% \interval is used to provide better spacing after a [ that
% is used as a closing delimiter.
\newcommand{\interval}[1]{\mathinner{#1}}
% Enclose the argument in vert-bar delimiters:
\newcommand{\envert}[1]{\left\lvert#1\right\rvert}
\let\abs=\envert
\newcommand{\wt}{\widetilde}
\title[Somethin' there is about you\ldots]{Somethin' there is about you,\\
That I can't quite put my finger on}
\author{Chris Rowley}
\begin{Article}
I have been stimulated (goaded?) into penning these thoughts by Peter
Cameron's expression of concerns about the future utility of \TeX{} to
mathematicians. Many of these I share and, indeed, feel all the more
pressing since I have set myself up as being responsible for them (and
even, in some cases, I do directly bear that burden).
I shall first consider some particulars of speaking mathematics and,
in particular, voicing division; then conclude with some more general
thoughts about communicating with and via computers.
\section{Talking divisively}
Peter, in company with Don Knuth, wants to write his \TeX{} ``as close
as possible to the way [he] pronounces'' it (I added the ``he''
deliberately, see below).
Well, he may say `x over y' for $\frac{x}{y}$ or even `n-one-factorial
n-two-factorial n-three-factorial, over, n-one + n-two + n-three' for
$(n_1!\,n_2!\,n_3!)\,/\, (n_1+n_2+n_3)$ but I suspect that he would
not say `x overwithdelims \ldots' for
% $x \overwithdelims() y$
$ \genfrac{(}{)}{}{}{x}{y} $
(\verb|$x \overwithdelims() y$|)---and how would he cope with this (selected
``at random'' from an AMS paper)?
\[
\left\lvert\frac{\hat v(s)-\hat
v(t)}{\abs{\wt{D}u}(\interval{\left[t,s\right[})} -\frac{f(\hat
u(t)+\dfrac{\wt{D}u}{\abs{\wt{D}u}}(t)
\abs{\wt{D}u}(\interval{\left[t,s\right[}))-f(\hat
u(t))}{\abs{\wt{D}u}(\interval{\left[t,s\right[})}\right\rvert
\]
Of course, Peter may put only very simple fractions into his letters
but an average physicist is not so fortunate (or so communicatively
challenged?).
The problem with `over' (in the sense of `divided by') derives from
the development over time of the notation for division; the use of
built-up fractions is one of the more bizarre of the many usages that
historical accidents have bequeathed us. If the good old `division
sign' (whose \TeX{} name I have, I suspect, never known) had won out
then life would have been much easier for coders and typesetters of
mathematical documents, and possibly also for mathematicians. I also
have a feeling that our generation were perhaps the first to adopt
such a sloppy mode of mathematical speech---the phrase `quotient of x
by y' seems only a little old-fashioned to me.
In practice, mathematicians can often speak to each other in many
abbreviated forms just like `over'. For example, in context I could
say to Peter:
\begin{quote}
For t greater than $0$, $-1$ $0$ $1$ t is non-singular.
\end{quote}
and I would expect \emph{him} to very easily (sic) understand that
the `$-1$, $0$, $1$, t' should be formatted as:
$ (
\begin{smallmatrix}
-1 & 0 \\ 1 & t
\end{smallmatrix}
) $
My `day job' (but I do it evenings too) involves me in spending more
time than your average mathematician communicating notation over the
telephone, so I have become quite adept at inventing methods of
speaking math notation to a fairly wide variety of people---this often
involves private codes which I would not expect anyone else to
understand. This has little to do with talking or writing to either
computers or general mathematical audiences, nor should it have, but
for me it has illustrated very clearly the fact that any particular
convention, however well it works in a restricted context, is not a
good paradigm for a general way of making mathematical documents
portable. I would, of course, place the method that Don and Peter use
to talk mathematics to each other firmly in this category of ``private
codes''.
