Timeline for Is it possible to do calculus and differential geometry the old school way, without any ortho frames or axis? [closed]
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55 events
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| Jan 22, 2021 at 21:30 | comment | added | Yemon Choi | Other readers coming late to this long exchange of comments might wish to check out questions on math.SE by the same user and his or her replies to interlocutors: math.stackexchange.com/questions/3608556/… and math.stackexchange.com/questions/3956355/… | |
| Dec 22, 2020 at 21:16 | comment | added | Matko | But if my claim is right, although I didn't exactly explicitly say what you imply, who will care about that, unless I actually do some great breakthrough or unification, and who will care to check it... Thaugh you do have a point.. | |
| Dec 22, 2020 at 20:48 | comment | added | Yemon Choi | "I could wrote a book about c.free formulations and their merits, and especially on how exactly they leat to new advances and unifications" -- this sounds like a much more productive activity than constantly claiming research mathematicians here have some kind of deficient understanding | |
| Dec 22, 2020 at 18:43 | history | closed |
Liviu Nicolaescu abx user21349 Yemon Choi Deane Yang |
Needs details or clarity | |
| Dec 22, 2020 at 11:01 | history | edited | Matko | CC BY-SA 4.0 |
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| Dec 22, 2020 at 9:32 | history | edited | Matko | CC BY-SA 4.0 |
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| Dec 22, 2020 at 9:26 | history | edited | Matko | CC BY-SA 4.0 |
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| Dec 22, 2020 at 9:18 | comment | added | Matko | I do not idealise any man or work, the only reason I brought up principia is to save myself the trouble of answering unending strings of questions on how will I practically calculate without a basis, so that's why I called upon the highest authority on that matter. | |
| Dec 22, 2020 at 9:12 | comment | added | Matko | "Modern work, however, has shown the value of coordinate-free geometric formulations. Likewise there is also a value in geometrically invariant derivations. Not only do they directly show that the result is coordinate independent they also serve to clarify certain assumptions." google.com/url?sa=t&source=web&rct=j&url=http://… | |
| Dec 22, 2020 at 9:08 | comment | added | Matko | Keep your basis tricks for yourself. I simply asked for a reference, I know what I want, and by some miracle I got a few good references and a nice answer. I don't wish to preach invariant philosophy and praxis nor am I against non invariant ways. Only problem I have with it is that people say it's that way or the highway, you can't do anything without a basis, yada yada.. As I said both ways have their drawbacks and benefits. I could wrote a book about c.free formulations and their merits, and especially on how exactly they leat to new advances and unifications. | |
| Dec 21, 2020 at 23:14 | comment | added | Deane Yang | By the way, a trick I learned from Robert Bryant is that, if you do your calculations on the orthonormal frame bundle (or sometimes on the full not-necessarily-orthonormal frame bundle), the all of the differential forms are defined globally and you do not have to prove invariance under change of frame. | |
| Dec 21, 2020 at 22:54 | history | became hot network question | |||
| Dec 21, 2020 at 22:08 | comment | added | Yemon Choi | 'when it comes to bridging different fields and tackling the biggest problems invariant reasoning is the most effective and practical" - citation needed. You are on a site which is read by and commented upon by research-level mathematicians, and yet you repeatedly make sweeping assertions like this without really backing them up. What is your actual mathematical experience (not physics!) which supports this? | |
| Dec 21, 2020 at 21:57 | comment | added | Yemon Choi | Also, why all the different user accounts? | |
| Dec 21, 2020 at 21:56 | comment | added | Yemon Choi | if you do not allow yourself the definition of a basis then what, pray, is your definition of finite-dimensional? How do you even know that you are working in the Euclidean plane as opposed to Euclidean 3-space? It seems to me that you are attracted by a dream of synthetic geometry but are being very dogmatic about what should be allowed in it | |
| Dec 21, 2020 at 20:25 | comment | added | Matko | I am thankful for the interesting links and books Iv got, and, if anyone is interested, first time I fell through this rabbit hole is due to baez and his gauge fields knots and gravity. There is also misners gravitation. Two of the most beautiful books Iv come across ever.. And both emphasize the synergy of coordinate and coordinate free approaches. I have just found that, athouh usually much harder and more impractical, purely invariant ways have their merits, that's all | |
