$$s(a+b)$$
Does it mean s times a+b, or is s a funtion and take a+b as arguments?
$$(s(a+b))(c+d)$$
Does it mean s(a+b) times (c+d), or is s a high order function and s(a+b) return a function which then take c+d as arguments?
As a reader, how do I distinguish these cases?
As a writer, how can I make my formula super clear when it's needed?
I don't believe I'm the first one who find them confusing. I expect to see references/literatures. Thanks.
The notation for the different things you mention are completely identical, and yes, without context it is certainly ambiguous.
However, such expressions (should) never exist in a vacuum. Before (or possibly immediately after) such an expression is written, it should be made clear whether $s$ is a function or a number (and it should also be made clear what $a$ and $b$ are). And with such a clarification, the expression is no longer ambiguous.
That being said, $(a+b)s$ is less prone to being misunderstood, as most mathematicians apply functions to whatever is to the right, not to the left.
