The short answer is nobody has a plausible upper bound on the size of the entire Universe, and there's no firm consensus on whether it's literally infinite. If it is finite, there is no "wall" or "center", since it's not a big sphere -- a finite Universe just means that if you go straight long enough, you end up in the same spot. It's analogous to how if you stood on the ground of featureless planet, the land you could inhabit is finite, but it has no borders and no point where you could stick a flag in the ground and say "this is the center of the planet's land".
Most of the time when people give hard numbers for the size of the Universe, they either mean A) the portion of the Universe we can see, B) the portion of the Universe we could ever see in principle, or C) the portion of the Universe which could, in principle, be causally influenced by early particle interactions that could have causally influenced us. If you suspect someone might be talking out of their ass about cosmology, ask them how big the Universe is -- if they give you a clean multiple of 14.6 billion light years, they're an idiot.
> [...] there's no firm consensus on whether it's literally infinite. If it is finite, there is no "wall" or "center", since it's not a big sphere -- a finite Universe just means that if you go straight long enough, you end up in the same spot.
My understanding is that for your interpretation here of "finite" to be true, there must be some curvature. If you only consider the geometry of the universe, then what we see with our instruments does allow us to put a bound on the size of the curvature.
If you want to say something along the lines of "outside of the observable universe is probably cheese whiz" then that's your perogative, and no-one can refute that. But (from the article) the geometry of the universe that we have measured is consistent with an infinite universe, and gives a minimum bound on its size which is many multiples of the observable universe.
It's mathematically simple for a space to loop without being curved locally, although I am under the impression that there are physics reasons to infer we don't exist in such a space.
The pac-man universe acts euclidean until you have universe-sized effects. For example, a circle will satisfy circumference = diameter * pi until its diameter is greater than the span of the universe. Basically the universe could 'hide' the fact that it is a torus from pac-man by simply being really large.
There isn't anything of importance in cosmology that's a clean multiple of 14.6 billion LY. Anything under discussion at that scale would by influenced by the Hubble expansion.
But, yeah, being misinformed about cosmology doesn't make you an idiot. Hyperbole.
Well, that blew me away. I would happily have gone for 13billion light years across.
This one little blog (that was a few hours work covering decades of hard work by hundreds of people) has reminded me of the vast importance of education, scientific inquiry and just plain old reading.
One recent suggestion here in the UK is to make maths study compulsory till age 18. If I can go till age 40 and not know how we know how big the universe is, and I am supposedly in the top 5% of educated people, then yes yes yes.
Is there any politican I can vote for who will double the science budget ?
Shouldn't it be 13 * 2 = 26 across?
13 light years that way, and 13 light years the opposite way?
Is there some "relativity trick" I'm missing? Is some space aliean on the edge of the universe going to see 13 light years in all directions?
This is a misconception. If you just want a satisfying mental image of the Universe: It's really big, it doesn't have any point defined as its center, and it doesn't have any wall-like borders. Aliens living 14.6 billion LY away from us see a Universe that looks just like what we see in all directions.
If it is limited in size at all, it's because it wraps around on itself ("if you go straight long enough you end up where you were"). Nobody has firm evidence that it actually does wrap around on itself, and there's reason to believe that if it does, it's on a scale that's much larger than we could ever hope to observe. We will likely never have any sign inconsistent with it being literally infinite regardless of whether or not it is.
If you want a better answer than that, you'll need a very firm grasp of special relativity and a basic understanding of general relativity and the metric expansion of spacetime. In which case, read this: http://arxiv.org/pdf/astro-ph/0310808.pdf
The parent post was trying to say "13 billion lightyears in <--- that direction and 13 billion lightyears in ---> this direction", because we happen to be at a relatively fixed point in space and can look in both directions. That doesn't assume we're at the center of the Universe, only that the light has reached us coming from both directions for as long as the Universe has been around.
The Schwarzschild radius of the observable universe is 10 billion light years. Me thinks that for the mass to be compacted into a smaller space at some point in the past, it would have either had to escape it's own event horizon (since under most current definitions it would be a black hole), or the definition of light-year had to have changed from what it is today. In that regard, the only thing making up the definition of light-year is distance and time; so space-time itself was radically different way back then (from our current perspective of relatively low energy densities).
Both! If the universe is flat, then it was always infinitely big. And, according to the inflationary universe theory, it expanded faster than the speed of light too.
In this case, you have to think of "expanding" as being like stretching. I.e., things within the infinitely big universe rapidly got farther apart from each other.
