I believe each time a new conceptualization arises in mathematics, once it is embraced and adopted, it is quickly put to use to resolve many long-standing issues. But then after a time, the conceptualization no longer is sufficient to resolve the issues that remain and so the conceptualization becomes simply another tool to reach for in a longer process, as new conceptualizations come along that repeat the sameprocesses.
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Oh this is beautiful. I hadn’t seen this post yet and I’d already written something in a comment on another post fiber bundles,I hope I didn’t put my foot in my mouth; chronology doesn’t show up properly on Facebook but it’s very much a reality for us humans.I’ll read this now – I’m 1/2 way through – but what I see so far is a very nice logical order of learning with expectations, then issues, then modifying expectations with new learning mitigating issues allowing one to do more.Beautiful so far.==see how many things we put on top of that poor little point there?==This was an enjoyable summary to read. The visuals that played in my brain as I was reading matched up with the visuals in my brain I’d had in the past when reading about these things; the big surprise for me was how compact your presentation was. It reminded me of encountering Category Theory and HoTT and reading through the materials I could find but better as you are not talking about general mathematical concepts but physics – and better still as it’s coming from you; two-way communication (or the possibility of it) is vastly superior to one-way communication such as from videos and things.
“So, see how many things we put on top of that poor little point there? A metric, a vector space, spin spaces, symmetry transformations across reference frames, parallel transport, and then on top of all this, inner products, and the observables became non-commuting matrices (operators)… So, the geometry got quite complicated, so much so that we needed something called a “fibre bundle” to describe it… “
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