A layman's questions about universes

mike the wiz asks various questions.The problem is that you haven't defined what you mean by "infinite." And in the process, you have equivocated among many possible definitions.The big one seems to be that you have confused "infinite" with the idea of "bounded." A mathematical example:We can say that there are an "infinite" number of numbers between 0 and 1. But what would be considered the "largest" number? That very much depends upon if we have bounded the interval or not. That is, when we say "between 0 and 1," do we include 1 or do we not? If we do include 1, then yes, there is a "largest" number in that interval: 1. Every other number in the interval is necessarily smaller.But what if we don't include 1 in that interval? Then there is no such thing as a largest number. The interval is "unbounded" and even though we know there is a line it cannot cross, it never manages to reach that line. No matter what number you give me, I can always find another number that is bigger.How does this relate to the universe? Well, it means we have to define what you mean by an "infinite" universe. Are you talking about the stuff inside, the boundary, or what may exist beyond the boundary? For example, if one is talking about the stuff inside, there is a reasonable claim that there is an "infinite" amount.Given the fact that universe is expanding (and for the moment, we will not worry about what it is expanding "into"), there is another reasonable claim that it is "infinite" in the sense that were you to start on a journey toward the "edge" of the universe, you would never reach it.So depending upon what you mean by "infinite," the answer to your first question could be no or it could be it already is. A better question would be, "Is the universe bounded?"Regarding your second question, that would depend upon the geometry of the "universe space," and I am using that in a mathematical sense, not a physical sense.Suppose we could model our universe as a line and similarly, any other universe as another line. Do those lines necessarily need to intersect? No, not really. If the geometry of the "universe space" is like a plane, it is quite easy to have a set of parallel universes that never intersect.But suppose our "universe space" were spherical. Then every single "universe line" would necessarily intersect with all the others.So the answer to your second question is, "It depends." Currently, we have no idea. There is an aspect of quantum physics that does hypothesize about what it may be like. A universe is embedded in what is called a "brane" (like a "membrane") When branes collide, the energy released creates a universe within the brane.This somewhat relates to your third question: Universe don't exist within universes. Instead, there would have to be a larger structure that contains universes.Your fourth question again is an equivocation as to what is meant by "infinite." It does not mean "unbounded." There can easily be an "outside" to an infinity depending upon what is meant by "infinity." If, indeed, brane-theory is correct, these Big Bangs are happening in a much larger structure than the universe.Regarding your last question, that also requires a definition of what you mean by nothingness. To get into it requires a great deal of mathematical abstraction and I'm not saying that it is inappropriate or beyond you, but it requires a fair amount of prep work and I'm not sure this specific post is the right place to do it. If you're interested, I will go into it. For now, I'll just say that just as "infinite" is a tricky concept where it can mean a whole bunch of things, each of which perfectly reasonable yet different (and even more problematic, something can have many of those characteristics at the same time) "nothingness" is right there alongside it.As for the idea that our universe is "inside" another universe, it doesn't seem that way. Rather, our universe seems to be inside a larger structure...something that isn't a universe (but that is only if we accept something like brane theory.)

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Rrhain
WWJD? JWRTFM!

buzsaw responds to me:

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Thanks for response rather than insult.

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Considering that I have never insulted you, one wonders where you get this impression that I would.

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The problem is that you haven't defined what you mean by "infinite." And in the process, you have equivocated among many possible definitions.

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Which specific statement of mine do you refer to here?

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All of them. That's why I said what I said, pointing out for each of your questions where you talk about "infinite," that you seem to be asking about thus-and-so, noting that it is different from the definition that was seemingly implied before.

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The interval is "unbounded" and even though we know there is a line it cannot cross, it never manages to reach that line. No matter what number you give me, I can always find another number that is bigger.

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Imo, herein lies your first problem. "...we know there is a line it cannot cross,..." You're contradicting yourself.

