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Those of us who grew up solving Newtonian mechanics, electrostatics, and calculus of 1-3 variables problems developed good intuition for understanding simple concepts. This helped me condition a problem, select a good coordinate system or frame of reference, and often provided a concrete example to more complex theories. For instance, it's impossible to picture a 100-dimensional gradient, but knowing a slope and a surface plane on a 3-D manifold makes it easy to imagine.

I'm getting to a point where the intuition is coming at a higher and higher cost as problems become more difficult. Differential forms are a good example, where the geometric interpretations are expensive to imagine and only really help in low dimensional space. I get the sense that quantum mechanics (or any "analogy" to it) becomes almost impossible to imagine.

For practitioners of physics: do you have to stop relying on geometric intuition, and just trust the mechanics/math? Or do you develop more mathematical intuition so visual analogies aren't necessary?

so860
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  • Related (and also closed): https://physics.stackexchange.com/q/27665/50583 This is primarily opinion-based because even expert physicists vary wildly in how much they rely on intuition vs. mathematical rigor. – ACuriousMind Mar 15 '19 at 17:35
  • I personally depend on visual analogies most of the time even after many years. But I mostly deal with pretty concrete, real things. A 100-dimensional gradient wouldn't be too bad, but a four-dimensional knot is beyond me. Some mathematicians have claimed to have trained themselves to visualize higher-dimensional forms, but I suspect that they really were visualizing analogies and knew the math well enough to know when the analogies were misleading. – S. McGrew Mar 15 '19 at 17:43
  • Depends on the person. I barely use any visual intuition at all -- my personal notes have less than one picture per 100 pages. But I have friends that, when asked to describe something about a high-dimensional fiber bundle, will start drawing pictures in the air with their hands as if they're seeing something. What is essential is that you have some intuition for the equations, but that could be, e.g. just formal intuition about their formal properties. – knzhou Mar 15 '19 at 17:46
  • @ACuriousMind Thanks for the link! That's an interesting conversation but only a few points seem relevant. Already some helpful comments have been posted here, however I'm happy to re-post on a different StackExchange site. Can you point me to the correct one? – so860 Mar 15 '19 at 17:48

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