1153: "Proof"

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TimXCampbell
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Re: 1153: "Proof"

Postby TimXCampbell » Sun Dec 30, 2012 4:31 pm UTC

Nooska wrote:So you are saying that at finitely small increments movement is indistinguishable from teleportation?

The release of energy is quantum, right? Why can't that also be the case with movement? If that's so, then Zeno's paradox ceases to be an issue, because at a certain level you no longer can say “half the distance.” Mind you, you can still say “half the time” but there's also such a thing as a Planck unit of time.

Please note that I have very little knowledge of physics, so I might be speaking rubbish. On the other hand, I've never read anything that says that movement and time are not quantum.

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Re: 1153: "Proof"

Postby Quake » Sun Dec 30, 2012 6:36 pm UTC

:arrow: . !

Yes siree ;

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Re: 1153: "Proof"

Postby Caledonian » Sun Dec 30, 2012 7:13 pm UTC

It's always somewhat shocking to realize just how terrible ancient philosophy was. Seriously, it didn't occur to them that you could divide time into infinite sections just as well?

Thank goodness philosophy gave birth to scientific inquiry and rational thought.

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Re: 1153: "Proof"

Postby madaco » Sun Dec 30, 2012 11:13 pm UTC

TimXCampbell wrote:
Nooska wrote:So you are saying that at finitely small increments movement is indistinguishable from teleportation?

The release of energy is quantum, right? Why can't that also be the case with movement? If that's so, then Zeno's paradox ceases to be an issue, because at a certain level you no longer can say “half the distance.” Mind you, you can still say “half the time” but there's also such a thing as a Planck unit of time.

Please note that I have very little knowledge of physics, so I might be speaking rubbish. On the other hand, I've never read anything that says that movement and time are not quantum.


before I say this, I should mention that I also have a poor understanding of this.
as a result I will put my confusion in spoiler tags so it doesn't take up as much room on the page.

that said:

Spoiler:
if a particle is moving at the plank speed (c), then one plank time later, it should have moved the plank length,
but if the particle is moving at less than c, then how far does it move is one plank time?

so if both time and space are quantized in the plank length and the plank time, that to me it seems like that would suggest that things would have to move at a integer multiple of c
and of course it cant move faster than c,
so that would mean that all particles at any instant must be either moving at c or at rest?

that does not seem reasonable to me.

So as a result I am confused.

it seems then, that either time and space are not both quantized at the plank length and plank time respectively
or the amount it goes forwards in one plank time is not as simple just some multiple of the plank length?

due to my lack of knowledge in the subject, I was wondering if maybe the particle would be in a superposition as to where it was?
or I guess maybe the uncertainty principle solves this problem because of not being able to know with that much accuracy which spot it is at?

hey can someone tell me the answer to this so that I would be less ignorant? (if it is convenient)
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Re: 1153: "Proof"

Postby speising » Sun Dec 30, 2012 11:52 pm UTC

That's easy,since time and space are essentally the same.
Everything is always moving at c. The quesion is only, how much of this is in the space or in the time direction. Something (like light), which moves at c through space does not move in time at all.

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Re: 1153: "Proof"

Postby rmsgrey » Mon Dec 31, 2012 2:27 am UTC

Caledonian wrote:It's always somewhat shocking to realize just how terrible ancient philosophy was. Seriously, it didn't occur to them that you could divide time into infinite sections just as well?

Thank goodness philosophy gave birth to scientific inquiry and rational thought.


Another of Zeno's paradoxes addresses the question of dividing time - an object in motion has a definite position at any instant, and doesn't move from that position during that instant. If there is no instant in which the object's position changes, then it cannot have moved.

It's not a question of how to divide time/space so much as realising that position and momentum are both needed (among other things) in order to describe the instantaneous state of a body

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Re: 1153: "Proof"

Postby TimXCampbell » Mon Dec 31, 2012 4:56 am UTC

speising wrote:[S]ince time and space are essentally the same ... [e]verything is always moving at c. The question is only, how much of this is in the space or in the time direction. Something (like light), which moves at c through space does not move in time at all.

Wow, you just blew my mind.

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Re: 1153: "Proof"

Postby webgiant » Mon Dec 31, 2012 5:57 am UTC

TimXCampbell wrote:
ijuin wrote:I think that the point of Zeno's paradox is that, prior to having an understanding of Calculus, it is difficult to wrap one's mind around how an infinite series can converge to a finite value.

[Emphasis added]

Indeed. Even in this day and age It took me until I was around 50 years old to finally understand (and accept) that .999… actually does equal 1. If I'd told that to a Pythagorean I probably would have been killed.

