The entire concept behind black holes is that they do in fact occupy 0 volume. They have a location, but not a size. Thats what it means to have a "singularity", where the forces that normally keep particles from coexisting in the same space are no longer sufficient to overcome the force of gravity pulling everything together.
Having never looked into a black hole to find out, it could be that our math describing them is incorrect and they do in fact occupy some space. But as far as we can tell from calculating mathematical forces, there shouldn't be any possible way for a black hole to occupy positive volume -- any particle that wasn't exactly superimposed over all the rest would inevitably be drawn closer to the rest until it was. In theory, the only thing that should stop the process of collapse would be the fundamental limit of the smallest possible particle size. But particles that small are believed to act more like waves than particles anyway, and there is, in theory, no smallest possible wave size. You can continue to make a wave smaller and smaller without limit, so that is most likely what happens inside a black hole.
Now, this shares more in common with the mathematical concept of lim(x->inf) 1/x = 0 then it does with normal 0, but considering how far limits are embroiled into this conversation to begin with, that shouldn't be a problem.
Your triangle analogy doesn't work, by the way. It's easier to explain if we view the numbers from the other side -- instead of counting the number of turns, lets count the length of the line segments. As the length of the line segments becomes smaller, meaning you are taking more and more turns to get to the end, the total distance walked remains exactly the same. So the limit of your walk as the line segment length approaches zero is still exactly 200. But if the line segment length is exactly 0, the problem is undefined -- you won't reach the other side at all. In both cases, 200 is still != 141.
<!--quoteo(post=1688831:date=Sep 25 2008, 02:54 AM:name=Hawkeye)--><div class='quotetop'>QUOTE(Hawkeye @ Sep 25 2008, 02:54 AM) <a href="index.php?act=findpost&pid=1688831"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->Take the example from the link on infinity for an example. If you have a chocolate bar and divide it into infinite pieces, where did all the pieces go? It's not that they're really really small pieces.. they no longer exist!<!--QuoteEnd--></div><!--QuoteEEnd-->
Its not that they no longer exist...its that they have no volume. They become points, rather than cubes, and you have infinitely many of them. You cannot simply jump out of the mathematical relm of things such to say "it does not exist".
It seems you are attempting to disprove the theory of infinity all together, but in order to do so, you must accept it as truth and prove it illogical.
In your chocolate argument, if you accept the theory of infinity, it is reasonable to say I can have an infinite large number of infinite small pieces.
<!--quoteo(post=1688831:date=Sep 25 2008, 02:54 AM:name=Hawkeye)--><div class='quotetop'>QUOTE(Hawkeye @ Sep 25 2008, 02:54 AM) <a href="index.php?act=findpost&pid=1688831"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->Tell me one thing that has 0 volume which everyone here will agree exists. One thing.<!--QuoteEnd--></div><!--QuoteEEnd-->
<!--quoteo(post=1688855:date=Sep 25 2008, 09:26 PM:name=homicide)--><div class='quotetop'>QUOTE(homicide @ Sep 25 2008, 09:26 PM) <a href="index.php?act=findpost&pid=1688855"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->A photon.<!--QuoteEnd--></div><!--QuoteEEnd--> That isn't true either. Photons have volume when volume is tested for. Of course when you test for waves, the photon appears to have no volume, but that is a completely different can of worms.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->Quantum particles. Everything.<!--QuoteEnd--></div><!--QuoteEEnd--> What quantum particles?
Subatomic particles have no volume? Which subatomic particles? Quarks have mass and volume. Gluons we know almost nothing about.
And last time I checked, "everything" simply can't have no volume because observation proves this incorrect. Take a graduated cylinder and tell me that it has 0 volume, with a straight face.
* x5 reads thread * * x5 tries to remember where his old anatomical diagrams are... *
So... are we talking about stuff like this from years ago on this forum but in terms of the marines only? <img src="http://www.xzianthia.net/images/onos_leftside.jpg" border="0" class="linked-image" />
Hmm... Well, come to think of it, I never theorized much about the marines, but if you all have any technical biologically-correct questions about the Kharaa, please ask! I love figuring out how it could work -- what would make it realistically plausible.
There <i>is</i> a balance that can be between immersive realism and fun gameplay. (and in truth often they compliment each other more than they conflict!)
Looks like you guys are grabbing some Philosophy, slapping on some numbers to qualify it as Mathematics, then trying to label it as Science. Such fun.
<!--quoteo(post=1688678:date=Sep 23 2008, 11:53 PM:name=Align)--><div class='quotetop'>QUOTE(Align @ Sep 23 2008, 11:53 PM) <a href="index.php?act=findpost&pid=1688678"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->But the universe contains math, not the other way around...<!--QuoteEnd--></div><!--QuoteEEnd--> Not according to Stephen Hawking.
<!--quoteo(post=1688861:date=Sep 25 2008, 04:23 PM:name=aNytiMe)--><div class='quotetop'>QUOTE(aNytiMe @ Sep 25 2008, 04:23 PM) <a href="index.php?act=findpost&pid=1688861"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->That isn't true either. Photons have volume when volume is tested for. Of course when you test for waves, the photon appears to have no volume, but that is a completely different can of worms. What quantum particles?
Subatomic particles have no volume? Which subatomic particles? Quarks have mass and volume. Gluons we know almost nothing about.<!--QuoteEnd--></div><!--QuoteEEnd-->
Quarks are shown to have mass, but this has nothing to do with volume. True.. they are wavelike as well as particle like, but they are "point like" particles. This does not imply that the particle must have a volume. Photon are the same.
<!--quoteo(post=1688861:date=Sep 25 2008, 04:23 PM:name=aNytiMe)--><div class='quotetop'>QUOTE(aNytiMe @ Sep 25 2008, 04:23 PM) <a href="index.php?act=findpost&pid=1688861"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->And last time I checked, "everything" simply can't have no volume because observation proves this incorrect. Take a graduated cylinder and tell me that it has 0 volume, with a straight face.<!--QuoteEnd--></div><!--QuoteEEnd-->
Are you nuts? A graduated cylinder SUPPORTS the theory that volume can be composed of many many very very small particles. Surely you agree that if I were to fill a graduated cylinder with water, that that water would be composed of atoms, which are composed of (almost) purely empty space. This is an example of how we can observe "volume" that is in reality composed of tiny particles.
In fact, I would have a hard time telling you that a graduated cylinder full of water DID have volume. Of course I would describe it that way to an elementary student simply because it is a good way to "model" the water, but I would know that in reality the water is composed of infinitely smaller particles.
While treating things as "solid" is intuitive, it has been shown false over and over.
<!--quoteo(post=1688876:date=Sep 26 2008, 03:37 AM:name=homicide)--><div class='quotetop'>QUOTE(homicide @ Sep 26 2008, 03:37 AM) <a href="index.php?act=findpost&pid=1688876"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->Quarks are shown to have mass, but this has nothing to do with volume. True.. they are wavelike as well as particle like, but they are "point like" particles. This does not imply that the particle must have a volume. Photon are the same. Are you nuts? A graduated cylinder SUPPORTS the theory that volume can be composed of many many very very small particles. Surely you agree that if I were to fill a graduated cylinder with water, that that water would be composed of atoms, which are composed of (almost) purely empty space. This is an example of how we can observe "volume" that is in reality composed of tiny particles.<!--QuoteEnd--></div><!--QuoteEEnd--> How can we test for sure if something has a volume? Well the smallest thing we can actually observe are the atomic nuclei. Take a tunneling microscope and look at an atom, you will observe volume. Was there any way we could have known whether protons had volume before we knew about electrons? We knew protons had mass, but we had no idea whether they had volume until we could actually observed them. What does this mean? Nothing. Further speculation has no scientific basis and as far as I'm concerned, if something has mass, it has volume until proven wrong, not the other way around.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->In fact, I would have a hard time telling you that a graduated cylinder full of water DID have volume. Of course I would describe it that way to an elementary student simply because it is a good way to "model" the water, but I would know that in reality the water is composed of infinitely smaller particles.<!--QuoteEnd--></div><!--QuoteEEnd--> In the case of your argument on whether a particle can be infinitely divided you have as much proof as I do therefore our opinions cancel out and we are left with a ?. Thanks to modern theory however, we assume something has volume simply because we haven't found any exceptions.
Also, Harimau, contribute something or stop trolling.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->But yea, I'd like us to scale back our nerdy discussions and go back to debating the mechanisms of space travel in NS, phase tech, fade blink and other stuff.<!--QuoteEnd--></div><!--QuoteEEnd-->
Sorry, Anytime! I suppose I should stop responding to these inquiries on infinity, but I enjoy it too much to stop. It's like potato chips for geeks.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->Its not that they no longer exist...its that they have no volume. They become points, rather than cubes, and you have infinitely many of them. You cannot simply jump out of the mathematical relm of things such to say "it does not exist".
It seems you are attempting to disprove the theory of infinity all together, but in order to do so, you must accept it as truth and prove it illogical.
