really have studied a great many systems that still have no known connection to reality.

Physicists and other scientists are taught and learn the parts of mathematics that physicists and other scientists have found useful when describing reality. Their self-selection of the relevant mathematics explains why mathematics, to them, has been so mysteriously successful in describing reality. It's because, by and large, they only know the mathematics that has proved useful, and ignore the mathematics that has no known connection to reality.

Continuous mathematics has proved useful when describing the reality of motion, whereas (as seen in this thread) ideological denials of continuous mathematics have proved counter-productive when describing that reality. That is why it is useful to assume continuous mathematics when talking about motion, and worse than useless to assume there's something fundamentally wrong with continuous mathematics.
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Old 11th September 2021, 08:11 AM   #203
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There are two different arguments being discussed here. Let me see if I can more clearly state them in a way that distinguishes better between them.

A. The Complete Task Argument

1. A path of movement can be subdivided into an infinite number of non-overlapping partial paths, each of nonzero finite length.

(Comment on A1: since I don't intend to challenge that premise, I will regard the manner of the subdivision as unimportant in this argument.)

2. Movement along the path requires traversing the partial paths one at a time.

3. Traversing a partial path of nonzero finite length requires a nonzero finite span of time.

4. An infinite number of finite quantities (in this case, spans of time) must have an infinite sum.

5. Therefore, the moving object cannot complete the traversing of any path.

B. The Precedence Argument

1. Any path of movement of nonzero finite length can be subdivided into two non-overlapping partial paths also of nonzero finite length.

2. In such a subdivision as (B1), the moving object must begin to traverse one of the subpaths (call it the "near" subpath) a finite span of time before beginning to traverse the other.

(Comment on B2: we will not speak of the summing of such spans of time, in this argument; doing so transforms it back into argument A.)

(Comment on B2: If you assume right-to-left left-to-right moment you can substitute "left" for "near," which might be clearer.)

3. The subdivision of the near subpath can be repeated indefinitely, producing an infinite set of near subpaths, each of finite size, in an ordered hierarchy of decreasing size.

4. Each of our infinite set of near subpaths requires a finite amount of time for the moving object to traverse, which is sufficient to maintain the strict chronological precedence posited in B2. That is, no subpath in our set of near subpaths can be completely traversed until strictly after the next shorter near subpath has been completely traversed.

5. By B3, there is no shortest near subpath.

6. Therefore, no traversal of any subpath in the set of near subpaths can ever begin.

7. Therefore, the object cannot move.

The flaw in A has been much discussed: A4 is simply false.

The flaw in B is also simple, but only because I was clear about the subdivision process and the chronological relationships involved. B6 is false, because as the set of near subpaths was defined, the moving object begins traversing all of them simultaneously when it begins moving. The near subpaths are not non-overlapping.

Surely that can be fixed, though. Even though Zeno didn't. (Note that this is about Zeno's Dichotomy Paradox, not the Arrow Paradox, despite the frequent mention of an arrow in discussion of it.) Let's try again:

B4a. Each of our infinite set of near subpaths requires a finite amount of time for the moving object to traverse, which is sufficient to maintain the strict chronological precedence posited in B2. That is, no subpath that is not in the set of near subpaths can be completely traversed until strictly after its corresponding near subpath (at the same level of the hierarchy of subdivided paths) has been completely traversed.

B5a. By B3, there is no shortest near subpath.

B6a. Therefore, no traversal of any subpath not in the set of near subpaths can ever be completed.

B7a. Therefore, the object cannot move.

Nope, that doesn't work either. Given the modification of B4a, B5a is simply irrelevant, and B6a and B7a don't follow.
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Old 11th September 2021, 10:42 AM   #204
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I wouldn't even say it's a matter of continuous maths being better or worse than discontinuous functions. I'd say it's more of a matter of choosing whatever maths actually works for whatever problem and approximation level you're solving.

