51–60 of 101 posts
I hope you will forgive me for this wall of text, but this is a topic that is quite close to my heart and that I've gone back and forth in many times before settling on my current perspective. I appreciate that especially for mathematicians and programmers, making a clean distinction between a function and its evaluation is a key conceptual point, and Leibniz notation obscures this fact. However, there are good reaso…
I don’t like the accepted answer here, because it criticises Leibniz notation without explaining what it means – and apparently without actually _understanding_ it, since the author several times writes words to the effect of “this is nonsense” in reference to expressions that in fact make perfect sense when interpreted correctly. The salient issue that the author of that answer seems not to have understood, which we…
Earlier quoted context omitted.
This specific notation makes no sense in a Physics context. The names of the variables encode information that is lost if you use purely positional arguments. If I take a state function F(V, T) and its partial derivative wrt T, things like notation ∂_2 F get much murkier and context dependent than ∂_T F or ∂F/∂T, which benefit from a general consistency of notations. Ultimately, the function F has a physical meaning…
FWIW, they start counting function arguments from 0, so _2 is indeed the velocity. But I do agree with your main point that the order of arguments is irrelevant, and it is a mistake to make it a first-class citizen of the notation.
Earlier quoted context omitted.
As a mathematician I like the positional notation, but once we assign meaning to the slots it gets weird, so I had the same objection. One could add some new notation to distinguish the element _T_ from the slot _T_. For example let’s write slots as _[T]_ (I’m not creative enough to come up with something good), then we can talk about ∂_[T]F or ∂F/∂[T]. One can think of the [.] operator as mapping from “semantic symb…
Or use the convention from differential geometry, where the letter denotes a coordinate on the function's domain.
Such discussions show that teaching of calculus often tends to be overly algebraic, for no good reason. Clearly, Leibniz notation does not intrinsically contain any deep insights since at one point Leibniz himself was erroneously induced by it to think that d(xy)= d(x)d(y), which is false. Newton on the other hand thought in terms of simple geometric concepts (areas), which make it crystal clear that d(xy) = x d(y) +…
The discussion in Sussman and Wisdom's "Structure and Interpretation of Classical Mechanics", about how the Euler-Lagrange equations don't literally make sense as traditionally written, has long resonated with me: https://mitp-content-server.mit.edu/books/content/sectbyfn/b... They also adopt a notation where partial derivatives are taken with respect to "argument slots".
This specific notation makes no sense in a Physics context. The names of the variables encode information that is lost if you use purely positional arguments. If I take a state function F(V, T) and its partial derivative wrt T, things like notation ∂_2 F get much murkier and context dependent than ∂_T F or ∂F/∂T, which benefit from a general consistency of notations. Ultimately, the function F has a physical meaning…
Earlier quoted context omitted.
This specific notation makes no sense in a Physics context. The names of the variables encode information that is lost if you use purely positional arguments. If I take a state function F(V, T) and its partial derivative wrt T, things like notation ∂_2 F get much murkier and context dependent than ∂_T F or ∂F/∂T, which benefit from a general consistency of notations. Ultimately, the function F has a physical meaning…
FWIW, they start counting function arguments from 0, so _2 is indeed the velocity. But I do agree with your main point that the order of arguments is irrelevant, and it is a mistake to make it a first-class citizen of the notation.
Dammit yes, you’re right! Well, it’s not a bit less confusing.
The most frustrating is that they have a point: we need to be stricter about disambiguating functions and numbers, and derivation really should be an operator.
But you don’t need to go all the way to zero-indexing (which is definitely not a thing in the fields I know) or positional arguments. This is putting abstract notation purity above practical concerns. It’s not surprising they like Scheme.
f(v_1, m_1, v_2, m_2)
For the derivatives I would prefer writing
d_v_2 f
instead of
d_3 f
when counting arguments from 1
Not just Leibniz notation, but for example https://en.wikipedia.org/wiki/Partial_derivative is total bonkers, every single equation on it means the total opposite than in the rest of the mathematics. Wtf is > the partial derivation of a function f(x,y,z...) or deriving a function respect to a variable? Functions don't have variables, named variables, that's Python, not math[0]. In math expressions can have free varia…
No, Python named arguments are a good analogy for Leibnitz notation.