Timeline for When are $\mathbb{R}^n$ and $\mathbb{R}^m$ essentially similar?
Current License: CC BY-SA 3.0
24 events
| when toggle format | what | by | license | comment | |
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| Apr 30, 2017 at 3:01 | review | Close votes | |||
| Apr 30, 2017 at 8:33 | |||||
| Apr 24, 2017 at 9:48 | comment | added | Sam Nead | To be clear, I think this question violates the principles listed here: mathoverflow.net/help/dont-ask -- the question is open ended, cannot have a "correct" answer, and invites discussion. Note that I do not think the poster is crazy, or the question is dumb, etc... It is just that the question doesn't have (and cannot have) a definite answer. | |
| Apr 24, 2017 at 9:46 | comment | added | Sam Nead | The original question was "how large does $n$ have to be for Euclidean spaces of dimension $\geq n$ all to behave essentially the same way?" This question can't mean anything, because we aren't told what "the same way" means or what "essentially" means. What is the notion of sameness that is of interest here? | |
| Apr 24, 2017 at 3:35 | answer | added | Wlodek Kuperberg | timeline score: 6 | |
| Apr 23, 2017 at 15:54 | answer | added | Mikhail Katz | timeline score: 1 | |
| Apr 23, 2017 at 15:47 | review | Close votes | |||
| Apr 23, 2017 at 16:51 | |||||
| Apr 23, 2017 at 15:39 | comment | added | Bruce Blackadar | This was indeed intended to be a serious question, and I don't think it is at all meaningless. It is ill-defined, and intentionally so; $\mathbb{R}^n$ is many things. I asked it to try to provoke the kind of discussion which has begun. As for the manuscript, it is not such a huge file (48MB), but it is long, although I have tried to make it easily searchable (I eventually hope to do better), and the relevant section can be easily found by a hyperlink from the table of contents. | |
| Apr 23, 2017 at 15:35 | comment | added | Franz Lemmermeyer | Of course $n$ has to be at least $42$, but everybody knows that. | |
| Apr 23, 2017 at 15:26 | comment | added | Peter Heinig | One of the infinitely many aspects in which high-dimensional products of the real numbers can be considered to be eventually similar is whether they are suitable for embedding a given class of objects into them (w.r.t. a given class of maps). There is a plethora of examples. Elementary example: w.r.t. smoothly embedding countable simple graphs, $\mathbb{R}^n$ is equivalent to $\mathbb{R}^m$ for any $m,n\geq 3$. Classical example: H. Whitney proved that for smoothly embedding a $d$-dimensional manifold, they are equivalent as soon $m,n\geq 2d$. | |
| Apr 23, 2017 at 15:26 | answer | added | Joel David Hamkins | timeline score: 10 | |
| Apr 23, 2017 at 14:22 | history | reopened |
Alain Valette R.P. Mikhail Katz user44143 Ulrich Pennig |
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| Apr 23, 2017 at 12:46 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Apr 23, 2017 at 7:54 | comment | added | Yemon Choi | @SamNead I am ambivalent at the moment about whether the question should be reopened, but "meaningless" seems too strong. Perhaps "ill-defined" might be a fairer criticism? (Having read some of the OP's work I am inclined to start from the position that he has serious intent and proven experience, to put it mildly.) | |
| Apr 23, 2017 at 7:48 | review | Reopen votes | |||
| Apr 23, 2017 at 14:11 | |||||
| Apr 23, 2017 at 7:12 | history | closed |
YCor GH from MO Qiaochu Yuan Franz Lemmermeyer Sam Nead |
Needs details or clarity | |
| Apr 23, 2017 at 7:12 | comment | added | Sam Nead | I took a (very brief) look at the PDF and the discussion within. I applaud your work in trying to come to grips with both the foundations of mathematics and some of the deeper questions beyond the foundations. But it seems to me that your question is meaningless. I am voting to close. | |
| Apr 23, 2017 at 0:09 | comment | added | Yemon Choi | The PDF of the full book seems to be a rather large file. Would it be possible to replace your link with a link to an excerpt? | |
| Apr 23, 2017 at 0:04 | comment | added | Danielle Ulrich | It's not clear to me that $\mathbb{R}^{2n}$ is all that similar to $\mathbb{R}^{2n+1}$. For instance, consider the following property of a vector space $V$: ``There exists a projection $\pi: V \to V$ and an isomorphism $\psi: V \to V$ such that $V = \pi(V) \oplus \psi(\pi(V))$." This characterizes the $\mathbb{R}^n$ for $n$ even. | |
| Apr 22, 2017 at 23:44 | comment | added | Oscar Cunningham | One could look at this question to find theorems which are true in low dimensions but false in high dimensions. The highest dimension involved seems to be $65$. So $\mathbb R^{65}$ is similar to all higher dimensional spaces in the sense that each property mentioned in that post either holds for all of them or fails for all of them | |
| Apr 22, 2017 at 23:28 | review | Close votes | |||
| Apr 23, 2017 at 7:15 | |||||
| Apr 22, 2017 at 23:13 | comment | added | user44191 | There's of course the simple answer: as sets, groups, and $\mathbb{Q}$ or $\bar{\mathbb{Q}}$-vector spaces, they are isomorphic. | |
| Apr 22, 2017 at 23:12 | comment | added | Geoffrey Irving | Couldn't this question be asked about the integers themselves? I.e., how large do integers have to be before they're basically the same? It's hard to imagine a more context dependent question. | |
| Apr 22, 2017 at 23:11 | comment | added | paul garrett | Although I don't have anything smart to say about this, I do see how the question could make sense: algebraic/differential topology behaves significantly differently in 1, 2, 3, 4, and then $\ge 5$ dimensions. As dimension goes up, spheres become (perhaps surprisingly) much smaller subsets of the circumscribing cube. Etc. | |
| Apr 22, 2017 at 22:59 | history | asked | Bruce Blackadar | CC BY-SA 3.0 |