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NeatNit
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Postby NeatNit » Thu Jul 11, 2013 10:46 pm

The mathematical model is wrong. On the same grounds I can say that first the arrow passes 0.0000...0001 of the distance, then again, then again, where the ... is of course an infinite amount of zeros. It is simply not the right way to model motion. Your (Zeno's) claim might as well be:
Let's take a flawless high-speed camera of infinite framerate, and shoot an arrow being fired from a bow to hit a target. Then, at playback, let's just keep slowing down the playback speed in such a way that the arrow keeps getting closer and closer to the target but never reaches it.

The fact that you think of it that way does not make it a paradox. It means your thought process is wrong. The mathematical model never ends, it keeps getting closer and closer to 2*, but it is not designed to examine the motion afterwards. You can examine the first second as closely as you want, study every inch, every particle in the air that the arrow comes in contact with, every imperfection of the arrow that causes slightly unexpected or unpredictable results, every goddamn particle in the universe applying its forces on every goddamn particle of the arrow... And it won't tell you anything about the second after that. Because that's not what you looked at. When you ignore everything ≥2 then OF COURSE you're never going to reach 2.


The only baffling part of this "paradox" (the one thing it is definitely NOT) is the universe's unbelievable precision and detail.


* I don't see why the accepted sequence doesn't start with 1/2 so that it would end up limiting up to 1, to correspond with the saying that it first travels half the distance, then half of the remainder (quarter), etc.
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Postby Feud » Thu Jul 11, 2013 11:44 pm

xander's arrow explanation makes sense to me. Things involving mathematical proofs, not so much.
The Zimmerman court rejecting a third degree felony murder charge based upon a child abuse statute based on the merger doctrine, or the habeas corpus issues that got discussed today at work.? I got that. Numbers though, are like cursive. I can do it, but that doesn't mean I like it.
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Postby Jackdapantyrip » Fri Jul 12, 2013 2:07 am

Feud wrote:xander's arrow explanation makes sense to me. Things involving mathematical proofs, not so much.
The Zimmerman court rejecting a third degree felony murder charge based upon a child abuse statute based on the merger doctrine, or the habeas corpus issues that got discussed today at work.? I got that. Numbers though, are like cursive. I can do it, but that doesn't mean I like it.


Well Zimmerman will get convicted of manslaughter.. but it should be 3rd degree murder, negligence. The prosecution was 'greedy' to go after 2nd and are unbelievably lucky the judge ruled to allow manslaughter be included in the verdict.. /sigh The prosecution should be held liable for their watering mouths and made an example of.. Beyond a reasonable doubt is there for a reason..

This is what happens when a small courtroom is trust into national television unfortunately..

and people making it a race issue are ill informed, it's a confrontation issue -
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Postby xander » Fri Jul 12, 2013 4:21 am

NeatNit wrote:The mathematical model is wrong.

Okay. You assert that the model is wrong. What is the error? Can you propose a better model? I would point out that by assuming that this model is correct, we get a useful notion of limits. With a useful notion of limits, we get derivatives, integrals, Taylor series, differential equations, chaotic dynamical systems, and basically all the rest of modern mathematics. If the model is a bad one, and we choose to scrap it, then we need to come up with some new model that allows us to send people to the moon, develop useful medications, and predict the weather.

NeatNit wrote:On the same grounds I can say that first the arrow passes 0.0000...0001 of the distance, then again, then again, where the ... is of course an infinite amount of zeros.

Nonsense. 0.000...0001 (with an infinite number of zeros) is 0. You can't have an infinite number of zeros followed by a one. Either you have a finite number of zeros followed by a one, or you have an infinite number of zeros.

NeatNit wrote:It is simply not the right way to model motion. Your (Zeno's) claim might as well be:
Let's take a flawless high-speed camera of infinite framerate, and shoot an arrow being fired from a bow to hit a target. Then, at playback, let's just keep slowing down the playback speed in such a way that the arrow keeps getting closer and closer to the target but never reaches it.

Oh, you are so very, very close!