Like, if I'm trying to explain the black-body spectrum of a incandescent lightbulb, I might have to go with some rather discontinuous stuff. It's, after all, what caused QM to be invented. Meanwhile, if I just need to figure out the temperature for a given power, I can just apply Stefan-Boltzmann (a continuous function,) and call it a day.

And then for some stuff it wouldn't even matter much. Like, if it turned out that space really is quantized, and everything is neatly aligned on integer multiples of a Planck length and Planck time unit (and let's ignore for a moment how that doesn't play nice even with SR, much less GR), well, so what? As long as you're not solving a problem where rounding to the nearest Planck unit actually causes observable effects, t=x/v works the same in integers.

What we have here isn't IMHO a matter of whether continuous or discontinuous maths are inherently better for everything. It's more like a matter of fitting the DSM-IV definition of a delusion. If X is consistently observed to happen in a certain way, and my hypothesis says that should be impossible, for anyone who's even heard of the scientific method it means my hypothesis is wrong. That's how the scientific method works. It might mean changing the maths. It might even mean going from continuous to quantized, if that's what solves the problem. (After all, that's literally what Max Planck did.) Or viceversa. Or changing something else. Until I find something that actually predicts the observed results. Going some version of "if observed reality disagrees with my ideas, then my my ideas are right, and observed reality is wrong", a la Zeno, is pretty much the definition of being delusional.
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Old 11th September 2021, 02:20 PM   #205
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`Of course, we don't know a whole lot about Zeno and how he approached things, and he could be an earlier example of the paradox producer who knows perfectly well that what he has proposed cannot work despite its apparent rightness, which is another way of saying "something is wrong about which we don't know enough yet." Hume did this when he proposed his radical empiricism, which separated cause and effect, stating, as I recall, something to the effect that despite what he so meticulously proved in his work, he still lived according to common sense. He openly invited followers to find a way to get around this (which is what Kant set about). And Schrodinger's cat would seem a similar situation.

So perhaps, while still observing that Zeno's paradox does not, especially in this day and age, tell us much we need to know, it was a pretty slick bit of paradox-making.
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Old 11th September 2021, 02:42 PM   #206
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Originally Posted by bruto View Post
`Of course, we don't know a whole lot about Zeno and how he approached things, and he could be an earlier example of the paradox producer who knows perfectly well that what he has proposed cannot work despite its apparent rightness, which is another way of saying "something is wrong about which we don't know enough yet." Hume did this when he proposed his radical empiricism, which separated cause and effect, stating, as I recall, something to the effect that despite what he so meticulously proved in his work, he still lived according to common sense. He openly invited followers to find a way to get around this (which is what Kant set about). And Schrodinger's cat would seem a similar situation.

So perhaps, while still observing that Zeno's paradox does not, especially in this day and age, tell us much we need to know, it was a pretty slick bit of paradox-making.
Yeah. For my money he was just having fun in a thought-provoking way.
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Old 11th September 2021, 02:44 PM   #207
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The problem is that we do know quite a bit about Zeno from secondary sources, including the counters they had to come to those ideas.

E.g., Aristotle feels a need to refute Zeno, in order to argue that movement does exist. That makes no sense at all if actually Zeno too was just proving that movement does exist. If some guy from Elea is already making your point, you cite him as a source, not refute him.

E.g., the atomism of Democritus seems to be thoroughly designed to counter specific ideas of the Eleatic school. And I don't mean just the fact that it introduces "indivisibles" (which is what "atom" meant) to stop paradoxes from infinitely dividing stuff, but also postulates the existence of void between the atoms to counter another Eleatic argument about the impossibility of motion. The Eleatics basically argued that you can't move if anything is in the way (so yes, basically I can't go to bed now because air is in the way), you need a void for that, but (based on a dumb word play) a void can't exist.

And that's just the hostile sources. There's also the fact that the Eleatic school was practically revering Zeno. You don't do that for someone who just disproved your whole philosophy, while still holding onto that philosophy.

