Why does higher level mathematics more often than not use Greek lettering?

Question

In high school, at least from what I've seen, mathematics courses never use Greek lettering in their description of concepts, with the notable exceptions of $\Sigma$ for summations, $\Delta$ for changes over time, $\pi$ as $3.14159\ldots$, $\tau$ in physics courses, and $\theta$ for basic sines, cosines, and tangents. This behavior is mirrored in typical college placement exams, such as the SAT or AP exams, which also do not typically use any Greek lettering.

Yet, when students enter college, classes and instructors do use Greek lettering, and use it without preamble; they assume students are familiar with such notation. Yet, typical freshman are not familiar with Greek lettering, and are not sure how to draw, pronounce, or think in terms of, such letters.

Is there a specific reason Greek lettering is deferred to the high school $\to$ university transition, and, more generally, for Greek lettering in the first place?

Answer 1

Maybe greek letters are now playing the role of the slide rule (you're too young to need them in High School, you are assumed to already know them first year of college)...

I honestly think that if the use of different letters made "things [you] considered easy difficult", then you didn't know them well enough (though you thought you did). I find that students who get confused in calculus when the function is not called $f$ but is called something else don't really understand what is going on, and if the same sort of thing happened to you with algebra when switching from $a$, $b$, $c$, to $\alpha$, $\beta$, $\gamma$, then there was a gap in your understanding that went beyond not knowing the greek alphabet.

Now, there are only so many letters around; and in order to try to give some order to the use of letters, certain letters tend to be used for specific purposes. We generally use $a$, $b$, $c$, etc for algebra constants; we tend to use $f$, $g$, $h$ for functions; $i$, $j$, $k$ for indices (and $i$ gets reserved for the imaginary unit in some contexts); $m$ and $n$ usually denote integers. Lower case $o$ is too easy to confuse with $0$; $t$, $u$, $v$, $w$, $x$, $y$, $z$ are often used for variables; etc. There are only so many letters to go around, and soon you start needing new letters to make things easier. The use of greek letters is not designed to confuse, it's designed to clarify, by leaving other letters to their "standard" uses.

(Of course, you could simply have looked up the Greek alphabet, or requested the instructor to help you with it; I remember when I took Algebraic Number Theory in grad school, the professor distributed on the first day a sheet with the handwritten fraktur alphabet so we would know that $\frak{P}$ was a capital $P$, etc.)

Answer 2

Hugh Montgomery once did some thinking-out-loud about the possibility of writing a paper where, whenever he needed a new symbol, he would just take the first letter of the alphabet he hadn't already used. The title would be, On the Riemann $a$-function, and the paper would begin, Let $a(b)=\sum_{c=1}^{\infty}c^{-b}$.... He concluded that the paper would be unreadable.

The point is that mathematicians have adopted conventions. The convention adopted may not make sense, or may not make any more sense than any of the possible alternative conventions, but once it is adopted it is of enormous value in communication, which is what mathematics is about. Once you have been inducted into the conventions, you can instantly grasp $\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$ because you have so many associations with it, whereas it takes a great effort to understand $a(b)=\sum_{c=1}^{\infty}c^{-b}$.

Answer 3

Historically, part of a classical education used to be learning Ancient Greek and Latin, so most college students and above were expected to know the Greek alphabet.

A pure example of this history is the naming of college fraternities with Greek letters.

Today, very few English-speakers learn Greek, so there might be a value to having instructors at least present to students a list of the Greek alphabet, with pronunciations, to make acclimation easier.

Typically, the first Greek letter we learn is $\pi$, followed by $\epsilon$ and $\delta$ in calculus. Maybe $\Sigma$ and $\Delta$. But we certainly don't have a systematic intro. I know I couldn't tell you the entire Greek alphabet in order, and still forget the names of some of them, particularly $\xi$ for some reason. (Note - we don't tend to use the Greek letters that look like their Roman alternates, precisely because we are using the change in alphabet to represent types, and so using those letters would hardly be helpful.)