Zeno fails to account for the velocity. Here's how the model can be made to make sense. Let us, for simplicity sake, say that the arrow is traveling at a rate of 128 m/s and that the target is 128 m from the shooter (this is just a bullshit number to give us easy division problems as we halve things). For the arrow to get to the target, it has to first get to the halfway point. But it is traveling at 128 m/s, so it only takes half a second to get to the halfway point. From there, it has to get to 96 m, but that only takes 1/4 second. Then it has to get to 112 m, but that only takes 1/8 second. And so on.

The arrow has to hit each of an infinity of points, but the time it takes to get from one point to the next is proportional to the distance between those points. Thus we can resolve the paradox. Do do this formally, we need to define limits, and a derivative would be nice, too, but the model is fine!

NeatNit wrote:The fact that you think of it that way does not make it a paradox. It means your thought process is wrong.

Um... no. A paradox, in mathematics, is a surprising result. The result, in mathematics, is that one can make sense of the statement "here is an infinite process which terminates." The entire study of calculus deals with infinite processes which terminate, hence these kinds of paradoxes are very important to understand.

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Postby NeatNit » Fri Jul 12, 2013 11:44 am

Why do you think the infinite process terminates? The infinite process examines the motion of the arrow in deeper and deeper detail the closer it gets to its target. It doesn't have to terminate - the fact that the arrow continues beyond the scope of your infinite series doesn't mean that it terminated. In fact, just between the inifnite series' start point and limit there are infinitely more points that the arrow travels through - why don't those pose a problem to you?
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Postby xander » Fri Jul 12, 2013 4:34 pm

NeatNit wrote:Why do you think the infinite process terminates? The infinite process examines the motion of the arrow in deeper and deeper detail the closer it gets to its target. It doesn't have to terminate - the fact that the arrow continues beyond the scope of your infinite series doesn't mean that it terminated. In fact, just between the inifnite series' start point and limit there are infinitely more points that the arrow travels through - why don't those pose a problem to you?

They don't pose a problem for me because none of this problem bothers me. I grok limits. I don't have a problem with an infinite process having a finite result. It is you who is complaining that the math is wrong, or that the paradox is silly because "Derp! Arrows move! Duh!"

Perhaps we need to move away from the analogy, and actually look at the abstraction. Suppose I told you that I had an infinite sum of positive numbers. That is, I am going to add together an infinite number of positive (strictly greater than zero) numbers. What should the result be? A finite number? Infinity? Something else?

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Postby NeatNit » Fri Jul 12, 2013 4:54 pm

Well, since you neglected to point out which numbers you were going to add up, I am going to have to say "infinity, probably" and obviously there's no way of actually knowing without knowing which positives we're talking about.

As for the [1, 1/2, 1/4, 1/8, 1/16...] series, clearly they would amount to 2... If you do it an "infinite" amount of times. But you can't really do that. You can keep adding more and more and you'd get closer to 2, but you already know that.

Thing is, reality CAN add things up infinitely. It has no problem doing so, and did it an infinite amount of times since you started reading this word.





Edit: The first time I heard of Zeno, I was sure his paradoxes were being mocked and ridiculed by modern science.* There was just no way that these "paradoxes" could be baffling or problematic these days... It's just too simple. I still can't believe that's true. For someone to refer to it as "baffling" is just... weird to me. They're all rather stupid paradoxes.



* It certainly didn't help that I heard of it from here, I guess.
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Postby xander » Fri Jul 12, 2013 5:36 pm

NeatNit wrote:Well, since you neglected to point out which numbers you were going to add up, I am going to have to say "infinity, probably" and obviously there's no way of actually knowing without knowing which positives we're talking about.

I didn't neglect anything. I very intentionally didn't mention what numbers are being added up. And your intuition of "infinity, probably" is exactly the intuition that most people have. Add up an infinite number of positive numbers, and it should be infinite. That is what makes naive, intuitive sense.

NeatNit wrote:As for the [1, 1/2, 1/4, 1/8, 1/16...] series, clearly they would amount to 2...

Why? You say it is clear, but you have not explained why it is clear. What about 1+1/2+1/3+1/4+1/5+...? Is that finite? And if either sum is finite, doesn't that contradict the intuition that it should be infinite? Obviously, we have the benefit of living in an age where these questions have already been worked out, but, a priori, there is something non-intuitive going on.

