'Freezing' symbols in mathematical notation |
October 22nd, 2009 |
notation [html] |
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 (whereMathematics 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
, andy
, andz
, 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 "propertyL
" -- 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 demensionp
with indexi
, wherasx^i_p
is a co-chain of dimensionp
wiith indexi
. Question: what isx^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.
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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