You can see how mathematical notation evolved during the last centuries here.
I think everyone here knows that a bad notation can change an otherwise elementar problem into a difficult problem. Just try to do basic arithmetics with roman numbers, for example.
As a computer programmer I know that in some situations programming language notation plays a critical rule because some algorithms are better expressed in a particular language than in other languages even considering they all have the same basis: Lambda Calculus, Turing machines, etc
The linguists has their so-called Sapir–Whorf hypothesis which "...holds that the structure of a language affects the ways in which its respective speakers conceptualize their world, i.e. their world view, or otherwise influences their cognitive processes."
Then, I ask: is there any field in Math that studies Math's notation and its influence for good or for bad in Math itself?
Modifying the fragment on the paragraph above: is it possible that the notation, the symbols and the language used in Math affects the ways in which Mathematicians conceptualize their world and influences their cognitive processes?
In my opinion, this is one of the exciting promises of intuitionistic mathematics and topos theory.
By discarding the law of the excluded middle $\neg\neg P\implies P$ (of course, we can add it back later if we want), much more of the structure of our axioms becomes evident in our theorems, because we can no longer label arbitrary statements as true-or-false.
For example, in topos theory, one no longer speaks of "the" real numbers, but of "a" real numbers object in a topos. When the topos is $Set$, nothing special happens. But in a different setting, we may have a countable real numbers object, or nontrivial subobjects without points. My understanding is that there are also topoi for which every function $\mathbb{R}\to\mathbb{R}$ is differentiable (which should be a relief to any physicists who pretend this all the time).
I like this view because it makes certain properties of "the" real numbers—its cardinality, for example—appear to be artifacts of the language of sets, which relies upon a firm notion of membership. But since the vast majority of real numbers cannot be pinned down in any meaningful sense, it is not hard to argue that treating $\mathbb{R}$ as a set is, at the very least, a choice of perspective.
And this can actually be useful—the now-classic quantum physics paper What is a Thing? has argued that the standard $\mathbb{R}$ is inadequate for non-classical theories of physics. After all, if the number of particles in a system is dependent on how we measure the system, then why should we expect anything in the universe to behave like a set?