NeatNit wrote:If you do it an "infinite" amount of times. But you can't really do that.

Obviously, one cannot actually do anything an infinite number of times. But we are not talking about actually doing anything until the end of time and beyond. We are examining an abstraction. Infinity itself is an abstraction, so abstractions really are the only way to handle it, anyway.

NeatNit wrote:You can keep adding more and more and you'd get closer to 2, but you already know that.

I know that, and you claim to know it, but why? Why does it add up to 2 and not 3? or 1? or pi?

NeatNit wrote:Thing is, reality CAN add things up infinitely. It has no problem doing so, and did it an infinite amount of times since you started reading this word.

Can it? What is infinite? How does it relate to reality? What does it mean to say that "reality can add things up infinitely?" Those are deep questions, and highlight the fact that mathematics is not reality, and that reality isn't built on mathematics.

NeatNit wrote:Edit: The first time I heard of Zeno, I was sure his paradoxes were being mocked and ridiculed by modern science.* There was just no way that these "paradoxes" could be baffling or problematic these days... It's just too simple. I still can't believe that's true.

The paradoxes are not baffling if you define your way around them. Limits are how mathematicians "solve" the paradox. But limits are an abstraction. As long as you don't try to think too hard about it, or trust the results of Newton, Leibniz, and their ilk, then you are just fine. But try to pretend for a minute that you are faced with this dilemma, and no previous person has ever explained how an infinite process can resolve (i.e. how can an arrow pass through a continuum of points---there are an infinity of points, the arrow must pass through each one, thus it can't move, because movement would imply that an infinite process terminates, which doesn't make sense!). How do you reason about that?

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Postby NeatNit » Fri Jul 12, 2013 9:45 pm

I apologize for not answering the rest of your post but let's just focus on this:

xander wrote:how can an arrow pass through a continuum of points---there are an infinity of points, the arrow must pass through each one, thus it can't move, because movement would imply that an infinite process terminates, which doesn't make sense! How do you reason about that?

xander

Slowly.
1. There are an infinity of points.
Okay, that's clear enough.
2. The arrow must pass through each one.
Yes, that is our definition.
3. It can't move (movement would imply that an infinite process terminates)

That's where I think you're being tied down by previous knowledge and can't just think outside of it for a minute.

Take the function f(x) = x. How many points are between x=0 and x=1? Infinite points.
Take the function f(x) = 2x. How many points between x=0 and x=1? Also infinite.

Take the motion of the arrow. How many points does it pass in one second? Infinite points.

But a point is exactly that - a point. It has no dimensions, and points exist in infinite density. Speed - any speed - would pass through infinite points per second. Maybe points aren't the right way to count things? The very definition of points makes it so everything finite, even the smallest, has either exactly one or an infinite amount. But everything is analog. For every point A on the path of the tip of the arrow, you could find a time B when the tip was at point A. And the chances that is passes at that point again in the future are almost zero. Infinity is not some crazy thing that doesn't work with anything else, you just gotta wrap your head around it, and I'm genuinely disappointed in you xander that you don't get it.


Now if you'll excuse me, I should probably go to bed and look back here in the morning to read about how I failed to express myself.

*goes off to play something on Steam*


Edit: just to be clearer, there's a reason we don't measure things in points. If we did, then everything (including length, area and volume) would have infinity points. I have absolutely no problem wrapping my head around the infinite density of reality.
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Postby xander » Fri Jul 12, 2013 10:22 pm

NeatNit wrote:I apologize for not answering the rest of your post but let's just focus on this:

xander wrote:how can an arrow pass through a continuum of points---there are an infinity of points, the arrow must pass through each one, thus it can't move, because movement would imply that an infinite process terminates, which doesn't make sense! How do you reason about that?

xander

Slowly.
1. There are an infinity of points.
Okay, that's clear enough.
2. The arrow must pass through each one.
Yes, that is our definition.
3. It can't move (movement would imply that an infinite process terminates)

That's where I think you're being tied down by previous knowledge and can't just think outside of it for a minute.

