Conceptual Difficulties in the Roles of Variables - Comments

[haruspex]

haruspex submitted a new PF Insights post

Conceptual Difficulties in the Roles of Variables

Continue reading the Original PF Insights Post.

 

[Greg Bernhardt]

Nice work! Love the first statement!

 

[burakumin]

haruspex submitted a new PF Insights post

Conceptual Difficulties in the Roles of Variables

View attachment 178528Continue reading the Original PF Insights Post.

I think there are still important pitfalls that might deserve deeper explanations and details:

1. The opposition variable/constant is not a mathematical one but a meta-mathematical/logical one. Variables and constants are not mathematical objects but notation tools to describes mathematical objects (that's why for example you can say at the same time that $x$ is a variable and $x$ is a real number). A big pitfall with the ubiquitous Leibnitz notations ($\frac{d f}{d x}$, $\frac{\partial f}{\partial x}$, ...) is that it mixes mathematical objects (like function $f$) with notation devices (like variable $x$ which represents one of the function "slots", i.e. argument position and absolutely not a number here).
2. As mentioned it is a common abuse of notation to use the same function symbol for related but distinct functions when their output has the same semantical meaning but the arguments (= the coordinate system) are different. One of the big issue with this abuse of notation is this one:
   Consider $(x, y)$ and $(x, u)$ two coordinate systems with the common coordinate $x$ and a smooth function $f$. I can then compute $(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})$ and $(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial u})$. But the two objects $\frac{\partial f}{\partial x}$ are in fact entirely different functions : $\left. \frac{\partial f}{\partial x}\right|_y$ and $\left. \frac{\partial f}{\partial x}\right|_u$. The fact that derivation along a single coordinate depends on the entire coordinate system would really deserve to be insisted upon.

 

[UsableThought]

Nice article, though I can't follow the section on derivatives as I haven't reached calculus yet. At a much simpler level (which is where I'm at as an older adult/amateur learner revisiting high school math), I sometimes invent "temporary variables" just for purposes of calculating; e.g. there was a homework question recently about a word problem in basic algebra that stated, among other things, "$d$ is $k$ percent less than $c$". Solving it myself for fun, rather than work directly with $d = (1-k)c$, I made a temporary variable $k' = 1-k$ so I could get rid of the parens and just say $d = k'c$, postponing the subtraction until after everything else was done. But that's just me working on my own; I have no idea if "temporary variables" are used by other people.