I would if I could but it's not my server. Hopped onto jabber.org.uk (which got far more takers than the hoster originally intended). So I just need to wait. I'm in no hurry, anyway:)
Okay. Let's go through this one section at a time.
And what triggered the spontaneity? Where did the "stuff" come from? How can you have a creation without a Creator? "Stuff" does not simply materialize out of nothing, all by itself. If you acknowledge there was some kind of matter in the universe and then it spontaneously became the universe as we know it, then you're just back to a different flavor of evolutionism.
Well, first of all, an interesting part quantum theory dictates that matter materialises out of nothing on a regular basis, but that is largely beside the point. Your entire argument is based upon assumptions that you do not back up. You say the Universe cannot spontaneously be created out of nothing because, well, the you can't create stuff out of nothing. It's a circular argument!
The reason that you cannot have moral absolutism without appealing to God is that I can disagree with you, and you have no basis with which to refute me. Suppose that you say murder is evil. Suppose that I were to argue that no, murder is good! Without any absolute standard, we could argue all day back and forth about it, and neither of us would be any more right than the other, because we're both just humans. It's the old, "What is right for you might not be right for me" business. "Who are you to say that murder is bad? I say I'm the exception, and my opionion is just as valid as yours!" We know that this is wrong, because we do have moral absolutes set out for us. It's just that not all of us recognize where those absolutes came from.
Um, sorry? Again, you fail to justify your arguments, and ironically, you appear to claim that this is because you assume that beliefs have to be justified! If I claim that it is immoral to kill, and claim that such is an base property of the Universe (in the same way that large clumps of matter seem to pull other large clumps of matter together), then this claim is no less valid then a Creator making it as such.
As a practical example, Buddhism is a belief that includes both atheism and moral absolutism. Karma (though I'm told this is a frequently misunderstood concept of Buddhism, and doubtless I am butchering the concept of it) deals out punishment to those who do wicked deeds simply because that's the way the Universe works.
You argue that any base for morality needs a conscious creator, yet provide no logical proof for such a case. In other words, back up your arguments!
I should have worded my thoughts before better. You are correct, atheism itself just believes there is no God. But you cannot believe something in a vacuum, all by itself. If you are to remain consistent, atheism requires you to believe in a multitude of other things, because a multitude of other things are dependent upon God. So by not believing in God, you must take on a plethora of other beliefs, by default. Otherwise, as I already stated, you quickly become very inconsistent.
Whilst I can accept that the belief that there is no god may logically imply other beliefs, you have not provided sound logical arguments for any of the things you claim that atheism implies. If you make a claim, for instance that atheism implies moral relativism, you must prove that claim through logical consitancy. In other words, the ball's in your court, not mine.
Nobody created God, He is eternal. You are setting up a false comparison here. Morals are not equal to God. Morals have no conscious thought, they do not live, they are just a set of rules. Morals, just like the universe, do not simply appear out of nowhere.
Gah! Again with the assumptions. "Morals... do not simply appear out of nowhere". That's a fine theory. Now back it up. Why cannot morals be eternal? Why can a conscious being exist without creation, and yet something that is not sentiant has to be created? Unless you show that consciousness is logically necessary for an entity to exist without creation (and good luck on that one!), then the ent
Jabber is moderately P2P, or at least as P2P as email is. There is no central server that jabber clients have to use. Two jabber clients can be on separate servers, but still communicate. In theory, one can have a server per client, and thus true P2P, but it is rather less efficient.
The only reason I can think of having a true P2P IM is for anonymity. Otherwise, if you just want decentralisation, use Jabber.
As an added bonus, many jabber servers have "gateways", something built into the jabber protocol that allows intercommication between other IM services. Ideally, that also means that one only has to patch the server to keep up with stunts like the one MS has pulled. The clients don't need to worry.
That said, the MSN gateway on my jabber server is down. D'oh! Doubtless it'll be patched in a while though:)
Well, I meant that if the definition in the Patriot act of "communications provider" was only to apply to proper ISPs, then there would be no problem. If the FBI's definition is right, then obviously you're more or less screwed. You can't set up a limited company for a mailing list or something trivial like that. Well, not unless you're a big corperation.
And, interestingly, the US does have a lot more specific laws than the UK. As I understand it (and IANAL, but I've had people try to explain this to me), British law has a lot of ambiguous statements in like "to the best of your ability". In fact, if I recall right, though I might be completely wrong (I frequently am), UK accounting practise says that accounts must be a "best estimate". Many of Enron's creative accounting practices, for example, were legal in the US, but probably not in the UK, due to the "best estimate" clause.
