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May 1, 2019 at 21:35 history edited Konstantinos Kanakoglou CC BY-SA 4.0
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Apr 8, 2019 at 2:33 history edited Konstantinos Kanakoglou CC BY-SA 4.0
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Apr 6, 2019 at 22:56 history edited Konstantinos Kanakoglou CC BY-SA 4.0
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Apr 6, 2019 at 22:36 history edited Konstantinos Kanakoglou CC BY-SA 4.0
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Apr 1, 2019 at 0:38 comment added Konstantinos Kanakoglou At the same time its mathematical formulation suggested that vectors are not enough: geometry had to be generalized to keep up with the phenomenological description suggested by a series of experiments. So, covectors or covariant and contravariant vectors, 1-forms, dualities, contractions, metric tensors, came into the picture, while phenomenology was generating pure theory, a route which has led to GR. This is the frame where the "physicists" or "standard" or "pre-1930's mathematician's" definition of a tensor in arbitrary dimensions was born.
Apr 1, 2019 at 0:34 comment added Konstantinos Kanakoglou By the end of the 19th century, lots of experiments suggested a very strange thing: the rules of transformation of velocity vectors between different reference frames should leave one particular value of magnitude unaltered, that is the speed of light in vacuum $c$. Einstein posed this suggestion as an axiom and started the search for the correct transformation rules which turned out to be the Lorentz transformations. Special relativity rised as the triumph of phenomenology (in my opinion).
Mar 31, 2019 at 21:55 comment added Timothy Chow Perhaps a useful example to consider: Electric and magnetic fields transform in a subtle way under a Lorentz transformation. One doesn't really know a priori how they'll transform; one must rely on experimental observation (phenomenology). In some sense, the "computational" or "physicist" version of the transformation law is the true starting point, since it's what nature presents us with phenomenologically. Of course we then develop our basis-independent theoretical account to explain things. But if you're experimentally-minded then the computational version is closer to actual observations.
Mar 31, 2019 at 3:24 history edited Konstantinos Kanakoglou CC BY-SA 4.0
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Mar 31, 2019 at 3:17 history edited Konstantinos Kanakoglou CC BY-SA 4.0
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Mar 31, 2019 at 3:05 history edited Konstantinos Kanakoglou CC BY-SA 4.0
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Mar 31, 2019 at 2:57 history edited Konstantinos Kanakoglou CC BY-SA 4.0
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Mar 31, 2019 at 2:51 history edited Konstantinos Kanakoglou CC BY-SA 4.0
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Mar 30, 2019 at 17:11 comment added KConrad @MichaelBächtold you are correct that mathematicians also used the ugly coordinate-dependent definition of tensors until the 1930s. And it's not true that all of physics is plagued with the "physicist's definition" of tensors: current textbooks on general relativity define tensors close to how the mathematicians do. See my discussion of tensor products in physics in the last section of kconrad.math.uconn.edu/blurbs/linmultialg/tensorprod.pdf.
Mar 30, 2019 at 11:15 vote accept Arthur
Mar 30, 2019 at 10:12 comment added Michael Bächtold Historically this "physicists definition" was probably also mathematicians definition up to ~1930. So I prefer to think of it as "pre 1930 vs post 1930 mathematicians" instead of "physicists vs. mathematicians".
Mar 30, 2019 at 5:05 history edited Konstantinos Kanakoglou CC BY-SA 4.0
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Mar 30, 2019 at 4:59 history edited Konstantinos Kanakoglou CC BY-SA 4.0
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Mar 29, 2019 at 23:24 comment added Konstantinos Kanakoglou maybe it would be useful to distinguish between undergaduate and graduate education at that point. Since there are different objectives and requirements in each.
Mar 29, 2019 at 22:44 comment added Sergei Akbarov @DeaneYang , I, on the contrary, would add critical notes to the observation of Konstantinos. I do not like this solipsism, in which physicists live (and not only they, this is a common phenomenon everywhere, including mathematics). Instead of seeking connections between disciplines that would make easier for people to understand the essence of things, scientists, on the contrary, seem to try to divide science, as if pursuing the goal of making it as incomprehensible as possible. I would say, that is unfair.
Mar 29, 2019 at 22:25 comment added Konstantinos Kanakoglou @Deane Yang, yes i would sign every word you say. I completely agree. By no means did i want to use the term "physicists educational culture" in a derogatory way.(see also my comment after David Roberts' comment at the OP).
Mar 29, 2019 at 22:20 comment added Deane Yang I think it's a little unfair to call it the physicists' educational culture. Physicists care mostly about how things work and not the abstract concepts we use to explain why things work the way they do. So they focus mostly on the quantities that need to be calculated and the rules the calculations have to follow. They need to know how to use physical information to set up the inputs to a calculation and at the end interpret the outputs of the calculation as physical consequences. They have their own plausibiity arguments for deriving the right rules of calculations from physical principles.
Mar 29, 2019 at 21:50 comment added Konstantinos Kanakoglou Hi Sergei! Nice to see you here! Of course i agree with your comment. However, this would require a "multidisciplinary" approach from the professors teaching different courses even in the grad school: for example the "indices" definition is usually pursued in introductory courses on differentiable manifolds and riemannian geometry while the algebraic definition is usually pursued in general abstract algebra courses.
Mar 29, 2019 at 21:36 comment added Sergei Akbarov Konstantinos, however this lack of connection between the two definitions in textbooks creates a problem. Pedagogically it would be correct to discuss the different definitions and relations between them from the very begining.
Mar 29, 2019 at 21:21 history answered Konstantinos Kanakoglou CC BY-SA 4.0