Furthermore, it's the way these ideas work together. Like, sure, you could imagine that the paradoxes alone might be some ad absurdum proof that movement must exist after all, but then the other arguments aren't as easy to see as an ad absurdum. Like the one I just gave above that needed a void.


But basically, look, yes, ancient sources can be unreliable. It's possible that everyone misunderstood Zeno. But if all sources tell you X, while you have exactly zero sources that say not-X, then it's still not enough reason to assume that non-X is the real thing. Even if you dismiss the former as unreliable, you still end up with the latter being not supported by anything.
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Old 12th September 2021, 04:16 PM   #208
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Originally Posted by Robin View Post
Everybody seems to be trying to say why it is wrong instead of asking if it is right or wrong.
Originally Posted by HansMustermann View Post
Because it IS wrong.
I hate how many multi-page "discussions" there are because someone can't get past this.
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Old 12th September 2021, 04:21 PM   #209
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Hi,

I just want to point out that I have put on ignore the people who were telling me that the sum of an infinite convergent series is something that could be calculated in real time, without even attempting to give any reason why you think so and evading any on-topic question.

The only person to agree with me only did so because he had come in late and had absolutely no idea what was being discussed and didn't realise he was agreeing with me.

I think it is unreasonable that in my own thread people will disagree with me on things whether they are true or false.

Let me stress that this is not out of animosity but because it is simply impossible to discuss things in circumstances like that. Many of you say very sensible things in other contexts.

Again the only people who I put on the ignore list were the people who repeatedly, over a number of pages, that the sum of an infinite convergent series is something that could be calculated in real time.

So from this point on I will only be discussing things with people who realise that the sum of an infinite convergent series can only be arrived at by calculus and cannot even in principle be arrived at as a calculation in real time, which a straightforwardly true fact about high school maths.

Consequently though I am not seeing very many posts.
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Old 12th September 2021, 04:37 PM   #210
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Originally Posted by W.D.Clinger View Post
Although Robin's arguments have changed over time (!), all of Robin's arguments beg the question Begging_the_question in the original sense of that phrase: assuming the conclusion by assuming motion through a continuum is impossible.
Which shows that you don't understand the argument, or indeed the original.

You appear to be unaware of the form of argument called Reductio ad absurdum of which Zeno's argument is an early example.

For the purposes of this argument, for example, Zeno assumes that motion through a continuous path is possible and then show that it leads to a contradiction, which it does.

It is common that people misunderstand this and assume that Zeno is actually saying that things can't move when he says that.

I have not made any sort of assumption about the impossibility of motion in any of the arguments.

Here is what you don't understand. An infinite sequence of numbers does not have any final term.

So there is no final finite part of the path. So the object cannot pass through any path that, by definition.

That is the entire point of any of the Zeno dichotomy arguments and I am sorry that I didn't realise that I had to explain that, I was under the impression that this was quite a well known argument. However I realise form your previous post that many are not aware of what the argument is.

You will appreciate that since I have gone through pages of everyone else in the thread insisting that the sum of an infinite convergent series is something that could be calculated in real time I am a bit weary of anyone who does not pay attention to what the argument actually says.

Now if you could point out something that is actually wrong with the argument, that would be nice.
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Old 12th September 2021, 04:49 PM   #211
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Originally Posted by Robin View Post
I think it is unreasonable that in my own thread people will disagree with me on things whether they are true or false.
Well, get over it because that's not how reality works.

You're wrong. You don't get to be "not wrong" just because it's "your" thread.
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Old 12th September 2021, 04:52 PM   #212
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Originally Posted by W.D.Clinger View Post
Your statement 3 does not actually follow from your statements 1 and 2.
Again I will draw your attention to the fact that there is, by definition, no end to the infinite regress.

I don't have time to actually put the inference into symbolic logic as to make it explicit, one normally does not have to.

