… A mathematician is not a man who can readily manipulate figures; often he cannot. He is not even a man who can readily perform the transformations of equations by the use of calculus. He is primarily an individual who is skilled in the use of symbolic logic on a high plane, and especially he is a man of intuitive judgment in the choice of the manipulative processes he employs.
All else he should be able to turn over to his mechanism, just as confidently as he turns over the propelling of his car to the intricate mechanism under the hood. …
from As we may think – Vannevar Bush.
You'll find much better pages on mathematics at Imperial, at mathworld (where I am a very minor contributor), in Eric's Treasure Trove or among Gregory Chaitin's papers; integer sequences in Sloane's database; better teaching materials at Cut the Knot; St. Andrews covers the history of mathematics; and Volker Runde collects mathematical jokes. Colin Wright writes and blogs about mathematics, somewhat more orthodoxly than me, among other things.
Please read my caveat and apologia about these pages before trying to tell me how to keep them. On the other hand, if you can see how to fit all these fragments together, or want to point me to somewhere which covers related material or does its page-design well, I'll be delighted to hear about it: and when you notice my mistakes (or catch me using a term in a way which conflicts with that of some pertinent orthodoxy – thanks Jeremy ;^) please tell me about them. The odds on my fixing them are then greatly improved. My e-mail address is email@example.com.
I've decided to write my mathematics in plain text, so that it can be read using any browser. In practice I use Opera as my review browser, so those using other browsers may run into problems at times; let me know if that happens, so I have some chance of fixing it. I also (since 2006/Summer) use the W3C's validator to help me make my pages conform to relevant specifications, which should ease cross-browser compatibility.
In devising denotations to replace squiggles outside the ISO 8859 Latin-1 character set, I've leant heavily on the accumulated wisdom of programming language designers, notably those in the tradition of Ponder (design, type system) and Haskell. I gave up on waiting for HTML to standardise mathematical mark-up back in 1995; now that it has a standard (MathML, about which I'm less than enthusiastic) I've lost interest thanks to having my own denotations and liking them better than orthodoxy's.
The primary sub-sections of this ramshackle assembly of writings about mathematics are:
Here are some hook-in points to bits and pieces I've written, many of which could use some further sorting out and tidy-up:
Meanwhile, those of you whose browsers do background images can enjoy a pictorial proof of Pythagoras' theorem: the picture suffices, if you think about it, but messes up the text. This is a wilful feature. Those with simpler browsers (and those who turn off image-loading), on the other hand, will be able to read the associated text.
Activity on this section of my site is somewhat sporadic. The notes below may help readers to understand what's going on; however, they only give highlights – plenty of further mess goes on without any comment here ! Other delays are site-wide.
Extensive linkage upgrade; moved many links to primary sub-topic pages, added links for everything else in this directory.
Learned (from hixie) how to use XHTML and a DTD hack in the DOCTYPE to map mnemonic character entities to their right Unicode code points (and, hence, a suitable glyph if your browser can find one).
Creation of a fresh area in which to start on a project with more emphasis on editorial cohesion. Haphazard material will still arrive here: when I have a coherent handle on stuff, it can join the queue to migrate into that more orderly area.
Time to separate the naturals from
foundation. This is going to be a major up-heaval: if you hit
broken links, try inserting
~eddy/math, or removing either, or replacing it with the
math/ground. Sorry if
even these fail !
I spent a lot of time on a low-level toolset with which to describe relations, mappings, collections and lists (of which pairs are a significant example). When I've finished sorting that out, I aim to build up the structures that yield scalars and linearity, all in a single fluid notation, shared (with common meaning) across all branches of the toolset. My hope is that it'll make it easier to see how the toolsets needed for gravity and quantum chromodynamics relate to one another. Spring/Summer 1998: Yoneda, Autumn: the naturals.