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For example, $\epsilon$ and $\delta$ are used in Real Analysis for proving limits... $\phi$, $\nu$ and $\mu$ were introduced to me in the context of number theory... and then $\pi$ is used for purposes other than its canonical $3.14...$ value, such as to represent the equation of a plane, or as a function name.

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    $\begingroup$ This isn't specific to Greek letters. Why do $f$ and $g$ always stand for functions? Why is $i = \sqrt{-1}$? Why is entropy always represented by $H$ or $S$? Why is $\aleph_0$ the cardinality of countable sets? $\endgroup$ – Peter Shor Sep 28 '13 at 18:29
  • $\begingroup$ en.wikipedia.org/wiki/Greek_letters_used_in_mathematics $\endgroup$ – Don Larynx Sep 28 '13 at 18:30
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    $\begingroup$ I think one of the reasons was that the greek alphabet is the closest one to the latin alphabet but different...and we needed different since there're lots of things to denote and our alphabet is way too small. $\endgroup$ – DonAntonio Sep 28 '13 at 18:31
  • $\begingroup$ @DonLarynx , I don't think that link explains why greek letters are used...does it? $\endgroup$ – DonAntonio Sep 28 '13 at 18:31
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    $\begingroup$ As I undersand it the Greeks did not have separate numerals but used letters to code for numbers. Euclid may also be responsible, as would (in the UK) the fact that "the ancient universities" in the 19th century insisted on theology, and the original texts of the New Testament for the Church of England were not the Latin Vulgate, but the Greek texts. $\endgroup$ – Mark Bennet Sep 28 '13 at 18:36
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If we all use the same letter for the same thing, then we can skip the explanation. Imagine if you had to preface every post with "Let $π$ be the ratio of a circle's circumference to its diameter." When you approach an $\epsilon$-$\delta$ proof, if the author uses the familiar letters in their familiar roles then the reader familiar with the proof technique is already halfway there understanding the proof. That explains the consistency.

As for why any particular letter is used, such as why do we use $\pi$ instead of $\rho$ or $\psi$, the short answer is "tradition". Folk etymology abounds, but it takes serious scholarship in the history of mathematics to get real answers. For new coinages, the limiting factor is which letters aren't already being used. That's why nowadays new symbols sometimes come from exotic alphabets.

It's worth noting that a letter will often suggest its own meaning. For example, in most languages in which mathematics is written, the word for plane has a prominent P sound, and the Greek letter for that is $\pi$. Maybe that's why so many authors will use the symbol $\pi$ to refer to a plane.

If your question is, why is the Greek alphabet used instead of any other, the answer is that until relatively recently in history, all educated people were expected to be able to read at least a little Greek. Therefore aside from the Latin alphabet, the Greek alphabet was the next-most easily recognized alphabet.

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