Which is what I mean by ambiguous laws, which obviously have up sides and down sides.
At least in the UK, a limited company is a separate legal entity to whomever is in it, so one would imagine that if you do run an ISP, it would be set up as a limited company and thus stand on it's own.
As for being "A provider of wire or electronic communication services", it strikes me that the law could be written a lot more unambiguously. Especially for a country like the US, that likes unambiguous laws.
Even with exponential growth, I doubt we'll need to assign any more ip addresses, then, say 4 billion a second. Even if we needed to assign 18 million million million addresses per second, it would still last us nearly 600 billion years. As such, within the foreseeable future, this is not a worry at all.
Because then the path length is greater than n * 2w where n is the number of mirrors and w is the wavelength of the photon (which is a constant). Thus your path length is now infinite and takes an infinite time for a photon to traverse it.
Yep; I realised that after I posted. D'oh!:)
But it's irrelevant anyway. This is an argument for indivisible, discrete things within the Universe. In effect, Zeno was arguing that some things have to be indivisible, like photons, otherwise you'd have a paradox.
The difficulty only arises when you attempt to divide the world into an infinite number of slices. Of course that won't work, because infinity doesn't exist!
Why?
You might not be able to create such a structure, but in an infinitely divisible Universe, such a structure could exist. Of course, you're right that the Universe isn't infinitely divisible- not just because Zeno's argument demands it, but because we have solid theoretical reasons why.
Besides, singularities are generally accepted as being real, so infinities exist even in a non-continuous reality.
Please show a way in which calculus does not completely describe the results of the Achilles & the tortoise paradox. You can calculate the time by summing a simple limit, likewise the distance. You get the same answer you do when you don't divide things into an infinite number of slices. Sounds pretty complete to me.
I have. The set of infinite mirrors thought experiment cannot be explained by calculas if reality is totally continuous.
When the first thing you do in a thought experiment is postulate an impossibility, you've gone wrong. It would be just as reasonable to say that I have a beam of light bouncing between two mirrors, without loss. Now I wait an infinite amount of time... which direction is the light travelling? Obviously, by contradiction, this implies light can't bounce between mirrors. Wait, you say you don't have an infinite amount of time to wait? I say you don't have an infinite number of mirrors, and can't postulate a way in which it's even theoretically possible to build them.
Your counter example is flawed, as you cannot wait an infinite amount of time. However, in a universe that is infinitely divisible, there is no theoretical reason why such a maze of mirrors could not exist. Assuming the mirrors are perfect, then the light beam loses no energy. The distance the light beam travels is finite. The matter it takes to create this setup is finite. The time it takes for the light beam to travel through the maze of mirrors is finite.
On the other hand, light does interact in the universe. It makes perfect sense to analyze how light "experiences" the universe.
So if I were travelling at the speed of light, watching a light beam travelling toward me, I would see that it- no, wait, I can't see what it would do. The set up of an observer travelling at the speed of light is impossible. Is, therefore, the thought experiment debunked?
To put it another way, what reason do you have that such a system cannot exist in a totally continuous Universe? Forget about how it's created; this is a theoretical struction, like Einstein's lightbeam-riding-chair. In a continuous, infinitely divisible Universe, could this mirror structure be created?
The link you gave looks to me like a lot of handwaving as to what Zeno's Stadium paradox said - I can find no reference that claims to have a reasonably accurate translation, the most they can get to is that it involves bodies moving towards each other and somehow the velocity equals half the velocity. These sources (including your own) say as much
It was a bit hand-wavy, wasn't it? But I noticed some key points. First of all, the Arrow argument implies that an arrow at rest is physically different from an arrow in flight. Special Relativity tells us that that is true. And "somehow the velocity equals half the velocity"? Sounds suspiciously like relative motion to me!
I'm not saying Zeno would have somehow discovered Special Relativity on his own. The ancient greek philosophers like creating paradoxes to impress their peers, but didn't like solving them nearly as much. But Zeno did touch upon relative motion and the interconnectivity of time and space, certainly more than anyone did until Newton. And even Newton didn't accept the relativity of space implied by his own theories- something which Zeno would probably have had no trouble questioning.
While an inventive thought experiment, you neglected to mention that the reflecting beam of light possesses a wavelength. When the separation between the two mirrors drops below this wavelength, your assumption about beam reflection from a surface breaks down.
Why not have the distance between the mirrors as (1/(n^2) + 2w) where w is the wavelength of the beam of light?
Right on the first point, but the last point is not at all clear to me. Special relativity shows that simple-minded Galilean transforms cause electrodynamics to break, but that a Lorentz transform preserves these relationships. Could you elaborate on why you think special relativity is involved here?