Everything we can see (which is called our "Hubble sphere") started off as a tiny little (and very massive) speck of stuff at the beginning of time. There were infinitely many tiny little specks, and they all became different Hubble spheres. But all the Hubble spheres all overlap each other, forming one continuous space. The same thing was true about the little massive specks at the Big Bang.
As I said above, at the time of the Big Bang, the universe was like a very dense sheet of rubber, and then it started rapidly stretching, getting less and less dense over time, until it is now the density that we see around us.
Infinities can definitely give you a headache, though. For instance, there are just as many odd integers as there are integers.
Space is big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist's, but that's just peanuts to space. ~ Douglas Adams, The Hitchhiker's Guide to the Galaxy
>Aliens living 14.6 billion LY away from us see a Universe that looks just like what we see in all directions.
How's that? Won't they see a completely different universe due to the speed of light? If they're 14.6 billion light years away from us, and they in our direction, won't they see what was happening in this area 14.6 billion years ago?
He doesn't mean the aliens will see an exact duplicate of what we see but that the universe is uniform enough that what they see would have the same properties (density, etc.) as what we see.
I guess this depends on how you define "across". Our Hubble sphere is all the space around us for which light has arrived. If the universe is infinitely big, then yes, there are things that we can see that are 26 billion light years from each other. However, they can't see each other, and so they are not causally connected to each other, even though we are causally connected to both.
>Is some space aliean on the edge of the universe going to see 13 light years in all directions?
An alien at the edge of our Hubble sphere will live in a different, overlapping Hubble sphere, from which they will see 13 light years in all directions around themselves.
that would assume that there is a center to the universe. I'm not sure there is (speaking as the lay-est of laypeople). There are articles on the topic which I won't embarras myself by summarising in a HN comment...
Perhaps the most disturbing of which would be the fact that, by definition, at some distance, there would have to be a duplicate of yourself.
Wouldn't it also mean that there are an infinite number of duplicates out there? Including one typing this very comment, but who typos and leaves the "s" off "duplicates"?
I'm very much not a mathematician nor physicist, can this really be true if the universe is infinite in size? Asking as a layman, at what point when dealing with infinite possibilities and probabilities does something become certain? That's what really terrifies me.
If the universe is flat, and therefore infinite, yes it means that most probably there are an infinite number of identical duplicates of you. It does not mean, however, that there are any duplicates of you that make a particular typo. I should think that there are an infinite number of such duplicates for any given typo, BUT it may also be the case that such worlds are impossible. E.g., physics is deterministic enough and chaotic enough that there is no slight nudge to the initial conditions of a world that will result in such a minor eventual difference to occur.
I.e., just because there are an infinite number of parallel worlds, doesn't mean that every imaginable thing occurs in them. What occurs in a parallel world, must be possible.
I think you make a very important point. Even if the universe was infinitely large, there might be a point where past that the conditions can't exist that would allow something like our planet to function. For infinity to grant you duplicates of yourself the possibility must exist within that "zone of infinity". Gravity for example might be stronger or weaker.
What if the universe is empty beyond a certain distance from the origin/central point, like a tiny explosion in the middle of an unending emptiness? Then there wouldn't have to be duplicates of you.
(This is about the same thing as the question asked below about whether infinite size implies infinite mass.)
In a flat universe, the universe starts off infinitely large and infinitely dense at time 0. As the universe expands, it gets less dense. Imagine an infinitely dense rubber sheet that starts to stretch at time 0, and after time 0 it is no longer infinitely dense, and it keeps slowly stretching forever, getting less and less dense over time. That's the universe as it is currently envisioned.
But our "Hubble sphere" is in no way special compared to other Hubble spheres. Our Hubble sphere is the part of the universe surrounding us from which light has arrived. Outside that sphere, nothing has a causal connection to us, since nothing can travel faster than the speed of light.
I.e., it just wouldn't be the case that our Hubble sphere is full of stuff and all the other Hubble spheres are are so different as to be empty.
If the universe is infinite in size, does that automatically imply that it is infinite in mass?
That is, I can imagine a mass distribution that decreases as it moves away from the center, such that the total amount of mass is finite. In which case these problems don't apply.
I'm not sure whether that's possible though. If space-time is flat, that means there's a specific mass density in that universe, in which case infinite mass exists. For an infinite universe to have finite mass would mean a density that tends to zero, in which case we should observe a saddle shape in the experiments described. That is, unless space is just -locally- (in 14.6 billion light years) flat.
