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Old 9th September 2021, 06:07 AM   #1
Robin
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Zeno was right! We can't move. Stay glued to your seats

Actually I just realised that Zeno was right in at least two of those dichotomy arguments. The values in the infinite series represents positions that must have been passed through if the object had travelled further. So it doesn't depend on any naive pre-scientific view of space, it is true for any assumption that an object travels some distance at a certain velocity on some continuous path.

The convergent series doesn't harm the argument, it makes no difference if infinitely many things can be done in a finite time time because all of the positions mapped on those convergent series are at a finite distance from the finish.

So the concept of motion itself leads to a contradiction.

It has been built into our mathematics and science since at least Newton.

It makes no sense to go on pretending he was wrong because, well we have known for a long time that 'motion' as we experience it is an approximation of something that is somewhat different. So it does not matter that he was right.

And you don't have to tart up those arguments, they are pretty much just fine as they are. It is the objections that are wrong.

Here is the argument, pretty much as Zeno left it:

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 further than them.

3. S is an infinite series and all of the terms are greater than zero 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.

There. Don't bother with "how did it get half way if it didn't move?" That makes no difference because the assumption that it moves is made for the purposes of contradiction it only matters if that this assumption leads to a contradiction, which it does.

convergent series is not a problem because the argument does not say that infinitely many things cannot happen in a finite time, because all of the terms are non zero then even after infinitely many iterations the object is still not at zero.

But the sum of the series says the object gets to 1 and therefore causes another contradiction with itself.

Happy hunting.
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Old 9th September 2021, 06:12 AM   #2
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*Throws a rock at your head, safe in the knowledge that you'll never be able to complete the distance between me and you to slap me*
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Old 9th September 2021, 06:24 AM   #3
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Maybe I'm missing something profound here, but doesn't this all come down to a confusion between multiplication and addition? The fact that, when I walk from point a to point b, you can measure by fractional distance, does not mean that I walk that way.
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Old 9th September 2021, 06:29 AM   #4
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Quantum Mechanics solved all of this.

It's also assigning arbitrary "start and stop points"

Okay so you walk across the room. The "Zeno Paradox" is getting confused by infinite recursions, the idea that you have to walk half the distance, then half that distance, then half that distance, forever. There's an infinite number of half distances, therefore you are completing an infinite number of tasks in a finite amount of time, ergo paradox.

But there isn't an infinite number of "halfways." You hit the plank distance and with quantum mechanics on a very, very small level you can in two places at once (vast simplification.)
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Old 9th September 2021, 06:58 AM   #5
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Not sure why it needed a second thread, but here goes again: the only constant across all Zeno's paradoxes is just that the dude didn't even start to understand infinite series. Or really understand infinity, period.

In the dichotomy "paradox" about movement, his argument was literally that if you add an infinite number of finite terms (those times for the half distance, then quarter, etc) you must get infinity as a result. Those times must add up to an infinite time.

In another dichotomy "paradox", he talks about if an object is infinity divisible (as the opposing school of philosophy maintained about matter; remember this was not only before QM, but even before the atomism of Democritus), you can divide it until you have basically a small heap of zero-sized points. Because (and this is his actual argument), if you have something that's not zero-sized, you're not done dividing it yet. So now when you're done dividing it into a bunch of zero-sized things, you add them up, and get that the original thing also had zero volume. You know, because adding any number of zeroes, can't give you anything else than zero.

I find that this latter one is actually the best illustration of basically what was wrong with Zeno. The very notion that at some point you could be done dividing infinitely is all you need to know, if you ever planned to take his maths seriously. The fact that his whole argument boils down to (1/∞)*∞=0 is also obviously not sinking in for him.

And QM may or may not present a solution. The idea isn't even new or QM related. The atomism of Democritus was also essentially trying to be a hard counter to the above idea of keeping dividing infinitely until you get a paradox, and that's more than 2000 years before QM.