See here. Zeno's achilles argument isn't the only part of Zeno's paradox.
I suck at physics, but won't this experiment break when the vertical distance between two consecutive mirrors is shorter than the diameter of a photon?
Well, you could always have the distance between the mirrors as being (1/(2^n) + d) where d is the diameter of a photon:)
The mirror setup is impossible, unless you are capable of producing infinitessimally small mirrors.
I believe that was the point:)
The mirror paradox is an extension of Zeno's achilles argument. By agreeing that you cannot produce infintessimally small mirrors you're essencially agreeing with him that space is discrete:)
Calculus does in fact completely solve the problem of summing the infinite number of ever-smaller timeslices that Achilles takes to pass the tortoise.
No it doesn't. The mirror thought experiment is an extension of Zeno's achilles argument, and calculas obviously doesn't explain that! How can the direction of the light beam be determined by the last mirror in the sequence if there is no last mirror?
And you have no understanding of relativity if you think that Zeno's Paradox implies relativity. Special relativity is nothing more and nothing less than exploring the complete implications (ignoring acceleration and gravity) of the fact that the speed of light is a constant regardless of your velocity. Zeno didn't know the speed of light was a constant, and without that piece of information there are no grounds to postulate special relativity.
I think we're talking about two different things there. Zeno had four arguments, of which the Achilles argument is only one of. Essencially, Zeno was arguing that reality must be discrete, and yet it must be continuous (Achilles argues for discreteness). The Stadium arguement involves too bodies passing each other at speed (already this starts to sound familiar, no?). See here for more information on that (it does a better, and more accurate job at explaning that then I ever could!)
An "infinite number" of mirrors can't exist, so not being able to determine what they would do is not a failing of physics; it's a failing of your thought experiment.
Um... Well, firstly, you don't really understand the concept of a thought experiment. For instance, it's impossible to ride a light beam and for us to travel at the speed of light (well, so far as we know), but by reasoning out what would happen under such circumstances, Einstein formulated the Special Theory of Relativity. The 'infinite mirrors' extention of Zeno's Achilles argument is proof by contradiction.
There is a tradition of using calculas to solve this problem, and my explanation of the method behind the Achilles example was, well, iffy:)
That said, take the mirrors example (or a similar set up like Zeno's maze). That's essencially the same problem, but set up so that the direction of the moving object (ie. the light beam) depends on the last mirror. Calculas doesn't explain this.
Having shown that reality is discrete (though I find it dubious that this proves all reality is discrete), he then goes on with his 'Arrow' thought experiment to say that if there's no difference between a moving object and still object at a discrete interval in time, what makes the moving one move and the still one stay still? The paper seems to argue that this is because the discrete intervals are between certain times- that the invervals aren't infinitely small.
Again though, that's fairly obvious, and it all seems a bit iffy to me. I'm not even sure I've argued it all correctly without mistakes, though at least no-one has yet called me a moron for disparging the great name of Zeno. Check out here for a more in depth explanation.
How can there be infinitely small if you have a definite endpoint (zero)? Plus, if you go too small the distance between mirrors will be less than the size of a photon, so how can light bounce back and forth any longer?
Eheh. I realised once I posted there was a much more simple reply I could have made. I should have just pointed out that any infinite series by definition has no end. Therefore there is no end mirror, but at the same time the light only travels a finite distance before escaping, and that the direction of the light after it leaves depends on the last mirror, ie. it's a paradox.
In real life, matter is not infinitely divisible. Photons have size, as do the mirrors. But that was Zeno's point. The fact that it's wrong is the whole point- it's reducto ad absurdum (sp?) - proof by contradiction.
Light actually can stand still.
Yes, but it wouldn't. Assuming the light is of zero thickness, then the light beam would be continuously bouncing between the mirrors. It wouldn't stop, as the mirrors never converge to zero.
At this point, the distance between the mirrors is zero, which means the light must be standing still. If light cannot stand still, this must mean the distance between the mirrors is not zero, and hence does the example not apply (because it makes the assumption that mirrors can have zero distance).
Infinity isn't a number, and something infintesimately small is not zero. However far you go with the mirrors, the light is still bouncing. Assuming a light beam has no width to speak of, there is no point at which the light beam is not bouncing. And yet the distance the light beam travels is finite, even if the mirrors are infinite! Therefore the light will emerge in a finite space of time, and at every point will be bouncing and zigzagging along. You're right that it's impossible, of course, but that's why it's a paradox!
