I think that those were basic design decision that were made at the very early stages of the prehistory of Mathematica, three decades ago. Some functions protected their internal variables:

x = 1;
Plot[x^2, {x, 0, 1}]

and some others didn't:

x = 1;
D[x^2, x]

I am so used to how it works that I can't quite foresee the ramifications of doing otherwise. It is easier for a user to protect an unprotected variable when needed, than undoing a built-in protection. Perhaps some expert can comment?

Thanks. In my attempts to use debugging functions, I did see something like the code you wrote as an example of how it went wrong.

I'm too new at Mathematica to be sure of my opinion, but it seems to me it would be better if the program automatically added Module's functionality to D, Integrate, FourierTransform, and similar function-operators. Because in math outside Mathematica, "D[x^2, x]" has a definite meaning -- it means 2x -- and nothing going on outside that expression can change that expression's meaning.

What result would you expect from the brief dust-up below?

x = 5;
D[x^2, x]

(* During evaluation of In[89]:= General::ivar: 5 is not a valid variable. *)

(* Out[90]= D[25, 5] *)

I'd naively expect it give this result:

In[1]:= x = 5

Out[1]= 5

In[2]:= Module[{x}, D[x^2, x]]

Out[2]= 2 x$811

Despite my earlier, wording, my concern isn't so much that Mathematica's design has a flaw; my concern is that Mathematica's design in this respect offers some flexibility that's over my head, and I'm trying to grasp why this design-decision was made, so that I can better utilize this aspect of Mathematica. I don't expect Wolfram to alter their design because of comments; I'm just trying to get oriented. I'm coming to Mathematica as a LISP programmer, and I've also programmed (in Java) my own LISP interpreter with a complete kernel. When Mathematica works similar to LISP, it makes sense to me; when Mathematica does something different from LISP, I wonder what benefits are there to Mathematica's way. Of course LISP doesn't include primitives for FourierTransform, but if I were to make a LISP that did (I'm not planning to), I'd instinctively implement default lexical scoping like I naively expected here.

Thanks for your valuable insight and clarification.