1 + 1 = 3, for extremely large values of 1

(1.4 + 1.4 = 2.8, which rounds to 3)

It's one of the few things I occasionally write on the wall in a college bathroom stall. Writing it in the applied tech department bathrooms gets approving comments. Writing it in math department bathrooms gets vitriolic responses. I haven't been able to develop the courage to write it in philosophy department bathrooms. I suspect if I ever did they'd hunt me down and use the hemlock.

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Re: 1153: "Proof"

Postby TimXCampbell » Mon Dec 31, 2012 6:25 am UTC

webgiant wrote:1 + 1 = 3, for extremely large values of 1 [since] (1.4 + 1.4 = 2.8, which rounds to 3)

This is cute! I've never seen this before. I like it! However, I was referring to something that didn't involve rounding.

For years I rebelled at the notion that .999… was equal to 1. It didn't matter how many clever math experts told me it did, I just didn't get it. Then one day I was noodling around with a calculator and realized this:

1/3 = 0.33333333…

2/3 = 0.66666666…

3/3 = 0.99999999…


Actually, the calculator stubbornly insisted that 3/3 equalled 1, but by now I'd seen the progression and finally reached the inevitable though unpalatable conclusion.

Sometimes I have a sneaking suspicion that this universe isn't quite right.

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Re: 1153: "Proof"

Postby Vroomfundel » Mon Dec 31, 2012 10:19 am UTC

madaco wrote:if a particle is moving at the plank speed (c), then one plank time later, it should have moved the plank length,
but if the particle is moving at less than c, then how far does it move is one plank time?

so if both time and space are quantized in the plank length and the plank time, that to me it seems like that would suggest that things would have to move at a integer multiple of c
and of course it cant move faster than c,
so that would mean that all particles at any instant must be either moving at c or at rest?



That's actually a very good question and I don't think science can answer it conclusively at this point. My understanding is that both Planck length and Planck time are hypothesized to be indivisible units of space and time, respectively - but this doesn't mean that both hypothesis have to be true. If the grain of space is the Planck length and time is infinitely divisible - no problem, and vice versa. On the other hand, when space and time are viewed as just dimensions of the pseudo-Riemannian manifold that spacetime is so it wouldn't feel natural for just one of them to be infinitely divisible.

If both are true then indeed motion with speeds of less than c are problematic. This can be covered by the uncertainty principle - the universe is probably not even theoretically observable at such scales.

I can understand Zeno's frustration with the apparent impossibility of motion. No matter how you look at it, it doesn't seem possible to complete an infinite number of steps in finite time. If the number of steps is finite, then, we run into the other Zeno paradox.
Even nowadays, if we assume finite divisibility we run into all kinds of counter-intuitive implications, even if it's at finer grain than the Planck units. If both space and time are finitely divisible, then we have, first, a lower bound for speed and second, a smallest possible increment; speed must then be an integer multiple of slowest possible speed (1 atomic unit of length per 1 atomic unit of time)
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Re: 1153: "Proof"

Postby Klear » Mon Dec 31, 2012 11:58 am UTC

Hey- another non-physic here to add to the confusion!

I remember reading somewhere* that if you try to measure 1/2 of planck length, you'll actually measure 2 planck lengths, measuring 1/10 of planck length gives you 10 planck length etc. I have no idea why that should be the case, but it makes smaller distances impossible (and, I guess, irrelevant) to measure.

Also - what speising wrote is correct - everything is moving at c all the time. If something is not moving in space at all, it moves through time at c. If it moves a bit, its speed in the direction of time is somewhat lower. That is why moving a considerable fraction of c makes time go slower for you.

*The Elegant Universe by Brian Greene

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Re: 1153: "Proof"

Postby rmsgrey » Mon Dec 31, 2012 6:06 pm UTC

Klear wrote:Also - what speising wrote is correct - everything is moving at c all the time. If something is not moving in space at all, it moves through time at c. If it moves a bit, its speed in the direction of time is somewhat lower. That is why moving a considerable fraction of c makes time go slower for you.


When you start considering time as another dimension, nothing is moving - it just extends at a given angle into the "future" direction like a ladder leaned against a wall extends at a certain angle into the "up" direction. "Speed" is only a meaningful concept if you have separate "time" and "space" and compare how the two vary along an object's history...

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Re: 1153: "Proof"

Postby e=hv » Tue Jan 01, 2013 2:49 pm UTC

because of Heisenberg principle
sigma x * sigma p < h-bar/2
(where x is location and p is momentum and delta. a pair of conjugate variables) The more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa.
It must be emphasized that measurement does not mean only a process in which a physicist-observer takes part, but rather any interaction between classical and quantum objects regardless of any observer.

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Re: 1153: "Proof"

Postby orthogon » Tue Jan 01, 2013 4:16 pm UTC

rmsgrey wrote:
Klear wrote:Also - what speising wrote is correct - everything is moving at c all the time. If something is not moving in space at all, it moves through time at c. If it moves a bit, its speed in the direction of time is somewhat lower. That is why moving a considerable fraction of c makes time go slower for you.