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Incorrect.You're saying that the mathematics of set theory contradicts itself in the concept of "unbounded"?We have notation for it, buzsaw. When you're dealing with a continuous numeric interval, if you wish to indicate that the boundary is included, you write a bracket. If you wish to indicate that the boundary is excluded, you write a parenthesis.Thus, the bounded interval from 0 to 1 where we include 0 and 1 is written as:[0, 1]The unbounded interval from 0 to 1 where we do not include 0 nor 1 is written as:(0, 1)You can even mix and match. If you want to write the interval from 0 to 1 where 0 is included but 1 is not, you write:[0, 1)

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If it is infinitely boundless, why not just say so and leave it at that.

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Because that isn't what it is. Things can be finite and yet unbounded.This is what I was talking about when I said that the subject is extremely complex and requires a fair amount of mathematical prep work. I don't expect you to believe me just because I say so, but I would hope that you would recognize that you are contradicting a great deal of well-understood mathematics.One of the mathematical definitions of "unbounded" (actually "open") is that for every element of the set, you can draw a neighborhood around it such that all elements in the neighborhood are included in the set. A bounded set (actually "closed") is such that there exists at least one element for which no neighborhood can be drawn that doesn't include elements from both inside and outside the set. The set of points that meet that criterion is the boundary.In our example of using the interval from 0 to 1, exclusive (meaning neither 0 nor 1 are included), there is no boundary point. For every element you give me inside the interval, I can draw you a neighborhood that includes only other points inside the set. Here's the formula:If x <= 1/2, draw a neighborhood of x +/- x/2
If x > 1/2, draw a neighborhood of x +/- (1 - x)/2That is, if x is less than or equal to 1/2, draw a neighborhood of radius one-half the distance between x and 0. If x is greater than 1/2, draw a neighborhood of radius one-half the distance between x and 1.But notice, neither 0 nor 1 is in the set. Those two points define te boundary. You cannot draw a neighborhood around those two points without including elements both inside the set and outside the set.The interval is finite in that it does not extend to infinity, but it is unbounded in that you can never reach the boundary.

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What gives you the idea there's a line/boundary it cannot cross?

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Because we have mathematically defined it as such. The interval of 0 to 1 has a set of points that meet the mathematical definition of a boundary. Specifically, 0 and 1.

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That like saying, for example eternity is not eternal.

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But that's just it: Without a definition of those terms, there's no reason to think it isn't. In fact, even with a definition, there's no reason that it can't be.Take the empty set. Is it open or closed? Given the above definition of a boundary, we say that a set is open if and only if it does not contain its boundary. Similarly, a set is closed if and only if its complement is open. That is, a set is closed if and only if the set of all points not in the set does not contain the boundary.Well, the empty set fits both of those definitions. The empty set has no elements, thus does not contain its boundary and thus is open. But the complement of the empty set is the universal set, which also doesn't contain its boundary. Thus, the universal set is open and the complement of an open set is closed. Thus, the empty set is closed.And thus, because the empty set is both open and closed, that means the universal set is both open and closed, too.The point I am trying to make is that you are treading into waters where "common sense" simply does not apply. It is perfectly logical for a set to be both open and closed at the same time.

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"There has to be a beginning." But there doesn't.

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Precisely.But the thing is, if time goes away, it becomes nonsensical to talk about "before."

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Nevertheless, just as time is a segment of eternity, so matter is part of the universe which, being everthing includes all matter, rays and infinite space/area.

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Not necessarily. There might be something beyond that. Our "common sense" definitions of what matter is don't seem to work when we're looking at the very small. That's one of the things that Einstein discovered when he showed that matter and energy are equivalent. Not just "convertible" but are actually the same thing.F'rinstance, if you shine light through a diffraction grating, you get an interference pattern which is what we expect from light as a wave, a type of energy. But if you do the same thing with electrons, you get an interference pattern. But electrons are made of matter, right? Surely they shouldn't be behaving like a wave. But they do.And similarly, if you shine light on metal of an appropriate frequency, you can knock electrons off...just as if light were not waves but particles. That's what Einstein won his Nobel Prize for: The explanation of the photoelectric effect.Light, which we consider to be energy, is both a wave and a particle. Electrons, which we consider to be matter, are both particles and waves. The two are exactly the same.

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Given the fact that universe is expanding (and for the moment, we will not worry about what it is expanding "into"), there is another reasonable claim that it is "infinite" in the sense that were you to start on a journey toward the "edge" of the universe, you would never reach it.