I don't claim to be right. But, you can use this possibility to ease your existential dread at being one of infinitely many near-identical copies. Of course you still have to deal with many-world quantum physics...
Right, as long as you look through an amount of space that scales superexponentially with the complexity of the entity you want to find a copy of. (With stronger assumptions, like attractors in development of matter according to physical laws, you might be able to improve the bound.)
But don't worry, you get to search in three dimensions, so its distance only scales with the cube root of that!
The way I've heard this describes is with an analogy. If you have an infinite number of decks of cards, shuffled randomly, the same shufflings will show up an infinite number of times. There are only so many ways to put together an infinite number of particles, do the shufflings get reused.
You don't. There are bounds on how much matter can be contained within your past light cone before it turns out you are simply in a black hole.
The number of possible past light cones that could originate from the same Big Bang using reasonable assumptions about QM is staggeringly large, but nevertheless, finite.
Very cool, but I'm still a little confused. The earth's surface more or less just extends out in two dimensions, but doesn't space extend out in three? How does all of this work when you have take into account the z-axis as well as x and y? How would a three-dimensional object "close" in around itself?
There are different kinds of dimensions. There are simple, Euclidian dimensions where each dimension is perfectly orthogonal to the others, where parallel lines never intersect, etc. But there are other possibilities, non-Euclidean spaces and volumes. This is where the example of the sphere comes in. If you were a 2-dimensional being you could be on a plane or you could be on the surface of a sphere. Similarly, if you are a 3-dimensional being, such as us, you could be within a simple R^3 volume, or you could be on a hyper-sphere or another non-euclidian volume. The math is a little more complex, but similar.
To imagine a 2d surface closing in on itself, you imagine it in 3d. To imagine a 3d object closing in on itself, you unfortunately have to try to imagine it in 4d.
Another factoid that causes the mind to reel is that a flat universe, which is of infinite size, also has zero total energy. This means that it is conceivable that someday we might figure out how to manufacture entire new universes.
Who says that there's no such thing as a free lunch!
It doesn't follow that we will be able to do it, nor even that it is possible, but before the understanding that flat universes contain zero energy, the claim that you might manufacture a universe would seem to completely absurd. Where would you get all that energy? Especially if energy is conserved?
As it turns out, it doesn't matter if energy is conserved, because a flat universe doesn't have any net energy, and consequently they can come into existence without violating known conservation laws.
In any case, you don't have to take my word for it. Look up Alan Guth, the MIT cosmologist who invented the inflationary universe theory. This is his claim.
There was a talk about this issue at the Midwest Science of Origins Conference on 31 March 2012 by Marco Peloso of the University of Minnesota. Peloso reviewed the evidence available for the condition of the Universe just after the Big Bang. He also mentioned that current observations are consistent with a "flat" geometry of the entire universe, but pointed out other lines of evidence, not mentioned in the blog post submitted here, consistent with a finite (although very, very large) size for the universe.
Finite size is consistent with current observations and theories, and an issue that the submitted article doesn't have a lot of space to address. Per a Wikipedia article,
consistent with what I heard in the lecture earlier this year, "The universe appears to have no net electric charge, and therefore gravity appears to be the dominant interaction on cosmological length scales. The universe also appears to have neither net momentum nor angular momentum. The absence of net charge and momentum would follow from accepted physical laws (Gauss's law and the non-divergence of the stress-energy-momentum pseudotensor, respectively), if the universe were finite." Wikipedia cites the Landau and Lifshitz physics textbook from the Soviet Union, Landau, Lev, Lifshitz, E.M. (1975). The Classical Theory of Fields (Course of Theoretical Physics, Vol. 2) (revised 4th English ed.). New York: Pergamon Press. pp. 358–397. ISBN 978-0-08-018176-9, for this statement.
Some multiverse theories such as those mentioned in another comment posted before this one can be overlaid on a simpler theory of a single finite "observable" (in principle) universe with the properties we know from human observation. Testing theories like those, or like the "level I multiverse" theory mentioned in the other comment, still needs more work.
started off its first unit with the students reproducing the effort of Eratosthenes of Cyrene to measure the circumference of the earth more than 2,200 years ago. On reasonable assumptions known to the ancient Greeks, it was possible to get a surprisingly accurate estimate of the earth's circumference, with a major source of error being simply measuring distances between one city and another a few days' journey away.
