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

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?

-
I would expect anyone who is able to grasp the trig identities to be able to learn 24 letters... – t.b. Nov 18 '11 at 20:56
Surely the professor says the word "alpha" as he writes $\alpha$ on the board. What else can be done? Even if you devoted an entire day to the Greek alphabet, would it really sink in after such a short time? It's simply one of those things that becomes natural after a time. – Austin Mohr Nov 18 '11 at 20:57
One big problem is that we only have 26 letters in our alphabet. We often use other alphabets to distinguish types of variables - so we'll write $\delta$ and $\epsilon$ in calculus to represent small increments in the domain and range of a function, but use $x$ when discussing an element in the range. This is essentially visual typing. – Thomas Andrews Nov 18 '11 at 20:58
Note, you've skewed far from your original, essentially neutral question, "why?" to complaining about your misfortune at nobody preparing you or helping you. That's not productive. The "why" is a complex mix of reasons, from simplicity, to convention, to history (Greek was a common part of classical education into the 20th century). The "why me" is another matter entirely. – Thomas Andrews Nov 18 '11 at 21:20
On a more serious note, the reason it happens at the university level is that high school teachers tend to obsess over the difficulty that these little notational details could cause their students, while college professors expect their students to be able to digest ideas presented with any reasonable form of notation. This is related to the fact that high school teachers teach the same subjects, often from the same textbook, year after year and do not read research artciles that requires the ability to transition between different notations, while the opposite is generally true of professors. – Michael Joyce Nov 18 '11 at 22:19

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.)

-
+1, okay this is reasonable. Perhaps I should have clarified what I meant by "difficult". I'm thinking more in terms of mechanical errors (e.g. misplaced something because a letter was confused) than conceptual problems. (e.g. "t" has this problem in that it's often confused with tau or + when hand written) – Billy ONeal Nov 18 '11 at 21:21
@Billy: That kind of issue shows up with regular letters too! I've seen too many students confuse t with +, as you mention, or confuse x with y because they are written too similarly (not to mention $a$ with $0$). I forced myself, when I got to college, to stop writing x and y and to start writing $x$ and $y$ instead, precisely to avoid confusion. Now it comes naturally. – Arturo Magidin Nov 18 '11 at 21:30
Yea; I started to strike the vertical line of my q for similar reasons. Actually, visual ambiguities are a big reason to use (e.g.) greek letters alongside the roman. As you'll notice, nobody uses kappa and omicron but rather those more discernible. Sadly though, small omega is often used near w, which drives me crazy. – Raphael Nov 18 '11 at 23:25
It's a shame you CW'd this. :( Accepted. – Billy ONeal Nov 19 '11 at 5:39
@BillyONeal: I CW'd it because I feel the entire question should be CW's. I flagged it for moderator attention, but clearly they disagreed. – Arturo Magidin Nov 19 '11 at 19:59

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 $a(b)=\sum_{c=1}^{\infty}c^{-b}$.

-
Considering that the OP is a programmer, it might be noteworthy that conventions are not only essential to understanding mathematics. Nobody forces you to call your methods getX, setX or isX and your look counters i and j, after all. – Raphael Nov 18 '11 at 23:31
@Raphael: Funny you should mention that. I don't think most such rigid conventions make sense in programmer land either :P But I can see your point +1. – Billy ONeal Nov 19 '11 at 3:48
@BillyONeal You may be interested in this analysis of the topic by Terence Tao. The essence is that the differing notation is actually there to help you. Whether they mean to or not, the characters we use begin to become attributed with implicit meanings, and while we could use different letters than we were accustomed to it would actually hurt our understanding rather than help it. – process91 Nov 19 '11 at 13:51

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.)

-
+1. I didn't see ϵ until late calc 2 -- in high school $x_0$ was used instead. δ wasn't used until calc 3. Completely forgot about pi, delta, and sigma though -- I've fixed those in my question. – Billy ONeal Nov 18 '11 at 22:06
I suspect a lot of people learn $\Sigma$ without really knowing it is a letter in another alphabet. There are plenty of math symbols, like $\exists$ and $\forall$, that are not letters. – Thomas Andrews Nov 18 '11 at 22:10
@Billy I don't think anyone would use $x_0$ in place of $\epsilon$. My guess is that $x_0$ would've been the point to which $x$ approached (i.e., $x \to x_0$). – Srivatsan Nov 18 '11 at 22:47
@Srivatsan: Hmm.. maybe it was delta X. E.g. $\lim_{\Delta x \rightarrow 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$ – Billy ONeal Nov 18 '11 at 23:03
@Raphael: Printers had those letters because most educated people were taught Greek, so I think you have the time frames wrong. See en.wikipedia.org/wiki/Classical_education_movement – Thomas Andrews Nov 19 '11 at 17:06