Let me ask you a question. Can the object travel through a part of the path that, by definition, does not exist?
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Old 12th September 2021, 05:01 PM   #213
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In case anyone else makes the same mistake, consider that an argument that includes a premise about an object travelling half way in order to show a contradictory consequence must include the premise that the object can travel through continuous space.

If such an argument assumed that an object couldn't travel through continuous space then it wouldn't work.

Again, I didn't think I needed to explain that.
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Old 12th September 2021, 05:04 PM   #214
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Your statement is both nonsensical and has nothing to do with Zeno's paradox.

Stop trying to bring the discussion to a halt going "I demand someone prove it isn't technically true in a contextless bubble."
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Old 12th September 2021, 05:05 PM   #215
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On the other hand I do assume that an object cannot travel through a part of a continuous path that does not exist.

I think this is a reaonable assumption.
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Old 12th September 2021, 05:18 PM   #216
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Originally Posted by Robin View Post
Hi,

I just want to point out that I have put on ignore the people who were telling me that the sum of an infinite convergent series is something that could be calculated in real time, without even attempting to give any reason why you think so and evading any on-topic question.

The only person to agree with me only did so because he had come in late and had absolutely no idea what was being discussed and didn't realise he was agreeing with me.

I think it is unreasonable that in my own thread people will disagree with me on things whether they are true or false.

Let me stress that this is not out of animosity but because it is simply impossible to discuss things in circumstances like that. Many of you say very sensible things in other contexts.

Again the only people who I put on the ignore list were the people who repeatedly, over a number of pages, that the sum of an infinite convergent series is something that could be calculated in real time.

So from this point on I will only be discussing things with people who realise that the sum of an infinite convergent series can only be arrived at by calculus and cannot even in principle be arrived at as a calculation in real time, which a straightforwardly true fact about high school maths.

Consequently though I am not seeing very many posts.

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

No calculus required. Also note the finite number of steps involved.

It appears Robin has me on ignore, but this makes sense to everyone else, right?
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Old 12th September 2021, 05:21 PM   #217
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Honestly I don't even get to the math part, her argument falls apart way before you starting counting up things.

Her premise is just a self-contradictory "If a set contains everything but doesn't have something it in..." bad wording.

If it doesn't have everything, then it's not everything. If it's everything, you can't then point at something that's not in it and try to factor it in.
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Old 12th September 2021, 05:33 PM   #218
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Originally Posted by JoeMorgue View Post
Honestly I don't even get to the math part, her argument falls apart way before you starting counting up things.

Her premise is just a self-contradictory "If a set contains everything but doesn't have something it in..." bad wording.

If it doesn't have everything, then it's not everything. If it's everything, you can't then point at something that's not in it and try to factor it in.

I was only addressing the claims that the sum of an infinite convergent series can only be arrived at using calculus, and in an infinite number of steps. (Added highlighting to the quote in my previous post to clarify that.) I addressed the dichotomy paradox itself in somewhat rigorous detail a few posts back.
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Old 12th September 2021, 05:35 PM   #219
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I'm just trying to sidestep her constant demands about "If such and such is blahblah than is this..."

If she wants to waste her time trying to frost Gabriel's Cake let her, it just has nothing to do with her actual claim.
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Old 12th September 2021, 05:54 PM   #220
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Originally Posted by Robin View Post
I think it is unreasonable that in my own thread people will disagree with me on things whether they are true or false.
Starting a thread does not convey ownership of the thread, nor does it make you the final arbiter of which statements are true or false.

Originally Posted by Robin View Post
Originally Posted by W.D.Clinger View Post
Although Robin's arguments have changed over time (!), all of Robin's arguments beg the question Begging_the_question in the original sense of that phrase: assuming the conclusion by assuming motion through a continuum is impossible.
Which shows that you don't understand the argument, or indeed the original.

You appear to be unaware of the form of argument called Reductio ad absurdum of which Zeno's argument is an early example.
It would not be hard to locate indications that I am familiar with that form of argument.