| Dec 21, 2020 at 20:20 | comment | added | Matko | No, obviously from the description, a basis is a big no no, as it is the most obvious example of pasting non canonical structure. Again..., I'm not saying this is bad in general but simply as I explained it should not be the only way | |
| Dec 21, 2020 at 20:17 | comment | added | Matko | @Deane Yang I asked this question to see what the current state of research is in terms of invariant geometry. It is not a matter of one over the other. In everyday math work is is easier and more practical to use whatever particular structure is convinient,instead on delving too deep on the very fabric of foundations, but in a more global view, when it comes to bridging different fields and tackling the biggest problems invariant reasoning is the most effective and practical. The reason I brought up Newton is to demistify what it means to be coordinate less. Look at his brachistochrone proof | |
| Dec 21, 2020 at 18:53 | comment | added | Yemon Choi | BTW there is a definition of the derivative for a function f: U \to E, where U is an open subset of a normed vector space X and E is another normed vector space, which never mentions partial derivatives/Jacobians etc. (I know many in this comment thread know this but I wonder if it is the kind of thing which the OP is seeking or finds acceptable.) | |
| Dec 21, 2020 at 18:49 | comment | added | Yemon Choi | That is: the present account keeps referring to Newton's Principia, but surprisingly enough I do not have a copy on my bookshelf or my hard drive. So constantly saying "look Newton did calculus without Cartesian coordinates so there should be diff geom without mentioning coordinates" still leaves it unclear for many readers what the OP precisely wants. (What kind of definition of "3-dimensional Euclidean space" can one use, for instance? Is one allowed to mention the notion of a basis for a vector space?) | |
| Dec 21, 2020 at 18:46 | comment | added | Yemon Choi | There have been several very similar questions to this one in recent years (coming from various user accounts), with an equal insistence that the questioner wants to do diff geom or calculus without mentioning coordinates at all, which is a much stronger requirement than demonstrating "coordinate independence". It might help if the present account wrote out an explicit indication or example of something in calculus which is defined-and-proved without using coordinates, rather than just telling respondents that what they propose does not meet the requirements. | |
| Dec 21, 2020 at 18:30 | comment | added | Denis Nardin | I'm fairly confident you can do linear algebra, multivariable calculus, differential geometry etc without ever choosing a basis (so "coordinate free"). On the other hand this is very far from what Euclid would do, of course. Would that satisfy you? | |
| Dec 21, 2020 at 16:41 | comment | added | Deane Yang | What people like Burago, Ivanov, Petrunin, and others are doing with metric geometry is very much in the spirit of how Euclide did geometry, but it requires a much deeper geometric insight than what I have. As others have said, you might want to investigate their work. If you can understand it, I believe it's quite beautiful. | |
| Dec 21, 2020 at 16:38 | comment | added | Deane Yang | I have to admit that I share some of your views. When I teach differential geometry, I do like to start by recalling what classical Euclidean geometry is, as well as the need to develop coordinate-independent physical laws, and emphasizing that differential geometry has the same ultimate goal but for curved spaces. And I do believe that often a more abstract approach is easier to understand than using coordinates. But in practice when you're actually trying to prove theorems or do calculations, coordinates or moving frames are almost always easier. | |
| Dec 21, 2020 at 15:45 | history | edited | Matko | CC BY-SA 4.0 |
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| Dec 21, 2020 at 15:42 | comment | added | Matko | I will shut this question down if I get one more comment on the justification of coordinates. I am not saying coordinates are bad, so you don't need to defend yourself. Maybe I phrased the question wrong. But you're right mark charts do give manifolds their algebraic structure like vectors, I'm just looking for a source that does it in a diffent way. I'm not condemning anybody | |
| Dec 21, 2020 at 15:38 | history | edited | Matko | CC BY-SA 4.0 |