But ultimately it's not really needed anyway. If you understand convergent series, or that ∞/∞ is a mathematical nonsense, all of Zeno's paradoxes just disappear anyway.
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Old 9th September 2021, 07:32 AM   #6
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"Infinity is NOT a number" was hammered in by my math teacher pretty well. I have never seen him proven wrong.
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Old 9th September 2021, 07:48 AM   #7
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I don't think step 3 is supported.
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Old 9th September 2021, 08:07 AM   #8
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OK for anyone not convinced tell me this.

Why are convergent series inconsistent if you try to use them as a description of actual motion?

The algebraic sum is one thing.

But if you were to regard them as as a description of an infinite sequence of values actually being summed in a finite amount of time then you get an answer that differs from the sum by a small undefined but finite amount.

Because, by their very definition the terms never reach the sum.

Hasn't that ever bothered anyone? Seriously tell me that it hasn't bothered you. You can't fudge and say, oh, they reach infinity and become one. It contradicts the definition of the sum. By the definition no term ever reaches zero and so if you could actually sum them in a finite amount of time then they would differ from the algebraic sum.

A convergent series becomes a perfectly consistent thing as soon as you drop the idea of something happening. Nice and neat, no small undefined value.

You can't deny that, when regarded as a process of actual summing they give a different value. Why? What other reason could it be than that the whole concept of things happening in time is an inconsistent idea.

Why do we persist in pretending that convergent series refute Zeno when they very obviously illustrate neatly what he is saying?

I mean think about it. 2,500 years of people trying to knock it down and the best we can do is convergent series that actually illustrate what Zeno said.
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Old 9th September 2021, 08:11 AM   #9
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Originally Posted by Robin View Post
But if you were to regard them as as a description of an infinite set values actually being summed in a finite amount of time then you get an answer that differs from the sum by a small undefined but finite amount.
No you don't. If you sum the infinite series, the "small undefined but finite amount" you refer to is in fact zero.

Dave
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Old 9th September 2021, 08:14 AM   #10
Robin
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Indeed, you don't really need Zeno any more, you can just say look at convergent series, they illustrate what he said perfectly.
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Old 9th September 2021, 08:18 AM   #11
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Originally Posted by Meadmaker View Post
I don't think step 3 is supported.
I'd go beyond that and say that step 3 is mathematically equivalent to stating, as a premise, that Zeno's Paradox is valid. As such, it means that the argument is circular.

Dave
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Old 9th September 2021, 08:20 AM   #12
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Okay fine then if you've proved motion is impossible then how is anything actually ******* moving?

Why does reality have to prove itself to the ponderous philosophical argument and not the other way around?

I'd say the burden of proof is on Zeno, not us.
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Old 9th September 2021, 08:20 AM   #13
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Originally Posted by Robin View Post
OK for anyone not convinced tell me this.

Why are convergent series inconsistent if you try to use them as a description of actual motion?
Why aren't my gloves working as shoes? There must be some bigger mystery, I tell you

Originally Posted by Robin View Post
The algebraic sum is one thing.

But if you were to regard them as as a description of an infinite set values actually being summed in a finite amount of time then you get an answer that differs from the sum by a small undefined but finite amount.

Because, by their very definition the terms never reach the sum.
Because it's not a description of the whole phenomenon. It's just a series you've arbitrarily defined on a subset of the space-time events (i.e., 4d poits) that describe that movement. If your model doesn't include the final point, that's just an issue with your model, that's all.

To see why that's not saying anything, consider me running 2 miles, not 1. Yet I still define my series only up to 1. I have to first run to the 1/2 mile point, then to 3/4, then to 7/8, etc. Not only the series doesn't include the 1 mile point, it doesn't include anything from the second half of my jog.

Actually, I can do even better: let's ditch the space axis together, and do the same problem purely one axis.

Let's say my alarm clock wakes me up at 8 AM and I have to be at work at 9 AM. Simple problem, right? Well, obviously first half an hour must pass, bringing me to 8:30. Then half the rest must pass, bringing me to 8:45. And so on. It's still Zeno, but only on one axis. Obviously my series will never reach 9 AM, so wooohoo, I'll never need to work. So I go back to sleep.

But that actually says not much about the passage of time. It's my series that converges that way, not time itself. Time can still get to 9 AM, or for that matter 10 AM, and so on. Just because I defined a series that behave in a particular way, it doesn't mean much for anything else than that series.