No, unfortunately it doesn't work like that. It's not like the big bang took place "within space" -- like a grenade in the center of a stadium, but the bang is the rapid expansion of space(-time) itself. So, the universe could have even been infinite and very dense, and then BOOM!, it's still infinite but less dense.
While it's really enjoyable to talk/think about these topics from a purely scientific point of view, there's another side to all this, which is the sheer beauty of such a thing.
There's a great show on Discovery called How the Universe Works. Season 1 had an episode called Galaxies that sorta touches on some of this. The graphics for the show, first off, are awesome. When they show the super zoomed out view of all the galaxy clusters and how everything is connected, immediately I thought, "wow, those look like neurons".
And that's where the beauty of nature comes in. You can start to see how systems arrange and the patterns used from the micro to the macro. The distances on either level are incomprehensible (strings, atoms, molecules, galaxies, universe), but the patterns are visible.
And this is why I love science. While I'm not a physicist, I always appreciated Feynman because you could see he had that same love for the extremely complex and the simple observations - and he had a great way of describing it. I think the same goes for Sagan, although, in a totally different field. But Sagan too just had that love for the beauty of nature, on top of his accomplishments of the Mariner missions and other planetary science.
I have a 6mo son right now and I hope to instill that love of the beauty of nature through science. We're each an insignificant speck on an insignificant blue speck in an unfathomable universe. But, our connections to each other and our appreciation for the beauty of it all are a privilege. And posts like this, shows like How the Universe Works will be my way of conveying to him his place in it all. I hope he'll appreciate it someday like I do.
For the layperson (like me), the below is an intriguing and related TED talk from Brian Greene. A big conclusion is that we now believe the universe is expanding at an accelerating rate based upon the edge of the observable universe. But Greene posits that 1000s of years in the future, this accelerating edge will be too far away for us to observe and the universe will look to future mankind to be more static and small. I'm not physicist so I can't criticize this claim but I found it intriguing.
This article and the science behind it at this point is making some very interesting assumptions, such as that the Universe is closed (like the Earth is in their analogy).
But I believe at this point, the real answer is that we don't yet know. We're looking at a universe that looks flat and basically saying "well, it could be flat and finite, or it could be closed and curved, or it could be flat and infinite... but last time we thought our plane of existence was flat we ended up way off, so we better not make that assumption again."
Point being: take this conclusion about the size of the Universe with a grain of salt.
As an example of a flat but finite topology, think of the game "Asteroids" where, if the player's spaceship flies beyond the edge of the screen, it reappears at the opposite edge. This topology actually forms a torus! And, even though a torus definitely seems curved when embedded in 3D, its surface is actually flat - the angles of any triangle drawn on the surface always add up to exactly 180 degrees.
Cosmologist do usually take a flat universe to indicate an infinite universe. They also take "negative curvature" to indicate an infinite universe. Only a universe with "positive curvature" is taken to indicate a finite universe.
It's an analogy, because while it's easy to visualize how a two-dimensional surface can be curved, it's hard to visualize how a three-dimensional surface can be curved, because we think of it as a "volume".
I believe you're an reasonable, intelligent layman. Do you believe we're reasonable, intelligent people too? Are you willing to consider the possibility that the article did not make an "appalling oversight", the article merely phrased things confusingly?
You and I both have a good idea of what we think of as a two-dimensional surface, and as a three-dimensional space. In particular, we have an idea of how many real numbers uniquely identify any point in the surface or space (2 and 3, respectively), and we call these coordinates. Now comes the important part: we also have an idea of how to measure the length of any path in the surface or space, and as a derivative idea, the idea that the distance between any two points is the length of the shortest path between them. On a flat two-dimensional surface--among the English-speakers who converse often about such things, this is called a "plane"--it turns out that the distance between any two points is exactly the square root of the sum of the squares of the differences between the coordinates of each point (in other words, sqrt( (x1-x0)^2 + (y1-y0)^2 )). There is a similar relationship between distances between points and their coordinates in three-dimensional space, right? I'm sure you've seen all this before.
Now, there are two-dimensional surfaces for which that relationship is untrue, for example, the surface of a sphere or a surface that is "saddle-shaped" (like in the diagram, although they don't (and can't) show it extending infinitely). In these two examples, it turns out we can describe such surfaces, rather than as a set of points each uniquely determined by 2 coordinates, instead as a set of points uniquely determined by 3 coordinates and one constraint on what those coordinates can be, where the 3D distance has the relationship with coordinates that we're used to; for example, the surface of a sphere can both be thought of as being uniquely determined by longitude and latitude, and as being uniquely determined by x,y, and z coordinates, subject to the constraint that every point in the surface is the same 3D distance from some point in the 3D space. When described this way, things that would be non-obvious if I told you the relationship between distance and latitude and longitude, such as the fact that the 3D sphere is closed, become very obvious, because we have a good intuition for distances like that.
