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author | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
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committer | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
commit | e0c6872cf40896c7be36b11dcc744620f10adf1d (patch) | |
tree | 60335e10d2f4354b0674ec22d7b53f0f8abee672 /usergrps/uktug/baskervi/5_4/rowley.tex |
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diff --git a/usergrps/uktug/baskervi/5_4/rowley.tex b/usergrps/uktug/baskervi/5_4/rowley.tex new file mode 100644 index 0000000000..324e6e75f4 --- /dev/null +++ b/usergrps/uktug/baskervi/5_4/rowley.tex @@ -0,0 +1,185 @@ +% \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} |