## 'Freezing' symbols in mathematical notation |
October 22nd, 2009 |

notation |

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`

, 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.

`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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