Now consider the following: let's say you're trying to figure out how the world works. An approach that's been quite successful is describing all the possible ways you can think of that are consistent with what we know, and then conducting experiments to try to rule each possibility out. One possibility you're describing involves the points being uniquely identified by 3 coordinates, but the distance between points doesn't have the usual square-root-of-the-sum-of-the-squares-of-the-differences relationship with the coordinates, and is instead just-so-slightly off. Let's say it is off in such a way that by describing the same set of points with 4 coordinates and 1 constraint (similar to how we described the sphere as 3 coordinates and 1 constraint), where the 4D distance between points uniquely described by those 4 coordinates obeys a relationship to the coordinates that is very similar to intuitive distances for the flat 2D surface and 3D space, so similar that almost all the same reasoning applies (continually increasing a coordinate, for example, will eventually continually increase distance in these cases, but not for longitude and latitude of a sphere).
Don't you think describing this as a "curved three-dimensional surface" is a sensible and accurate description?
First, thank you very much for taking the time to explain things to me. I really appreciate it, and I learned a lot from your post.
Regarding the "appalling oversight." To be fair, I would probably say this about 99% of all physics articles written for the layman. Practically all of them contain abstractions which the intelligent layman cannot connect to actual reality. I certainly encountered this in physics classes in college. This infuriates me to no end. I suspect the "average person" (I am not actually that average) doesn't really mind, and perhaps doesn't even really notice. The thing about it which actually infuriates me, is that I think a lot of scientists actually kind of delight in this.
Of course, a large reason for this whole problem is that the nature of scientific truths about reality has not yet been established among scientists (or really anyone in general, modulo a small group of philosophers I happen to agree with). So there is a sense in which it is "not their fault."
Regarding calling it a "curved three-dimensional surface." No, I do not think that is an acceptable description, because a surface is, by definition, defined as a 2D space within a 3D volume. You can't just throw out the definition. It's important to maintain the integrity of our concepts (including our definitions). The practical consequence of throwing out the integrity of our concepts is, for one thing, that someone like myself can't actually understand what is going on. (I have an MS and am about half way though a PhD). Now that you have explained the whole reason for describing it that way to me, I understand what it is - but I could not have understood it before that, and neither could the vast majority of readers of that article (unless they happened to be physicists). The practical consequences of this are possibly much wider than just confusing non-physicists, though.
To clarify my reason for defining a "surface" as I did. First, I think that's the actual everyday definition. More fundamentally - that is the only concrete thing in reality that people actually encounter. In other words, we do not encounter "3D surfaces" in reality, because that would require four dimensions. That fact is precisely why "surface" means what it does in English, and not what it (apparently) means in Physics.
It may be that Physics has accepted that certain words mean certain things, distinct from what they do in English. If that is the case, it was a mistake historically, and physicists today should be keenly aware of the issue when writing and teaching, so that they do not confuse ordinary people. Ordinary people need to push back harder on physicists to explain things in terms of concretes they can relate to in reality or (where needed) explicit mathemetics (as you did in your explanation).
Woah. Thank you for taking the time to read what I wrote. I hoped but hadn't expected that.
The thing about it which actually infuriates me, is that I think a lot of scientists actually kind of delight in this.
Have you ever heard the dictum "never ascribe to malice that which is adequately explained by incompetence"? A lot of experts really, really suck at explaining things, because explaining well is really, really hard.
It doesn't help that articles like the OP are aimed towards a mainstream audience assumed to have the attention span of a 5-year-old with ADD. The result is quick-and-dirty explanations that make the explainer feel like they explained something, and the reader feel like they learned something, but communication didn't actually happen. (My explanation was almost the length of the article. In order to cut it down to fit inside that article, it would have ended up equally as fruitful an explanation and drawn just as much ire as you.)
Aside:
Of course, a large reason for this whole problem is that the nature of scientific truths about reality has not yet been established among scientists (or really anyone in general, modulo a small group of philosophers I happen to agree with). So there is a sense in which it is "not their fault."