What do you mean "there cannot"? If time is discrete, Zenon's paradox does not apply, because it talks about timeslices smaller than what the actual ones would be.
Sorry; I meant you cannot have a discrete slice of time if time is continuous. However, I've said elsewhere that the paper in the article seemed, well, dubious. I'm not saying I agree with the paper, or that the paper is of any import as the article seems to suggest. Just that the original poster misunderstood what the paper was proposing.
Zeno's paradox does not claim Achilles can never catch up with the tortoise; making such a claim would require talking about infinite time -- Zeno's paradox does only talk about the time before Achilles catches up with the tortoise, hence the correct conclusion is "Achilles cannot possibly catch up with the turtle in the timeframe before he catches up with the turtle".
Well, the quote for Zeno's Achilles paradox I have is: "The slower will never be overtaken by the quicker, for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must always be some distance ahead."
The 'solution' seems a little obvious, too. I mean, I see where he's coming from, but the solution, seems, well, odd for something of such proclaimed import. The paper seems to be saying that you cannot take an instance of time in real life, just a specific interval.
Unless I'm missing something, that's something that's really quite obvious- I mean, exact measurement is obviously impossible in the real world. Everything's going to have an error ratio. Besides, Planck specifically put a lower limit on the duration of time possible to observe. Infinitely divisible reality is a discredited ancient greek theory, and something that Zeno's paradoxes specifically discredit.
I personally can't see any difference between Zeno's implication that time and space cannot be infinitely divided, and this new paper that seems to just proclaim what Zeno was implying all along.
I think you're missing the point. Let's take a more practical example; assume there's a set of mirrors facing each other, with the mirrors getting ever closer in a geometric sequence:
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If a lightbeam is fired in it'll zigzag between mirrors, with the zigzag getting smaller and smaller. Now, obviously if, say, the distance between each set of mirrors halves each time, then we have the sum of a geometric sequence. If the lightbeam travels 1m between mirrors, then 0.5m, then 0.25m and so forth, then the light beam will travel 2m in total. So the lightbeam will eventually emerge from these infinite mirrors. So what direction will the lightbeam be heading in? Up, or down? It depends on the last mirror- but there is no last mirror! Thus we have a paradox.
Of course, today we know that matter is not infinitely divisable, but that was Zeno's point! You cannot have a continuous function in real life and divide it into discrete segments! In fact, 'poor Zeno' was well ahead of his time, not only arguing against infinitely divisible, but also touching on Relativity! His 'stadium' paradox of two bodies of objects passing each other essencially begs the solution of Special Relatively.
In the archilles paradox, the runner will always have further to go. If time and space can be divided into discrete slices, then the runner will have to transverse an infinite number of slices to get to his destination, which is impossible. Infinity isn't a number, it's a position which is unreachable through finite additions. Therefore, the runner cannot overtake the tortoise, because he has to go through and infinite amount of 'time-slices' to get there. The solution in the article is that time is continuous; there cannot be a discrete slice of time, only a duration of time between two points.
I'm not going to say you are catagorically wrong because I'm not a lawyer, but I think your suggestion of no fair use is misleading. My understanding is that there is such a requirement implicit within copyright law.
Well, it's my understanding that there's allowances in UK law. You're allowed to quote samples of text and suchlike for critical review. But you can't make entire copies of copyrighted material for whatever reason.
If this was not the case then how could they justify placing an additioanl royalty cost on audio cassesttes, MDs and audio CDRs.
Uhm, as far as I know they don't, at least not by law. You can sell cassettes, MDs and audio CDRs for whatever price you want.
Whilst the UK may lack the draconian DMCA, and, hopefully, will resist such similar laws being created by the EU (though I doubt it), the UK also doesn't have the same Fair Use laws as in the US (IIRC). Here it's illegal to make tapes/mp3s from CDs even if it's just for your own use. Fair Use laws would be very nice over here:)
Though, to be fair, there hasn't really been any complaining by the music industry here about personal copying; so even if it is illegal, no-one's bothering to do anything about it. One person I knew pointed out that it's somewhat of a tradition for the British to overlook laws they view as too trivial to enforce, like underage drinking for instance. Pubs might get fined if an officer happens to notice an obvious young teenager getting a pint of beer, but the police don't go looking for such things. After all, there are larger problems about.
That doesn't stop traffic police from being just as picky as anywhere else though:)
Slashdot Mirror
Good point :)
:)
It must be said that I enjoy arguing religion and philosophy for the sake of it. So I'll hop over there and take you up on your offer. Thanks
I would if I could but it's not my server. Hopped onto jabber.org.uk (which got far more takers than the hoster originally intended). So I just need to wait. I'm in no hurry, anyway :)
Okay. Let's go through this one section at a time.