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Herein, imo, lies problem two with you people. Imo, since by definition the universe includes everything, it is impossible for it to expand. Why? Because the area/space where expansion occurs is already inclusive by definition as part of IT, THE UNVERSE.

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That's why I said that for the moment, we will not worry about what it is expanding "into."You are trying to use a "common sense" definition of "expansion" that does not apply. In essence, the universe is not expanding "into" anything. Here's where the balloon analogy comes in handy. Suppose the universe is the surface of a balloon. Now, blow up the balloon. When you do this, all the points in the rubber of the balloon expand away from each other. No matter where you are on the balloon, you see every other point moving away from you...and the points further away are receding more quickly than the ones closer.But the rubber of the balloon isn't expanding "into" anything that the balloon contains. From the surface of the balloon, you cannot see where the balloon is managing to come up with this extra space.Now, we are not a balloon. For one thing, the balloon is a two-dimensional surface and we live in a three-dimensional space. To mathematically expand this into our universe requires a great deal of mathematics.

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Rrhain
WWJD? JWRTFM!

buzsaw responds to NosyNed:

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Ned, but numbers can have bounds. How can infinite space/area possibly have a bound?

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The same way. Fractal geometry provides a wonderful example of this:Take a cube. In each of the three directions, drill out the center square. In each of the remaining sections, drill out the center square. Continue this infinitely. What do you get?You get an object that has a finite volume but infinite surface area.

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By definition, infinite space/area can have no bounds.

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Why not? Why can't it have both? You're trying to use a "common sense" definition where it doesn't apply.

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Rrhain
WWJD? JWRTFM!

mike the wiz responds to me:

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So please DON"T give no mathematics.

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And yet...

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surely a Big Bang can only create an finite universe?

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No. It can easily create an infinite universe. But to explain it requires extremely hefty mathematics which not even I really want to go into. Barbie was right...math is hard. To get into this requires a deep understanding of multidimensional topography and to be honest, I never studied that far. There's a reason that it's the Ph.D. physicists who are doing this work and not the undergraduates. It literally takes a good 10 years simply to become competent enough to even begin asking the appropriate questions, let alone trying to answer them.

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Yet how can a larger structure exist outside a boundless infinite universe?

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Because there are things bigger than infinity.Let's go back to the comparison of the rationals and the reals.I hope you can just trust me that we can put the rationals into a one-to-one correspondance with the natural numbers. That is, if we had a list of 1, 2, 3, ... and a list of every single fraction out there, we could put this list of fractions right up alongside the list of natural numbers and every single one would pair up without there being anything left over. Mathematically, it is called "denumerable."But what about the irrationals? As I described in another post (Message 35 of "International High IQ Society" thread), the irrationals cannot be put into this one-to-one correspondance. There is always an irrational number left over. The set of irrationals is bigger than the set of rationals, even though the rationals are infinitely large.The mathematician Georg Cantor came up with a hierarchy of infinities. He used the Hebrew letter aleph to distinguish them. The size of the rationals he called "aleph-null." The next infinity in the series, "aleph-one," is equal to 2^aleph-null. The next one, "aleph-two," is 2^aleph-one.One of the biggest questions in mathematics is just how big the real numbers are. We know that they're bigger than the rationals, but are they equal to aleph-one? Aleph-two? Any of the alephs? Turns out we don't know. In fact, this question is one of the famous undecideable questions that are predicted by the Incompleteness Theorems.But let's pull back from that and try to come up with another example.What does it mean to be a "boundary"? Mathematically, it means that if you draw a neighborhood around the point, you will necessarily grab elements both inside and outside the set, no matter what the radius size of the neighborhood.With a bit more mathematical coaxing, we come up with a definition of an "open" set meaning that it is a set that does not contain its boundary. Every point in the set is an "interior" point of the set. A "closed" set is one where its complement is "open."What we end up discovering is that by this definition, the universal set is both open and closed. Similarly, the empty set (being the complement of the universal set) is both open and closed.My point is that when you're talking about infinity and nothingness, things do not behave in "common sense" ways. How can something be both open and closed? Well, by the definitions that we have come up with that work so well for so much lead us to conclude that there are at least two things that are both.And in the end, why not? Why can't something be both? So much of what we have found out about our universe is that our distinctions of "or" really don't apply when carefully scrutinized.

