What a terrible article. Can anyone who is not a mathematician tell me one thing they learned from this?
The naked term "manifold" in its modern usage, refers to a topological manifold, loosely a locally euclidean hausdorff topological space, which has no geometry intrinsic to it at all. The hyperbolic plane and the euclidean plane are different geometries you can put on the same topological manifold, and even does not depend on the smooth structure. In order to add a geometry to such a thing, you must actually add a geometry to it, and there are many inequivalent ways to do this systematically, none of which work for all topological manifolds.
Well as a non mathematician all I saw in your description was opaque jargon. "locally euclidean hausdorff topological space" means nothing to me. It'd be like if I asked what the Spanish word "¡hola!" meant and the answer was in evocative Spanish poetry. Extremely unlikely to be helpful to that person who doesn't know basic greetings.
This article breaks that loop and it's refreshing to see a large topic not explained as an amalgamation of arcane jargon
I'm pretty confident I'm not going to give a definition that's suitable for you and I don't wish to engage with diminutive minutiae or arcane jargon in a hostile combative atmosphere.
> Can anyone who is not a mathematician tell me one thing they learned from this?
I can share my two take-aways.
- in the geometric sense, manifolds are spaces analogous to curved 2d surfaces in 3d that extend to an arbitrary number of dimensions
- manifolds are locally Euclidean
If I were to extrapolate from the above, i'd say that:
- we can map a Euclidean space to every point on a manifold and figure out the general transformation rules that can take us from one point's Euclidean space to another point's.
- manifolds enable us to discuss curved spaces without looking at their higher-dimension parent spaces (e.g. in the case of a sphere surface we can be content with just two dimensions without working in 3d).
Naturally, I may be totally wrong about all this since I have no knowledge on the subject...
The more language is allowed to drift, the harder it becomes to read old language. I think this is a particularly silly case, but in general, the complaint that people are misusing words shouldn't be met with "It's impossible to misuse words", which this argument implicitly is.
No one allows or disallows language to drift, there are no language enforcers. This argument is not “it’s impossible” but rather it’s pedantic to claim a word is misused, when it’s been used this way for hundreds of years and so the original definition is no longer applicable.
Someone could of course institute language enforcers for English, but I'm very skeptical about both the enforcement mechanisms, and the usefulness of even a successful enforcement.
Bodies like the Académie Française do try to promote language standards ('enforce' is probably not the right word). But I'm not sure how successful they are.
This seems like an object lesson in making sure that the right hand does not know what the left is doing. Yes, if you have two departments working on two mutually exclusive architectures, one of them will necessarily fail. In exchange, however, you can guarantee that it will be the worse one. This is undervalued as a principle since the wasted labor is more easily measured, and therefore decision making is biased towards it.
I agree with you, but perhaps this is very hard (impossible?) to pull off. Invariably, politics will result in various outcomes being favored in management and the moment that groups realize the game is rigged, the whole fair market devolves into the usual political in-fighting.
> This was not a immigration operation where agents went into the premises, rounded up folks, and put them on buses
This quote implies that there are immigration operations where people do these things, and that this particular instance is not an example of that type of operation.
