Looking at things from abstract view does allow us not to worry about how we visualize the geometry which is actually hard and sometimes counter intuitive.
In special relativity, for example, a huge amount of attention is typically given to the Lorenz transformations required when coordinates change. However, the (Minkowski) space that is the setting for special relativity is well defined without reference to any particular coordinate system, as an affine space with a particular (pseudo-)metric. It's not conceptually very complicated, and I never properly understood special relativity until I saw it explained in those terms in the amazing book Special Relativity in General Frames by Eric Gourgoulhon.
For tensors, the basis-independent notion is a multilinear map from a selection of vectors in a vector space and forms (covectors) in its dual space to a real number. The transformation properties drop out of that, and I find it much more comfortable mentally to have that basis-independent idea there, rather than just coordinate representations and transformations between them.
The issue is the level of mathematical sophistication one has when a certain concept is introduced. That often defines or at least heavily influences how one thinks about it forever.
The basics of special relativity came up in my first year of university, and the rest didn't really get focused on until my second year.
The first time around I was still encountering linear algebra and vector spaces, while for the second I was a lot more comfortable deriving things myself just given something like the Minkowski "inner product".
(As an aside: I really love abstract index notation for dealing with tensors)
Eric Gourgoulhon is a product of the French education system, and I often think I would have done better studying there than in the UK.
I had started in a theoretical physics degree which was jointly taught by the maths and physics department. By my final year I had changed into an ostensibly pure maths degree, although I did it mainly to take more advanced theoretical/mathematical physics courses (which were taught by the maths department), and avoid having to do any lab work—a torsion pendulum experiment was my final straw on that one, I don't know what caused it to fuck up, but fuck that.
In the end I took on more TP courses than the TP students, nearly burnt out by the end of the year, and... didn't exactly come out with the best exam results.
That was one of the most interesting things of my EE/CS dual-degree and the exact concept you're describing has stuck with me for a very long time... and very much influences how I teach things when I'm in that role.
EE taught basic linear algebra in 1st year as a necessity. We didn't understand how or why anything worked, we were just taught how to turn the crank and get answers out. Eigenvectors, determinants, Gauss-Jordan elimination, Cramer's rule, etc. weren't taught with any kind of theoretical underpinnings. My CS degree required me to take an upper years linear algebra course from the math department; after taking that, my EE skills improved dramatically.
CS taught algorithms early and often. EE didn't really touch on them at all, except when a specific one was needed to solve a specific problem. I remember sitting in a 4th year Digital Communications course where we were learning about Viterbi decoders. The professor was having a hard time explaining it by drawing a lattice and showing how you do the computations, the students were completely lost. My friend and I were looking at what was going on and both had this lightbulb moment at the same time. "Oh, this is just a dynamic programming problem."
EE taught us way more calculus than CS did. In a CS systems modelling course we were learning about continuous-time and discrete-time state-space models. Most of the students were having a super hard time with dx/dt = A*x (x as a real vector, A as a matrix)... which makes sense since they'd only ever done single-variable calculus. The prof taught some specific technique that applied to a specific form of the problem and that was enough for students to be able to turn the crank, but no one understood why it worked.
Having studied physics, I would disagree rather strongly. I only really started understanding Special Relativity once I had a clear understanding of the math. (And then it becomes almost trivial.) Those of my fellow class mates, however, who didn't take the time to take those additional (completely optional) math classes, ended up not really understanding it at all. They still got confused by what it all meant, by the different paradoxes, etc.
I saw the same effect when, later, I was a teaching assistant for a General Relativity class.
(The comment I replied to mentioned both.)
Kip Thorne was also heavily influenced by this geometric approach. Modern Classical Physics by Thorne & Blandford uses a frame invariant, geometric approach throughout, which (imo) makes for much simpler and more intuitive representations. It allows you to separate out the internal physics from the effect of choosing a particular coordinate system.
Um, yes it is. "A foo is an object that transforms as a foo" is a circular definition because it refers to the thing being defined in the definition. That is what "circular definition" means.
When people say "a tensor is a thing that transforms like a tensor" they're using a convenient shorthand for the bit that I put in angle brackets above.