Originally Posted by Robin View Post
Here is what you don't understand. An infinite sequence of numbers does not have any final term.
Believe it or not, I understand that an infinite sequence whose order type is that of the natural numbers does not end with a final element.

I also understand that an infinite sequence whose order type is that of the negative integers does end with a final element. I mention this second example to emphasize the importance of definitions: Although we often assume the phrase "infinite sequence" refers to sequences whose order type is that of the natural numbers, we can't reasonably make such assumptions when we are addressing someone whose arguments appear to be founded upon idiosyncratic definitions.

Originally Posted by Robin View Post
So there is no final finite part of the path.
If I knew what you meant by the word "path", I might agree with that. As it is, however, I have no opinion on that.

Originally Posted by Robin View Post
So the object cannot pass through any path that, by definition.
Grammar might improve the intelligibility of that utterance.

Originally Posted by Robin View Post
That is the entire point of any of the Zeno dichotomy arguments
I don't think Zeno would have agreed that his entire point could be summed up by an utterance that, being grammatically incorrect, doesn't actually say anything.

Originally Posted by Robin View Post
and I am sorry that I didn't realise that I had to explain that, I was under the impression that this was quite a well known argument. However I realise form your previous post that many are not aware of what the argument is.

You will appreciate that since I have gone through pages of everyone else in the thread insisting that the sum of an infinite convergent series is something that could be calculated in real time I am a bit weary of anyone who does not pay attention to what the argument actually says.

Now if you could point out something that is actually wrong with the argument, that would be nice.
I did that yesterday, and expanded upon that remark a few hours later. This morning you disagreed with my remark by appealing to an infinite regress that does not actually repair the flaw in your argument. You also noted that you didn't "have time to actually put the inference into symbolic logic as to make it explicit".

You concluded with this question:
Originally Posted by Robin View Post
Can the object travel through a part of the path that, by definition, does not exist?
Your question appears to assume an idiosyncratic definition of "path" that you have consistently failed to state with the precision necessary to support a rigorous discussion of whether such paths exist. In particular, you never replied to the question I asked of you concerning whether you would like to repair your definition of "traversable" so as to rule out examples such as the one I gave in which a totally ordered set is traversable according to the definition you had given, even though almost all pairs of set elements bound an interval that contains an uncountable infinity of elements.

Your question also appears to assume an idiosyncratic notion of "travel" in which Atalanta's travel from one end of a path to another does not count as "travel", even though similar travels along similar paths are a familiar sight.

Please note that I am not saying your idiosyncratic notions of "travel" or "path" differ in any meaningful way from Zeno's idiosyncratic notions. Indeed, I suspect your idiosyncratic notions might align fairly well with Zeno's if you ever get around to defining your idiosyncratic notions carefully enough for us to consider that question. If and when that turns out to be the case, we might then be able to conclude that your argument fails for the same reasons Zeno's arguments failedórecognizing, of course, that Zeno might well have intended for his arguments to fail, inasmuch as he may have been posing his arguments as a way to expose the flawed thinking of his contemporaries.

It seems to me that your chances of participating in a meaningful conversation about Zeno would improve if
  • You understand that threads are not owned by the person who starts the thread.
  • You understand that refusing to read anything written by people who disagree with you is not the best way to gather constructive feedback that might improve the precision and rigor of your argument.
  • You understand that definitions you refuse to state precisely may not be as commonly accepted as you believe.
  • You understand that your diagnoses of why people are disagreeing with you may not always be accurate.
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Old 12th September 2021, 06:12 PM   #221
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Robin has reversed Zeno's dichotomy paradox to subdivide the end of the span of motion instead of the start. But that has the same flaw I already pointed out. The dichotomized sub-paths at the end overlap. The final 1/4 path is part of the final 1/2 path, the final 1/8 path is part of the final 1/2 and the final 1/4, and so forth. The way they overlap means that all of them finish at the same position and time.

