〈 Berger | Dillon 〉

I actually think it is a kinda of a priori reason in a way because I think this really is implicitly defining what we mean by "force".

hmm i think i disagree with you on this one about this being a purely mathematical statement.

It is a consequence of the convention F = - dU/dx. That minus sign is what caries through to make the minus sign in L = T-U.

right but the *relative* minus sign, no matter where it goes, is physical. That's the empirical fact i'm talking about. The fact that nature wants to "go downhill", as it were.

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

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.