My favourite explanation is that "Tensors are the facts of the universe" which comes from Lillian Lieber, and is a reference to the idea that the reality of the tensor (eg the stress in a steel beam or something) is independent of the coordinate system chosen by the observer. The transformation characteristic means that no matter how you choose your coordinates, the bases of the tensor will transform such that it "means" the same thing in your new coordinates as it did in the old ones, which is pretty nifty.
https://www.youtube.com/watch?v=f5liqUk0ZTw&pp=ygURdGVuc29yc...
Yes, but the "convenient shorthand" only makes sense if you already know what a tensor is. That renders the "definition" useless as an explanation or as pedagogy. It's only useful as a social signal to let others know that you understand what a tensor is (or at least you think you do).
> My favourite explanation is that "Tensors are the facts of the universe"
That's not much better. "The earth revolves around the sun" is a fact of the universe, but that doesn't help me understand what a tensor is.
What matters about tensors are the properties that distinguish them from other mathematical objects, and in particular, what distinguishes them from closely related mathematical objects like vectors and arrays. Finding a cogent description of that on the internet is nearly impossible.
> the reality of the tensor ... is independent of the coordinate system chosen by the observer
Now you're getting closer, but this still misses the mark. What is "the reality of a tensor"? Tensors are mathematical objects. They don't have "reality" any more than numbers do.
> no matter how you choose your coordinates, the bases of the tensor will transform such that it "means" the same thing in your new coordinates as it did in the old ones
That is closer still. But I would go with something more like: tensors are a way to represent vectors so that the representation of a given vector is the same no matter what basis (or coordinate system) you choose for your vector space.
That's just incorrect though for a couple of reasons. Firstly, a vector in the sense in which it is used in physics is a rank 1 tensor so it has this transformation behaviour just like other higher order tensors. Secondly the representation is the thing that changes, but the meaning of that representation in the old basis and the new basis is the same. For example, if I take the displacement from me to the top of the Eiffel tower, I can represent that in xyz Cartesian coordinates or in spherical or cylindrical coordinates, or I can measure it relative to an origin that starts with me or at sea level at 0 latlong. The representation will be very different in each case, but the actual displacement from me to the top of the Eiffel tower doesn't change. What has happened is the basis vectors transform in exactly such a way as to make that happen. It's a rank 1 tensor in 3 dimensions because there is a magnitude and one direction (one set of 3 basis vectors) in whatever case.
Now if I want an example of a rank 2 tensor think about a stress tensor. I have a steel beam which is clamped at both ends and a weight is on top of it. This is a tensor field. For every point in the beam there are different forces acting in each direction. So you could imagine the beam as made up of a grid of little rubik's cubes. On each face of each cube you have different net forces. (eg at the middle of the beam the forces are mainly downwards due to gravity, at the ends of the beam the fact that the middle of the beam is bowing downards will lead to the "faces" that point to the middle of the beam to be being pulled towards the middle (transverse to the beam and slightly downwards) whereas the opposite face is pulled in the opposite direction because the ends of the beam are clamped. So I need two sets of basis vectors. One set indicates the "face" experiencing the force, one set indicates the direction of the force. Now just like the vector/rank one tensor case I can represent those in whatever coordinate system I want, and my representation will be different in each case, but will mean the same sets of forces in the same directions and applied to the same directions because both sets of basis vectors will transform to make that true. I would call that a rank 2 tensor field because I would express it as a function from a set of spatial coordinates to a thing which has a magnitude and 2 directions (that's what I think of as the tensor). However I understand physicists and civil engineers and stuff just call the whole thing the stress tensor (not the stress tensor field). I could be wrong.
So what I mean when I talk about the reality of the tensor I mean whatever it is the tensor is expressing in the physical universe (eg the displacement from me to the tower or the stress in the beam). From a mathematical point of view I agree of course, mathematical objects themselves are purely arbitrary and abstract. But if you have a bridge and you want to make sure it doesn't buckle and fall down, the stress tensor in the bridge is a real and important fact of the universe that you need to have a decent understanding of.
Quite possible. But that's in no small measure because I have yet to find an actual cogent definition of "tensor" that distinguishes a tensor from an array. (I have a similar problem with monads.)
> So what I mean when I talk about the reality of the tensor I mean whatever it is the tensor is expressing in the physical universe
OK, but then "the reality of a tensor" not depending on the coordinate system has nothing to do with tensors, and becomes a vacuous observation. It is simply a fact that actual physical quantities don't depend on how you write them down, and hence don't change when you write them down in different ways.
That's recursive, not circular. These are not the same thing despite being closely related. Recursive definitions are useful. Circular ones are not.