Paul Lutus wrote:
> Mathematical notation is extremely strict and slow to change. Apart from
> some recent window dressing, the last significant change was the adoption
> of Liebniz' Calculus notation over that used by Newton in the late 17th
> century.

Ah, but a variant of Newton's notation is still in wide use for ordinary
differential equations:

y'(x) = y(x); y(0) = 1


> Consequently, mathematical notation has the widest audience of any
> formal symbolic language. And programs that purport to be able to fluently
> read and write mathematical notation are in great demand and fetch high
> prices (Mathematica, Maple, Matlab, IDL).

I wouldn't call Matlab a "symbolic" language, unless it's changed a lot
over the years. And concerning the high prices, there are two or three
open-source Matlab-like environments, Octave being the most well known.
For purely numerical computing with an emphasis on statistics, there is,
of course, R as well.

In the symbolic realm, there is Axiom and Maxima, both open source, in
the general-purpose category. In addition, there are a number of
open-source high-speed special-purpose tools like GiNaC, Pari, GAP, and
Singular.

And let's not forget TeX and mathematical typesetting and the notions of
"literate programming" and "reproducible research". I know there are
some high-priced commercial tools to do this, but most everybody I know
uses things like LyX, TeXmacs, noweb and such rather than "the
high-priced spread". (or Word.) :)

Finally, I think there's a formal symbolic language with a wider
audience than mathematics. Can you guess what it is? I'll give you a
hint -- Google for "lilypond". :)