Why I always put parentheses around the arguments to any function.

[Ackbach]

Once upon a time, yours truly was taking junior-level classical mechanics. The textbook was the standard Classical Dynamics of Particles and Systems, by Marion and Thornton. In one homework problem from the book, there was a trig function that looked something like this:
$$ \sin \; \text{stuff}_{1} \; \text{stuff}_{2} $$
Clearly, $ \text{stuff}_{1}$ was in the argument of the trig function. But what about $ \text{stuff}_{2}$? I can't remember which assumption I went with, but it ended up being the wrong one.

Ever since then, I have ALWAYS put parentheses around the arguments to any function, whether it is sine, cosine, logarithm, etc. Then there can be no misunderstanding.

Don't write so that you can be understood. Write so that you can't be misunderstood. Don't write $ \sin x$, but $ \sin(x)$. Don't write $ \ln x$, but $ \ln(x)$. That way, you can follow the function with anything you please, with no danger of misunderstanding. Does it really take that much longer to type? Think about the time you might save some other poor soul who's reading your stuff. You might save him time.

 

[Turgul]

I agree with the spirit of your advice, but there does come a point where additional parenthesis (and other "clarifying" notation) actually impair clarity. For a (seemingly) silly example, note that multiplication is a binary operation so, in some sense, $a \cdot b \cdot c$ is not a sensible expression and should be written either as $(a \cdot b) \cdot c$ or $a \cdot (b \cdot c)$. But so long as multiplication associates, it really seems best to drop the parenthesis (and the dots!).

In a more serious example, if one were to try to explicitly keep track of all of the notation for trekking through the behavior of different natural transformations between some functors (say to prove Yoneda's lemma) in a completely unambiguous way, one would be drifting in a sea of symbols. In such a situation, you really do want to simplify your notation (even at the risk of being actually ambiguous!) or no one will understand what you are trying to say at all.

As a similar comparison, take the use of the serial comma in English. Some sentences are made additionally clear with its use, some more ambiguous. Only context distinguishes between the cases. As a cop-out, style guides and English classes will stress the use of one convention, but to be unaware of the pitfalls of that choice is a problem, too. One must always strive for clarity in context.

Now, there is a big difference between a trig class and a course in homological algebra. Part of the context is understanding the level of the ideas one is trying to express. Math is a language--and just as in any "normal" language, one always has to balance being precise and muddling one's sentences with excess verbiage.

Unfortunately, unlike other languages, the proper context can be rather difficult to learn or identify in mathematics. It can easily take years (or lifetimes!) to master.

 

[Ackbach]

I agree with the spirit of your advice, but there does come a point where additional parenthesis (and other "clarifying" notation) actually impair clarity. For a (seemingly) silly example, note that multiplication is a binary operation so, in some sense, $a \cdot b \cdot c$ is not a sensible expression and should be written either as $(a \cdot b) \cdot c$ or $a \cdot (b \cdot c)$. But so long as multiplication associates, it really seems best to drop the parenthesis (and the dots!).

In a more serious example, if one were to try to explicitly keep track of all of the notation for trekking through the behavior of different natural transformations between some functors (say to prove Yoneda's lemma) in a completely unambiguous way, one would be drifting in a sea of symbols. In such a situation, you really do want to simplify your notation (even at the risk of being actually ambiguous!) or no one will understand what you are trying to say at all.

As a similar comparison, take the use of the serial comma in English. Some sentences are made additionally clear with its use, some more ambiguous. Only context distinguishes between the cases. As a cop-out, style guides and English classes will stress the use of one convention, but to be unaware of the pitfalls of that choice is a problem, too. One must always strive for clarity in context.

Now, there is a big difference between a trig class and a course in homological algebra. Part of the context is understanding the level of the ideas one is trying to express. Math is a language--and just as in any "normal" language, one always has to balance being precise and muddling one's sentences with excess verbiage.

Unfortunately, unlike other languages, the proper context can be rather difficult to learn or identify in mathematics. It can easily take years (or lifetimes!) to master.

Thank you for the thoughtful response! Really quite good. I would definitely agree with you. The goal is clarity, and easing the burden of the reader, however that looks in different situations. Although, come to think of it, sometimes you do want the reader to put forth the effort and reach out to grab a hold of your ideas. Sometimes that very effort is what makes the reader remember the things you want him to remember.

Generally, though, making your expression of the ideas straight-forward and clear is preferable, even if the ideas are deep.