When you start considering time as another dimension, nothing is moving - it just extends at a given angle into the "future" direction like a ladder leaned against a wall extends at a certain angle into the "up" direction. "Speed" is only a meaningful concept if you have separate "time" and "space" and compare how the two vary along an object's history...

I was going to say that "proper time" was a separate axis with respect to which you can be said to be moving at c. But of course proper time is just the time axis in a frame defined such that the observer is stationary. If I'm moving relative to you, our co-ordinate systems are at an angle to one another, so that one second of my time projects onto less than one second of your time (and, weirdly, vice versa), distance is foreshortened because some of your spatial dimension is resolving into my time dimension, and events on my hyperplane of simultaneity, which is perpendicular to my time axis, aren't simultaneous for you because our time axes aren't parallel.

Anyway, if spacetime is quantised, how would that work in more than one dimension? Surely there can't be a rectangular grid, since that would privilege certain spatial directions over others, and certain inertial frames over others. And/or different particles' grids would not be aligned, making the cosmos like one big Microsoft drawing.
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Re: 1153: "Proof"

Postby Klear » Tue Jan 01, 2013 4:53 pm UTC

orthogon wrote:Anyway, if spacetime is quantised, how would that work in more than one dimension? Surely there can't be a rectangular grid, since that would privilege certain spatial directions over others, and certain inertial frames over others. And/or different particles' grids would not be aligned, making the cosmos like one big Microsoft drawing.


I was wondering about this myself. With all the planck units, it seems that the universe is digital as opposed to analog, but I have a hard time wrapping my mind around how it should work.

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Re: 1153: "Proof"

Postby mathmannix » Wed Jan 02, 2013 4:28 pm UTC

TimXCampbell wrote:
webgiant wrote:1 + 1 = 3, for extremely large values of 1 [since] (1.4 + 1.4 = 2.8, which rounds to 3)

This is cute! I've never seen this before. I like it! However, I was referring to something that didn't involve rounding.

For years I rebelled at the notion that .999… was equal to 1. It didn't matter how many clever math experts told me it did, I just didn't get it. Then one day I was noodling around with a calculator and realized this:

1/3 = 0.33333333…

2/3 = 0.66666666…

3/3 = 0.99999999…


Actually, the calculator stubbornly insisted that 3/3 equalled 1, but by now I'd seen the progression and finally reached the inevitable though unpalatable conclusion.

Sometimes I have a sneaking suspicion that this universe isn't quite right.


I would adamantly argue against my math teachers that the difference between 0.999 and 1 is exactly 0.0001 (I am underlining instead of overlining because I don't know how to make the text do the latter.)

I never had any problem with adding digits after an infinite number of digits, either... like 0.6667.
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Re: 1153: "Proof"

Postby dudiobugtron » Wed Jan 02, 2013 8:03 pm UTC

mathmannix wrote:I would adamantly argue against my math teachers that the difference between 0.999 and 1 is exactly 0.0001 (I am underlining instead of overlining because I don't know how to make the text do the latter.)

I never had any problem with adding digits after an infinite number of digits, either... like 0.6667.


Using your definition, 0.999 + 0.0001 = 0.9991
Is that the same as 1?
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Re: 1153: "Proof"

Postby Pfhorrest » Wed Jan 02, 2013 8:41 pm UTC

mathmannix wrote:I would adamantly argue against my math teachers that the difference between 0.999 and 1 is exactly 0.0001 (I am underlining instead of overlining because I don't know how to make the text do the latter.

That's fine so long as it's understood that 0.0001 = 0 (so the difference between 0.999 and 1 is 0, and there are thus equal). The principle is exactly the same as why 0.999 = 1: as you add more nines to the latter, you get a number arbitrarily closer and closer to 1, so if you did that "forever", the number you would "eventually" get would be 1. Likewise, as you add more zeroes to the former, you get a number arbitrarily closer to 0, so if you did that "forever", the number you would "eventually" get would be 0. Scare quotes around "forever" and "eventually" because of course a process of incrementally inserting digits into these decimal representations would never actually end, but we could plot the course of that process's progress and see where it's headed, and it turns out the string of nines is headed to 1, and the string of zeroes followed by a one is headed to 0, so the full expansions of those numbers would be equal to 1 and 0, respectively.
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Re: 1153: "Proof"

Postby oauitam » Thu Jan 03, 2013 11:44 am UTC

mathmannix wrote:I would adamantly argue against my math teachers that the difference between 0.999 and 1 is exactly 0.0001 (I am underlining instead of overlining because I don't know how to make the text do the latter.)


