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".