Originally Posted by Robin View Post
Hasn't that ever bothered anyone? Seriously tell me that it hasn't bothered you.
Nope, I'm trying hard to remember when it bothered me, and I'm drawing blanks. Can't remember a single instance of being bothered that reality doesn't get changed by my picking the wrong model for it.

Originally Posted by Robin View Post
[...]
I mean think about it.
Well, yes, try to actually THINK for a change. I'm sure you'll figure it out.

Free hint: Zeno is trying to give you a lower bound for the time, which he believed to be infinite time, not a general theory of motion. It's not a formula for where the runner will be at a given time t or anything similar. It it's missing the final point, but then it's also missing most points in between. Because that was not the POINT. The point was to say "you can't finish in less than infinite time", no more, no less.
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Old 9th September 2021, 08:23 AM   #14
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Originally Posted by Dave Rogers View Post
No you don't. If you sum the infinite series, the "small undefined but finite amount" you refer to is in fact zero.

Dave
Really? Are you saying that if you were to algorithmically sum the elements in n infinite number of times the result would be the same as the algebraic sum?

Is there any point in the series that the sum so far would be equal to the algebraic sum?

If not then the algorithmic sum of all the elements would not be equal to the algebraic sum.

If at any time in the series the running total sum became the same as the algebraic sum then the value it returned would be larger than the algebraic sum.
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Old 9th September 2021, 08:26 AM   #15
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Originally Posted by JoeMorgue View Post
Okay fine then if you've proved motion is impossible then how is anything actually ******* moving?

Why does reality have to prove itself to the ponderous philosophical argument and not the other way around?

I'd say the burden of proof is on Zeno, not us.
Hey, Zeno didn't say the perception of motion didn't happen.

And guess what, as I said before, we have long ditched the idea that our perception of motion actually tells us what is really happening.
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Old 9th September 2021, 08:28 AM   #16
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If you were to sum the convergent series for some finite interval you obviously don't get the algebraic sum. This will be the case for the entire set of natural numbers.

So if there is an infinite number of values that does not equal a certain value then how do you get that value out of it.
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Old 9th September 2021, 08:29 AM   #17
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It is like saying that you could go through an infinite collection of values, none of which are three and find the number 3. How does that work?
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Old 9th September 2021, 08:31 AM   #18
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Someone tell me really.

If you were able to go through an infinite set of values, none of which was 3, would you find 3?

Or maybe you wouldn't find three.
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Old 9th September 2021, 08:32 AM   #19
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Originally Posted by Robin View Post
Someone tell me really.

If you were able to go through an infinite set of values, none of which was 3, would you find 3?

Or maybe you wouldn't find three.
"Throws another rock at your head"

Wow that rock made it through all the infinite recursion of halfway points and somehow still made it to your head. What a paradox.
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Old 9th September 2021, 08:33 AM   #20
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In the set of all the partial sums of the series, would you find the algebraic sum of the infinite series?
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Old 9th September 2021, 08:33 AM   #21
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Originally Posted by Robin View Post
It is like saying that you could go through an infinite collection of values, none of which are three and find the number 3. How does that work?
And again: if your model doesn't fit the problem, then the only problem is with your model. I'm not even sure what kind of confusion of mind is needed to pick an utterly inappropriate model for movement, and then be confused by its actually being not an appropriate model, or think it has any higher meaning than that.

I mean, if you're trying to calculate how long it takes for a train to get from New York to Washington DC, and your model excludes Washington DC, so it can't calculate that... well? What do you think is the problem? Can it be that you just picked the wrong model?
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Old 9th September 2021, 08:33 AM   #22
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Originally Posted by HansMustermann View Post
Not sure why it needed a second thread,
If you move an idea from "science" to "philosophy" the intellectual standards drop like a rock is why.
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Old 9th September 2021, 08:34 AM   #23
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Originally Posted by JoeMorgue View Post
"Throws another rock at your head"