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I suppose I meant the universe with no boundary. Infinite without one.

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Just because something has no boundary doesn't mean it is "infinite" in the sense that it contains "everything." The interval (0,1) is unbounded, but it is also considered "finite" in the sense that it doesn't go all the way out to infinity (see what I mean about the word "infinity" having so many definitions?)The universe can be infinite and unbounded and still not contain everything. The set of rational numbers from negative infinity to positive infinite is infinite, unbounded, and still doesn't contain everything.You need to be very careful about trying to use "common sense" definitions of what "infinity" means. We have to be extremely formal and pedantic. It's tedious, but necessary.

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Rrhain
WWJD? JWRTFM!

mike the wiz responds to me:

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As surely there is only evidence of this universe. The "bigger structure" and multiple Big Bang's, I'm guessing there's no evidence for.

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Yep. String theory is an absolutely wonderful and elegant set of equations for describing the nature of the very big and the very small. It holds the promise of the Holy Grail of Physics: Tying gravity into quantum theory and providing a "theory of everything."But it has no experimental evidence. Strings are so small that they cannot be detected by any method we can conceive of for the moment. The reason it is taken seriously is that it is such an elegant solution to the questions that we have and most other paths of inquiry have proven to be disastrous. There does appear to be work that can be done that holds the promise of being able to create experiments that could help us test the concept, but we aren't there yet.I think you would agree that if someone is trying to talk about the how of the creation of the universe and wanting to make sure every i is dotted and every t is crossed, "god did it" really isn't satisfying. It's too simplistic. It's too reduced. We need much more detail and to really treat that detail right, it requires a great amount of theological study. I'm not talking about the faith that "god did it"...I'm talking about the process by which god did it.Well, scientific inquiry into the physical universe is just as complicated. It can be reduced to some very simplistic phrases that, when you know a bit about the process, are quite helpful in visualizing what is going on, but are absolutely worthless when trying to explain it in detail. You can have the faith that "strings are kings," but it's going to take a lot of work to really understand it well.I'm sure if we were all to knuckle down and delve into the subject, we'd all manage to come out a lot more knowledgeable about the subject, but I'm not qualified to teach it (I'm a mathematician, not a physicist) and this isn't really the place to do it.

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Rrhain
WWJD? JWRTFM!

buzsaw responds to me:

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But in your cube model you already begin with/assuming bounds exist.

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You're missing the point. I am not "beginning with/assuming bounds." I am simply showing you a method by which the shape can be constructed.You asked about how you could have infinity in a finite object. And I showed you. The sponge has a finite volume but an infinite surface area.

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With your cube model you have visible surfaces, but with space/area you have none.

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Only because you are incapable of viewing them. When I was in my prime as a mathematician, I had no problem visualizing four-dimensional surfaces and if I tried hard enough, I could get fleeting glimpses of five-dimensional ones, too. We live in a three-dimensional world and it is hard to conceive of what a 4D space looks like.Do not confuse your inability to visualize something with the idea that it cannot be done.

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If you could somehow travel all the way out to the end of space

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But that's just it: You can't. No matter how fast you try to go, the universe will always have more for you to traverse.You are also assuming a spatial geometry as if it were a simple sphere. Why? On the surface of a Mobius strip, traveling in a single direction will bring you back to the same spot you started at...and reversed.

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Why not? Why can't it have both? You're trying to use a "common sense" definition where it doesn't apply.

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Mmmm, but common sense makes a whole lot more sense!

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That's why it's so appealing. But as someone wiser than I said, "common sense" is neither common nor sensible.

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Imo, you hadn't oughta have let the ejukaters talked you outa all that money you spent to replace yours with sofistikated theerees.