In your chocolate argument, if you accept the theory of infinity, it is reasonable to say I can have an infinite large number of infinite small pieces.<!--QuoteEnd--></div><!--QuoteEEnd-->
That's precisely why I say infinity does not exist in any sense in this world. To accept infinity, you have to accept certain realities that are rather difficult to comprehend, and while I cannot prove infinity does not exist, I can show what silly reality we'd be living in if our universe had no borders.
And yes, in my chocolate argument, It is reasonable to say you have infinite large number of infinite small pieces, all of which have 0 volume and have 0 mass. If you think infinity can exist, you must also accept that you can make a chocolate bar disappear by dividing it infinite number of times. You could claim it would be impossible to do so because of the mechanics of it, but that's the point, really, isn't it? You can't divide it into infinite pieces because you can't have infinite pieces of chocolate bar with 0 volume and 0 mass.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->A photon. Quantum particles. Everything.<!--QuoteEnd--></div><!--QuoteEEnd--> Photon has mass. That's a well-established fact, though you're welcome to disagree with that fact.
Most quantum particles have been proven to have a mass and/or charge. The few that don't are things like electrons which have a strong charge and seem to not have any volume at all. Though due to the nature of the fact that it has a strong charge, it makes an accurate reading very difficult. Besides, Anytime had a good point. At those scales, it is difficult to even know whether such a 'mass' measurement is its true value or if it is the inaccuracy factor of the instrument you're using to measure.
As for everything, once you get down to that level, the basic understanding of 'volume' breaks down entirely. Techncially there's no such thing as volume, but merely the cumulation of buffer spaces between atoms and between electrons and their nuclei, as there's nothing in it. However we call it volume. That's cheating a bit. What I implied was that you couldn't divide anything so much that it would have 0 mass AND 0 volume (that is to say, existing but not having any influence on the universe). You could say points are such, but points are non-existent. They exist only in concept, just like infinity. What is a point afterall but the inverse of infinite space?
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->The entire concept behind black holes is that they do in fact occupy 0 volume. They have a location, but not a size. Thats what it means to have a "singularity", where the forces that normally keep particles from coexisting in the same space are no longer sufficient to overcome the force of gravity pulling everything together.<!--QuoteEnd--></div><!--QuoteEEnd-->
Not true. Black holes are merely compressed atoms. Granted, the amount of compression is tremendous. If an atom were the size of a football stadium, they say an electron would be the size of a ping pong ball whizzing at unimaginable speeds and the nucleus would be the size of a tennis ball in the very center. However, that's not to say that if you brought the ping pong ball and the tennis ball within close proximity that they'd vanish or something of that nature. A black hole is a very real object, having very little volume and very large mass.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->Your triangle analogy doesn't work, by the way. It's easier to explain if we view the numbers from the other side -- instead of counting the number of turns, lets count the length of the line segments. As the length of the line segments becomes smaller, meaning you are taking more and more turns to get to the end, the total distance walked remains exactly the same. So the limit of your walk as the line segment length approaches zero is still exactly 200. But if the line segment length is exactly 0, the problem is undefined -- you won't reach the other side at all. In both cases, 200 is still != 141.<!--QuoteEnd--></div><!--QuoteEEnd-->
You said it yourself. The problem is undefined. You can't do that operation to achieve a correct result because the answer is undefined. Using infinity results in undefined or incorrect answers (or correct for infinity, depending on how you look at it). I think you've proven that infinity cannot truly exist better than I could have. To reiterate, I'll get into the mathematics of it.
The equation is this:
x (100 / x + 100 / x) = y
x represents the number of 'stops' and 100 / x represents the distance of each stop. For each stop, you make a width pass and a length pass. (100 / x + 100 / x). You multiply that times the number of passes to determine how far you go, so x * (100 / x + 100 / x) = y.
Graph that and see what you get. 200 = y, essentially. It's a horizontal line that goes off into the horizon where pigs fly and fat ladies sing. It's the place of impossibilities and concepts which do not really exist in any shape or form. It's also the place where y becomes 141.421 meters.
If 141.421 != 200, then you must therefore conclude that the means by which I used to get that number is flawed. Though the square root of (100^2 + 100^2) is 141.421 and the limit of 200 = y as x approaches infinity is 200, so where did I go wrong? If infinity exists, then this is a perfectly legal operation (unless you want to dispute one of my other two computations). If we were making the same discussion for the limit of x as x approaches 32, we'd find that in fact 200 = 200, and that we're still in sane country. The error comes in assuming infinity is like any other number. It isn't. Finite real numbers in operations with infinity allows you to create scenarios otherwise not feasible in mathematics and the world in general.
<!--quoteo(post=1688880:date=Sep 25 2008, 09:49 PM:name=aNytiMe)--><div class='quotetop'>QUOTE(aNytiMe @ Sep 25 2008, 09:49 PM) <a href="index.php?act=findpost&pid=1688880"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->In the case of your argument on whether a particle can be infinitely divided you have as much proof as I do therefore our opinions cancel out and we are left with a ?. Thanks to modern theory however, we assume something has volume simply because we haven't found any exceptions.<!--QuoteEnd--></div><!--QuoteEEnd-->
No... we don't, you have it backwards. Modern physics assume the exact opposite and model subatomic particles as infinitely small points, simply because this is the simplest explanation and any other model would be speculation.
However, Hawkeye attempted to "prove" that the universe is finite but relied on the the premise that "everything must have volume." Because this premise has never been proven, this proof holds no ground.
<!--quoteo(post=1688880:date=Sep 25 2008, 09:49 PM:name=aNytiMe)--><div class='quotetop'>QUOTE(aNytiMe @ Sep 25 2008, 09:49 PM) <a href="index.php?act=findpost&pid=1688880"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->Further speculation has no scientific basis and as far as I'm concerned, if something has mass, it has volume until proven wrong, not the other way around.<!--QuoteEnd--></div><!--QuoteEEnd-->
What form of science are you following? Until any theory has been proven wrong, we just don't know. From there, the most reasonable approach is the simplest approach.
By definition, to state something has volume is to say it can be divide. It is simpler to assume something cannot be divided.
This is the general approach taken in all modern physics. Until we successfully divide a particle we model it as an infinitesimal point because any other model including volume would be speculation.
-
<b> By stating something must have volume to exist you are effectively stating it must be divisible. It follows then, that anything that exists...can be divided an infinite number of times, yet you are claiming infinite is an unreachable feat. This theory in itself contradicts.
It is simpler and more logical to assume there are fundamental particles that cannot be divided, infinitesimal, by definition. This theory is sound and has no logical contradiction. </b>
A photon, by definition has no mass. More specifically, as defined by special relatively a photon is a particle which is defined to have no "rest mass". Thus, given the law of relativistic masses the photon is the only particle which can be accelerated to the speed of light (the speed of a photon), c. This is not a something neither you not I can "agree with", this is how a photon is "defined".
more importantly, as it relates to this debate...MASS != VOLUME. The presence of mass doesn't even intel volume. It is perfectly reasonable that an object without volume can have mass.
Anyways...to say a photon has no volume is to say it cannot be divided into a particle which is smaller within the first three dimensions of space. Because this has never been done, I along with most of the scientific community assumes this to be true because it is the simplest explanation. Although this is the commonly accepted theory, no one would base a "proof" based on it.
You.. surprisingly, have actually based a proof that rests on the commonly accepted theory as being false.
<!--quoteo(post=1688890:date=Sep 26 2008, 08:07 AM:name=homicide)--><div class='quotetop'>QUOTE(homicide @ Sep 26 2008, 08:07 AM) <a href="index.php?act=findpost&pid=1688890"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->No... we don't, you have it backwards. Modern physics assume the exact opposite and model subatomic particles as infinitely small points, simply because this is the simplest explanation and any other model would be speculation.<!--QuoteEnd--></div><!--QuoteEEnd--> Really? In chemistry I've been taught that it is not the simplest explanation because it involves having infinite density etc...
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->By definition, to state something has volume is to say it can be divide. It is simpler to assume something cannot be divided.<!--QuoteEnd--></div><!--QuoteEEnd--> When something is infinitely dense, it can be proven that something can hypothetically have infinite mass. If something had infinite mass, it would crush everything, even Arnold Schwarzenegger.
<!--quoteo(post=1688886:date=Sep 26 2008, 02:45 AM:name=Hawkeye)--><div class='quotetop'>QUOTE(Hawkeye @ Sep 26 2008, 02:45 AM) <a href="index.php?act=findpost&pid=1688886"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->And yes, in my chocolate argument, It is reasonable to say you have infinite large number of infinite small pieces, all of which have 0 volume and have 0 mass. If you think infinity can exist, you must also accept that you can make a chocolate bar disappear by dividing it infinite number of times. You could claim it would be impossible to do so because of the mechanics of it, but that's the point, really, isn't it? You can't divide it into infinite pieces because you can't have infinite pieces of chocolate bar with 0 volume and 0 mass.<!--QuoteEnd--></div><!--QuoteEEnd-->
Not quite. The pieces don't have 0 mass and volume, they just have an infinitesimally small mass and volume. Basically its infinity in the other direction.