Wow that rock made it through all the infinite recursion of halfway points and somehow still made it to your head. What a paradox.
If you were to search through an infinite collection of values, none of which were 3, would you find 3?
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Old 9th September 2021, 08:35 AM   #24
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Originally Posted by HansMustermann View Post
And again: if your model doesn't fit the problem, then the only problem is with your model. I'm not even sure what kind of confusion of mind is needed to pick an utterly inappropriate model for movement, and then be confused by its actually being not an appropriate model, or think it has any higher meaning than that.
If you were to search through an infinite set of values, none of which were 3, would you find 3?
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Old 9th September 2021, 08:35 AM   #25
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Originally Posted by Robin View Post
If you were to search through an infinite collection of values, none of which were 3, would you find 3?
Meaningless gibberish.
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Old 9th September 2021, 08:36 AM   #26
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Wouldn't it be fun if someone actually answered that question.
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Old 9th September 2021, 08:36 AM   #27
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Originally Posted by Robin View Post
Wouldn't it be fun if someone actually answered that question.
Then ask a non-stupid question.
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Old 9th September 2021, 08:37 AM   #28
Robin
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Originally Posted by JoeMorgue View Post
Meaningless gibberish.
Simple question. I am surprised you didn't answer it. Oh, actually I am not.

If you searched through an infinite set of values, none of which were 3, would you find 3?
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Old 9th September 2021, 08:38 AM   #29
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Originally Posted by JoeMorgue View Post
Then ask a non-stupid question.
OK, I will.

If you searched through an infinite set of values, none of which were 3, would you find 3?

It is not exactly a brain teaser.
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The non-theoretical character of metaphysics would not be in itself a defect; all arts have this non-theoretical character without thereby losing their high value for personal as well as for social life. The danger lies in the deceptive character of metaphysics; it gives the illusion of knowledge without actually giving any knowledge. This is the reason why we reject it. - Rudolf Carnap "Philosophy and Logical Syntax"
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Old 9th September 2021, 08:38 AM   #30
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Originally Posted by Robin View Post
Simple question. I am surprised you didn't answer it. Oh, actually I am not.

If you searched through an infinite set of values, none of which were 3, would you find 3?
If I ate an infinite number of cheesecakes, none of which were lemon-lime, would I taste lemon-lime?
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Old 9th September 2021, 08:39 AM   #31
Robin
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Oh dear, the question that must not be answered. I feel like I am in Question Time at Parliament.
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The non-theoretical character of metaphysics would not be in itself a defect; all arts have this non-theoretical character without thereby losing their high value for personal as well as for social life. The danger lies in the deceptive character of metaphysics; it gives the illusion of knowledge without actually giving any knowledge. This is the reason why we reject it. - Rudolf Carnap "Philosophy and Logical Syntax"
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Old 9th September 2021, 08:39 AM   #32
Robin
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Originally Posted by JoeMorgue View Post
If I ate an infinite number of cheesecakes, none of which were lemon-lime, would I taste lemon-lime?
If you searched through an infinite set of values, none of which were 3, would you find 3?

You finding it a bit of a tough one, or are you seeing the point?
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The non-theoretical character of metaphysics would not be in itself a defect; all arts have this non-theoretical character without thereby losing their high value for personal as well as for social life. The danger lies in the deceptive character of metaphysics; it gives the illusion of knowledge without actually giving any knowledge. This is the reason why we reject it. - Rudolf Carnap "Philosophy and Logical Syntax"
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Old 9th September 2021, 08:41 AM   #33
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I mean seriously, it is not a stupid question, it is not a difficult question.

Do you think that there is some reason why people are avoiding answer it?
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The non-theoretical character of metaphysics would not be in itself a defect; all arts have this non-theoretical character without thereby losing their high value for personal as well as for social life. The danger lies in the deceptive character of metaphysics; it gives the illusion of knowledge without actually giving any knowledge. This is the reason why we reject it. - Rudolf Carnap "Philosophy and Logical Syntax"
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Old 9th September 2021, 08:42 AM   #34
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Originally Posted by Robin View Post
If you were to search through an infinite set of values, none of which were 3, would you find 3?
And what I'm saying is that if your problem is determining movement from the 0 mile point to the 3 mile point, and you chose a model that excludes 3, it's just a case of YOU choosing the wrong model for the problem. It's not saying anything profound about the problem. It's just saying that you should pick a model that works instead of a nonsense one.

