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The geometry of our universe is, on human scale, Euclidean. This means that our everyday experience gives us a good understanding of $\mathbb{R}^2$ and $\mathbb{R}^3$, which in turn is of great help to develop Mathematics and Sciences. For instance, our experience of $\mathbb{R}^2$ makes easy to draw (sufficiently regular) $\mathbb{R} \to \mathbb{R}$ functions, and to represent complex numbers with a geometry interpretation of their sum and product. Moreover, our experience of $\mathbb{R}^2$ and $\mathbb{R}^3$ makes easy to develop linear algebra in such vector spaces, and to generalize it to more general vector spaces such as $\mathbb{R}^n$.

But suppose that we evolved instead in a universe that, on human-scale, is non-Euclidean, so that our everyday experience is, for example, a geometry with positive or negative curvature. Suppose also that this does not mess up too much with physics so that human life is not particularly different.

How would Mathematics and Sciences had developed, with humans lacking the intuition of $\mathbb{R}^2$ and $\mathbb{R}^3$ (but with a probably strong intuition of curved spaces) ?

rosan98
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    Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. – Community May 02 '23 at 14:41
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    if you think a universe with more or fewer dimensions than we have means that this non euclidean geometry is going to result in some sort of fundamental change to mathematic realities then I think that euclidean geometry isn't what you think it is, two plus two will always be four regardless of the number of dimensions any 'universe' has .. I think you must have fallen victim to some form of magical thinking? – Pelinore May 02 '23 at 19:18
  • "Suppose also that this does not mess up too much with physics": Unfortunately this is very hard to arrange; our physics just doesn't work at all in a non-flat universe. If the universe has a human-noticeable positive curvature then it is small. Very small. Things get weird very fast if Earth is inside the sun. If the universe has a human noticeable negative curvature then things get even weirder even faster. For example, if the universe has a human-noticeable negative curvature, how does gravitation work? – AlexP May 02 '23 at 20:05
  • Hello @rosan98, welcome to [worldbuilding.se]. Whenever you see that first comment from CommunityBOT, that's a red flag. That comment is generated automatically when the software can't discern a specific, focused question with the possibility of an objective answer. The problem is this, you're asking what we call a High Concept Question, and they're off-topic. HCQs are open-ended, opinion-based, too broad, hypothetical, and lead to all answers being equally valid - all of which is prohibited by Stack Exchange (see [help/dont-ask]). – JBH May 02 '23 at 22:01
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    To summarize and personalize, you've asked us to re-invent all mathematics since the creation of Euclidean Geometry (300BC) until today. That's not just a tall order, that might be too much for a PhD dissertation, and that assumes that whatever assumptions we make performing that gargantuan effort made any sense (the opinion-based part). Is there a question you can ask that's (much) more specific and possible to answer with a single page answer? If not, either we'll need to close the question. – JBH May 02 '23 at 22:07

1 Answers1

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The Same

Science and Maths would have developed in the same manner. Experiments and reason, logic and imagination, used to suss out the way the universe works.

Find something you expect to be true; design an experiment that would disprove it if if wasn't true; then run the experiment. If the thing is not disproven, then you have some evidence. Lots of evidence you have a Theory.

The only difference would be the sorts of things they are sussing out. A different universe has different rules. So their strongly curved LHC is proving different things to our Euclidean LHC.

To get into more detail about how the rules are different, this discussion runs the risk of failing "the book test": If you can imagine an entire book that answers your question, you’re asking too much.

For examples of such books check out Orthogonal by Greg Egan.

Daron
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  • I'm 98.5% sure you're not answering the question the OP asked... at least not until the 3rd paragraph, which is a meaningless generalization, and the 4th paragraph, where you admit the question shouldn't have been answered. – JBH May 02 '23 at 22:08
  • @JBH I'm pretty much saying "this is a bad question". For a more experienced user, I would have left a comment. But the user is new, and I wish to be more lenient. My advice is too long for a comment. – Daron May 03 '23 at 08:22
  • Believe me, I get it, but posting a comment as an answer is literally a reason to flag an answer for deletion (click "flag" and read the explanation under "Not an answer"). Please consider leaving multiple comments to express more complex issues about the question. Posting answers to bad questions legitimizes the bad question, no matter the intent. – JBH May 03 '23 at 15:00
  • @JBH Okay, I have flagged the answer for moderator intervention. My glorious leader, do what you gotta do. It has been a few days so I suspect rosan98 has already read the answer. – Daron May 04 '23 at 12:00
  • I wonder if the OP did read the answer. The good fellow hasn't so much as taken the tour or left a comment. I wonder periodically about people who appear to swoop in, toss a random hand grenade, then swoop out... never to return. The Internet has spawned odd behavior in its lifetime. – JBH May 04 '23 at 13:57
  • @JBH An enigma who also knows TeX commands. – Daron May 04 '23 at 14:50
  • @Daron (sorry for the late reply). I don't believe that mathematics would have developed in the same way. The first paragraph of my question already explains why (your answer ignores it). There's probably some kind of misunderstanding. I don't see how it can be claimed that in a non-Euclidean on human-scale universe mathematics would have developed in the same way as in our universe. For example, would ancient Greeks had focused so much on Euclidean geometry? I don't think so. – rosan98 May 07 '23 at 14:44
  • Anyway, it seems to me that my question does not interest this community. I'm a bit surprised since a very similar (and perhaps more extreme) question was very welcome in the past https://worldbuilding.stackexchange.com/questions/2966/a-world-without-mathematics – rosan98 May 07 '23 at 14:46
  • @rosan98 I suspect their ancient Greeks would be more interested in whatever geometry their world has rather that the abstract (to them) zero curvature geometry. But being more specific than that is impossible – Daron May 07 '23 at 15:22
  • @rosan98 actually, Euclidean geometry would still probably have been studied because non-Euclidean spaces can still be treated as locally Euclidean (that is the basis of differential geometry, where you investigate spaces locally homeomorphic to $\mathbb{R}^n$). So actually, a first step to answering your question would be: at what scale do you start seeing the effects of non-Euclidean geometry? – Azur Jun 18 '23 at 11:34
  • @rosan98 this is certainly not at the small scale of a planet (assuming that planets had formed). One could probably cook up some valid solar system in a space-time which is $1$-dimension of time + $3$-dimensions of already-curved space; though you'd probably have to get into the equations to see how that would look. More likely though, the non-Euclidean effects would start appearing at very large scales (like a galaxy), and everything at the scale of a solar system or less would appear perfectly Euclidean to us. Things would simply get weird if we look out to far-away stars. – Azur Jun 18 '23 at 11:37
  • @Azur You are mixing up the geometry and the topology of a space. The top of a sphere is "locally Euclidean" in the topological sense. But this does not help you measure how much land you have if you live on that sphere. You need to know the geometry for that. And the sphere is not locally Euclidean-geometric. – Daron Jun 18 '23 at 16:32
  • @Daron what I was saying is that, locally, the rules of Euclidean geometry apply. If a farmer lives on Earth and wants to measure they land's surface, they don't need to bother about the fact that we live on a sphere. They'll just trace a bunch of rectangles, and apply the classic $\text{width}\times\text{height}$ rule. Now, of course, they'll realize that they live on a sphere eventually. But maybe the universe looks very Euclidean in a local neighbourhood of the planet's solar system, and they'll never realize anything is off – Azur Jun 19 '23 at 17:11
  • @Daron but I do see your point: it is true that the equivalence is only topological. And it may also be the case that even at our scale, the rules of geometry look really messed up. – Azur Jun 19 '23 at 17:14