And its entirely acceptable that you can make a chocolate bar disappear by dividing it. You don't even have to invoke infinity for that. You can divide it just a few hundred times and you won't be able to keep track of the shards anymore. You can divide it a much larger but still finite number of times, and the pieces will be small enough that they don't reflect light anymore, meaning the bar has quite literally disappeared. (The pieces aren't really chocolate anymore by this point, but thats another issue.)
On a subatomic scale, there is no particular reason to assume we CANT divide it infinitely. Of course, we don't know for sure that we CAN either, but its possible. We don't know enough about very small subatomic particles to know if theres a point at which you cant divide them any further. We don't know enough about the nature of space to say whether an infinitely small volume is even possible, or whether there is some smallest possible unit of volume that is either "filled" or "not filled" with no in between.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->What I implied was that you couldn't divide anything so much that it would have 0 mass AND 0 volume (that is to say, existing but not having any influence on the universe). You could say points are such, but points are non-existent. They exist only in concept, just like infinity. What is a point afterall but the inverse of infinite space?<!--QuoteEnd--></div><!--QuoteEEnd-->
Again, not 0, just infinitesimally small. 1 / infinity is not actually equal to zero -- its just <i>approximately</i> equal to zero.
And now that I think about it, this is a good spot to break off from answering you point by point and go back to your original question.
You wanted to tell us that the universe couldn't be infinite in size because we would then occupy an infinitely small portion of that space. But "infinitely small" is not synonymous with "zero". Infinity multiplied by true zero is still zero, but infinity multiplied by an infinitesimal can produce a real number.
Suppose the size of the universe is represented by X, and the size of earth is Y. The portion of the universe that we occupy can then be described as Y/X. If you know the portion we occupy, but not our size, you can find it by taking (Y/X) * X. So what happens if the universe is infinite in size? Lim(x->inf) x * (y/x) = y We still ocupy Y size, no matter how big the universe gets. Even if it is truly boundless, we still come out to size Y.
I have other problems with the universe being infinite in size -- I personally think it has a very definite size. But being unable to figure out what portion of the universe we occupy is not one of my problems with infinity.
If we stick with the pattern presented to us, we should have 0.999999 (0.9 bar). So if 0.9 bar isn't 1, then why the difference? What changes? I thought 3 / 3 was 1.. I guess we'll have to completely tear down the fundamentals of mathematics because 3 / 3 is not 1 but 0.9 bar. Right?
What I want to know is, what is 1 - 0.9 bar? What is the difference? According to you, they aren't the same number, so I should be able to have a non-zero number by subtracting the two, right? Are you really going to tell me it is 0 / infinity? If so, you're going to have to prove to me why 0 / infinity is not zero. I don't think you're able to, because in my opinion it's not possible.
The concept of infinity is just that, a concept. If x / infinity were anything other than zero (say y), I'd be able to perform the operation 1 / y and get a finite number. But that's just it, it couldn't possibly be finite, because I can arrive at any finite number performing x / z where z is a number that can produce z / x = 1 / y.
Seems too much like sweeping it under the rug to say that something divided by infinity is an infinitely small number. It's not a new idea you're introducing. An infinitely small number is zero.
If we stick with the pattern presented to us, we should have 0.999999 (0.9 bar). So if 0.9 bar isn't 1, then why the difference? What changes? I thought 3 / 3 was 1.. I guess we'll have to completely tear down the fundamentals of mathematics because 3 / 3 is not 1 but 0.9 bar. Right?
What I want to know is, what is 1 - 0.9 bar? What is the difference? According to you, they aren't the same number, so I should be able to have a non-zero number by subtracting the two, right?<!--QuoteEnd--></div><!--QuoteEEnd-->
What? Where did you get the idea I would think they aren't the same number? .9 bar IS equal to 1, so if you subtract the two, you will get exactly zero. I don't know what you think you are proving with this.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->The concept of infinity is just that, a concept. If x / infinity were anything other than zero (say y), I'd be able to perform the operation 1 / y and get a finite number.<!--QuoteEnd--></div><!--QuoteEEnd-->
That isn't what you would get at all. You would get infinity / x, which is A number, but its not a FINITE number. That doesn't mean you can't perform operations on it.
Do you recall the use of the letter h from calculus? h was used to represent a number that was approximately zero -- but it couldn't <i>actually</i> be zero because the equations involving h all required you to divide by h at some point. If h was equal to zero, you would have a divide-by-zero error and couldn't complete the analysis, but if h is equal to anything BUT zero, you don't get the right answer. And yet we still managed to come up with an answer, usually by reaching the point where we could say h / h = 1 even though that equation looks remarkably like 0 / 0. Or perhaps 2h / h = 2, even though its still 0 / 0. In both cases you get a finite number, and it can be several different finite numbers depending on the circumstance, but its never an uncertain number. It's never 1 AND 2, just one or the other.
If you all really get stuck you could always use <a href="http://askanexpert.web.cern.ch/AskAnExpert/" target="_blank">this web-form</a> to ask an expert at CERN.
I for one would like to see a debate about super-symmetry and dark matter.
HL2 has already supported this theory by talking about dark-energy reactors (HL2: Episode 1 primarily when you get to have fun re-stabilizing one) and the my favorite weapon the is the dark energy orb alternate fire on the AR2 (aka. Overwatch Standard Issue Pulse Rifle). Granted this is all fiction, but it's based on a real work, next-generation physics theory called super symmetry.
[<a href="http://half-life.wikia.com/wiki/Dark_energy" target="_blank">click to view a wiki topic discussing dark energy in HL2</a>]
The Large Hadron Collider was supposed to start exploring some particles that might give experimental data for these and other theories but of course the helium leak has it temporarily shut down, probably until 2009. [<a href="http://press.web.cern.ch/press/PressReleases/Releases2008/PR10.08E.html" target="_blank">press release here</a>]
Still, to get the topic back on track here you all COULD try to shift the argument into how such theories might be adapted to NS2's TSA technology like Infantry Portals and Phasegates. This might lead to tips for Jason and Cory on how the dynamic sprites should look and so-such. (or just provide plausible backing to the content for immersive purposes)
I've heard of h, though we called it by a different name. It came up a lot in theoretical computer science when trying to minimize error in floating point calculations. h is practical because it is a finite number, albeit small. The idea was that if you were in a loop in a program doing the same add calculation between numbers with h as an error margin, you were suming or even multiplying your error margin, so while h is small, it would quickly grow if you weren't careful.
(For example, for you programmers, try summing 1.0F a million times together and then dividing the result by a million. What you get won't be 1.0F because of the inability for a computer to represent 1.0F with exact precision)
However, h even if it is an unnamed value, it is still finite. You could proceed to tell me that h is really really <i>really</i> small and that I could even pretend it were just like an infinitely small number, but you'd still be inaccurate. So long as I can multiply h * 1/h and arrive at 1, it is not infinitely small. I wish I could give some sort of better proof for it, but the reasons for why you cannot do that are precisely the opposite of the reasons you can with h being a finite number. I'd have to show you with infinity, which by the sake of our argument is exactly what I'm trying to prove does not exist. You could argue that infinity * 42 = 7, and I wouldn't have a leg to stand on because gosh darn it, where is infinity when you need to prove that it doesn't exist anyway?
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->What? Where did you get the idea I would think they aren't the same number? .9 bar IS equal to 1, so if you subtract the two, you will get exactly zero. I don't know what you think you are proving with this.<!--QuoteEnd--></div><!--QuoteEEnd-->
If an infinitely small number is not zero, what number do you get when you subtract 1 from that number? I thought for sure you would have said it is 0.9 bar, though if you can't say it's 1 because 1 - 1 = 0 and you argued that an infinitely small number is not 0, and you can't say it is 0.9 bar because you just stated that 0.9 bar *is* 1. So I ask, what is the number you get when you subtract an infinitely small number from 1?
I think the problem here is that the universe (space) isn't infinite in size, but it simply doesn't exist. Space is just a realm of coordinates, it is a concept and isn't an actual substance. This is why you can move in a certain direction for as long as you can (if there was no gravity). Is there anything else which can be considered "infinite?"
<!--quoteo(post=1689225:date=Oct 2 2008, 12:48 AM:name=Hawkeye)--><div class='quotetop'>QUOTE(Hawkeye @ Oct 2 2008, 12:48 AM) <a href="index.php?act=findpost&pid=1689225"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->If an infinitely small number is not zero, what number do you get when you subtract 1 from that number? I thought for sure you would have said it is 0.9 bar, though if you can't say it's 1 because 1 - 1 = 0 and you argued that an infinitely small number is not 0, and you can't say it is 0.9 bar because you just stated that 0.9 bar *is* 1. So I ask, what is the number you get when you subtract an infinitely small number from 1?<!--QuoteEnd--></div><!--QuoteEEnd-->
You get 1 - h. You can't combine them really, because infinitesimals and infinities don't interact well with real numbers. But depending on what you want to DO with that 1 - h, you might still have to pay attention to that h hanging off the end.
