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'Freezing' symbols in mathematical notation

October 22nd, 2009
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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 fstarts 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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