Why do mathematicians use single-letter variables?

Solution 1:

I think one reason is that often one does not want to remember what the variable names really represent.

As an example, when we choose to talk about the matrix $(a_{ij})$ instead of the matrix $(\mathrm{TransitionProbability}_{ij})$, this expresses the important fact that once we have formulated our problem in terms of matrices, it is perfectly safe to forget where the problem came from originally -- in fact, remembering what the matrix "really" describes might only be unnecessary psychological baggage that prevents us from applying all linear-algebraic tools at our disposal.

(As an aside, have you ever seen code written by a mathematician? It very often looks exactly like your first example.)

Solution 2:

We are very, very lazy. I am very, very serious about this.

NB1: The history is told in Florian Cajori's book on the history of notation. In very old times, there were no variables (and no formulas, really) and everything was incredibly verbose. Cajori's book beautifully shows the very long and tortuous way from that to modern day notation for variables; there are several sections regarding the notation of unknowns and of their powers.

NB2: Additionally, we usually deal with very complicated expressions, so using verbose names for variables you render things almost impossible. Writing down the formula for Gaussian curvature in terms of $E$, $F$, $G$ and the Christoffel symbols if we wrote $\mathsf{Christoffel}^i_{jk}$ instead of $\Gamma^{i}_{jk}$ would turn differential geometry into a dead subject very soon :P

Solution 3:

Perhaps the most compelling reason for using single character variables is that it enables the usual convention of omitting the multiplication sign in products. This enables great conciseness in notating polynomials - which is important since polynomials are ubiquitous in mathematics, so any convention that simplifies their notation, comprehension, etc is surely worthwhile. Thus we can write $\rm\ xyz\ $ to mean $\rm\ x\cdot y\cdot z\ $ without any worry that it will be mistaken for a variable name.

While having to insert the multiplication signs doesn't reduce conciseness much for a monomial, it can greatly increase complexity for a polynomial of many terms. For it may cause equations to overflow the line/page length, etc, greatly hindering comprehension. Moreover, as many cognitive studies show, humans read words by their shape (e.g. cover up the top/bottom half of a line of text and note how you can still easily read it), so any convention that alters shapes (or increases their visual complexity) may inhibit visual parsing, pattern-matching, and global inference of key structural characteristics.