What we can say about that infinite set of shorter and shorter final sub-paths is that the moving object begins traversing them at different times. The final 1/4 begins before the final 1/8, and so forth. There are an infinite number of such beginnings, but that doesn't present any problem, because while traversing a path requires time and a distance moved, beginning to do so is instantaneous. Beginnings take zero time and cover zero distance, so it doesn't matter that the number of beginnings of final sub-paths that occur is infinite. It's not even an infinite series of finite values; it's an infinite series of zeros.

The paradox is phrased as if to imply that those final increasingly tiny sub-paths must be traversed one at a time, but at the end of the path they're all completed simultaneously. The dichotomy paradox does not point out any actual contradiction.
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Old 12th September 2021, 06:51 PM   #222
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Originally Posted by Myriad View Post
s = 1/2 + 1/4 + 1/8 + 1/16 + ...
2s = 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...
2s = 1 + s
s = 1

No calculus required. Also note the finite number of steps involved.

It appears Robin has me on ignore, but this makes sense to everyone else, right?
Yes.
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Old 12th September 2021, 07:02 PM   #223
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Originally Posted by Myriad View Post
s = 1/2 + 1/4 + 1/8 + 1/16 + ...
2s = 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...
2s = 1 + s
s = 1

No calculus required. Also note the finite number of steps involved.

It appears Robin has me on ignore, but this makes sense to everyone else, right?
I was in fact about to post the same proof, so yeah, of course it makes sense.
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Old 12th September 2021, 07:14 PM   #224
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Originally Posted by W.D.Clinger View Post
Believe it or not, I understand that an infinite sequence whose order type is that of the natural numbers does not end with a final element.

I also understand that an infinite sequence whose order type is that of the negative integers does end with a final element. I mention this second example to emphasize the importance of definitions: Although we often assume the phrase "infinite sequence" refers to sequences whose order type is that of the natural numbers, we can't reasonably make such assumptions when we are addressing someone whose arguments appear to be founded upon idiosyncratic definitions.
The actual irony there is that Robin's latest redefinition of the problem, in which the FIRST half of the path is being divided instead of Zeno's dividing the second half of the path, is the negative integer order type (you can't list it from the beginning of the motion, you can only list it from the end.) Which sabotaged that argument in a major way.

In fact, taking it to mean something or another about the object not being able to get to be in some point because the arbitrary model (that series) doesn't include it, the point that the latest redefinition of the series never actually contained is the START point. Resulting in the much dafter 'paradox' that the object can't even NOT move, since it can't be in the starting point. You know, because that series doesn't contain that point.

It's why I said it's no longer Zeno in response to that redefinition.
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Old 13th September 2021, 03:55 AM   #225
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Originally Posted by Robin View Post
I just want to point out that I have put on ignore the people who were telling me that the sum of an infinite convergent series is something that could be calculated in real time, without even attempting to give any reason why you think so and evading any on-topic question.
Let me explain to you why that is an utterly stupid position to take, then you can put me on ignore or actually try to lean something. Your choice.

First: It is possible to calculate the sum of an infinite convergent series in finite time. Myriad has provided a perfect example. I suggest you take him off ignore if you have him on it, because he's one of the posters here that one can most reliably learn useful things from.

Second: You may note that this method of calculating the sum of an infinite convergent series is by some method other than adding each term in succession and summing every one of an infinite number of terms. I think this is what you mistakenly describe as "calculus".

Third: An object in motion is not performing mathematical operations to your exact personal specifications; there is no rational reason to suppose it is doing so. It is rather engaging in a physical process, the result of which can be modelled as the sum of an infinite series.

So, if you can explain why an object in motion is required to determine the sum of an infinite series by an impossible method you have arbitrarily chosen, rather than by a much simpler method that works in finite time, then you have a point. But since, in fact, objects in motion do not do that, then clearly you don't.

One final thing: "I've put everybody who says this on ignore" is the cry of everyone who seeks not to have their dogma challenged, rather than those who genuinely seek to learn whether or not it's valid.

