Let's take a simple example:
f(x)Even if I don't say anything about it, people will often agree that it is a function "f of x".
But what about this?
t(1)Now here's an ambiguity (even if you don't realize it immediately): is that supposed to be "t times 1" (a product of t and 1) or "t of 1" (a function t evaluated at 1)?
Chances are that people don't encounter this very often, since it is often explained whether t is a function or not. However, in physics it is common to mix functions and numbers, because equations work for both.
There are other less noticeable issues as well. Consider the following:
x2Most people will say this is "x squared", but if you've done any tensor algebra, you'll know that it can also mean "the value of vector x at index 2"
That's the problem: upper indices use the same notation as for powers, which can be ambiguous sometimes. Although powers don't usually occur with tensors, it is still important to keep the notation consistent and unambiguous.
So far I haven't thought of a reasonable solution for this superscript problem, but I have been thinking about the function/bracket issue mentioned earlier.
One thing I had noticed was that parentheses, square brackets, and also curly braces are often used to denote the order of operation. However, I think it's redundant and unnecessary to use all three different kinds of brackets just to do the same thing. It's a "waste", I would call it. Perhaps we could use square brackets for functions instead of the usual parentheses? It is compatible with the typical use of brackets (since functions are sometimes indicated with square brackets, though not as often). So I could say "f(x)" to mean the product of f and x (of course it would be rare to write it like this; it's better to say "f x") while "f[x]" to mean the function f acting on x. I wouldn't call this convention-breaking (since it's fairly compatible with the original notation of functions), but it certainly would be unusual.
There are other cases of ambiguities in math & science. For example, consider "e". "e" is the natural base, but it's also the symbol for the elementary charge, so how do I differentiate them? (Again, it's unlikely that an equation would use these 2 at the same time, but it's still possible.) My subtle solution is to use italics for elementary charge ("e") and upright for the natural base ("e"). It's a very minute difference, but I prefer it this way to avoid confusion.
Similarly, I would use the upright form of "π" to represent the circular constant, and italics for anything else (unfortunately, sometimes it is very difficult to distinguish between upright and italic Greek letters in some fonts). The reason why "π" and "e" get an upright font is because they are very special mathematical constants.
Besides constants, I also prefer to use the upright "d" for derivatives to avoid confusion with variables or functions that are named "d". There are quite a few textbooks that actually use this convention.
I really wish there was a specific "committee" of some kind that regulates mathematical syntax and conventions (sort of like the IUPAC for chemistry, or SI for scientific units).
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