And what triggered the spontaneity? Where did the "stuff" come from? How can you have a creation without a Creator? "Stuff" does not simply materialize out of nothing, all by itself. If you acknowledge there was some kind of matter in the universe and then it spontaneously became the universe as we know it, then you're just back to a different flavor of evolutionism.
Well, first of all, an interesting part quantum theory dictates that matter materialises out of nothing on a regular basis, but that is largely beside the point. Your entire argument is based upon assumptions that you do not back up. You say the Universe cannot spontaneously be created out of nothing because, well, the you can't create stuff out of nothing. It's a circular argument!
The reason that you cannot have moral absolutism without appealing to God is that I can disagree with you, and you have no basis with which to refute me. Suppose that you say murder is evil. Suppose that I were to argue that no, murder is good! Without any absolute standard, we could argue all day back and forth about it, and neither of us would be any more right than the other, because we're both just humans. It's the old, "What is right for you might not be right for me" business. "Who are you to say that murder is bad? I say I'm the exception, and my opionion is just as valid as yours!" We know that this is wrong, because we do have moral absolutes set out for us. It's just that not all of us recognize where those absolutes came from.
Um, sorry? Again, you fail to justify your arguments, and ironically, you appear to claim that this is because you assume that beliefs have to be justified! If I claim that it is immoral to kill, and claim that such is an base property of the Universe (in the same way that large clumps of matter seem to pull other large clumps of matter together), then this claim is no less valid then a Creator making it as such.
As a practical example, Buddhism is a belief that includes both atheism and moral absolutism. Karma (though I'm told this is a frequently misunderstood concept of Buddhism, and doubtless I am butchering the concept of it) deals out punishment to those who do wicked deeds simply because that's the way the Universe works.
You argue that any base for morality needs a conscious creator, yet provide no logical proof for such a case. In other words, back up your arguments!
I should have worded my thoughts before better. You are correct, atheism itself just believes there is no God. But you cannot believe something in a vacuum, all by itself. If you are to remain consistent, atheism requires you to believe in a multitude of other things, because a multitude of other things are dependent upon God. So by not believing in God, you must take on a plethora of other beliefs, by default. Otherwise, as I already stated, you quickly become very inconsistent.
Whilst I can accept that the belief that there is no god may logically imply other beliefs, you have not provided sound logical arguments for any of the things you claim that atheism implies. If you make a claim, for instance that atheism implies moral relativism, you must prove that claim through logical consitancy. In other words, the ball's in your court, not mine.
Nobody created God, He is eternal. You are setting up a false comparison here. Morals are not equal to God. Morals have no conscious thought, they do not live, they are just a set of rules. Morals, just like the universe, do not simply appear out of nowhere.
Gah! Again with the assumptions. "Morals... do not simply appear out of nowhere". That's a fine theory. Now back it up. Why cannot morals be eternal? Why can a conscious being exist without creation, and yet something that is not sentiant has to be created? Unless you show that consciousness is logically necessary for an entity to exist without creation (and good luck on that one!), then the ent
Jabber is moderately P2P, or at least as P2P as email is. There is no central server that jabber clients have to use. Two jabber clients can be on separate servers, but still communicate. In theory, one can have a server per client, and thus true P2P, but it is rather less efficient.
:)
The only reason I can think of having a true P2P IM is for anonymity. Otherwise, if you just want decentralisation, use Jabber.
As an added bonus, many jabber servers have "gateways", something built into the jabber protocol that allows intercommication between other IM services. Ideally, that also means that one only has to patch the server to keep up with stunts like the one MS has pulled. The clients don't need to worry.
That said, the MSN gateway on my jabber server is down. D'oh! Doubtless it'll be patched in a while though
You are required to *not* believe in God.
True.
You are required to *not* believe in creation ex nihilo because that would require the existence of God.
Why? One could believe in the spontaneous creation of the Universe, as the enlightened AC pointed out.
You are required to *not* believe in moral absolutism, because absolutes in morality requires a God to set them.
Why?
Atheism does have opinions on all of the things you mentioned.
No it doesn't. Atheism is the belief that there is no God. Nothing more, nothing less.
Who gets to set the morals?
Who created God? That such morals were created by an entity is as much a supposition as there being a creater of God.
You cannot have an absolute without having God to set that absolute.
Gah! Why why why? You don't offer up any argument for this! Besides, that's a contradictary statement, as "God" is an absolute!