I think it's "not their fault" for the reasons I laid out in my previous two paragraphs. Every single elementary science student I have ever spoken to has put in some thought into the "nature of scientific truths about reality", enough that we feel like it's quite established among ourselves. I suspect you have a very specific, narrow notion of "[establishing] the nature of scientific truths about reality" that isn't what I have in mind, and that isn't relevant to why physics articles written for the layman have terrible explanations, either.
No, I do not think that is an acceptable description, because a surface is, by definition, defined as a 2D space within a 3D volume. You can't just throw out the definition.
Then what is an acceptable description?
What are you getting a PhD in? In every single academic field, people can and do extend definitions of terms that occur in plain English, because they study things laymen don't think about ("laymen" being our linguistic ideal of a "plain English speaker"), which is what the other commenter was trying to say about "plain English" being incapable of expressing abstract concepts in physics. Many academic fields also use technical terminology, but using jargon for every single thing that doesn't mean exactly what it means in plain English would require using a whole new language, I don't think that's what you're trying to say. But I don't understand what you are trying to say.
It's important to maintain the integrity of our concepts (including our definitions).
Have you heard of the "map-territory relation"? I agree that maintaining the integrity of our concepts is very important, and I agree that maintaining the integrity of our definitions is very important, but they're maintained in very different ways because they're not the same thing. Specifically, concepts themselves can't be allowed to be like one thing now and be like another thing later. We have to be able to state truths about a concept that are true in a way that can't change. Definitions, however, can change as long as they remain internally consistent and backwards-compatible with all previous important statements using such the definitions. This is done in every academic field, as explained in my previous paragraph.
Now that you have explained the whole reason for describing it that way to me, I understand what it is - but I could not have understood it before that, and neither could the vast majority of readers of that article.
I'm glad I was able to make you understand vaguely how 3D spaces could be curved. And I did just assert above that the article's explanation sucked. Now I'm going to argue that even so, it was fine: what did I explain, actually? I didn't actually explain in any more detail or clarity than the article how the universe might be curved? No, I explained what physicists and mathematicians mean when they talk about 3D space being curved (that distance doesn't bear the usual relationship to coordinates). Your understanding of the article is the same: it's been suggested that the universe might be curved, by analogy with how 2D surfaces can be curved, physicists measured some stuff, found that the universe is as close to flat as we can measure.
When you first read the article, you got super hung up on the problem that the universe isn't 2D, so why are they talking describing curvature of the universe in terms of curvature of 2D surfaces? And maybe this is an oversight by the authors, and they should've clarified that they were making analogy between curvature of 3D and 2D spaces. Or maybe most people who read the article don't get super hung up on this. The point stands that at worst it's a minor oversight.
To clarify my reason for defining a "surface" as I did. First, I think that's the actual everyday definition. More fundamentally - that is the only concrete thing in reality that people actually encounter. In other words, we do not encounter "3D surfaces" in reality, because that would require four dimensions. That fact is precisely why "surface" means what it does in English, and not what it (apparently) means in Physics.
"First...More fundamentally..." These do not appear to be distinct points. Am I misreading to interpret these as the same point: the "plain English" definition of surfaces is that they're 2D? That doesn't require clarification.
Although, re: "the only concrete thing in reality that people actually encounter...we do not encounter '3D surfaces' in reality", I'd agree with you more if you said "real life" rather than "reality".
If you consider the universe to be a well-defined volume, then there must be some boundary to it. And if there's a boundary, then you can ask the question "What's on the other side of the boundary?". Thinking of the universe as a 2D surface in a 3D space is a way to picture how it's possible to have a finite space with no well-defined boundary.
Well, it's not clear to me why we would assume there is a boundary. And I don't know what it means to think of a 3D area as a 2D surface anyway. And I don't think it's possible to have a finite space with no well-defined boundary.
Oh. Yes, now that you have mentioned it, I do recall "space" being used in math classes in a unique way. It never occured to me that that is how "space" was being used.
Most of the time when people give hard numbers for the size of the Universe, they either mean A) the portion of the Universe we can see, B) the portion of the Universe we could ever see in principle, or C) the portion of the Universe which could, in principle, be causally influenced by early particle interactions that could have causally influenced us. If you suspect someone might be talking out of their ass about cosmology, ask them how big the Universe is -- if they give you a clean multiple of 14.6 billion light years, they're an idiot.
Technically inclined readers will find this more enlightening than the article above: http://arxiv.org/pdf/astro-ph/0310808.pdf