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| Dec 21, 2020 at 15:31 | comment | added | Mark Wildon | I find I can persuade myself that coordinates aren't ugly if instead of thinking of a vector in $\mathbb{R}^n$ as a tuple of coordinates I regard it as an intrinsic geometry object that (alas) has to be represented in this way after a choice of basis. So really a vector is an equivalence class of pairs (basis, coordinates), one transforming contravariantly to the other, and no single coordinate system has to be preferred. I'm no expert, but I think much the same works replacing 'vector' with 'chart'. | |
| Dec 21, 2020 at 15:11 | comment | added | user21349 | related: mathoverflow.net/questions/14877/… math.stackexchange.com/questions/53021/… | |
| Dec 21, 2020 at 15:07 | vote | accept | Matko | ||
| Dec 21, 2020 at 15:01 | answer | added | Gabe K | timeline score: 5 | |
| Dec 21, 2020 at 14:48 | comment | added | Matko | I'm not here to debate which is better, I'm just asking for an alternative to coordinates, that's all | |
| Dec 21, 2020 at 14:20 | comment | added | Deane Yang | By the way, there is a reason why we all abandoned Euclid's approach to geometry and prefer working with Cartesian coordinates. It's a lot easier to prove theorems using the latter, even with the additional step of proving coordinate independence. | |
| Dec 21, 2020 at 14:14 | answer | added | Deane Yang | timeline score: 13 | |
| Dec 21, 2020 at 14:10 | comment | added | Michael Bächtold | Coordinates are variables. In which sense are these different concepts? | |
| Dec 21, 2020 at 14:01 | comment | added | Faris | And here I thought the old-school way was avoiding topoi... | |
| Dec 21, 2020 at 14:01 | comment | added | Matko | I don't know what you mean, ever looked at principia, or even for example vector calculus, as long as you don't express vectors as triples of scalar, you can still prove and calculate stuff and have variables without coordinates | |
| Dec 21, 2020 at 13:56 | comment | added | Michael Bächtold | I don't see the difference between coordinate-less and variable-less geometry. I doubt the second one is possible, hence to my mind also the first one. | |
| Dec 21, 2020 at 13:54 | comment | added | Matko | @michael yes that is what I consider coordinate less, it was the only geometry available for thousands of years, from the time of Aristotle to the time of Newton, up until descartes | |
| Dec 21, 2020 at 13:52 | comment | added | Matko | The difference is in that while basis free and coordinate free might only imply independence in regards to a particular choice of basis, frames or coordinates, coordinate less means explicitly(manifestly) independent of coordinates, simply because no coordinates are used | |
| Dec 21, 2020 at 13:50 | answer | added | user44143 | timeline score: 17 | |
| Dec 21, 2020 at 13:46 | comment | added | Michael Bächtold | If you describe a geometric figure by introducing variables that denote some properties of the figure (like arc lengths, areas, distances between special points etc.) would you then be doing coordinate less geometry? I'm not an expert in pre Descarte geometry, but I do believe they introduced variables this way. | |
| Dec 21, 2020 at 13:43 | comment | added | Deane Yang | What's the difference between "coordinate free" and "coordinate less"? | |
| Dec 21, 2020 at 13:32 | comment | added | Matko | I don't speak French, and I'm not looking just for a coordinate free text, coordinate less text, there is a difference | |
| Dec 21, 2020 at 13:30 | comment | added | Pietro Majer | For a coordinate-free treatment of différentiel calculus see for instance Henri Cartan's Cours de calcul différentiel | |
| Dec 21, 2020 at 12:22 | comment | added | Matko | Why the dislikes? | |
| Dec 21, 2020 at 11:57 | review | Close votes | |||
| Dec 22, 2020 at 18:47 | |||||
| Dec 21, 2020 at 11:32 | comment | added | Matko | I don't see why newtons calculations couldnt be more polished to formalise analogous calculus concepts, and then generalized to the case where the fifth postulate doesn't hold | |
| Dec 21, 2020 at 11:29 | comment | added | Matko | But maybe appearances are in some way deceiving, or the machinery could be more canonical | |
| Dec 21, 2020 at 11:28 | comment | added | Matko | I agree, this is exactly what appears to be the case. | |
| Dec 21, 2020 at 11:24 | comment | added | Ben McKay | Newton assumes that the space he works in is Euclidean, giving him symmetries and uniquely determined operators with those symmetries. If you want to work with something less symmetrical, you need a way to grab hold of its invariant differential operators, something which, it appears, must involve heavier machinery. | |
| Dec 21, 2020 at 11:23 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
probably a typo in the title
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| Dec 21, 2020 at 11:22 | history | edited | Matko | CC BY-SA 4.0 |
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| Dec 21, 2020 at 10:47 | review | First posts | |||
| Dec 21, 2020 at 14:41 | |||||
| Dec 21, 2020 at 10:42 | history | asked | Matko | CC BY-SA 4.0 |