For most purposes, you can treat 1 - h as being equal to 1. If you were to subtract 1 from that though, then you have to remember your answer is -h instead of 0. And then depending on what you want to DO with that -h, you can probably treat it just as if it were 0 anyway, until you try to divide by h. But when you do that, I hope you've been keeping track of what you did to that infinitesimal all this time up till now, so you will know what number pops out at the end. It might be 7. <img src="style_emoticons/<#EMO_DIR#>/smile-fix.gif" style="vertical-align:middle" emoid=":)" border="0" alt="smile-fix.gif" />
Your programming example is slightly different, due to the limitations of writing numbers in binary form. Computers can't exactly store infinity as a number. In calculus, h is a number larger than 0 but smaller than every possible number larger than 0.
<!--quoteo(post=1689242:date=Oct 2 2008, 08:39 AM:name=aNytiMe)--><div class='quotetop'>QUOTE(aNytiMe @ Oct 2 2008, 08:39 AM) <a href="index.php?act=findpost&pid=1689242"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->Is there anything else which can be considered "infinite?"<!--QuoteEnd--></div><!--QuoteEEnd--> If something is infinitely large, could it also no be said that there is infinitely small? (limit in the microscopic direction instead of macroscopic, if you will)
It's important not to confuse infinity with undefined. While I agree with you that space is an abstract of coordinates and distances relative to each other, much like time is an abstract of observance of change, it almost seemed like you were arguing space is not infinite because it is an undefine-able abstract. (unless I'm misunderstanding what you meant to say) Space is infinite, not undefined; but I totally agree with your definition of it being an abstraction of how our minds observe (or cope with) the fundamental concept.
In my opinion, root measurements that all measurements are derived from -- such as time and mass -- all seem to be a relative abstract for a higher concept our minds simply can't understand. Implications of this include why I do not believe you can time travel backwards. Sure you can accelerate or slow time, but what is time really? Is it not fair to say that time is nothing more that the observance of change? We describe how quickly or slowly something changed in terms of time but time has to relative to reliable constant that can be reproduced experimentally. Today we label time with the SI unit called a second and define it experimentally as: "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom." This is a highly accurate and reliable constant, but my point is that the constant is still in terms of the period of a change. It's an abstract. FYI, the other so-called SI base units are the meter for distance, the kilogram for mass, the second for time, the ampere for electric current, the kelvin for temperature, the mole for amount of substance, and the candela for intensity of light. Yet many of these definitions are a measurement based on mass and time.
Anytime, you talked about space being an abstract of coordinates. The meter (SI unit of distance) is defined in terms of <i>c</i> which is the velocity of light in a vacuum, equal to exactly 299,792,458 meters per <!--coloro:#FFFF00--><span style="color:#FFFF00"><!--/coloro-->second<!--colorc--></span><!--/colorc-->. Ok, see here we go again, time in terms of a second. In mathematical terms, isn't this just like how velocity is the integral of acceleration, if you keep on integrating, at some point all you are left with is the abstract, right? So what are we left with? An abstract called time, an abstract called mass, and an infinite abstract called space where it all exists?
<!--QuoteBegin-Cxwf+--><div class='quotetop'>QUOTE(Cxwf)</div><div class='quotemain'><!--QuoteEBegin-->You get 1 - h. You can't combine them really, because infinitesimals and infinities don't interact well with real numbers. But depending on what you want to DO with that 1 - h, you might still have to pay attention to that h hanging off the end.<!--QuoteEnd--></div><!--QuoteEEnd--> Fair enough, but is it not possible that the real limitation is our method of describing nature in mathematical terms and computational logic? Whole numbers, real numbers, imaginary numbers, abstracts, infinitesimals, infinites, etc. all described and computed with operands like addition, subtraction, multiplication, division, exponentiation, derivation, integration, etc. If we are having trouble computing, perhaps there's a higher operand and/or a larger set of numeric values that we have not yet conceived with our collective brains.
Through out all of historic scientific discovery, has mathematics not had a direct relationship with observing experiments and learning how to describe them? I would argue we, as humans of the planet earth relative to October 2nd in 2008 CE, haven't discovered a wider picture on the concept yet.
PS: Again I ask, how such things could relate to NS2, much like I hinted at in the dark energy (super-symmetry theory) in HL2 reference. This topic has become fascinatingly off-topic.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->This topic has become fascinatingly off topic<!--QuoteEnd--></div><!--QuoteEEnd-->
Indeed it has, which is what makes it so fascinating. <img src="style_emoticons/<#EMO_DIR#>/smile-fix.gif" style="vertical-align:middle" emoid=":)" border="0" alt="smile-fix.gif" />
Incidentally, I don't believe the universe IS actually infinite. I'm just defending the concept of infinity in general, but I suspect our current universe has a definate size. A while back, someone convinced me that Cosmic Background Radiation was residue from the Big Bang, which makes absolutely no sense unless space is curved. But there's no reason space <i>can't</i> be curved, given that we know Gravity curves space to some extent, so it seems likely to me that our universe is a hollow sphere in 4D space, with the entirety of our 3D space being the surface of that sphere. The size of the universe is therefore limited by the radius of that sphere, and the sphere expands as the universe grows. Theoretically the sphere had zero radius at the time of the Big Bang.
I'm not saying space is an abstract, I'm saying that space simply doesn't exist. It is an illusion perceived by our senses. What really exists though are coordinates. Every single particle has a coordinate as a function. Nothing is observed to have a coordinate of infinity.
I would argue that h is not truly existent. It is a tool for engineers who have been told a thousand times that they cannot divide by zero and want to cheat a little bit. It is the value of 1/x as x approaches minus infinity. Theoretically, that value is zero, but for all practical intents and purposes, it is a really really really really small number.
So what do you get when you divide a number by a really really really really small number? A really really really really big number. Theoretically, this is the same thing as infinity, but in practice a number so large that you won't have to worry about anything being bigger than it.
Like imaginary numbers, we use these tools to help our calculations, because rather than throwing your hands up in the air because your answer produces imaginary numbers and 'imaginary numbers don't exist' we use it as a tool for finding the answer to our problem. Likewise, you use h to represent a really small number that doesn't have a number smaller than itself before zero even if that doesn't exist either.
What are the rules of the real number system anyway?
1) Between real number x and real number y, there exists a real number z such that x < z < y.
2) For all other rules, see integer number system.
h fails your test, since there's no number you can put between h and zero. Therefore, it doesn't exist as a number. It is a concept like infinity. Showing that infinity exists by giving me an example of another concept doesn't convince me, I'm afraid.
Nothing is infinite in this world, not even the coordinate system. Unless you've got a pencil which doesn't wear down and a piece of paper of infinite size or a computer with a magical infinite memory size, you cannot possibly represent it. The reality is that we label it infinity and deal with numbers within our capability of dealing with. That's no closer to infinity than a hand full of sand is an approximate representation of all the sand on the planet.
As for infinite smallness (anything can be divided further), where's your proof for that? Burden of proof is not on me but those of you who want to convince me that infinity exists in some real sense.
<!--quoteo(post=1689307:date=Oct 3 2008, 07:15 AM:name=Hawkeye)--><div class='quotetop'>QUOTE(Hawkeye @ Oct 3 2008, 07:15 AM) <a href="index.php?act=findpost&pid=1689307"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->h fails your test, since there's no number you can put between h and zero.<!--QuoteEnd--></div><!--QuoteEEnd-->
Sure there is: h/2. That's about halfway between h and zero, and if you divide h by h/2, you'll get the very real number 2, even though both of the numbers you started with were indescribable.
This is getting very abstract though, what were we arguing about again? I think you were trying to convince me that the universe couldn't be boundless, because you could no longer mathematically represent the portion of the universe that we would occupy?
To that I would respond that the integer number system is also boundless, and we routinely identify both locations and sizes within the number system. All you need to do is select one point to serve as a "zero" reference point, and identify all other locations relevant to that.
Your definition of h was that it was a number 'next to' zero. A number which does not have another number between itself and zero. If that's the case, and you say h/2 exists, you've already disproved h as a number. Can an infinitely small number be divided by 2? o.O If it doesn't make any sense, it's probably because it's not something you can do.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->This is getting very abstract though, what were we arguing about again? I think you were trying to convince me that the universe couldn't be boundless, because you could no longer mathematically represent the portion of the universe that we would occupy?<!--QuoteEnd--></div><!--QuoteEEnd-->
Pretty much, yep.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->To that I would respond that the integer number system is also boundless, and we routinely identify both locations and sizes within the number system. All you need to do is select one point to serve as a "zero" reference point, and identify all other locations relevant to that.<!--QuoteEnd--></div><!--QuoteEEnd-->
Like infinity, the integer number system is a concept as well. It doesn't really exist. The universe isn't mapped out on a grid with numbers hovering in midair. For it to *truly* exist, in some shape or form, every point would have to be capable of being mapped out, including the ones with coordinates so large, there are less atoms in the universe than the numbers for such coordinates.
It's really the same argument all over again. If the universe were boundless, our space in the universe would have no reference, size, shape, or volume.
By the way, a neat math problem I saw the other day (to further derail the thread):
You work at a factory which creates lottery balls, the ones they draw from a cage on a random basis.
