What will mathematicians do when they run out of letters in the Greek and English alphabets?

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Like $x$, $y$, $z$ are commonly understood to be dimensions and $\theta$ is an angle, $\pi$ is a specific irrational constant, and $\tau$ is two times $\pi$, et cetera.

They must be running out of letters by now. Is that a problem? What's the solution?

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This happened a long time ago. Which is why modern mathematics reads the way it does: We declare what our symbols will represent before proceeding with the rest of whatever it is we're doing.

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P. Halmos addressed the problem in his highly recommended paper How to write mathematics. Let me quote his advice (from the end of Section 6).

As history progresses, more and more symbols get frozen. The standard examples are e, i and π, and, of course, 0,1,2,3,... (Who would dare write “Let 6 be a group.”?) A few other letters are almost frozen: many readers would fell offended if “n” were used for a complex number, “ε” for a positive integer, and “z” for a topological space. (A mathematician’s nightmare is a sequence nε that tends to 0 as ε becomes infinite.)

Moral: do not increase the rigid frigidity. Think about the alphabet. It’s a nuisance, but it’s worth it. To save time and trouble later, think about the alphabet for an hour now; then start writing.