u/Blakut

I'm an astrophysicist turned data scientist so bear with me as I butcher terms or make some assumptions while asking this question. I learned math for physics, so my courses were light on proofs. However I do enjoy reading about maths, solving the occasional problem for fun, or when I have time more rigorously revisiting the maths from gradschool. I chose the analysis flair because I had to choose a flair.

From my understanding while lurking on this sub, of how maths at your level works, for a certain field, it's basically based on certain axioms and all the theorems that derive from them, or from possibile conjectures. Then a body of knowledge of useful statements is built and so on.

From my side, I see a lot of mathematics that was discovered or invented (whichever term you prefer haha) to solve certain problems. Sometimes existing maths was sometimes coopted, for example imaginary numbers were discovered way before they were used in physics.

Now to get to the meat of my question. Let's take for example the equation of motion, the laws of gravity, and Newton's work. From what I've read so far, at that point in history, calculus was being developed to solve physical problems. Newton's Principia used this rather geometric approach to prove and develop his theory. Later on, however, the methods used, for example, to calculate motions were refined. We got the Lagrangian and the Hamiltonian. Then in GR or QM other powerful tools are used, vectors, matrices and tensors etc.

While looking at how certain problems are dealt with mathematically in physics I'm wondering, how many of the methods used exist the way they are because that is how they were first formulated, caught on, and remained unchanged, maybe because further developments were based on them, or because they just worked? And how often did a better more efficient way, maybe simpler and more elegant method, replaced it?

I guess my question can also be framed as, can the development and application of certain mathematical methods and tools be viewed as traveling along a path, to get to a set of working, useful solutions or methods? And how often were shortcuts or more efficient paths were discovered? How often does this happen nowadays?

This question I could also be valid for mathematics in general as well, I guess, as geometry was done in a certain way in ancient times, with a straight edge and compass, but later on algebra was introduced into geometry. Sorry for possible mistakes or wrong assumptions, but this is the clearest way I could formulate my questions so far.

I got many hours in Qud, but this tops everything so far. There's a historical site in my run (I'm almost level 40 true kin) populated by chrome pyramids. How am I supposed to handle it? I'm considering getting quantum pages to increase rep, coz I can't beat them all.