You are given the instructions to unwrap a package with the following contents: a ball and two strips of number stickers from 0 to 9. Your task is to place the stickers on the ball to create the numbers starting from 1 and then up. However, the stickers you do not use in that package, you place aside to use later. So for example, I start by taking the '1' sticker and placing it on the ball, and the 0, 2, 3, 4, 5, 6, 7, 8, 9 and the other 0 - 9 number sticker strip go to a pile. Then I open another package and I take the '2' sticker and place it on the ball, placing the 0, 1, 3, 4, 5.. and the other 0 - 9 number sticker to the same pile.
At what point do I run out of numbers using this system? I'll give you a hint, it's not a small number.
Well, most lottery systems only go up to about 50, so you'd most likely stop after ball number 50. But if you really wanted to just keep going until you ran out of numbers, you'll need to start generating balls with 21-digit numbers before your stack of spares starts getting smaller instead of bigger. I estimate you would run out at the 21-digit ball numbered entirely with 2s.
Well, you're always going to be more short of 1s than any other number, seeing how it is the first number you start with. If I had a number of 22 digits of 2s, wouldn't I have to pass the number with 22 digits of 1s first?
In any case, that's not the solution either, but you're on the right track I suppose.
<!--quoteo(post=1689631:date=Oct 7 2008, 03:51 PM:name=Hawkeye)--><div class='quotetop'>QUOTE(Hawkeye @ Oct 7 2008, 03:51 PM) <a href="index.php?act=findpost&pid=1689631"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->Well, you're always going to be more short of 1s than any other number, seeing how it is the first number you start with. If I had a number of 22 digits of 2s, wouldn't I have to pass the number with 22 digits of 1s first?
In any case, that's not the solution either, but you're on the right track I suppose.<!--QuoteEnd--></div><!--QuoteEEnd-->
I don't think you'd be more short of 1's than anything else. They all cycle in order at the same frequency. The one number that does show up at a different rate is 0, since leading 0s aren't used, so you would have a surplus of 0s when you run out of everything else.
Also, you can't possibly reach 22-digit numbers, since there are 10 times as many 21-digit numbers as there were all numbers before that, and 21-digit numbers all drain your stack of reserves.
If you could figure out how many extra zeros you have left over at the 21-digit all 2s, and count backwards from that, you could probably find the point at which you run out of numbers that aren't zero.
Come to think of it, I believe the number of extra zeros you have is the same number I calculated for total spare numbers available when you reach 21-digit number territory. After all, all of the spares should have been leading zeros that weren't used, right? So the number where your reserves run out might very well be the very first 21-digit number, or 1 followed by 20 zeros. Except that can't possibly be the last number, because the previous number didn't use numbers at the same rate you opened them, and so numbers 2 through 9 which should have a reserve of "0" now must have had a reserve of "-1" last number.
So what we're really looking for is the point where the instantaneous demand for a digit is higher than the average supply of that digit up to this point, even though the average demand for that digit will equal the average supply eventually. And while all digits have a number at which their instantaneous demand is at a peak, the 1s digit tends to reach that peak before any other digit. However, the peak of maximum instantaneous demand is not likely to be the same point at which demand outstrips supply.
The supply of 1s is equal to two 1s per number, while the demand varies anywhere from zero to twenty. So how do you figure out the total demand for 1s up to a given number?
Comments
Having never looked into a black hole to find out, it could be that our math describing them is incorrect and they do in fact occupy some space. But as far as we can tell from calculating mathematical forces, there shouldn't be any possible way for a black hole to occupy positive volume -- any particle that wasn't exactly superimposed over all the rest would inevitably be drawn closer to the rest until it was. In theory, the only thing that should stop the process of collapse would be the fundamental limit of the smallest possible particle size. But particles that small are believed to act more like waves than particles anyway, and there is, in theory, no smallest possible wave size. You can continue to make a wave smaller and smaller without limit, so that is most likely what happens inside a black hole.
Now, this shares more in common with the mathematical concept of lim(x->inf) 1/x = 0 then it does with normal 0, but considering how far limits are embroiled into this conversation to begin with, that shouldn't be a problem.
Your triangle analogy doesn't work, by the way. It's easier to explain if we view the numbers from the other side -- instead of counting the number of turns, lets count the length of the line segments. As the length of the line segments becomes smaller, meaning you are taking more and more turns to get to the end, the total distance walked remains exactly the same. So the limit of your walk as the line segment length approaches zero is still exactly 200. But if the line segment length is exactly 0, the problem is undefined -- you won't reach the other side at all. In both cases, 200 is still != 141.
Singularities are still subatomic particles.
Edit: Are you talking about waves of probability?
Its not that they no longer exist...its that they have no volume. They become points, rather than cubes, and you have infinitely many of them. You cannot simply jump out of the mathematical relm of things such to say "it does not exist".
It seems you are attempting to disprove the theory of infinity all together, but in order to do so, you must accept it as truth and prove it illogical.
In your chocolate argument, if you accept the theory of infinity, it is reasonable to say I can have an infinite large number of infinite small pieces.
<!--quoteo(post=1688831:date=Sep 25 2008, 02:54 AM:name=Hawkeye)--><div class='quotetop'>QUOTE(Hawkeye @ Sep 25 2008, 02:54 AM) <a href="index.php?act=findpost&pid=1688831"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->Tell me one thing that has 0 volume which everyone here will agree exists. One thing.<!--QuoteEnd--></div><!--QuoteEEnd-->
A photon.
Quantum particles.
Everything.
That isn't true either. Photons have volume when volume is tested for. Of course when you test for waves, the photon appears to have no volume, but that is a completely different can of worms.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->Quantum particles.
Everything.<!--QuoteEnd--></div><!--QuoteEEnd-->
What quantum particles?
Subatomic particles have no volume?
Which subatomic particles? Quarks have mass and volume. Gluons we know almost nothing about.
And last time I checked, "everything" simply can't have no volume because observation proves this incorrect. Take a graduated cylinder and tell me that it has 0 volume, with a straight face.
* x5 tries to remember where his old anatomical diagrams are... *
So... are we talking about stuff like this from years ago on this forum but in terms of the marines only?
<img src="http://www.xzianthia.net/images/onos_leftside.jpg" border="0" class="linked-image" />
Hmm... Well, come to think of it, I never theorized much about the marines, but if you all have any technical biologically-correct questions about the Kharaa, please ask! I love figuring out how it could work -- what would make it realistically plausible.
<img src="style_emoticons/<#EMO_DIR#>/smile-fix.gif" style="vertical-align:middle" emoid=":)" border="0" alt="smile-fix.gif" />
There <i>is</i> a balance that can be between immersive realism and fun gameplay.
(and in truth often they compliment each other more than they conflict!)
Looks like you guys are grabbing some Philosophy, slapping on some numbers to qualify it as Mathematics, then trying to label it as Science. Such fun.
<!--quoteo(post=1688678:date=Sep 23 2008, 11:53 PM:name=Align)--><div class='quotetop'>QUOTE(Align @ Sep 23 2008, 11:53 PM) <a href="index.php?act=findpost&pid=1688678"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->But the universe contains math, not the other way around...<!--QuoteEnd--></div><!--QuoteEEnd-->
Not according to Stephen Hawking.
What quantum particles?
Subatomic particles have no volume?
Which subatomic particles? Quarks have mass and volume. Gluons we know almost nothing about.<!--QuoteEnd--></div><!--QuoteEEnd-->
Quarks are shown to have mass, but this has nothing to do with volume. True.. they are wavelike as well as particle like, but they are "point like" particles. This does not imply that the particle must have a volume. Photon are the same.
<!--quoteo(post=1688861:date=Sep 25 2008, 04:23 PM:name=aNytiMe)--><div class='quotetop'>QUOTE(aNytiMe @ Sep 25 2008, 04:23 PM) <a href="index.php?act=findpost&pid=1688861"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->And last time I checked, "everything" simply can't have no volume because observation proves this incorrect. Take a graduated cylinder and tell me that it has 0 volume, with a straight face.<!--QuoteEnd--></div><!--QuoteEEnd-->
Are you nuts? A graduated cylinder SUPPORTS the theory that volume can be composed of many many very very small particles. Surely you agree that if I were to fill a graduated cylinder with water, that that water would be composed of atoms, which are composed of (almost) purely empty space. This is an example of how we can observe "volume" that is in reality composed of tiny particles.
In fact, I would have a hard time telling you that a graduated cylinder full of water DID have volume. Of course I would describe it that way to an elementary student simply because it is a good way to "model" the water, but I would know that in reality the water is composed of infinitely smaller particles.
While treating things as "solid" is intuitive, it has been shown false over and over.
