[xg15]

Yeah, I think that's a good point and, I think, an important differecne between math notation and programming languages.

I think mathematicians thend to use a lot of "notational slang" or "ad-hoc syntactic sugar", if you will. This will make the notation often look imcomprehensible for people not familiar with the exact domain - however, there is usually a consistent meaning behind it. If one wanted, the notation could be "desugared" into a more rigid (but more verbose) form that expresses the same.

To take big-O-notation as an example, if you write something crazy-looking like

  x + 5 = O(x)

What you mean is

The function "f(x) := x + 5" is a member of the set of functions "O(x)".

Where "O(x)" is itself shorthand for a convoluted set expression.

Similarly, if you write

  x^3 + O(x)

You mean Take x cubed, then add the result of some function from the set O(x). (Whether you mean any function or one specific function is again context-dependent but usually it's made clear which of the two is meant.)

Contrast that with programming languages, where you often have a rigid syntax but higher-level semantics being quite fuzzy.

E.g., equals() and toString() in java are straight-forward to write but they can "mean" quite a lot of different things depending which kinds of objects you call them.

e.g., the "obvious" meaning of equals (value equality) works only with immutable value types - yet the method is defined on any kind of object. So it might also mean instance equality - or even entity equality if you deal with ORM proxies - or even wilder things...