Well, I meant that if the definition in the Patriot act of "communications provider" was only to apply to proper ISPs, then there would be no problem. If the FBI's definition is right, then obviously you're more or less screwed. You can't set up a limited company for a mailing list or something trivial like that. Well, not unless you're a big corperation.
And, interestingly, the US does have a lot more specific laws than the UK. As I understand it (and IANAL, but I've had people try to explain this to me), British law has a lot of ambiguous statements in like "to the best of your ability". In fact, if I recall right, though I might be completely wrong (I frequently am), UK accounting practise says that accounts must be a "best estimate". Many of Enron's creative accounting practices, for example, were legal in the US, but probably not in the UK, due to the "best estimate" clause.
Which is what I mean by ambiguous laws, which obviously have up sides and down sides.
At last, I've found a line worthy of my .sig! :)
Can I quote you on that?
At least in the UK, a limited company is a separate legal entity to whomever is in it, so one would imagine that if you do run an ISP, it would be set up as a limited company and thus stand on it's own.
As for being "A provider of wire or electronic communication services", it strikes me that the law could be written a lot more unambiguously. Especially for a country like the US, that likes unambiguous laws.
"Cast in the name of God: Ye not guilty."
:)
Hee hee
So...McBride reads /. eh? I'm onto you Pavan_Gupta (623567), or should I say Darl McBride! That's right, you can't fool us!
Even with exponential growth, I doubt we'll need to assign any more ip addresses, then, say 4 billion a second. Even if we needed to assign 18 million million million addresses per second, it would still last us nearly 600 billion years. As such, within the foreseeable future, this is not a worry at all.
Because then the path length is greater than n * 2w where n is the number of mirrors and w is the wavelength of the photon (which is a constant). Thus your path length is now infinite and takes an infinite time for a photon to traverse it.
:)
Yep; I realised that after I posted. D'oh!
But it's irrelevant anyway. This is an argument for indivisible, discrete things within the Universe. In effect, Zeno was arguing that some things have to be indivisible, like photons, otherwise you'd have a paradox.
The difficulty only arises when you attempt to divide the world into an infinite number of slices. Of course that won't work, because infinity doesn't exist!
Why?
You might not be able to create such a structure, but in an infinitely divisible Universe, such a structure could exist. Of course, you're right that the Universe isn't infinitely divisible- not just because Zeno's argument demands it, but because we have solid theoretical reasons why.
Besides, singularities are generally accepted as being real, so infinities exist even in a non-continuous reality.
Please show a way in which calculus does not completely describe the results of the Achilles & the tortoise paradox. You can calculate the time by summing a simple limit, likewise the distance. You get the same answer you do when you don't divide things into an infinite number of slices. Sounds pretty complete to me.
I have. The set of infinite mirrors thought experiment cannot be explained by calculas if reality is totally continuous.
When the first thing you do in a thought experiment is postulate an impossibility, you've gone wrong. It would be just as reasonable to say that I have a beam of light bouncing between two mirrors, without loss. Now I wait an infinite amount of time... which direction is the light travelling? Obviously, by contradiction, this implies light can't bounce between mirrors. Wait, you say you don't have an infinite amount of time to wait? I say you don't have an infinite number of mirrors, and can't postulate a way in which it's even theoretically possible to build them.
Your counter example is flawed, as you cannot wait an infinite amount of time. However, in a universe that is infinitely divisible, there is no theoretical reason why such a maze of mirrors could not exist. Assuming the mirrors are perfect, then the light beam loses no energy. The distance the light beam travels is finite. The matter it takes to create this setup is finite. The time it takes for the light beam to travel through the maze of mirrors is finite.
On the other hand, light does interact in the universe. It makes perfect sense to analyze how light "experiences" the universe.
So if I were travelling at the speed of light, watching a light beam travelling toward me, I would see that it- no, wait, I can't see what it would do. The set up of an observer travelling at the speed of light is impossible. Is, therefore, the thought experiment debunked?
To put it another way, what reason do you have that such a system cannot exist in a totally continuous Universe? Forget about how it's created; this is a theoretical struction, like Einstein's lightbeam-riding-chair. In a continuous, infinitely divisible Universe, could this mirror structure be created?
The link you gave looks to me like a lot of handwaving as to what Zeno's Stadium paradox said - I can find no reference that claims to have a reasonably accurate translation, the most they can get to is that it involves bodies moving towards each other and somehow the velocity equals half the velocity. These sources (including your own) say as much
It was a bit hand-wavy, wasn't it? But I noticed some key points. First of all, the Arrow argument implies that an arrow at rest is physically different from an arrow in flight. Special Relativity tells us that that is true. And "somehow the velocity equals half the velocity"? Sounds suspiciously like relative motion to me!
