A note on mathematical and programming notation

In the immediate prior post I wrote of musical notation. I have only casual knowledge of that notation, but I do know a bit about some other notations: mathematics and programming.

Wikipedia has a good overview of mathematical notation. Wikipedia supports the display of mathematical notation, Help:Displaying a formula.

Mathematics just visually can be an art in its own right. For example, one of the most memorable stories in mathematics is that of the collaboration of G.H. Hardy and Srinivasa Aiyangar Ramanujan. Ramanujan made excptional contributions to the theory of continued fractions. For a brief example, see Ramanujan Continued Fractions and
Rogers-Ramanujan Continued Fraction
. By the way, these last two examples come from wolfram.com, the authors of Mathematica, a commercial software product for mathematics. Mathematca provides another way to display and manipulate mathematical formulas.

An extended example of mathematical notation, in the form of a proof consisting of about 15000 pages, can be found in the multiple papers on the solvability of finite groups. [1] See Classification Theorem of Finite Groups.

There is also a close linkage between Unix and the display of information. The original Unix from Bell Labs included many software packages concerned with text formatting and laser printing. The fundamental work in creating software for displaying mathematics is the work of Donald E. Knuth, as mentioned in my previous post On mathematics and programming: Donald E. Knuth. Software packages such as Tex and LaTex are widely used throughout the world to display mathematics.

My IBM colleage Bob Sutor, IBM’s VP of Open Standards and Open Source, did some work in displaying mathematics in a browser back in the 90’s. Indeed, that’s how I first came to know his name, as his Techexplorer plugin was one of the first alphaWorks offerings, in September 1996. I recall coming across the download stats for it when I perused the download stats for Jikes after it went out. Bob has written about this in his blog recently, I’m a document guy.

I’m familiar with other notations. Each programming language defines a notation for writing code. Here for example is the code that prints the “Blog stats” as part of the WordPress Dashboard, from the file index.php:


<?php _e(‘Blog Stats’); ?></h3>
<?php
$numposts = $wpdb->get_var(“SELECT COUNT(*) FROM $wpdb->posts WHERE post_status = ‘publish'”);
if (0 < $numposts) $numposts = number_format($numposts);

$numcomms = $wpdb->get_var(“SELECT COUNT(*) FROM $wpdb->comments WHERE comment_approved = ‘1’”);
if (0 < $numcomms) $numcomms = number_format($numcomms);

$numcats = $wpdb->get_var(“SELECT COUNT(*) FROM $wpdb->categories”);
if (0 < $numcats) $numcats = number_format($numcats);
?>
<p><?php printf(__(‘There are currently %1$s <a href=”%2$s” \
title=”Posts”>posts</a> and %3$s <a href=”%4$s” title=”Comments”>comments</a>, \
contained within %5$s <a href=”%6$s” title=”categories”>categories</a>.’),
$numposts, ‘edit.php’, $numcomms, ‘edit-comments.php’, $numcats,
‘categories.php’); ?></p>

To fully appreciate this code you have to know something about both HTML and the programming language PHP. I’ll attempt to provide a brief introduction to each in forthcoming posts.

Notes:

1. One of my classmates from Caltech, Michael Aschbacher, made fundamental contributions to this work. He is currently a professor of mathematics at Caltech. He was also a fellow waiter at the Caltech faculty club, the Athaneum, during our undergraduate days.

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