Are you nuts? A graduated cylinder SUPPORTS the theory that volume can be composed of many many very very small particles. Surely you agree that if I were to fill a graduated cylinder with water, that that water would be composed of atoms, which are composed of (almost) purely empty space. This is an example of how we can observe "volume" that is in reality composed of tiny particles.<!--QuoteEnd--></div><!--QuoteEEnd-->
How can we test for sure if something has a volume? Well the smallest thing we can actually observe are the atomic nuclei. Take a tunneling microscope and look at an atom, you will observe volume. Was there any way we could have known whether protons had volume before we knew about electrons? We knew protons had mass, but we had no idea whether they had volume until we could actually observed them. What does this mean? Nothing. Further speculation has no scientific basis and as far as I'm concerned, if something has mass, it has volume until proven wrong, not the other way around.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->In fact, I would have a hard time telling you that a graduated cylinder full of water DID have volume. Of course I would describe it that way to an elementary student simply because it is a good way to "model" the water, but I would know that in reality the water is composed of infinitely smaller particles.<!--QuoteEnd--></div><!--QuoteEEnd-->
In the case of your argument on whether a particle can be infinitely divided you have as much proof as I do therefore our opinions cancel out and we are left with a ?. Thanks to modern theory however, we assume something has volume simply because we haven't found any exceptions.
Also, Harimau, contribute something or stop trolling.
Sorry, Anytime! I suppose I should stop responding to these inquiries on infinity, but I enjoy it too much to stop. It's like potato chips for geeks.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->Its not that they no longer exist...its that they have no volume. They become points, rather than cubes, and you have infinitely many of them. You cannot simply jump out of the mathematical relm of things such to say "it does not exist".
It seems you are attempting to disprove the theory of infinity all together, but in order to do so, you must accept it as truth and prove it illogical.
In your chocolate argument, if you accept the theory of infinity, it is reasonable to say I can have an infinite large number of infinite small pieces.<!--QuoteEnd--></div><!--QuoteEEnd-->
That's precisely why I say infinity does not exist in any sense in this world. To accept infinity, you have to accept certain realities that are rather difficult to comprehend, and while I cannot prove infinity does not exist, I can show what silly reality we'd be living in if our universe had no borders.
And yes, in my chocolate argument, It is reasonable to say you have infinite large number of infinite small pieces, all of which have 0 volume and have 0 mass. If you think infinity can exist, you must also accept that you can make a chocolate bar disappear by dividing it infinite number of times. You could claim it would be impossible to do so because of the mechanics of it, but that's the point, really, isn't it? You can't divide it into infinite pieces because you can't have infinite pieces of chocolate bar with 0 volume and 0 mass.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->A photon.
Quantum particles.
Everything.<!--QuoteEnd--></div><!--QuoteEEnd-->
Photon has mass. That's a well-established fact, though you're welcome to disagree with that fact.
Most quantum particles have been proven to have a mass and/or charge. The few that don't are things like electrons which have a strong charge and seem to not have any volume at all. Though due to the nature of the fact that it has a strong charge, it makes an accurate reading very difficult. Besides, Anytime had a good point. At those scales, it is difficult to even know whether such a 'mass' measurement is its true value or if it is the inaccuracy factor of the instrument you're using to measure.
As for everything, once you get down to that level, the basic understanding of 'volume' breaks down entirely. Techncially there's no such thing as volume, but merely the cumulation of buffer spaces between atoms and between electrons and their nuclei, as there's nothing in it. However we call it volume. That's cheating a bit. What I implied was that you couldn't divide anything so much that it would have 0 mass AND 0 volume (that is to say, existing but not having any influence on the universe). You could say points are such, but points are non-existent. They exist only in concept, just like infinity. What is a point afterall but the inverse of infinite space?
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->The entire concept behind black holes is that they do in fact occupy 0 volume. They have a location, but not a size. Thats what it means to have a "singularity", where the forces that normally keep particles from coexisting in the same space are no longer sufficient to overcome the force of gravity pulling everything together.<!--QuoteEnd--></div><!--QuoteEEnd-->
Not true. Black holes are merely compressed atoms. Granted, the amount of compression is tremendous. If an atom were the size of a football stadium, they say an electron would be the size of a ping pong ball whizzing at unimaginable speeds and the nucleus would be the size of a tennis ball in the very center. However, that's not to say that if you brought the ping pong ball and the tennis ball within close proximity that they'd vanish or something of that nature. A black hole is a very real object, having very little volume and very large mass.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->Your triangle analogy doesn't work, by the way. It's easier to explain if we view the numbers from the other side -- instead of counting the number of turns, lets count the length of the line segments. As the length of the line segments becomes smaller, meaning you are taking more and more turns to get to the end, the total distance walked remains exactly the same. So the limit of your walk as the line segment length approaches zero is still exactly 200. But if the line segment length is exactly 0, the problem is undefined -- you won't reach the other side at all. In both cases, 200 is still != 141.<!--QuoteEnd--></div><!--QuoteEEnd-->
You said it yourself. The problem is undefined. You can't do that operation to achieve a correct result because the answer is undefined. Using infinity results in undefined or incorrect answers (or correct for infinity, depending on how you look at it). I think you've proven that infinity cannot truly exist better than I could have. To reiterate, I'll get into the mathematics of it.
The equation is this:
x (100 / x + 100 / x) = y
x represents the number of 'stops' and 100 / x represents the distance of each stop. For each stop, you make a width pass and a length pass. (100 / x + 100 / x). You multiply that times the number of passes to determine how far you go, so x * (100 / x + 100 / x) = y.
Graph that and see what you get. 200 = y, essentially. It's a horizontal line that goes off into the horizon where pigs fly and fat ladies sing. It's the place of impossibilities and concepts which do not really exist in any shape or form. It's also the place where y becomes 141.421 meters.
If 141.421 != 200, then you must therefore conclude that the means by which I used to get that number is flawed. Though the square root of (100^2 + 100^2) is 141.421 and the limit of 200 = y as x approaches infinity is 200, so where did I go wrong? If infinity exists, then this is a perfectly legal operation (unless you want to dispute one of my other two computations). If we were making the same discussion for the limit of x as x approaches 32, we'd find that in fact 200 = 200, and that we're still in sane country. The error comes in assuming infinity is like any other number. It isn't. Finite real numbers in operations with infinity allows you to create scenarios otherwise not feasible in mathematics and the world in general.
No... we don't, you have it backwards. Modern physics assume the exact opposite and model subatomic particles as infinitely small points, simply because this is the simplest explanation and any other model would be speculation.
However, Hawkeye attempted to "prove" that the universe is finite but relied on the the premise that "everything must have volume." Because this premise has never been proven, this proof holds no ground.
<!--quoteo(post=1688880:date=Sep 25 2008, 09:49 PM:name=aNytiMe)--><div class='quotetop'>QUOTE(aNytiMe @ Sep 25 2008, 09:49 PM) <a href="index.php?act=findpost&pid=1688880"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->Further speculation has no scientific basis and as far as I'm concerned, if something has mass, it has volume until proven wrong, not the other way around.<!--QuoteEnd--></div><!--QuoteEEnd-->
What form of science are you following? Until any theory has been proven wrong, we just don't know. From there, the most reasonable approach is the simplest approach.
By definition, to state something has volume is to say it can be divide. It is simpler to assume something cannot be divided.
This is the general approach taken in all modern physics. Until we successfully divide a particle we model it as an infinitesimal point because any other model including volume would be speculation.
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By stating something must have volume to exist you are effectively stating it must be divisible. It follows then, that anything that exists...can be divided an infinite number of times, yet you are claiming infinite is an unreachable feat. This theory in itself contradicts.
It is simpler and more logical to assume there are fundamental particles that cannot be divided, infinitesimal, by definition. This theory is sound and has no logical contradiction.
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A photon, by definition has no mass. More specifically, as defined by special relatively a photon is a particle which is defined to have no "rest mass". Thus, given the law of relativistic masses the photon is the only particle which can be accelerated to the speed of light (the speed of a photon), c. This is not a something neither you not I can "agree with", this is how a photon is "defined".
more importantly, as it relates to this debate...MASS != VOLUME. The presence of mass doesn't even intel volume. It is perfectly reasonable that an object without volume can have mass.
Anyways...to say a photon has no volume is to say it cannot be divided into a particle which is smaller within the first three dimensions of space. Because this has never been done, I along with most of the scientific community assumes this to be true because it is the simplest explanation. Although this is the commonly accepted theory, no one would base a "proof" based on it.
You.. surprisingly, have actually based a proof that rests on the commonly accepted theory as being false.
Really? In chemistry I've been taught that it is not the simplest explanation because it involves having infinite density etc...
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->By definition, to state something has volume is to say it can be divide. It is simpler to assume something cannot be divided.<!--QuoteEnd--></div><!--QuoteEEnd-->
When something is infinitely dense, it can be proven that something can hypothetically have infinite mass. If something had infinite mass, it would crush everything, even Arnold Schwarzenegger.
Not quite. The pieces don't have 0 mass and volume, they just have an infinitesimally small mass and volume. Basically its infinity in the other direction.
And its entirely acceptable that you can make a chocolate bar disappear by dividing it. You don't even have to invoke infinity for that. You can divide it just a few hundred times and you won't be able to keep track of the shards anymore. You can divide it a much larger but still finite number of times, and the pieces will be small enough that they don't reflect light anymore, meaning the bar has quite literally disappeared. (The pieces aren't really chocolate anymore by this point, but thats another issue.)
