Vocabulary and Notation

Between my quantum mechanics class and my directed reading project on Fourier analysis, I have experienced an incredible synergy of functional analysis, real analysis, algebra, and physics. Fourier analysis is a really interesting amalgam of these areas, and quantum mechanics takes on its full beauty in the terminology of Hilbert spaces and Dirac notation. While these subjects have been extremely fulfilling, I have experienced some minor but annoying problems parsing the zoo of verbiage and notation associated with these topics. Moreover, there is an incredible inconsistency in terminology and notation between physics and mathematics. This raises the paramount issue of the role of notation in math and physics.

This may initially seem like a trivial issue. After all, notation merely stands for some idea or operation; how one indicates the ideas seems to be irrelevant. However, I contend that notation actually affects the way in which a person can visualize and understand an idea. There is a very good reason, for example, that physicists have a strong appreciation for the Dirac notation in quantum mechanics.

This raises the question of what characteristics make notation good. I haven't come up with a lot of specifics, but I have some ideas. Perhaps I should start with what makes bad notation and bad terminology, since that is the area in which I have recently had experience. The first thing is inconsistency. Physics and mathematics have an incredibly gross amount of inconsistency in notation and terminology. Inner products in mathematics are represented usually with corner brackets and commas between entries. Occasionally they use paretheses. But physics always uses corner brackets, and uses the horizontal bar between entries only in quantum mechanics. Some of this is due to specialized (and elegant) notation for quantum mechanics, but these excessive standards are silly. The parenthesis notation is also particularly bad because it generates confusion with notation for vectors and coordinates. Complex conjugation is denoted by a bar above the argument in math but a superscript asterisk in physics. Fourier coefficients are sometimes denoted by hats above the name of the function, but other times they are just written as a or b with a subscript for the index. Laplace transforms are represented by several notations. The Laplacian is represented by a delta in some branches of mathematics, but it is represented by a nabla with a superscript 2 in other branches. The superscript convention in economics is disastrous. Rather than using subscripts on variables, economists will occasionally (but not always!) use superscripts, leading to catastrophic confusion with exponents (this is particularly bad because economists like to use the superscripts a and b, which are also frequently used as exponents).

Terminology is just as bad. Mathematicians call the space of square-integrable functions L². Physicists call it Hilbert space. But mathematicians call complete normed inner-product spaces Hilbert spaces. So L¹, L²,..., L^∞ are all Hilbert spaces. This may seem trivial, but it makes an enormous difference. The self-adjoint operators in mathematics become Hermitian operators in physics, and adjoints become Hermitian conjugates. It's one big disaster.

So what makes good notation? Clearly consistency is a major necessity, as is uniformity. Notation and terminology should also explicitly reflect the ideas they represent. Saying L² and self-adjoint is much better than referring to Hilbert and Hermite, regardless of how famous they were. Naming mathematical objects after people is bound to generate confusion. A less obvious example of good notation versus bad occurs in limits. Although it is longer, the "lim" notation is desirable because it is obvious what is means ("lim" is very close to "limit") and the subscript indicates the limiting variable and what it approaches. On the other hand the convention of writing an arrow to denote a limit is hideous. It is far less explicit and potentially generates confusion with the symbol for "implies."

Still, the limit notation is very bulky. Notation should be as brief as possible. I think the integral is a good example of this. It is easy to write and read, versatile in indicating boundaries and variables, and it has an obvious intuitive significance.

Notation is clearly one of these areas, like typesetting or displaying data, which is very much artistic but essential to facilitating understanding. The current state of notation is absolutely horrific, but I believe that common sense can go a long way in improving the way mathematics is written.