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Why not? It's because of them "ejukaters" that you have that computer sitting in front of you.Why on earth would I cling to something that was proven to be false? The mathematical description of the universe is not as you think it is. As another wiser man said, the universe is not only queerer than we imagine, it is queerer than we can imagine.Why on earth would there be a speed limit to the universe? Why is it no matter how much energy you pour into it, you can never go faster than the speed of light? Everything about our "common sense" tells us that if I am on a train going 50 miles an hour and I throw a ball at 50 miles an hour in the same direction of the train, that ball is going 100 miles an hour with respect to the ground because 50 + 50 = 100.But it isn't. It's going a little bit slower than that. The universe isn't linear but relative.Or do you deny relativity, too? One of them "ejukated" things?

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Rrhain
WWJD? JWRTFM!

buzsaw responds to Beercules:

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The sillier thing to do is to block off and disallow common sense and logic in trying to understand the universe, complicating the complicated, if you will, in so doing.

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(*blink!*)You did not just say that, did you?Are you seriously saying that we should maintain this "common sense" view of the world even when it's WRONG?The "common sense" and "logical" view of the world says that the universe is linear. Velocities add through a simple addition.The universe, however, laughs at that concept. If you were to try to use it for things like GPS, you'd fail miserably. That's right...the calculation required to determine your position on the earth...something that can be done by a tiny little box in your car...cannot be done with your "common sense" view of how things are supposed to be.They simply aren't.We're sorry that this now means you have to learn calculus in order to understand how the universe works, but the universe doesn't care about you.Don't you think that "common sense" would tell you that if the model you are using isn't working, you should discard it for a more accurate model, even if that means you have to make it much more complex?I find it amazing that you are saying that this simplistic, naive "common sense" should prevail over actual reality for no other reason than it's hard for you to understand the real thing.That computer sitting in front of you could not function if it weren't for that anti-"common sense" physics you are railing against. How can you possibly claim that it is a crock when it's staring you right in the face?

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Yah, that's funny! Boundless, but finite!

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Yep.(0,1)Unbounded, yet finite. There are definitely numbers that are not in the interval, but no matter what number you choose in the interval, you will always find another number that's just a little bit closer to the boundary.What on earth do you think the terms "bounded" and "finite" mean?Why is it millions of mathematicians can understand this concept and actually use it to create real-world applications? Why does your incredulity get to trump reality?

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Rrhain
WWJD? JWRTFM!

buzsaw responds to me:

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am fully aware that my logical arguments are to you very contrary to what you've been taught and sincerely believe.

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Incorrect.While I was certainly taught relativistic physics, it isn't something I "believe." It is something that I have directly experienced. There are actual experiments that you can do that directly show the predictions made by the theories of relativity.This isn't something I "believe." I can actually make it manifest right in front of everybody's eyes. Don't take my word for it: Run the experiments yourself. If you come up with something different, we'd all love to hear about it.

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You asked about how you could have infinity in a finite object. And I showed you. The sponge has a finite volume but an infinite surface area.

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1. Here again, logic and common sense says anything having finite volumn capacity must needs have also bounds inclusive of a finite surface and is in no way analogous to 2d.

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Incorrect. Your logic is obviously false because there it is, right in front of you: An object that has infinite surface area yet finite volume.I have to ask you again: Are you seriously saying that we should stick with a "common sense" that is obviously [I][B]WRONG[/i][/b] simply because you don't like it? Because it requires more effort to understand the more accurate model?Have you considered the possibility that the problem isn't a violation of logic or "common sense" but that you simply do not understand? Why should anybody follow your "common sense" when we can directly prove it to be fallacious?

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2. Using a sponge analogy seems to be adding a new dimension to our discussion about an alleged 3d universe isn't it?

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Not at all. A sponge is a 3D object. And yet, it has an infinite surface area contained in a finite volume.There was a wonderful Scientific American Frontiers on PBS last night that was about all of this: Origins of the universe, the concept of the universe being "infinite," and how can the universe be expanding when there is nothing for it to expand "into"?Alan Alda had a wonderful statement about it as he was talking to the physicist: We think of the Big Bang as something that happened at a point and expanded outward, so if we were going to look backward in time [from me: which is what we do when we look at the sky given the nature of light...when you see a star, you do not see it as it is now but rather how it was years ago because it takes the light that long to reach here], we should be looking toward the center. You're telling me that what we really do is look in every direction?And that was the point the physicist was making: That's exactly right. The idea that the Big Bang happened at a point and expanded outward is completely wrong. As Hawking pointed out, the Big Bang did not happen at a point: It happened everywhere at once.

