Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.

P.A.M. Dirac, 1930/May/29.

Mathemagogical Ramblings

I would dearly love to sit down and write up all the mathematics I have learned, and as much more as I learn along the way, as web pages. This is a part of our heritage which provides us with good toolsets for understanding what we observe. Although its teaching in schools is all too often dull and poorly motivated, I personally believe that a clear understanding of mathematics liberates minds from – and inoculates them against – confusion and superstition.

I consequently want the toolset of mathematics to be widely available, and the web offers the opportunity to make it so: but it also brings constraints. Making squiggles and signs widely used by mathematicians isn't half so easy in HTML as on a chalkboard, nor are all the ways to present them either widely or consistently supported by web browsers. Fortunately, the truths of mathematics are fairly insensitive to how they are expressed: none the less, the task of designing plaintext notation requires care.

Furthermore, I only have a few hours a day to play with (I'm not employed to do this) and my primary concern is to prepare a toolset sufficient to the task of expressing the two principal branches of theoretical physicsquantum mechanics and general relativity – preferrably succinctly but, above all else, in mutually intelligible terms. Without such a toolset I, at least, do not expect to be able to understand any attempt at unifying the two.

In view of this focus, I take, from the foundations, the bare minimum of relations: and build thereon the tools I want, both conceptual and denotational. My focus in this last combines: accessibility to crude web browsers with; brevity of what I have to type to express the information. Where it most blatantly fails is in its relationship with orthodoxy: where convenient, I have used orthodox denotations, but not where the latter involve a large vocabulary of arcane signs and squiggles; nor, even for the sake of a familiar and simple-looking appearance in what browsers display, am I willing to type anything vastly more complex than what appears there, especially if what I must type, in the plain text of the HTML, bears little resemblance to what gets displayed. There are, among designers of programming languages, orthodoxies which I have plundered more extensively – they have, after all, been working out how to express mathematical notions in plaintext with a limited character set, and been working diligently for some time, with no little success.

With the tools I introduce, I do pause to describe notions familiar to mathematicians (and even, sometimes, to express the same in orthodox notations). I do hope that these descriptions will serve as useful illustrations: but I provide them more out of my inate habit of digression, and my curiosity, than as the object of my writing. Consequently, such discourses tend to hang unfinished and ill-maintained off the side of the work, dreaming of the day when I can afford to devote more time to them. Please forgive the mess. I shall, however, aim to keep the central story – the construction of tools needed in physics – reasonably coherent and well-edited: and if the mess impinges there please be good enough to warn me of any confusions it causes, or of any actual errors.

There is a text, popular among mathematicians and usually attributed to Saint Augustine, that would have the reader suppose he thought mathematicians were in league with The Devil. A closer examination reveals it to be a mis-translation, really talking about astrologers and their ilk (thanks to Colin Wright for bringing this to my attention). I used to quote it here, so it seems only fair to refer to the correction in its place !

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