How many distinguishable handwritten symbols do physicists have available to use?

This is not really a physics problem, but it does matter to physicists. We have 26 letters in the Roman alphabet, upper and lower case, a total of 52 symbols. .. TODO: exclude symbols like e that can’t be used as variables We also have 24 Greek letters, upper and lower case. However, many of the uppercase Greek letters look too much like Roman letters, such as:

• alpha (Α≠A)
• beta (Β≠B)
• epsilon (Ɛ≠E)
• zeta (Ζ≠Z)
• eta (Η≠H)
• kappa (Κ≠K)
• mu (Μ≠M)
• nu(Ν≠N)
• omicron(Ο≠O)
• rho (Ρ≠P)
• upsilon(Υ≠Y)
• tau (Τ≠T)
• chi (Χ≠X)

The uppercase sigma (Σ) and pi (Π) are taken for sum (\(\sum\)) and products (\(\prod\)) of a sequence. If you use the Christoffel symbols, capital gamma (Γ) is taken, too, but let’s ignore that (and other specialized uses) for now.

Also, the lowercase omicron (ο≠o), iota(ι≠i), and upsilon (υ≠u) look too much like Roman letters.

Arguably, the lowercase kappa (κ≠k), nu (ν≠v), and chi (χ≠x) look too much like Roman letters, also, but that doesn’t seem to stop people from using them anyway (grr).

So we have (24 - 13) + (24 - 3) = 32 usable Greek symbols.

The Hebrew alphabet is another option (e.g. the cardinality of natural numbers, \(\aleph_0\)) although at this point we begin to encounter diminishing returns, since we must be able to hand-write Hebrew letters without confusing them with each other or with Latin or Greek letters. Gimel looks a bit like lambda (ג≠λ), Heth looks like pi (ח≠π), Shin looks like omega (ש≠ω), and so on.

What about the Cyrillic alphabet? It overlaps with other alphabets, but Cyrillic characters do get used occasionally in mainstream mathematics journals.

I have read numerous papers written in Russian :) and:
Yes, Cyrillic letters are _occasionally_ used for subscripts. This
seems to be more common with physics and economics papers than with
pure math papers. I don't recall _ever_ seeing Cyrillic letters used
for variables, unless they happen to look like Greek letters. One
person I know always write \Gamma for group (gruppa) and \Pi for
semigroup (polugruppa).

[ . . . ]

The Russian letter Sha has occurred (very rarely) in mathematics in
papers published within the last few years at the AMS. It was used to
represent something called `Shafarevich group' if I remember correctly.

https://www.tug.org/twg/mfg/mail-html/1993-08/msg00278.html

One thing that’s very curious is that, almost without exception, only Latin and Greek characters are ever used. Well, Cantor introduced a Hebrew aleph for his infinite cardinal numbers. And some people say that a partial derivative is a Russian d, though I think historically it really isn’t. But there are no other characters that have really gotten imported from other languages.

http://www.stephenwolfram.com/publications/mathematical-notation-past-future/

At any rate, if \(n\) is the number of letters we use, we can get \(n^2\) subscripts. Of course, subscript like \(a_a\) are not particularly meaningful, but if it makes you feel better, we skipped integer subscripts like \(n_1\), \(n_2\), and so forth.

Even so, judging by the frequent re-use of symbols in many math and physics textbooks, this are enough to cover all the use cases people would like to have.

Frequently, it is convenient to have multiple indices for different things, which can be confusing, and tensor notation just makes matters worse.

So we also use hand-written modifiers like hats (\(\hat{x}\)), dots (\(\dot{x}\)), arrows (\(\vec{x}\)), script/caligraphic (\(\mathcal{Q}\)), doubled (\(\mathbb{R}\)), and so forth.

What a mess.

Addendum 2017-05-09:

Engineers and scientists are not immune to feelings. Therefore, if a storm named after a letter of the Greek alphabet is retired, many engineers and scientists may be reluctant to use those letters again. This could wreak havoc. There may no longer be gamma rays. Instead we might have gan rays (Georgian), gim rays (Armenian), or gimel rays (Hebrew), or, if a name starting with the third letter of the Latin alphabet (C) is used, scientists might designate a “Charles Ray”!

[ … ]

An even stranger scenario would occur if the 37th name on the storm list (pi, the 16th letter of the Greek alphabet) is retired. Textbooks would need to be rewritten. Instead of “area = πr²” we might see the 16th letter of the Hebrew alphabet (nun), used in the formula. Wouldn’t that be confusing. Th us, the area of a circle would be “nun,” regardless of the radius! On the other hand, it is true that for circles … none are squared!

— Howard I. Epstein, “Retiring the Greek Alphabet.”, June 2006

https://doi.org/10.1175/BAMS-87-6-760

https://www.jstor.org/stable/26217181

<https://ui.adsabs.harvard.edu/abs/2006BAMS…87..760E/abstract>