I'm not saying Zeno would have somehow discovered Special Relativity on his own. The ancient greek philosophers like creating paradoxes to impress their peers, but didn't like solving them nearly as much. But Zeno did touch upon relative motion and the interconnectivity of time and space, certainly more than anyone did until Newton. And even Newton didn't accept the relativity of space implied by his own theories- something which Zeno would probably have had no trouble questioning.
While an inventive thought experiment, you neglected to mention that the reflecting beam of light possesses a wavelength. When the separation between the two mirrors drops below this wavelength, your assumption about beam reflection from a surface breaks down.
Why not have the distance between the mirrors as (1/(n^2) + 2w) where w is the wavelength of the beam of light?
Right on the first point, but the last point is not at all clear to me. Special relativity shows that simple-minded Galilean transforms cause electrodynamics to break, but that a Lorentz transform preserves these relationships. Could you elaborate on why you think special relativity is involved here?
See here. Zeno's achilles argument isn't the only part of Zeno's paradox.
I suck at physics, but won't this experiment break when the vertical distance between two consecutive mirrors is shorter than the diameter of a photon?
:)
Well, you could always have the distance between the mirrors as being (1/(2^n) + d) where d is the diameter of a photon
The mirror setup is impossible, unless you are capable of producing infinitessimally small mirrors.
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I believe that was the point
The mirror paradox is an extension of Zeno's achilles argument. By agreeing that you cannot produce infintessimally small mirrors you're essencially agreeing with him that space is discrete
How did this get a 5?
;)
That's what I want to know
Calculus does in fact completely solve the problem of summing the infinite number of ever-smaller timeslices that Achilles takes to pass the tortoise.
No it doesn't. The mirror thought experiment is an extension of Zeno's achilles argument, and calculas obviously doesn't explain that! How can the direction of the light beam be determined by the last mirror in the sequence if there is no last mirror?
And you have no understanding of relativity if you think that Zeno's Paradox implies relativity. Special relativity is nothing more and nothing less than exploring the complete implications (ignoring acceleration and gravity) of the fact that the speed of light is a constant regardless of your velocity. Zeno didn't know the speed of light was a constant, and without that piece of information there are no grounds to postulate special relativity.
I think we're talking about two different things there. Zeno had four arguments, of which the Achilles argument is only one of. Essencially, Zeno was arguing that reality must be discrete, and yet it must be continuous (Achilles argues for discreteness). The Stadium arguement involves too bodies passing each other at speed (already this starts to sound familiar, no?). See here for more information on that (it does a better, and more accurate job at explaning that then I ever could!)
An "infinite number" of mirrors can't exist, so not being able to determine what they would do is not a failing of physics; it's a failing of your thought experiment.
Um... Well, firstly, you don't really understand the concept of a thought experiment. For instance, it's impossible to ride a light beam and for us to travel at the speed of light (well, so far as we know), but by reasoning out what would happen under such circumstances, Einstein formulated the Special Theory of Relativity. The 'infinite mirrors' extention of Zeno's Achilles argument is proof by contradiction.
There is a tradition of using calculas to solve this problem, and my explanation of the method behind the Achilles example was, well, iffy :)
That said, take the mirrors example (or a similar set up like Zeno's maze). That's essencially the same problem, but set up so that the direction of the moving object (ie. the light beam) depends on the last mirror. Calculas doesn't explain this.
Having shown that reality is discrete (though I find it dubious that this proves all reality is discrete), he then goes on with his 'Arrow' thought experiment to say that if there's no difference between a moving object and still object at a discrete interval in time, what makes the moving one move and the still one stay still? The paper seems to argue that this is because the discrete intervals are between certain times- that the invervals aren't infinitely small.
Again though, that's fairly obvious, and it all seems a bit iffy to me. I'm not even sure I've argued it all correctly without mistakes, though at least no-one has yet called me a moron for disparging the great name of Zeno. Check out here for a more in depth explanation.
How can there be infinitely small if you have a definite endpoint (zero)? Plus, if you go too small the distance between mirrors will be less than the size of a photon, so how can light bounce back and forth any longer?
Eheh. I realised once I posted there was a much more simple reply I could have made. I should have just pointed out that any infinite series by definition has no end. Therefore there is no end mirror, but at the same time the light only travels a finite distance before escaping, and that the direction of the light after it leaves depends on the last mirror, ie. it's a paradox.
In real life, matter is not infinitely divisible. Photons have size, as do the mirrors. But that was Zeno's point. The fact that it's wrong is the whole point- it's reducto ad absurdum (sp?) - proof by contradiction.
