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Thank you for this very insightful answer! I've always been more interested in the structure of problems than in their specific detail, so your observations and

Thank you for this very insightful answer! I've always been more interested in the structure of problems than in their specific detail, so your observations and recommendations seem right up my alley. When it comes to mathematical foundations, do you believe that differential equations are the best approach to gain a grounded understanding of physics? Although I've been introduced to them, I lack a deeper, intuitive grasp. I'm curious if attaining such an understanding would markedly improve my comprehension of physical concepts.

PaulHoule 3y ago
The idea of "understanding" is problematic in that we can't understand" everything but can get understandings about particular situation; certain sorts of problems are provably impossible to solve such as predicting the weather three years from now or being able to prove that any computer program either will or will not complete in finite problems. You can understand that a problem is in such a class yet an insolvable problem always leaves the answer to that problem mysterious and knowingly "not understood". Then there is the things nobody understands, like what dark matter is. People have all kinds of ideas, and the math is usually a lot simpler than the math for problems like "stars". What we're lacking is the experiment or observation that proves one thing or another. After that somebody gets the Nobel prize for furthering our understanding of the universe. The line of understanding an area passes through solving problems, you prove what you understand through solving problems and discover gaps in what you understand to explore with by postulating more problems. Differential equations turn up in undergrad physics at the third year level with electromagnetism and then with the theory of the hydrogen atom with the Schrodinger equation. It's a special case because of the spherical symmetry which lets it be solved exactly but it teaches you the other problems can have symmetry break in a way that gives an understanding about how the structure of the system effects the energy spectrum and other behaviors. Differential equations might not be the ultimate nature of the universe and they do have that problem of not having a single strategy for solving all of them, many people think the differential equations course in undergrad math is disorganized and teaches a bunch of random methods but no unifying theory. They also behave can behave badly in like the Theory of General Relativity in which the singularities probably mean the theory is wrong or just misbehave in a computer. If you're trying to answer harder problems (say how does air flow around a space shuttle that is reentering the atmosphere) it tends to come down to differential equations.

PaulHoule 3y ago

The idea of "understanding" is problematic in that we can't understand" everything but can get understandings about particular situation; certain sorts of problems are provably impossible to solve such as predicting the weather three years from now or being able to prove that any computer program either will or will not complete in finite problems. You can understand that a problem is in such a class yet an insolvable problem always leaves the answer to that problem mysterious and knowingly "not understood". Then there is the things nobody understands, like what dark matter is. People have all kinds of ideas, and the math is usually a lot simpler than the math for problems like "stars". What we're lacking is the experiment or observation that proves one thing or another. After that somebody gets the Nobel prize for furthering our understanding of the universe. The line of understanding an area passes through solving problems, you prove what you understand through solving problems and discover gaps in what you understand to explore with by postulating more problems. Differential equations turn up in undergrad physics at the third year level with electromagnetism and then with the theory of the hydrogen atom with the Schrodinger equation. It's a special case because of the spherical symmetry which lets it be solved exactly but it teaches you the other problems can have symmetry break in a way that gives an understanding about how the structure of the system effects the energy spectrum and other behaviors. Differential equations might not be the ultimate nature of the universe and they do have that problem of not having a single strategy for solving all of them, many people think the differential equations course in undergrad math is disorganized and teaches a bunch of random methods but no unifying theory. They also behave can behave badly in like the Theory of General Relativity in which the singularities probably mean the theory is wrong or just misbehave in a computer. If you're trying to answer harder problems (say how does air flow around a space shuttle that is reentering the atmosphere) it tends to come down to differential equations.