|October 22nd, 2009|
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
Mathematics has access to a potentially infinite alphabet (c.g
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"
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_iis a chain of demension
x^i_pis a co-chain of dimension
i. Question: what is
As history progresses, more and more symbols get frozen. The standard examples are
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.
Ais a number,
$Ais 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
mis 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
f in new writing, but
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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