It’s actually much more well written than the majority or articles we usually come across.
Besides not treating readers like idiots, they take themselves seriously, hire smart people, tell good stories but aren't afraid to stay technical, and simply skip all the clickbait garbage. Right now from the Scientific American front page: "Type 1 Diabetes science is having a moment". Or from Nature: "'Biotech Barbie' says ..". Granted I cherry-picked these offensive headlines pandering to facebook/twitter from many other options that might be legitimately interesting reads, but on Quanta there's also no paywalls, no cookie pop-ups, no thinly-veiled political rage-baiting either
I like this honestly because this shows that I learned something intelligent. On the other hand, if I don't feel exhausted after reading, it is a strong sign that the article was below my intellectual capacity, i.e. I would have loved it if I could have learned more.
For other publications they are beholden to people who haven't figured out ad-block, and your bar needs to be pretty low to capture that revenue.
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.
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.
> The term “manifold” comes from Riemann’s Mannigfaltigkeit, which is German for “variety” or “multiplicity.”
Initially I recoiled at the thought of the stiffness of the CD, but of course your absolutely right, at least for 2d manifolds.
You're thinking of open sets.
hshdhdhehd•4h ago
dvt•4h ago
andycrellin•3h ago
It becomes a bigger problem when the etymology is actually a chain of almost arbitrary naming decisions - how far back do I go?!
Enginerrrd•2h ago
Names are important.
namibj•51m ago