Knuth's idea of writing mathematics as he and Peter would say it is
both impractical (as is well illustrated by most of plain \TeX{}), and
irrelevant to the real problem of communicating mathematics (not just
the notation, but the structures), both between people and computers
and inter-computer.
I shall pick over just one other point in Peter's article before
moving to more general matters: he complains that some of us are
``obsessed with the need for all operators to be prefix''.
All operators? No, at least not in the mathematical sense. All
commands, yes: but that is a consequence of yet another accident of
mathematical history---if you need general functions with an
arbitrary, and possibly not fixed, number of arguments then the
functions should be prefix, otherwise neither the computer (without a
lot of extra work) nor the user (often, remember, this is not a
mathematician) will be able to understand (in the sense of ``parse'')
them. As Leslie Lamport observed, this convention does also have the
advantage that the syntax of prefix commands often also makes it
necessary to delimit the arguments; this syntactical nicety is
essential for human readability but not much appreciated by most
mathematicians (see my remark about ``What are we summing?'' below).
If you do not understand the importance of writing commands in this
very inefficient (from the computer's viewpoint) prefix form, try
learning more than about a dozen Postscript commands with one to three
arguments and then try and read a file that uses them.
% Also, of course, to pursue Peter's argument about the paramount
% importance of how one says things: I do not say
\section{Talking to computers}
I shall now make some more general observations concerning maths,
communication and computers.
Knuth's bestiary of mathematical symbols and constructions is no
better or worse than any other: from the perspective of anyone from
outside mathematics they are all both mysterious and infuriating.
I have in other \emph{fora} argued strongly against too much formalism in the
definition of a language in which computers can communicate
mathematical notation. I now realise that, for general formatting
purposes, rather more structure needs to be expressed in the mark-up
than that which Knuth (and, hence, Lamport) thinks necessary. I say
`thinks' since recent reports from Florida suggest that Don is
unrepentant in thinking that he got it right---for example it is,
apparently, OK (\textsc{TM}) if the computer never knows what is being
summed by a summation sign, just like it does not need to know when it
is starting a quotient construction.
Computers need a lot more information than is provided by most schemes
in order to format notation properly; this is very eloquently and
dramatically illustrated by the work of T.~V.~Raman, who is getting
the computer to answer back (audibly!) so that he knows what it cannot
understand. I can assure Peter that Raman does \emph{not} want his
computer saying `over' at random places but rather needs it to be able
to efficiently distinguish and locate the beginning, end and ``type''
of all substructures. If it is ever sensible to use the phrase ``how
mathematics \emph{should} be spoken'' (when this is the only available
means of communication) then the only relevant answers must surely be
Raman's?
It has been apparent to me throughout my mathematical life that the
world would be a better place if mathematicians were more respectful
of an audience's intelligence (rather than of her knowledge of the
bizarre conventions of the subject itself) when writing about it; it
would be nice to think that training them to `talk to computers' would
make them more polite, but I doubt it.
I agree that it would be nice to talk informally about mathematics to
my computer, and I expect that to happen long before it can understand
(in any format) all the implicit conventions contained in the way I
write notation that is to be understood only by other mathematicians.
However, I am sure that it will never be so good to talk about the
subject, and others, with a computer as it is to do so with Peter.
\section{Talking to each other}
The article also touches on many other areas which contain genuine
problems (and I cannot see them becoming less numerous in the near
future). Amongst them are some that the new standard \LaTeX{}
attempts to tackle, and others that we know must be tackled by
\LaTeX3. One of the former is the ability to substitute for fonts
that you do not have.
Peter's thoughts are of great value to those of us who are actively
influencing the future of \TeX{}, both as a typesetting system and as
a mathematical lingua franca. Thus I hope we shall see many more
articles like this one---and not just from mathematicians, please! I
don't promise to argue with them all in print but I shall certainly
read them carefully and, who knows, they may goad someone else into
explaining a different viewpoint on a controversial issue.
\end{Article}