Re: expressiveness in notation

On Sep 11, 2004, at 5:21 AM, Gene Boggs wrote:

> This is a cool quote from the paper, "Parsing TEX into Mathematics":
>
>     We agree that there are good reasons for trying to come up with a
> single universal grammar and semantics for mathematics notation, but
> we are concerned that any efforts to find a fixed and complete
> notation must founder on the shoals of ambiguity. Even granted some
> oracle of disambiguation, it appears that total generality must
> require fairly substantial extensibility and the ability to
> incorporate context into interpretation. One simply cannot expect to
> represent all past and especially all future mathematics with a fixed
> set of notations. Therefore one must provide tools for extension that
> are sufficiently "universal" for all further work.

Math, like everything else, needs an enema.  Style and content are 
mixed up all over the place:

x is a scalar, but B<x> is a vector.

R stands for range, but ornate R (&real;, for those keeping score in 
HTML) stands for the set of real numbers.

Probability folks use middle-dot as a placeholder, like $_.  (OK, 
that's pretty cool.)

Inconsistency across disciplines: most fields use I<i> for the square 
root of -1, but electrical engineers already use i for current, so they 
use I<j> for the square root of -1.  I hope this has resulted in at 
least one electrocution.

Requires too much flexibility from typesetting packages (consider 
Eulerian notation, where the eighteenth derivative of "f" is "f" with 
eighteen dots over it, or cascading exponentials (2 to the 3 to the 4 
to the 5 to the 6 to the 7 makes for a pretty eentsy 7).

People who flip A upside down and turn E around left-to-right just for 
kicks deserve no mercy.

-Jon