I always found this principle a bit "strange". Is it about physics or about math? It looks to me like a funny way to write equations in, say, another form, very useful, but I am not sure about its "fundamentality" from the physics point of view.
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That is our convention of what "down" means.
yes, but it's the fact that nature chooses one thing over the other no matter what we call it, mathematically represented as a minus sign
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i.e. define "down" as the thing nature chooses , call it plus or minus or what you want it doesn't matter. But we need to distinguish between them somehow mathematically. But this i argue is physical content.
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F = ± dU/dx would cause L = T ± U. Is convention and math. What forces are needed to explain any given observation is physics.
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