… 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 eddy@chaos.org.uk.

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:

- A cursory look over the foundations and logic.
- Linearity is what you get when you
can scale things and add them, possibly with
diverse
directions

. - Smoothness is what you get when you don't have over-all linearity, but local structure feels almost like you do.
- Geometry is what you get when you can measure lengths; smoothness makes it interesting and linearity makes it simple.

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:

- The golden ratio
- and a cousin which involves solving cubic polynomial equations
- Information theory
- from a Bayesian perspective; and a weighing puzzle
- Benford's law
- why the first (non-zero) digits of numbers are disproportionately low.
- Chaos
- fractals and nonlinearities.
- Distributions
- some notes on the distributions that determine typical behaviour of common random processes.
- Statistics
- focussing on the use of linear algebra to help find and study correlations.
- Factorials
- simple combinatorics and relationship to the Gamma function; see also my extension of Pascal's triangle and the related tricks for summing sequences of naturals.
- Perfect numbers
- A whole number that's
equal to the sum of its proper factors is described as
perfect

. - Repayment mortgages
- and how house-price inflation interacts with interest on a loan.
- The birthday paradox
- in which certain kinds of coincidence are predicted to happen more often that some naïvely expect.
- Base 3 and trits
- plus how these relate to binary.
- Bézier curves
- piece-wise polynomial curves expressed in a form that makes it easy to select the control points that specify where the curve goes.
- Musical theory
- Explanations of the theory of music tend to assume you understand the jargon of the theory of music, which isn't possible until you understand the theory of music. So I'm having a stab at explaining the theory without assuming the jargon, as a framework within which to explain some of the jargon without requiring the reader to already know what it means.
- Remedial mathematics
- The beginnings of a long-term project to dispel some commonly-taught confusions.
- detritus
- further strays.

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.

- November 2007
Extensive linkage upgrade; moved many links to primary sub-topic pages, added links for everything else in this directory.

- Summer 2005
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).

- Spring 2000
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.

- November 1998
Time to separate the naturals from the

foundation

. This is going to be a major up-heaval: if you hit broken links, try inserting`/ground`

or`/finite`

after`~eddy/math`

, or removing either, or replacing it with the other. Turn`math/found`

into`math/ground`

. Sorry if even these fail !- In 1998
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.