On a subatomic scale, there is no particular reason to assume we CANT divide it infinitely. Of course, we don't know for sure that we CAN either, but its possible. We don't know enough about very small subatomic particles to know if theres a point at which you cant divide them any further. We don't know enough about the nature of space to say whether an infinitely small volume is even possible, or whether there is some smallest possible unit of volume that is either "filled" or "not filled" with no in between.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->What I implied was that you couldn't divide anything so much that it would have 0 mass AND 0 volume (that is to say, existing but not having any influence on the universe). You could say points are such, but points are non-existent. They exist only in concept, just like infinity. What is a point afterall but the inverse of infinite space?<!--QuoteEnd--></div><!--QuoteEEnd-->
Again, not 0, just infinitesimally small. 1 / infinity is not actually equal to zero -- its just <i>approximately</i> equal to zero.
And now that I think about it, this is a good spot to break off from answering you point by point and go back to your original question.
You wanted to tell us that the universe couldn't be infinite in size because we would then occupy an infinitely small portion of that space. But "infinitely small" is not synonymous with "zero". Infinity multiplied by true zero is still zero, but infinity multiplied by an infinitesimal can produce a real number.
Suppose the size of the universe is represented by X, and the size of earth is Y. The portion of the universe that we occupy can then be described as Y/X. If you know the portion we occupy, but not our size, you can find it by taking (Y/X) * X. So what happens if the universe is infinite in size?
Lim(x->inf) x * (y/x) = y
We still ocupy Y size, no matter how big the universe gets. Even if it is truly boundless, we still come out to size Y.
I have other problems with the universe being infinite in size -- I personally think it has a very definite size. But being unable to figure out what portion of the universe we occupy is not one of my problems with infinity.
Take this for an example.
1 / 3 = 0.3333333 (0.3 bar)
2 / 3 = 0.6666666 (0.6 bar)
3 / 3 = 0.9999999 (0.9 bar) ?
If we stick with the pattern presented to us, we should have 0.999999 (0.9 bar). So if 0.9 bar isn't 1, then why the difference? What changes? I thought 3 / 3 was 1.. I guess we'll have to completely tear down the fundamentals of mathematics because 3 / 3 is not 1 but 0.9 bar. Right?
What I want to know is, what is 1 - 0.9 bar? What is the difference? According to you, they aren't the same number, so I should be able to have a non-zero number by subtracting the two, right? Are you really going to tell me it is 0 / infinity? If so, you're going to have to prove to me why 0 / infinity is not zero. I don't think you're able to, because in my opinion it's not possible.
The concept of infinity is just that, a concept. If x / infinity were anything other than zero (say y), I'd be able to perform the operation 1 / y and get a finite number. But that's just it, it couldn't possibly be finite, because I can arrive at any finite number performing x / z where z is a number that can produce z / x = 1 / y.
Seems too much like sweeping it under the rug to say that something divided by infinity is an infinitely small number. It's not a new idea you're introducing. An infinitely small number is zero.
Take this for an example.
1 / 3 = 0.3333333 (0.3 bar)
2 / 3 = 0.6666666 (0.6 bar)
3 / 3 = 0.9999999 (0.9 bar) ?
If we stick with the pattern presented to us, we should have 0.999999 (0.9 bar). So if 0.9 bar isn't 1, then why the difference? What changes? I thought 3 / 3 was 1.. I guess we'll have to completely tear down the fundamentals of mathematics because 3 / 3 is not 1 but 0.9 bar. Right?
What I want to know is, what is 1 - 0.9 bar? What is the difference? According to you, they aren't the same number, so I should be able to have a non-zero number by subtracting the two, right?<!--QuoteEnd--></div><!--QuoteEEnd-->
What? Where did you get the idea I would think they aren't the same number? .9 bar IS equal to 1, so if you subtract the two, you will get exactly zero. I don't know what you think you are proving with this.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->The concept of infinity is just that, a concept. If x / infinity were anything other than zero (say y), I'd be able to perform the operation 1 / y and get a finite number.<!--QuoteEnd--></div><!--QuoteEEnd-->
That isn't what you would get at all. You would get infinity / x, which is A number, but its not a FINITE number. That doesn't mean you can't perform operations on it.
Do you recall the use of the letter h from calculus? h was used to represent a number that was approximately zero -- but it couldn't <i>actually</i> be zero because the equations involving h all required you to divide by h at some point. If h was equal to zero, you would have a divide-by-zero error and couldn't complete the analysis, but if h is equal to anything BUT zero, you don't get the right answer. And yet we still managed to come up with an answer, usually by reaching the point where we could say h / h = 1 even though that equation looks remarkably like 0 / 0. Or perhaps 2h / h = 2, even though its still 0 / 0. In both cases you get a finite number, and it can be several different finite numbers depending on the circumstance, but its never an uncertain number. It's never 1 AND 2, just one or the other.
If you all really get stuck you could always use <a href="http://askanexpert.web.cern.ch/AskAnExpert/" target="_blank">this web-form</a> to ask an expert at CERN.
I for one would like to see a debate about super-symmetry and dark matter.
HL2 has already supported this theory by talking about dark-energy reactors (HL2: Episode 1 primarily when you get to have fun re-stabilizing one) and the my favorite weapon the is the dark energy orb alternate fire on the AR2 (aka. Overwatch Standard Issue Pulse Rifle). Granted this is all fiction, but it's based on a real work, next-generation physics theory called super symmetry.
[<a href="http://half-life.wikia.com/wiki/Dark_energy" target="_blank">click to view a wiki topic discussing dark energy in HL2</a>]
<a href="http://images4.wikia.nocookie.net/half-life/en/images/thumb/1/10/Darkenergy.jpg/750px-Darkenergy.jpg" target="_blank"><img src="http://images3.wikia.nocookie.net/half-life/en/images/thumb/1/10/Darkenergy.jpg/250px-Darkenergy.jpg" border="0" class="linked-image" /></a>
The Large Hadron Collider was supposed to start exploring some particles that might give experimental data for these and other theories but of course the helium leak has it temporarily shut down, probably until 2009. [<a href="http://press.web.cern.ch/press/PressReleases/Releases2008/PR10.08E.html" target="_blank">press release here</a>]
Still, to get the topic back on track here you all COULD try to shift the argument into how such theories might be adapted to NS2's TSA technology like Infantry Portals and Phasegates. This might lead to tips for Jason and Cory on how the dynamic sprites should look and so-such. (or just provide plausible backing to the content for immersive purposes)
(For example, for you programmers, try summing 1.0F a million times together and then dividing the result by a million. What you get won't be 1.0F because of the inability for a computer to represent 1.0F with exact precision)
However, h even if it is an unnamed value, it is still finite. You could proceed to tell me that h is really really <i>really</i> small and that I could even pretend it were just like an infinitely small number, but you'd still be inaccurate. So long as I can multiply h * 1/h and arrive at 1, it is not infinitely small. I wish I could give some sort of better proof for it, but the reasons for why you cannot do that are precisely the opposite of the reasons you can with h being a finite number. I'd have to show you with infinity, which by the sake of our argument is exactly what I'm trying to prove does not exist. You could argue that infinity * 42 = 7, and I wouldn't have a leg to stand on because gosh darn it, where is infinity when you need to prove that it doesn't exist anyway?
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->What? Where did you get the idea I would think they aren't the same number? .9 bar IS equal to 1, so if you subtract the two, you will get exactly zero. I don't know what you think you are proving with this.<!--QuoteEnd--></div><!--QuoteEEnd-->
If an infinitely small number is not zero, what number do you get when you subtract 1 from that number? I thought for sure you would have said it is 0.9 bar, though if you can't say it's 1 because 1 - 1 = 0 and you argued that an infinitely small number is not 0, and you can't say it is 0.9 bar because you just stated that 0.9 bar *is* 1. So I ask, what is the number you get when you subtract an infinitely small number from 1?
You get 1 - h. You can't combine them really, because infinitesimals and infinities don't interact well with real numbers. But depending on what you want to DO with that 1 - h, you might still have to pay attention to that h hanging off the end.
For most purposes, you can treat 1 - h as being equal to 1. If you were to subtract 1 from that though, then you have to remember your answer is -h instead of 0. And then depending on what you want to DO with that -h, you can probably treat it just as if it were 0 anyway, until you try to divide by h. But when you do that, I hope you've been keeping track of what you did to that infinitesimal all this time up till now, so you will know what number pops out at the end. It might be 7. <img src="style_emoticons/<#EMO_DIR#>/smile-fix.gif" style="vertical-align:middle" emoid=":)" border="0" alt="smile-fix.gif" />
Your programming example is slightly different, due to the limitations of writing numbers in binary form. Computers can't exactly store infinity as a number. In calculus, h is a number larger than 0 but smaller than every possible number larger than 0.
