Why is 'abuse of notation' tolerated? I've personally tripped up on a few concepts that came down to an abuse of notation, and I've read of plenty more on stack exchange.  It seems to all be forgiven with a wave of the hand.  Why do we tolerate it at all?
I understand if later on in one's studies if things are assumed to be in place, but there are plenty of textbooks out there assuming certain things are known before teaching them.  This is a very soft question, but I think it ought to be asked.
 A: Personally (I can speak only for myself), I tolerate abuse when it helps keep things clear and simple (with regard to my subjective perspective). Sometimes it might be tolerated when there is not enough resources (e.g. time, space, etc.) available and the details are not that important.
A: When one writes/talks mathematics, in 99.99% of the cases the intended recipient of what one writes is a human, and humans are amazing machines: they are capable of using context, guessing, and all sorts of other information when decoding what we write/say. It is generally immensely more efficient to take advantage of this.
A: Done properly abusing notation makes things clearer. Suppose that $f \colon X \rightarrow Y$ and $A \subseteq X$. We can write


*

*$f(A)$

*$\{ f(x) \colon x \in A \} $

*Define $g \colon X \cup \mathcal{P}(X) \rightarrow Y \cup \mathcal{P}(Y)$ by $g(x) = f(x)$ if $x \in X$ and $g(A) = \{ f(x) \colon x \in A \} $ if $A \subseteq X$. Here we can use both $g(x)$ and $g(A)$. There are difficulties with this method if $X \cap \mathcal{P}(X) \neq \varnothing$.

A: Since Bourbaki is rather busy and is not (yet) a member of this site, I'm posting His answer (which He preemptively wrote about 70 years ago) on His behalf: 

As far as possible we have drawn attention in the text to abuse of language, without which any mathematical text runs the risk of pedantry not to say unreadability.

A: I came to realize that mathematics (in general all sciences) is a collection of strands of ideas; each strand is no more than a few inches long, i.e., incomplete on its own, but they connect with one another like neurons, and altogether they form a gigantic bridge several miles long. Not abusing notation is like trying to build a bridge using only a single strand -- a perfect, self-contained, single piece of idea. That is very counter-productive, and often hinders progress.
A: Abuse of notation is tolerated when the alternative is worse!
In some cases, abuse of notation isn't really abuse at all, but simply a lack of fleshing things out. For example, I'm sure many would consider
$$\arctan(+\infty) = \pi/2$$
an abuse of notation that is meant as shorthand for
$$ \lim_{x \to +\infty} \arctan(x) = \pi / 2 $$
But if you take a short trip into theory of the extended real line, the identity is seen to be a literally true fact about the $\arctan$ function on the extended real line (which is the continuous extension of the $\arctan$ function on the reals).
A: As I stated in my comment/question below your question, it seems that you are "abusing" (mis-using) the phrase "abuse of notation."

In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition (while being unlikely to introduce errors or cause confusion). Abuse of notation should be contrasted with misuse of notation, which should be avoided. A related concept is abuse of language or abuse of terminology, when not notation but a term is misused.

In particular, I'm referring to your observation:

I understand if later on in one's studies if things are assumed to be in place, but there are plenty of textbooks out there assuming certain things are known before teaching them.

Here, it seems to me that you are complaining that you are encountering the use of notation that you do not understand and have not yet encountered, and for which the author/instructor has not explicitly defined. This is NOT an abuse of notation. This is where you "speak up" and ASK what is meant (if in class). Alternatively, in such a situation, you need to take the initiative to understand the notation, to look to see if the text in question has an appendix or index defining the notation it uses, or you can appeal to some reference to better understand the symbols/notation and its various uses, which are usually context dependent.

That said, with respect what actually is meant by "abuse of notation": we are all human, and mathematical notation, like any language, is subject to ambiguity, perhaps less so than natural language, but nonetheless, it is still subject to ambiguity. 
Notation also provides a means to communicate, compactly, what would be laborious to try to communicate otherwise, even if at the cost of "abusing notation."
In any case, being human also means it's usually a good thing to avoid pedantry and to learn to tolerate the use//abuse/misuse of any language (mathematical or otherwise) by others. Certainly, you may want to it out when you take something to be an erroneous use of notation/language (and doing so in a helpful way), but deciding not to tolerate it is perhaps going too far. 
And I suspect that we all take "short-cuts", when handy and when we can safely assume the notation we may be "abusing" will be understood. Certainly, there is a "fine-line" between taking advantage of notational short-cuts, and full-fledged "abuse" of notation that fails to convey what was intended by its use.
A: I doubt I could put it better than this:

"The student of mathematics has to develop a tolerance for ambiguity. Pedantry can be the enemy of insight." - Gila Hanna

I also highly recommend Terence Tao's article describing the "pre-rigorous", "rigorous", and "post-rigorous" stages of a mathematician's development.
A: I think that as new ideas and new ways of thinking about things come to light, any language, lingo or symbolism needs to evolve also.  Languages would not be as sophisticated as they are today without some form of creativity element. People who come up with ideas don't always set the conventions (e.g. Latin squares in modern day combinatorial mathematics normally use letters or numbers to represent solutions, not the Latin symbols Euler used).  Would he be turning in his grave? Or would the underlying concepts of the mathematics, which are both symbolic and a-symbolic be the overriding factors.
I believe that what is needed, is a complicated compromise of tradition mixed with innovation and help for people  who don't get one, the other or both.
A: I believe it was proper mathematical notation that have birth to my love of mathematics, and I all now a mathematics professor. Also, may we be reminded that mathematics is a language, the most concise language today is truly universal.
