### 'Freezing' symbols in mathematical notation

October 22nd, 2009
notation
I just read How To Write Mathematics by paul halmos (1970). I agree with most of it, but one part (5: Thinking About The Alphabet) I'm not convinced on. He writes:

Mathematics has access to a potentially infinite alphabet (c.g `x'`, `x''`, `x'''`, ...), but, in practice, only a small finite fragment of it is usable. One reason is that a human being's ability to distinguish between symbols is very much more limited than his ability to conceive of new ones: another reason is the bad habit of freezing letters. Some ols-fashioned analysts would speak of "`xyz`-space", meaning, I think, 3-dimensional Euclidian space, plus the convention that a point of that space shall always be denoted by "`(x,y,z)`". This is bad: it "freezes" `x`, and `y`, and `z`, i.e., prohibits their use in another context, and, at the same time, it make it impossible (or, in any case, inconsistent) to use, say, "`(a,b,c)`" when "`(x,y,z)`" has been temporarily exhausted. Modern versions of the custom exist, and are no better. Example: matrices with "property `L`" -- a frozen and unsuggestive designation.

There are other awkward and unhelpful ways to use letters: "CW complexes" and "CCR groups" are examples. A related curiosity occurs in Lefschetz. There, `x^p_i` is a chain of demension `p` with index `i`, wheras `x^i_p` is a co-chain of dimension `p` wiith index `i`. Question: what is `x^2_3`?

As history progresses, more and more symbols get frozen. The standard examples are `e`, `i`, pi, 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 feel offended if "`n`" were used for a complex number, lowercase epsilon for a positive integer, and "`z`" for a topological space. (A mathematician's nightmare is a sequence n sub lowercase epsilon that tends to zero as epsilon 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.

The concept of "almost frozen" symbols started being interesting to me once I learned to program. I thought: this is neat; the mathematicians are putting type information in the variable names. Sort of like BASIC (where `A` is a number, `\$A` is a string, ...). I really like that I can pretty much count on '`n`' being a natural number because it makes expressions much easier to read. Someone can just write write "let ```m = 3n```" and without any messy type declarations ("where n is any natural number") I can see that `m` is divisible by 3. Hamos objects to this thing I'd always thought of as a neat way that mathematical communication was efficient, calling it "frigid rigidity". yikes.

The main part of "almost frozen" symbols that I like is that they make notation more consistent between writers. If everyone uses `f` to name an abstract function, then it's easier to interpret `f` in new writing, but `f`starts to freeze to that meaning. The reason hamos does not want us to "increase this rigid frigidity" is that "in practice, only a small finite fragment of [the infinite alphabet] is usable." I see this as a tradeoff between running out of symbols and consistency between authors. As long as we're willing to reclaim previously frozen symbols when the fall out of use (which his "`xyz`-space" example suggests we are) we shouldn't have to worry about running out of symbols.

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