Light actually can stand still.
Yes, but it wouldn't. Assuming the light is of zero thickness, then the light beam would be continuously bouncing between the mirrors. It wouldn't stop, as the mirrors never converge to zero.
At this point, the distance between the mirrors is zero, which means the light must be standing still. If light cannot stand still, this must mean the distance between the mirrors is not zero, and hence does the example not apply (because it makes the assumption that mirrors can have zero distance).
Infinity isn't a number, and something infintesimately small is not zero. However far you go with the mirrors, the light is still bouncing. Assuming a light beam has no width to speak of, there is no point at which the light beam is not bouncing. And yet the distance the light beam travels is finite, even if the mirrors are infinite! Therefore the light will emerge in a finite space of time, and at every point will be bouncing and zigzagging along. You're right that it's impossible, of course, but that's why it's a paradox!
What do you mean "there cannot"? If time is discrete, Zenon's paradox does not apply, because it talks about timeslices smaller than what the actual ones would be.
Sorry; I meant you cannot have a discrete slice of time if time is continuous. However, I've said elsewhere that the paper in the article seemed, well, dubious. I'm not saying I agree with the paper, or that the paper is of any import as the article seems to suggest. Just that the original poster misunderstood what the paper was proposing.
Zeno's paradox does not claim Achilles can never catch up with the tortoise; making such a claim would require talking about infinite time -- Zeno's paradox does only talk about the time before Achilles catches up with the tortoise, hence the correct conclusion is "Achilles cannot possibly catch up with the turtle in the timeframe before he catches up with the turtle".
Well, the quote for Zeno's Achilles paradox I have is: "The slower will never be overtaken by the quicker, for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must always be some distance ahead."
The 'solution' seems a little obvious, too. I mean, I see where he's coming from, but the solution, seems, well, odd for something of such proclaimed import. The paper seems to be saying that you cannot take an instance of time in real life, just a specific interval.
Unless I'm missing something, that's something that's really quite obvious- I mean, exact measurement is obviously impossible in the real world. Everything's going to have an error ratio. Besides, Planck specifically put a lower limit on the duration of time possible to observe. Infinitely divisible reality is a discredited ancient greek theory, and something that Zeno's paradoxes specifically discredit.
I personally can't see any difference between Zeno's implication that time and space cannot be infinitely divided, and this new paper that seems to just proclaim what Zeno was implying all along.
Of course, today we know that matter is not infinitely divisable, but that was Zeno's point! You cannot have a continuous function in real life and divide it into discrete segments! In fact, 'poor Zeno' was well ahead of his time, not only arguing against infinitely divisible, but also touching on Relativity! His 'stadium' paradox of two bodies of objects passing each other essencially begs the solution of Special Relatively.
In the archilles paradox, the runner will always have further to go. If time and space can be divided into discrete slices, then the runner will have to transverse an infinite number of slices to get to his destination, which is impossible. Infinity isn't a number, it's a position which is unreachable through finite additions. Therefore, the runner cannot overtake the tortoise, because he has to go through and infinite amount of 'time-slices' to get there. The solution in the article is that time is continuous; there cannot be a discrete slice of time, only a duration of time between two points.
I'm not going to say you are catagorically wrong because I'm not a lawyer, but I think your suggestion of no fair use is misleading. My understanding is that there is such a requirement implicit within copyright law.
Well, it's my understanding that there's allowances in UK law. You're allowed to quote samples of text and suchlike for critical review. But you can't make entire copies of copyrighted material for whatever reason.
If this was not the case then how could they justify placing an additioanl royalty cost on audio cassesttes, MDs and audio CDRs.
Uhm, as far as I know they don't, at least not by law. You can sell cassettes, MDs and audio CDRs for whatever price you want.
Whilst the UK may lack the draconian DMCA, and, hopefully, will resist such similar laws being created by the EU (though I doubt it), the UK also doesn't have the same Fair Use laws as in the US (IIRC). Here it's illegal to make tapes/mp3s from CDs even if it's just for your own use. Fair Use laws would be very nice over here :)
:)
Though, to be fair, there hasn't really been any complaining by the music industry here about personal copying; so even if it is illegal, no-one's bothering to do anything about it. One person I knew pointed out that it's somewhat of a tradition for the British to overlook laws they view as too trivial to enforce, like underage drinking for instance. Pubs might get fined if an officer happens to notice an obvious young teenager getting a pint of beer, but the police don't go looking for such things. After all, there are larger problems about.
That doesn't stop traffic police from being just as picky as anywhere else though