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It's probably because I'm an ignorant idiot, particularly when it comes to calculus, but this read like a revelation: "the concept of a function shouldn't

It's probably because I'm an ignorant idiot, particularly when it comes to calculus, but this read like a revelation: "the concept of a function shouldn't depend on what your favourite letter is!" Very helpful answer, thanks for posting.

enriquto 3y ago
> the concept of a function shouldn't depend on what your favourite letter is On the other hand, you can also argue that the concept of a function shouldn't depend on your favourite ordering of its variables. Thus, if you have a potential that oscillates in time, such as: V = (1 + sin(t))/sqrt(x^2+y^2) You may want to take derivatives with respect to each variable, such as dV/dt, dV/dx and so on. Their meaning is clear. What does D_2(V) mean, though? It does depend on your favourite ordering of the letters! Even if you prefer positional notation for derivatives (à la Spivak), it must be recognized that the naming-based notation has its merits, and sometimes is clearer.
- BruceEel 3y ago
  ah, interesting dark corner there. Also, I was thinking there must be merit in allowing for some discrepancies in use of different notations when dealing with "read" use, as in learning a new concept, vs. "write" use, as in working on something, like the SO OP..
- xigoi 3y ago
  In what you have written, V is not a function, but a value calculated from some other values, so it does not make sense to differentiate it. If you instead write: V : ℝ³ → ℝ V(t,x,y) = (1 + sin(t))/sqrt(x^2+y^2) then it’s clear what D_2(V) means.
  - archgoon 3y ago
    [dead]
  - enriquto 3y ago
    > not a function, but a value calculated from some other values The term of art is "an expression" [0]. And you can also differentiate expressions with respect to their variables. It's a perfectly supported construction in all symbolic computer algebra packages. No need to assign an (arbitrary) ordering to your variables in order to differentiate with respect to them. [0] https://en.wikipedia.org/wiki/Expression_(mathematics) https://en.wikipedia.org/wiki/Expression_(mathematics)
bikenaga 3y ago
Yeah, I noticed a thing students in math class would get confused is the difference between local and global variables. So I'd ask people if we're discussing functions from the reals to the reals, are these the same function? f(x) = x^2 vs. f(t) = t^2 They are, because they produce the same set of ordered pairs (input, output). Some of the confusion is the fault of math profs, because we get lazy. We could say "the function f: R -> R defined by f(x) = x^2", but we wind up saying "the function f(x)" (wrong - f is the function, f(x) is the value when x is plugged in) or worse "the function x^2" (really wrong). But if you always say the absolutely correct thing "the function f: R -> R defined by f(x) = x^2" you sound really pedantic, and the extra words make it hard for students to comprehend. Oh well. What was more confusing to people was if you had f(t) = t^2 and people thought "t" had some sort of independent existence outside the definition of f. Some guy was asking me about it and I was trying various explanations and he wasn't getting it. Then I remembered he was a CS major and I said "t is an argument to f and it's local to the function block" and he got it and nodded his head. So I wonder if these kinds of distinctions are easier for programmers than for typical math students - in the sense that programmers would know there's no difference between these: int f(int x) { return x * x; } int f(int t) { return t * t; }
- BruceEel 3y ago
  Brilliant. Love the programming analogy. I like that the above works particularly well in C as: int f(int x); int f(int t); will compile with no errors, while changing t to, f.i., "float" obviously won't. I wonder whether this could be pushed farther to come up with programmer-friendly descriptions of things like the chain rule starting from something like: int h(int x) { return f(g(x)); } etc.

enriquto 3y ago

> the concept of a function shouldn't depend on what your favourite letter is On the other hand, you can also argue that the concept of a function shouldn't depend on your favourite ordering of its variables. Thus, if you have a potential that oscillates in time, such as: V = (1 + sin(t))/sqrt(x^2+y^2) You may want to take derivatives with respect to each variable, such as dV/dt, dV/dx and so on. Their meaning is clear. What does D_2(V) mean, though? It does depend on your favourite ordering of the letters! Even if you prefer positional notation for derivatives (à la Spivak), it must be recognized that the naming-based notation has its merits, and sometimes is clearer.

BruceEel 3y ago
ah, interesting dark corner there. Also, I was thinking there must be merit in allowing for some discrepancies in use of different notations when dealing with "read" use, as in learning a new concept, vs. "write" use, as in working on something, like the SO OP..
xigoi 3y ago
In what you have written, V is not a function, but a value calculated from some other values, so it does not make sense to differentiate it. If you instead write: V : ℝ³ → ℝ V(t,x,y) = (1 + sin(t))/sqrt(x^2+y^2) then it’s clear what D_2(V) means.
- archgoon 3y ago
  [dead]
- enriquto 3y ago
  > not a function, but a value calculated from some other values The term of art is "an expression" [0]. And you can also differentiate expressions with respect to their variables. It's a perfectly supported construction in all symbolic computer algebra packages. No need to assign an (arbitrary) ordering to your variables in order to differentiate with respect to them. [0] https://en.wikipedia.org/wiki/Expression_(mathematics) https://en.wikipedia.org/wiki/Expression_(mathematics)

bikenaga 3y ago

Yeah, I noticed a thing students in math class would get confused is the difference between local and global variables. So I'd ask people if we're discussing functions from the reals to the reals, are these the same function? f(x) = x^2 vs. f(t) = t^2 They are, because they produce the same set of ordered pairs (input, output). Some of the confusion is the fault of math profs, because we get lazy. We could say "the function f: R -> R defined by f(x) = x^2", but we wind up saying "the function f(x)" (wrong - f is the function, f(x) is the value when x is plugged in) or worse "the function x^2" (really wrong). But if you always say the absolutely correct thing "the function f: R -> R defined by f(x) = x^2" you sound really pedantic, and the extra words make it hard for students to comprehend. Oh well. What was more confusing to people was if you had f(t) = t^2 and people thought "t" had some sort of independent existence outside the definition of f. Some guy was asking me about it and I was trying various explanations and he wasn't getting it. Then I remembered he was a CS major and I said "t is an argument to f and it's local to the function block" and he got it and nodded his head. So I wonder if these kinds of distinctions are easier for programmers than for typical math students - in the sense that programmers would know there's no difference between these: int f(int x) { return x * x; } int f(int t) { return t * t; }

BruceEel 3y ago
Brilliant. Love the programming analogy. I like that the above works particularly well in C as: int f(int x); int f(int t); will compile with no errors, while changing t to, f.i., "float" obviously won't. I wonder whether this could be pushed farther to come up with programmer-friendly descriptions of things like the chain rule starting from something like: int h(int x) { return f(g(x)); } etc.