If something is infinitely large, could it also no be said that there is infinitely small? (limit in the microscopic direction instead of macroscopic, if you will)
It's important not to confuse infinity with undefined. While I agree with you that space is an abstract of coordinates and distances relative to each other, much like time is an abstract of observance of change, it almost seemed like you were arguing space is not infinite because it is an undefine-able abstract. (unless I'm misunderstanding what you meant to say) Space is infinite, not undefined; but I totally agree with your definition of it being an abstraction of how our minds observe (or cope with) the fundamental concept.
In my opinion, root measurements that all measurements are derived from -- such as time and mass -- all seem to be a relative abstract for a higher concept our minds simply can't understand. Implications of this include why I do not believe you can time travel backwards. Sure you can accelerate or slow time, but what is time really? Is it not fair to say that time is nothing more that the observance of change? We describe how quickly or slowly something changed in terms of time but time has to relative to reliable constant that can be reproduced experimentally. Today we label time with the SI unit called a second and define it experimentally as: "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom." This is a highly accurate and reliable constant, but my point is that the constant is still in terms of the period of a change. It's an abstract. FYI, the other so-called SI base units are the meter for distance, the kilogram for mass, the second for time, the ampere for electric current, the kelvin for temperature, the mole for amount of substance, and the candela for intensity of light. Yet many of these definitions are a measurement based on mass and time.
Anytime, you talked about space being an abstract of coordinates. The meter (SI unit of distance) is defined in terms of <i>c</i> which is the velocity of light in a vacuum, equal to exactly 299,792,458 meters per <!--coloro:#FFFF00--><span style="color:#FFFF00"><!--/coloro-->second<!--colorc--></span><!--/colorc-->. Ok, see here we go again, time in terms of a second. In mathematical terms, isn't this just like how velocity is the integral of acceleration, if you keep on integrating, at some point all you are left with is the abstract, right? So what are we left with? An abstract called time, an abstract called mass, and an infinite abstract called space where it all exists?
<!--QuoteBegin-Cxwf+--><div class='quotetop'>QUOTE(Cxwf)</div><div class='quotemain'><!--QuoteEBegin-->You get 1 - h. You can't combine them really, because infinitesimals and infinities don't interact well with real numbers. But depending on what you want to DO with that 1 - h, you might still have to pay attention to that h hanging off the end.<!--QuoteEnd--></div><!--QuoteEEnd-->
Fair enough, but is it not possible that the real limitation is our method of describing nature in mathematical terms and computational logic? Whole numbers, real numbers, imaginary numbers, abstracts, infinitesimals, infinites, etc. all described and computed with operands like addition, subtraction, multiplication, division, exponentiation, derivation, integration, etc. If we are having trouble computing, perhaps there's a higher operand and/or a larger set of numeric values that we have not yet conceived with our collective brains.
Through out all of historic scientific discovery, has mathematics not had a direct relationship with observing experiments and learning how to describe them? I would argue we, as humans of the planet earth relative to October 2nd in 2008 CE, haven't discovered a wider picture on the concept yet.
PS: Again I ask, how such things could relate to NS2, much like I hinted at in the dark energy (super-symmetry theory) in HL2 reference. This topic has become fascinatingly off-topic.
Indeed it has, which is what makes it so fascinating. <img src="style_emoticons/<#EMO_DIR#>/smile-fix.gif" style="vertical-align:middle" emoid=":)" border="0" alt="smile-fix.gif" />
Incidentally, I don't believe the universe IS actually infinite. I'm just defending the concept of infinity in general, but I suspect our current universe has a definate size. A while back, someone convinced me that Cosmic Background Radiation was residue from the Big Bang, which makes absolutely no sense unless space is curved. But there's no reason space <i>can't</i> be curved, given that we know Gravity curves space to some extent, so it seems likely to me that our universe is a hollow sphere in 4D space, with the entirety of our 3D space being the surface of that sphere. The size of the universe is therefore limited by the radius of that sphere, and the sphere expands as the universe grows. Theoretically the sphere had zero radius at the time of the Big Bang.
So what do you get when you divide a number by a really really really really small number? A really really really really big number. Theoretically, this is the same thing as infinity, but in practice a number so large that you won't have to worry about anything being bigger than it.
Like imaginary numbers, we use these tools to help our calculations, because rather than throwing your hands up in the air because your answer produces imaginary numbers and 'imaginary numbers don't exist' we use it as a tool for finding the answer to our problem. Likewise, you use h to represent a really small number that doesn't have a number smaller than itself before zero even if that doesn't exist either.
What are the rules of the real number system anyway?
1) Between real number x and real number y, there exists a real number z such that x < z < y.
2) For all other rules, see integer number system.
h fails your test, since there's no number you can put between h and zero. Therefore, it doesn't exist as a number. It is a concept like infinity. Showing that infinity exists by giving me an example of another concept doesn't convince me, I'm afraid.
Nothing is infinite in this world, not even the coordinate system. Unless you've got a pencil which doesn't wear down and a piece of paper of infinite size or a computer with a magical infinite memory size, you cannot possibly represent it. The reality is that we label it infinity and deal with numbers within our capability of dealing with. That's no closer to infinity than a hand full of sand is an approximate representation of all the sand on the planet.
As for infinite smallness (anything can be divided further), where's your proof for that? Burden of proof is not on me but those of you who want to convince me that infinity exists in some real sense.
Sure there is: h/2. That's about halfway between h and zero, and if you divide h by h/2, you'll get the very real number 2, even though both of the numbers you started with were indescribable.
This is getting very abstract though, what were we arguing about again? I think you were trying to convince me that the universe couldn't be boundless, because you could no longer mathematically represent the portion of the universe that we would occupy?
To that I would respond that the integer number system is also boundless, and we routinely identify both locations and sizes within the number system. All you need to do is select one point to serve as a "zero" reference point, and identify all other locations relevant to that.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->This is getting very abstract though, what were we arguing about again? I think you were trying to convince me that the universe couldn't be boundless, because you could no longer mathematically represent the portion of the universe that we would occupy?<!--QuoteEnd--></div><!--QuoteEEnd-->
Pretty much, yep.
<!--quoteo--><div class='quotetop'>QUOTE</div><div class='quotemain'><!--quotec-->To that I would respond that the integer number system is also boundless, and we routinely identify both locations and sizes within the number system. All you need to do is select one point to serve as a "zero" reference point, and identify all other locations relevant to that.<!--QuoteEnd--></div><!--QuoteEEnd-->
Like infinity, the integer number system is a concept as well. It doesn't really exist. The universe isn't mapped out on a grid with numbers hovering in midair. For it to *truly* exist, in some shape or form, every point would have to be capable of being mapped out, including the ones with coordinates so large, there are less atoms in the universe than the numbers for such coordinates.
It's really the same argument all over again. If the universe were boundless, our space in the universe would have no reference, size, shape, or volume.
By the way, a neat math problem I saw the other day (to further derail the thread):
You work at a factory which creates lottery balls, the ones they draw from a cage on a random basis.
You are given the instructions to unwrap a package with the following contents: a ball and two strips of number stickers from 0 to 9. Your task is to place the stickers on the ball to create the numbers starting from 1 and then up. However, the stickers you do not use in that package, you place aside to use later. So for example, I start by taking the '1' sticker and placing it on the ball, and the 0, 2, 3, 4, 5, 6, 7, 8, 9 and the other 0 - 9 number sticker strip go to a pile. Then I open another package and I take the '2' sticker and place it on the ball, placing the 0, 1, 3, 4, 5.. and the other 0 - 9 number sticker to the same pile.
At what point do I run out of numbers using this system? I'll give you a hint, it's not a small number.
In any case, that's not the solution either, but you're on the right track I suppose.
In any case, that's not the solution either, but you're on the right track I suppose.<!--QuoteEnd--></div><!--QuoteEEnd-->
I don't think you'd be more short of 1's than anything else. They all cycle in order at the same frequency. The one number that does show up at a different rate is 0, since leading 0s aren't used, so you would have a surplus of 0s when you run out of everything else.
Also, you can't possibly reach 22-digit numbers, since there are 10 times as many 21-digit numbers as there were all numbers before that, and 21-digit numbers all drain your stack of reserves.
If you could figure out how many extra zeros you have left over at the 21-digit all 2s, and count backwards from that, you could probably find the point at which you run out of numbers that aren't zero.
Come to think of it, I believe the number of extra zeros you have is the same number I calculated for total spare numbers available when you reach 21-digit number territory. After all, all of the spares should have been leading zeros that weren't used, right? So the number where your reserves run out might very well be the very first 21-digit number, or 1 followed by 20 zeros. Except that can't possibly be the last number, because the previous number didn't use numbers at the same rate you opened them, and so numbers 2 through 9 which should have a reserve of "0" now must have had a reserve of "-1" last number.
So what we're really looking for is the point where the instantaneous demand for a digit is higher than the average supply of that digit up to this point, even though the average demand for that digit will equal the average supply eventually. And while all digits have a number at which their instantaneous demand is at a peak, the 1s digit tends to reach that peak before any other digit. However, the peak of maximum instantaneous demand is not likely to be the same point at which demand outstrips supply.
The supply of 1s is equal to two 1s per number, while the demand varies anywhere from zero to twenty. So how do you figure out the total demand for 1s up to a given number?