… 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.

Mathematics on the web

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.

Colin Wright writes and blogs about mathematics, somewhat more orthodoxly than me, among other things. Volker Runde collects mathematical jokes. Ben Orlin illustrates mathematics with bad drawings, and reminds us that other subjects are just as worthy. If all the mathematicians you've ever heard of were men, check out the #MathGals T-shirts and Power in Numbers: The Rebel Women of Mathematics by Dr. Talithia Williams.

This little corner of it all

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.

Back in the '90s I started writing mathematics in plain text, since that was the only thing that worked in all browsers. In devising denotations that are easy to type on a keyboard with few characters outside the ASCII repertoire, I leant heavily on the accumulated wisdom of programming language designers, notably those in the tradition of Ponder (design, type system) and Haskell. By the time browsers could display orthodox notation well, I'd written plenty of pages in a plain text notation that I like better than orthodoxy's; and I don't like what one has to type in the source file to get the orthodox appearance. So I still write in plain text.

In practice I use Vivaldi as my review browser, so those using other browsers might run into problems at times (although this is fairly unlikely these days); 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.

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 (and thus angles); smoothness makes it interesting and linearity makes it simple. It can be treated both pictorially and algebraically.

Rubble in the work-yard.

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 the binomial coefficients and my extension of Pascal's triangle, the related tricks for summing sequences of naturals and the multiplicities of primes as factors of factorials.
Brute Force & Ignorance: On the respective virtues of dumb and obvious approaches as compared to graceful and elegant (but sometimes harder to think of) ones.
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.
Estimating derivatives: A nice way to estimate the n-th derivative of a function from its values at 1+n distinct inputs, combined in a symmetric way.
Remedial mathematics: The beginnings of a long-term project to dispel some commonly-taught confusions.
On chosing canonical forms: Another beginning; a look at how I chose the exemplars of any given kind of thing to use as standard representatives.
Meta-philosophy: Is mathematics discovered or invented ? Why is it so good at describing the universe ?
detritus: further strays.

Meanwhile, if your borwser supports images, here's a preview of a pictorial proof of Pythagoras' theorem.

(Lack of) Structure

While there is some structure to my site, it's a bit haphazard and, all things considered, if I were starting afresh today I'd do it all differently. I mix discussion and proof rather sloppily: I want to include the proof in the discussion, but sometimes end up just proceeding from proof to proof. It might be better to have a network of pages devoted to formal definitions and proofs, that are referenced from a related network of discussions and explanations, that explore why the questions addressed are of interest and how we come by the approaches that resolve them. Ben Orlin has a nice essay on this that prompted some interesting discussion (see its comments section).

(In-)Activity

Activity on this section of my site, as on the rest, is somewhat sporadic. My thoughts of the moment index includes yearly diaries of activity, sketching what I've been up to.

Maintained by Eddy.