Dave
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Old 13th September 2021, 08:58 AM   #226
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Originally Posted by Kid Eager View Post
Remember, you have to say the name three times for a summoning to workÖ
And so, here is the third post in which I call upon the name of Doron Shadmi.
I'd love to see his colorful illustrations that Zenocate Mathematics.
Doron would be the one to agree with Robin that you can't complete an infinite number if iterations without treating Infinity as a completed whole. Then he would proceed to reject any Mathematics that does any such thing.
To wit the thread would ground to a halt because nobody wants to go there.

As for good old Zeno, what I learned in college was that the point of his paradoxes was to debunk the existence of individual objects as metaphysical entities. According to his teacher, Parmenides, Reality is a single, seamless Unity. Movement of objects is a mere appearance. If you posit inherently separate objects and look at that too closely, you come to absurd contradictions.
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Old 13th September 2021, 09:00 AM   #227
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2,500 years and we're still arguing about whether one Greek or another used a better metaphor when talking about reality.
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Old 13th September 2021, 09:04 AM   #228
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Originally Posted by JoeMorgue View Post
2,500 years and we're still arguing about whether one Greek or another used a better metaphor when talking about reality.
"The safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato."

Alfred North Whitehead
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Old 13th September 2021, 09:07 AM   #229
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It's like people who go on and on about being fans of a genre of music but when pressed they are fans of one group, the solo members of that group splitting off and forming new projects, groups that do covers of that original group, and groups who are homages to that group.
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Old 13th September 2021, 10:05 AM   #230
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Well, that and it's IMHO also a side effect of philosophy being a domain that's been for a while pretty much defined as whatever is left after the proper sciences took their share. I.e., it's whatever you can't decide whether it's right or wrong by just making a numerical prediction, running the damn experiment, and seeing if it produces the numbers you expected. It really can't do much more than cite some famous dead guy for why this or that has to be wrong. So unsurprisingly most debates tend to degenerate into "but <insert famous dead guy> said <insert really profound sounding thing>" (with the emphasis on "sounding") vs "yeah, but <insert other famous dead guy> said the opposite".

ROUND 1
FIGHT

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Old 13th September 2021, 12:35 PM   #231
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I think this thread illustrates very well why some people who a good at math nevertheless hate courses with proofs.

A lot of the rancor in this thread has been generated by people mixing up everyday uses of terms with mathematical definitions of terms. This includes Robin and some, but by no means all, of her detractors.

(I've probably been a little bit loose myself, but I've tried to be as accurate as I can manage.)
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Old 13th September 2021, 05:34 PM   #232
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Never got why Zeno's Paradox was supposed to be a paradox. Nothing stopped moving at the halfway point. The Observer pauses to take an ever decreasing measurement, till it was effectively measuring at the destination. For it to be some kind of mind bender, you'd have to assume the arrow could only travel half the destination, but no farther. Which is just grade-school level wrestling with the concept of infinity. Halving into infinity is not a paradox.
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Old 13th September 2021, 06:19 PM   #233
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Originally Posted by Thermal View Post
Never got why Zeno's Paradox was supposed to be a paradox. Nothing stopped moving at the halfway point. The Observer pauses to take an ever decreasing measurement, till it was effectively measuring at the destination. For it to be some kind of mind bender, you'd have to assume the arrow could only travel half the destination, but no farther. Which is just grade-school level wrestling with the concept of infinity. Halving into infinity is not a paradox.
But if you assume that the travel of the object follows the same rules as the calculation, it's a problem. It's not just "the halfway point," but an infinite series of halfway points. If you calculate the position of the object's travel by halves, of course you can keep cutting it in half infinitely and you'll never get to the end. Whatever distance is left, we can cut it in half again. We know of course that this is not the way the object moves, but this is not exactly a common sense issue.
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Old 13th September 2021, 07:16 PM   #234
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Originally Posted by bruto View Post
But if you assume that the travel of the object follows the same rules as the calculation, it's a problem. It's not just "the halfway point," but an infinite series of halfway points. If you calculate the position of the object's travel by halves, of course you can keep cutting it in half infinitely and you'll never get to the end. Whatever distance is left, we can cut it in half again. We know of course that this is not the way the object moves, but this is not exactly a common sense issue.
Yes, if you can only travel halfway to the destination at a clip, you will never quite reach it in practice or theory. That's just kind of how "stopping halfway" works; the destination by definition cannot be reached. The extrapolations from that truism are non-sequitors, though. The arrow is not compelled to stop at any halfway point or abide by that artificial premise. Similarly, the beginning movement is not restricted by an observer's attempts to quantify how far has been travelled. Motion is independent of an observer's measurements of that motion.