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Thanks to all for your patience and indulgence to ole Mr Logic's intrusion here into your lab.

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Um, when all the people who actually study this thing for a living are telling you that you're wrong, don't you think that you should at least consider the possibility that you might be mistaken?In other words, what on earth makes you think you're "Mr. Logic"?

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Rrhain
WWJD? JWRTFM!

jar asks:

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Way back when I was in high school we had to have Plane Geometry, Analytical Geometry, Eucledian and Non-Euclidian Geometry as requirements for graduation.

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None of the high schools I attended had a requirement for a specific level of mathematics. Instead, they all simply required that you had a certain number of credits in it. Thus, if you came in having a year of Pre-Algebra and a year of Algebra I, you were good to go.And to satisfy my curiosity, how does one do Analytical Geometry without Calculus? Perhaps what I'm thinking of (where you do things like derive the formula for the surface area of a sphere by integrating the formula for a circumference) isn't the same as what you are thinking of.I handily admit that it has been a couple decades since I had Geometry, so I can't recall if we had a section that was called "Analytical Geometry" that was more of a simple algebraic representation of geometrical concepts. Instead, "Analytical Geometry" was part of Calculus...all of my basic Calculus texts are titled "Calculus with Analytic Geometry."

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Rrhain
WWJD? JWRTFM!

buzsaw responds to me:

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Unbounded, yet finite. There are definitely numbers that are not in the interval, but no matter what number you choose in the interval, you will always find another number that's just a little bit closer to the boundary.

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Gotta get this one in before retiring.

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1. "Unbounded, yet............closer to the boundary??" The unbounded has a boundary??

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Yes.What you seem to be having difficulty with is recognizing that there are three things:The interior points, the exterior points, and the boundary points. Interior points are members of a set for which a neighborhood can be drawn that only includes other members of the set. Exterior points are those for which a neighborhood can be drawn that do not include any members of the set. Boundary points are those for which all neighborhoods necessarily include both interior and exterior points.So when describing an interval, you have a choice: Do you include the boundary or not? You don't have to. And depending upon the geometry, you can even include some parts of the boundary but not others.

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2. This's real confusing, Rrhain.

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I never said it wouldn't be. But I remind you that just because you find it confusing doesn't mean it isn't true. Consider the possibility that the problem is not that the claims are false but rather that you don't understand them.Would you rather I say something like this? Do you really want me going on about: And let's pull it all together: Is that really what you want?

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If you include the numbers outside of the bounded infinity of numbers for your analogy, you must also eliminate the boundary for a true analogy of the universe

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Not at all. The point that I am trying to impress upon you is that it is an equivocation to confuse "infinite" with "unbounded." The two are not the same. Things can be finite or infinite while at the same time being bounded or unbounded. Those two traits are not the same thing.

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It appears to be a bogus analogy for the universe, for the universe is one system.

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Only because you don't understand how infinity works.Time for a thought experiment. For this thought experiment, we need to assume a few things:1) Captain Marvel and Superman both exist.
2) They can move any finite distance in any finite amount of time (even if that means they move faster than the speed of light).
3) There are an infinite number of coconuts, all numbered, and a pit big enough to hold them.Now, Captain Marvel and Superman decide to play a game. At noon, CM tosses coconuts numbered 1 and 2 into the pit. Supes flies in, grabs coconut #1, and tosses it out.They sit around for half an hour discussing the superhero life and at 12:30, CM tosses in #3 and #4. Supes flies in, grabs #2, and tosses it out.Fifteen minutes later, in go #5 and #6 and out comes #3.They continue this process, each time halving the amount of time they wait before tossing their coconuts.When 1 o'clock comes around (and 1 o'clock always comes around), how many coconuts are in the pit?Answer: None. Why? Because for every coconut number you give me, I can give you a time when it was tossed out: #1 was tossed out at noon. #2 was tossed out at 12:30. #3 came out at 12:45. #4 came out at 12:52:30....The next day, they decide to play the game again, but with a twist. This time at noon, Supes throws in coconuts #1 and #2 and Cap tosses out #1. At 12:30, Supes throws in #3 and #4 and Cap tosses out #3. Again, they go faster and faster, halving the amount of time they wait.When 1 o'clock comes around (and 1 o'clock always comes around), how many coconuts are in the pit?Answer: An infinite number. Specifically, all the even-numbered ones.Even though the physical process is exactly the same on the surface (two go in, one comes out), the outcome is quite different.And it's all perfectly logical. It is easy to see why this has to be the case. Even though we might expect identical processes to achieve identical results, we can see that it can't be the same.Why? Because infinity doesn't work like other numbers. It is illogical to treat it the same when we can directly see that it isn't.