I mean, what the hell... the idea that you must choose a model that fits the problem you're trying to solve, was taught to me in physics class since middle school. Much as I like to be a snarky assh... err... hat about other people's education, I assume the same must apply to American schools, or really any schools in the world.
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Old 9th September 2021, 08:43 AM   #35
Robin
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Because everybody here is actually saying to me that they could search through an infinite set of numbers, none of which is 3 and find 3.
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The non-theoretical character of metaphysics would not be in itself a defect; all arts have this non-theoretical character without thereby losing their high value for personal as well as for social life. The danger lies in the deceptive character of metaphysics; it gives the illusion of knowledge without actually giving any knowledge. This is the reason why we reject it. - Rudolf Carnap "Philosophy and Logical Syntax"
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Old 9th September 2021, 08:43 AM   #36
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Originally Posted by Robin View Post
I mean seriously, it is not a stupid question, it is not a difficult question.

Do you think that there is some reason why people are avoiding answer it?
Nobody is avoiding it. They're just not giving you the pencils-up-the-nose idiotic nonsense that you seem to want. Well, tough beans. Your being stuck in nonsense mode doesn't constitute any obligation for anyone else to play along.
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Which part of "Ph'nglui mglw'nafh Cthulhu R'lyeh wgah'nagl fhtagn" don't you understand?
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Old 9th September 2021, 08:43 AM   #37
Robin
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Originally Posted by HansMustermann View Post
And what I'm saying is that if your problem is determining movement from the 0 mile point to the 3 mile point, and you chose a model that excludes 3, it's just a case of YOU choosing the wrong model for the problem. It's not saying anything profound about the problem. It's just saying that you should pick a model that works instead of a nonsense one.

I mean, what the hell... the idea that you must choose a model that fits the problem you're trying to solve, was taught to me in physics class since middle school. Much as I like to be a snarky assh... err... hat about other people's education, I assume the same must apply to American schools, or really any schools in the world.
If you were to search through an infinite set of values, none of which were 3, would you find 3?
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The non-theoretical character of metaphysics would not be in itself a defect; all arts have this non-theoretical character without thereby losing their high value for personal as well as for social life. The danger lies in the deceptive character of metaphysics; it gives the illusion of knowledge without actually giving any knowledge. This is the reason why we reject it. - Rudolf Carnap "Philosophy and Logical Syntax"
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Old 9th September 2021, 08:44 AM   #38
Robin
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Gosh this is fun.
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The non-theoretical character of metaphysics would not be in itself a defect; all arts have this non-theoretical character without thereby losing their high value for personal as well as for social life. The danger lies in the deceptive character of metaphysics; it gives the illusion of knowledge without actually giving any knowledge. This is the reason why we reject it. - Rudolf Carnap "Philosophy and Logical Syntax"
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Old 9th September 2021, 08:44 AM   #39
HansMustermann
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Originally Posted by Robin View Post
Because everybody here is actually saying to me that they could search through an infinite set of numbers, none of which is 3 and find 3.
No they're not. And if in all honesty that's what you're understanding, go see a neurologist.
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Which part of "Ph'nglui mglw'nafh Cthulhu R'lyeh wgah'nagl fhtagn" don't you understand?
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Old 9th September 2021, 08:45 AM   #40
Robin
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Originally Posted by HansMustermann View Post
Nobody is avoiding it. They're just not giving you the pencils-up-the-nose idiotic nonsense that you seem to want. Well, tough beans. Your being stuck in nonsense mode doesn't constitute any obligation for anyone else to play along.
If you were to search through an infinite set of values, none of which were 3, would you find 3?
__________________
The non-theoretical character of metaphysics would not be in itself a defect; all arts have this non-theoretical character without thereby losing their high value for personal as well as for social life. The danger lies in the deceptive character of metaphysics; it gives the illusion of knowledge without actually giving any knowledge. This is the reason why we reject it. - Rudolf Carnap "Philosophy and Logical Syntax"
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