Comment by rocqua | Hacker News Reader

Comment by rocqua | Hacker News Reader

The idea that the letter choice shouldn't matter fully makes sense to me. And in general, point free notation can be nice (point free meaning operating on the function rather than introducing an arbitrary point and operating on the function evaluated at the arbitrary point.

Despite this, I still believe Leibniz notation is superior for multi-argument functions. For multi argument functions, named arguments are much clearer than just depending on the order of arguments. Essentially I advocate for dropping point-freeness for clarity on the difference between function arguments.

Besides that, by expression equivalence it is clearly the case that the only correct interpretation of the derivative is the 'composition' where the change in t also counts for the change in x. Because replacing the f(x(t), t) with g(x) (where g is the composition, should not change the outcome of the derivative.

Named arguments are great, but mathematical notation doesn’t have them. x(t) is not a named argument to f(x(t), t), so notation like ∂f(x(t), t)/∂x(t) is nonsense.

If we did have real named arguments like f(a=x(t), b=t), where a and b are fixed by the definition of f rather than arbitrary names to be made up on each invocation, then maybe it would make sense to write something like (∂f/∂a)(a=x(t), b=t). Though that’s still pretty far from Leibniz notation, where ∂f/∂a is somehow supposed to be a numerical value depending on a, not a function to which arguments must be supplied.

But what I think is really lacking in mathematical notation is explicit lambda abstraction: we should be able to write

(λt. f(x(t), t))'(t) = (λa. f(a, t))'(x(t)) x'(t) + (λb. f(x(t), b))'(t),

which reuses the ordinary one-variable derivative and has none of these ambiguities.

This is a noble take for programming languages too, where point-free style in e.g. Haskell is a pain in the neck, and named arguments are a blessing.

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