Q: What does it mean for something to be infinite?
A: Something is infinite if it is not finite. It is without bound. It never ends. It doesn't stop.

Q: In order to move, mustn't an arrow pass through an infinity of points?
A: Yes. We can name a subset of those points, say 1/2 the distance, then 3/4 the distance, then 7/8 the distance, and so on.

Q: Does the arrow, in our experience of the universe, arrive at a target?
A: Yes. Not only that, but it passes through the target.

Conclusion: There exist infinite process that terminate in finite time (i.e. the arrow passing through infinitely many points.

You are trying way to hard to over complicate and over think the issue. A priori, without knowledge of mathematics invented only in the last 4-500 years, we have two seemingly true, yet seemingly contradictory statements (an arrow passes through an infinite number of points; something is infinite if it is without bound). The paradox is not that the statements really are contradictory, but that they seem that way to naive intuition. Within the proper framework, it is quite easy to understand how the statements are not contradictory, but it does require the correct framework.

NeatNit wrote:Take the function f(x) = x. How many points are between x=0 and x=1? Infinite points.
Take the function f(x) = 2x. How many points between x=0 and x=1? Also infinite.

I don't even know what you are trying to demonstrate here. There is a continuum of points between zero and one. So?

NeatNit wrote:Take the motion of the arrow. How many points does it pass in one second? Infinite points.

But a point is exactly that - a point. It has no dimensions, and points exist in infinite density. Speed - any speed - would pass through infinite points per second. Maybe points aren't the right way to count things? The very definition of points makes it so everything finite, even the smallest, has either exactly one or an infinite amount. But everything is analog. For every point A on the path of the tip of the arrow, you could find a time B when the tip was at point A. And the chances that is passes at that point again in the future are almost zero. Infinity is not some crazy thing that doesn't work with anything else, you just gotta wrap your head around it, and I'm genuinely disappointed in you xander that you don't get it.

What is a point? What is a dimension? What is speed? How does the definition of a point make everything finite (again, what is a point?)? What does any of this have to do with abstract measure theory? Do you actually understand the Wikipedia article that you cited?

Moreover, as I keep trying to get you to understand, you are overly fixated on the analogy of an arrow flying or a racer pacing a turtle. Go back to the abstraction. Take an infinite number of positive values and add them up. You would expect to get infinity (in fact, the subset of summable sequences in R is null (i.e. of measure zero) with respect to the set of sequences in R---pick a random sequence and, with probability one, it will not be summable). The fact that this isn't always true is surprising, unexpected, and not intuitive. Therefore it qualifies as a paradox.

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Postby NeatNit » Sat Jul 13, 2013 12:15 pm

The key sentence in my post lies here:
NeatNit wrote:Now if you'll excuse me, I should probably go to bed and look back here in the morning to read about how I failed to express myself.

*goes off to play something on Steam*



But more importantly this:
NeatNit wrote:just to be clearer, there's a reason we don't measure things in points. If we did, then everything (including length, area and volume) would have infinity points. I have absolutely no problem wrapping my head around the infinite density of reality.


If you take an infinite amount of a length infinitely short, what do you get? Just about any length you want...

My point is that points have absolutely no significance in the way you give them significance.


As for the abstraction - it is evidently simple to construct an infinite number of positive values that add up to <anything>. Okay. I was not aware of your definition for a paradox, which you posted a few posts back. In that case, it is a paradox that we exist and a paradox that Bruce Willis has been dead throughout the whole movie (a movie which I still have not seen, thank you Scrubs). And a paradox that this sentence ends with a comma,

MAYBE I'M JUST NOT IN THE MOOD FOR NONSENSE
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Postby Laika » Sat Jul 13, 2013 1:36 pm

xander wrote:Conclusion: There exist infinite process that terminate in finite time (i.e. the arrow passing through infinitely many points.