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If the system is "unbounded" it cannot have a "boundary"

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Incorrect.If the system is unbounded, it can easily have a boundary...it just doesn't include it.Think of your house. There is the space inside the walls, the space outside the walls, and the walls, themselves. The space inside the walls is not the same as the walls. When we generally think of your house, we include the walls. Thus, "outside" does not include the walls. Your house is bounded and includes the boundary. Outside is unbounded: Though it has the boundary of the walls, it does not include the walls...those are part of the house.I can get as close to your house as I want, but I am not touching your house until I touch the walls. There is a boundary, but the outside does not include it.

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Yes, yes, I know, with science we're not suppose to make any sense, common, that is.

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Have you considered the possibility that the problem is not "common sense" but that you simply do not understand the "common sense" of the situation?It makes perfect sense, buzsaw, and is perfectly logical. There is "inside," "outside," and "edge." The edge is not inherently part of the inside.

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Rrhain
WWJD? JWRTFM!

buzsaw writes:

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Quoting Rrhain:

If you were somehow able to switch off the gravitational field, "space" would disappear as well.

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I never said that. Beercules did (Message 66).

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I get the implication here that Rrhain has it in his mind that given the right conditions, space would have the capacity to disappear. Where'd I go wrong??

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By confusing gravity and space as separate things rather than being the same thing.

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Rrhain
WWJD? JWRTFM!

NosyNed responds to me:

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I'm pretty sure that Rrhain did, in fact, word that wrong.

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Unbounded, yet finite. There are definitely numbers that are not in the interval, but no matter what number you choose in the interval, you will always find another number that's just a little bit closer to the boundary

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Should have ended with ... just a little bit closer to the limit.

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No, I said "boundary" and I meant "boundary." That is what it is called in point-set topology.An interior point of a set is defined as all points where there exists a neighborhood such that all points in the neighborhood do not contain the boundary or any exterior points.Limits and boundaries are very similar. However, limits are for functions. Boundaries are for sets.

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Rrhain
WWJD? JWRTFM!

buzsaw writes:

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For the purpose of our discussion do these other dimensions solve anything?

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Yes. From string theory, the universe has 10 dimensions.

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Rrhain
WWJD? JWRTFM!

Tony650,No, I can't really describe it. How do you describe "red" to someone who hasn't seen it? It's really hard and I don't know how. The closest I can come is to artificially add a fourth dimension such as the temperature gradient of a solid object, but there's still the visual leap required to convert that non-spatial dimension into a spatial one.

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Rrhain
WWJD? JWRTFM!

Tony650 responds to me:

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Just to clarify, you are saying that you can actually visualize true four dimensional topology, aren't you?

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Yes, but without an actual four-dimensional object to show you that you could look at and verify for yourself, I have no way to prove it to you. I think I can. It certainly feels like I can. My work in mathematics seemed to follow intuitively from the visual models that I had in my head.Until we perfect that telepathy thing and I can project the image into your head, it's something you'll just have to agree that I claim.

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I guess what I'm asking is; can you tell me what I'll need to do in order to come to something even approaching your ability to visualize higher dimensions?

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I'll tell you what did it for me:Fundamental Concepts of Mathematics (often called "Real Analysis"), Differential Geometry, and Topology. When you spend six to eight hours a day, every day dealing with mathematical constructs of more than three dimensions, you brain starts coming up with ways to organize it. I don't know if there are other ways to do it...I only know that about midway through sophomore year, I realized that I was working through multi-dimensional problems visually in my head.

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Rrhain
WWJD? JWRTFM!