I think the problem is the idea that 0.0001 means something sensible.

I took it that the underlining means something along the lines of 'keep repeating this sequence forever'.
If so what can 'keep repeating these 0s forever, and then put a one after them' mean?
If we can put a 1 after the 0s then we didn't repeat the 0s forever - we only repeated them 'for a very long time indeed, until we got to the 1'.
And, if we do repeat the 0s forever then we never reach a time/place to put the 1.

0.0001 doesn't have a meaning any more than 0.00012377790 does.

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Re: 1153: "Proof"

Postby bmonk » Thu Jan 03, 2013 10:41 pm UTC

oauitam wrote:
mathmannix wrote:I would adamantly argue against my math teachers that the difference between 0.999 and 1 is exactly 0.0001 (I am underlining instead of overlining because I don't know how to make the text do the latter.)


I think the problem is the idea that 0.0001 means something sensible.

I took it that the underlining means something along the lines of 'keep repeating this sequence forever'.
If so what can 'keep repeating these 0s forever, and then put a one after them' mean?
If we can put a 1 after the 0s then we didn't repeat the 0s forever - we only repeated them 'for a very long time indeed, until we got to the 1'.
And, if we do repeat the 0s forever then we never reach a time/place to put the 1.

0.0001 doesn't have a meaning any more than 0.00012377790 does.


In calculus, the idea of a limit was developed precisely to avoid having to use such monstrosities. The basic idea is that, by limiting the input of a formula sufficiently, the result will also be limited. Thus for every e>0, you can calculate a d>0 so that whenever d>|x-a|>0, you have e>|f(x)-l|.

Here, what it means is that you can make 0.9999... arbitrarily close to 1. Want it within one millionth? Use 6 or more 9s. Within one trillionth? Use at least 12 nines. Within one part in a google? Use over 100 9s. It means you don't actually have to have infinitely long numbers--but you can see that, effectively, there "is no difference" that matters.
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Re: 1153: "Proof"

Postby dudiobugtron » Fri Jan 04, 2013 3:40 am UTC

oauitam wrote:I took it that the underlining means something along the lines of 'keep repeating this sequence forever'.
If so what can 'keep repeating these 0s forever, and then put a one after them' mean?

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Re: 1153: "Proof"

Postby mathmannix » Fri Jan 04, 2013 6:45 pm UTC

oauitam wrote:0.0001 doesn't have a meaning any more than 0.00012377790 does.


0.0001 only has ∞+1 significant digits. 0.00012377790 has 2∞+4 significant digits. The difference between them is an infinitesimal number, 0.0000237779.

Just because my math might not actually mean anything doesn't mean it's wrong.
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Re: 1153: "Proof"

Postby brenok » Fri Jan 04, 2013 6:47 pm UTC

How do you differentiate 0.000023 from 0.00023?

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Re: 1153: "Proof"

Postby mathmannix » Fri Jan 04, 2013 7:01 pm UTC

One is ten times the other. Since (I claim) they are not precisely zero, they cannot therefore be identical.
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Re: 1153: "Proof"

Postby Elmach » Sun Jan 06, 2013 5:43 am UTC

mathmannix wrote:
TimXCampbell wrote:
webgiant wrote:1 + 1 = 3, for extremely large values of 1 [since] (1.4 + 1.4 = 2.8, which rounds to 3)

This is cute! I've never seen this before. I like it! However, I was referring to something that didn't involve rounding.

For years I rebelled at the notion that .999… was equal to 1. It didn't matter how many clever math experts told me it did, I just didn't get it. Then one day I was noodling around with a calculator and realized this:

1/3 = 0.33333333…

2/3 = 0.66666666…

3/3 = 0.99999999…


Actually, the calculator stubbornly insisted that 3/3 equalled 1, but by now I'd seen the progression and finally reached the inevitable though unpalatable conclusion.

Sometimes I have a sneaking suspicion that this universe isn't quite right.


I would adamantly argue against my math teachers that the difference between 0.999 and 1 is exactly 0.0001 (I am underlining instead of overlining because I don't know how to make the text do the latter.)

I never had any problem with adding digits after an infinite number of digits, either... like 0.6667.


All right then. Let x be 0.0001 (base ten).

What is 10x? (What is 10 * 0.0001?) How would you represent it in base ten?
What is x10? (What is 0.0001 * 10?) How would you represent it in base ten?
What is 0.1x? (What is 0.1 * 0.0001?)
What is x0.1?
What is x in base 9?
Let y be 0.00019 -- the same construction mentioned earlier, but in base nine.)
(I am going to assume that 9y, y9, 0.19y, and y0.19 will have obvious representations in base nine at this point)
How would you represent y in base ten? x in base 9?
Which of x and y are greater? Is this a nonsensical question?


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