I, for example, assume definition of point as of a physical model, an ideal body with size close to zero (i.e. insignificant) in comparison to other sizes and/or distances we operate with. It can be an electron in atoms model; a projectile in its shooting range calculations. This assumption allows to differentiate what we understand as a point in real world, thus we can operate with abstract points that take 1 distance from arrow starting point to its end, or 1/2, or (...). An arrow can travel only through 1 point taking 1 distance exactly, 2 points of 1/2 of a distance (...). Conclusion is that while we can assume infinetely small point, in no case arrow has to travel through infinite amount of them, thus it is not an infinite process, and it's termination is not a paradox.
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Postby NeatNit » Sat Jul 13, 2013 1:49 pm

Rather, I don't see the paradox that the arrow successfully passes through infinitely many points. As I said in the beginning, that's the whole point of a continuum.

The abstraction that a sum of an infinite number of positive values is not infinity, is a different problem, and again solved by accepting that values can be infinitely small. This is also not surprising.
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Postby trickser » Sat Jul 13, 2013 6:07 pm

I don't see the paradox


To find a solution you need to have a problem.

However your solution is to see no problem, that's what I call a paradox.
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Postby xander » Sat Jul 13, 2013 6:25 pm

NeatNit wrote:If you take an infinite amount of a length infinitely short, what do you get? Just about any length you want...

What does that even mean? There is no such thing as "infinitely short" (well, unless you want to get into the alternative number system of infinitesimals, but that is something else entirely, and doesn't actually solve your problem). Either something has positive length, or it is of length zero.

NeatNit wrote:My point is that points have absolutely no significance in the way you give them significance.

I think that you are, perhaps, missing the point of an arrow traveling through an infinite number of points. Let me try again.

In order for an arrow to get from the shooter to the target (say the distance is 1 unit), it first has to get to the point 1/2. In order to do so, it must have traveled a length of 1/2 units. Then it has to half of the remaining distance, traveling through the point 3/4. In order to do so, it has to travel a distance of 1/4 units. And so on. At each step of the abstraction, the arrow must travel a positive distance. That is, the arrow must travel an infinite number of positive distances. Yet, somehow, this infinite number of positive distances adds up to a finite number.

Your own intuition, above, was that an unbounded number of positive numbers adds up to infinity. That is most people's intuition, and is the thing that you would want to be true. In general, it is true. Yet here is a model of motion that makes perfect sense, as well. We can divide the interval into an infinite number of sub-intervals of ever decreasing (but always positive) length, and the arrow must travel the distance of each of these intervals. The model and the intuition seem to clash. Hence there is a paradox.

The resolution of this paradox is to invent the notion of a limit. A consequence of inventing limits is the continuum. That is, the continuum is not something that exists a priori (unless you are a Platonist), but a direct result of the rules of logic applied to the invention of a limit. You keep appealing to the existence of a continuum of points in order to resolve the paradox, yet the continuum exists as a result of the resolution of the paradox.

NeatNit wrote:As for the abstraction - it is evidently simple to construct an infinite number of positive values that add up to <anything>.

It's like you aren't even engaging with the ideas... you are just asserting facts on the basis of a Wikipedia article or a math class that you took at some point in the past. You have claimed that 1+1/2+1/4+1/8+... adds up to something finite. How do you know that is true? Can you prove it? What about 1+1/2+1/3+1/4+1/5+...? Does that add up to something finite? If so, what? If I pick a random infinite sequence of positive numbers and add them up, will the result be finite? How does that gell with our intuition? If most infinite series sum to something infinite, what is special about series that sum to something finite? How can we detect these special cases?

NeatNit wrote:Okay. I was not aware of your definition for a paradox, which you posted a few posts back. In that case, it is a paradox that we exist and a paradox that Bruce Willis has been dead throughout the whole movie (a movie which I still have not seen, thank you Scrubs). And a paradox that this sentence ends with a comma,

Perhaps I have done a poor job of explaining what a mathematical paradox is, but none of your examples come even close to qualifying. In mathematics, we have a rigorous structure for determining whether or not a statement is true or false. We start by assuming that some very simple statements are true (in the current context, the ZFC axioms are sufficient---these are nine simple statements that we assume are true), then use the rules of first order logic to determine what those assumptions imply. A paradox occurs when a seemingly false result can be proved to be true, or when we can prove that two seemingly contradictory statements are both true.

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