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.
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. I want this toolset 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, nor is it either widely or consistently supported by web browsers. Fortunately, the truths of mathematics are fairly insensitive to the denotations used for their expression, but still the task of inventing 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, both denotational and conceptual, sufficient to the task of expressing the two principal branches of theoretical physics – quantum 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 the latter
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 huge vocabulary of signs and squiggles.
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 a limited character set, and been working
diligently for some time.
With the tools I introduce, I do pause to describe notions familiar to mathematicians (and even, sometimes, to express the same in orthodox notations), and do hope that these descriptions will serve as useful illustrations: but I do so 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.
The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and confine man in the bonds of Hell.
Saint Augustine.
Written by Eddy.