Dividing by half forever is the complement to adding one forever, hence the concept of being infinite. I don't get where the paradox lies, except in wordplay, presumably complete with Greek jazz hands.
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Old 13th September 2021, 08:12 PM   #235
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Originally Posted by Thermal View Post
Yes, if you can only travel halfway to the destination at a clip, you will never quite reach it in practice or theory. That's just kind of how "stopping halfway" works; the destination by definition cannot be reached. The extrapolations from that truism are non-sequitors, though. The arrow is not compelled to stop at any halfway point or abide by that artificial premise. Similarly, the beginning movement is not restricted by an observer's attempts to quantify how far has been travelled. Motion is independent of an observer's measurements of that motion.

Dividing by half forever is the complement to adding one forever, hence the concept of being infinite. I don't get where the paradox lies, except in wordplay, presumably complete with Greek jazz hands.
Well yes, there is no paradox in reality, but there is if you presume that the way one is measuring is somehow integral to the thing measured.
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Old 13th September 2021, 09:07 PM   #236
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Originally Posted by bruto View Post
Well yes, there is no paradox in reality, but there is if you presume that the way one is measuring is somehow integral to the thing measured.
Which is why I keep repeating that it's a model, not a voodoo doll
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Old 13th September 2021, 09:46 PM   #237
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Talk about overthinking something! I can see how in Ancient Greece, Zenoís paradoxes would seem profound. And sure, if Iím high as hell, Zeno is some deep ****. But any clear thinking modern person should be able to see right through that nonsense.

There lies my problem with a lot of philosophy. Sounds deep and maybe was deep at one time but now we know better. Itís entertaining to toss around but it isnít satisfying. The arrow reaches its destination. I get to my front door. Planes reach their destinations. Even if itís all an illusion, we canít escape the illusion so itís reality for all intents and purposes.
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Old 13th September 2021, 11:13 PM   #238
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As I've said before (though it may have been in the other thread), they already knew better even in Zeno's time. The Eleatic school (of which Zeno was one of the revered founders) were the ancient equivalent of flat-earthers. They were the guys arguing nonsense just to say it's the other way around than the existing knowledge was.
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Old 14th September 2021, 05:10 AM   #239
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Originally Posted by HansMustermann View Post
As I've said before (though it may have been in the other thread), they already knew better even in Zeno's time. The Eleatic school (of which Zeno was one of the revered founders) were the ancient equivalent of flat-earthers. They were the guys arguing nonsense just to say it's the other way around than the existing knowledge was.
I honestly do sometimes wish it were possible to go back to Greek, Roman, the Renaissance, the Enlightenment, etc and get a "feel" for how the "philosphering philosophers most philosophising" were actually seen by the general public.

Or as I once heard it put "People simply can't stop reminding me that the Greeks gave us Socrates yet never seem to remember that they also killed him."
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Old 14th September 2021, 05:15 AM   #240
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I do like that the idea that most ancient philosophers were just pre-internet trolls running on the same "If everyone says one thing but I say another I must be super-smart because at least I'm thinking for myself" delusion is gaining popularity.
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