“In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In their unhappiness over the events of this world, some immerse themselves in a kind of self-sufficiency in mathematics. (Some have engaged in it for this reason alone.)” — Stanislaw Ulam (1909-84)
Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry. What is best in mathematics deserves not merely to be learnt as a task, but to be assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement. Real life is, to most men, a long second-best, a perpetual compromise between the ideal and the possible; but the world of pure reason knows no compromise, no practical limitations, no barrier to the creative activity embodying in splendid edifices the passionate aspiration after the perfect from which all great work springs. Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from the dreary exile of the actual world. — Bertrand Russell, “The Study Of Mathematics” (1902)
The title of this incendiary intervention is of course a paronomasia on these lines from Led Zeppelin’s magisterial “Stairway to Heaven”:
“If there’s a bustle in your hedgerow, don’t be alarmed now:
It’s just a spring-clean for the May Queen…”
And “head-roe” is a kenning for “brain”.
The most powerful drug in the world is water. The second most powerful is language. But everyone’s on them, so nobody realizes how powerful they are. Well, you could stop drinking water. Then you’d soon realize its hold on the body and the brain.
But you can’t stop using language. Try it. No, the best way to realize the power of language is to learn a new one. Each is a feast with different flavours. New alphabets are good too. The Devanagari alphabet is one of the strongest, but if you want it in refined form, try the phonetic alphabet. It will transform the way you see the world. That’s because it will make you conscious of what you’re already subconsciously aware of.
But “language” is a bigger category that it used to be. Nowadays we have computer languages too. Learning one is another way of transforming the way you see the world. And like natural languages – French, Georgian, Tagalog – they come in different flavours. Pascal is not like Basic is not like C is not like Prolog. But all of them seem to put you in touch with some deeper aspect of reality. Computer languages are like mathemagick: a way to give commands to something immaterial and alter the world by the application of will.
That feeling is at its strongest when you program with machine code, the raw instructions used by the electronics of a computer. At its most fundamental, machine code is simply a series of binary numbers controlling how a computer processes other binary numbers. You can memorize and use those code-numbers, but it’s easier to use something like assembly language, which makes machine-code friendlier for human beings. But it still looks very odd to the uninitiated:
That’s almost at the binary bedrock. And machine code is fast. If a fast higher-level language like C feels like flying a Messerschmitt 262, which was a jet-plane, machine-code feels like flying a Messerschmitt 163, which was a rocket-plane. A very fast and very dangerous rocket-plane.
I’m not good at programming languages, least of all machine code, but they are fun to use, quite apart from the way they make you feel as though you’re in touch with a deeper aspect of reality. They do that because the world is mathematics at its most fundamental level, I think, and computer languages are a form of mathematics.
Their mathematical nature is disguised in a lot of what they’re used for, but I like to use them for recreational mathematics. Machine-code is useful when you need a lot of power and speed. For example, look at these digits:
1, 2, 3, 4, 5, 6, 7, 8, 9, 1*, 0*, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6*, 3*, 7, 3, 8, 3, 9, 4, 0, 4, 1, 4, 2, 4…
They’re what the Online Encyclopedia of Integer Sequences (OEIS) calls “the almost natural numbers” (sequence A007376) and you generate them by writing the standard integers – 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13… – and then separating each digit with a comma: 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3… The commas give them some interesting twists. In a list of the standard integers, the 1st entry is 1, the 10th entry is 10, the 213rd entry is 213, the 987,009,381th entry is 987,009,381, and so on.
But that doesn’t work with the almost natural numbers. The 10th entry is 1, not 10, and the 11th entry is 0, not 11. But the 10th entry does begin the sequence (1, 0). I wondered whether that happened again. It does. The 63rd entry in the almost natural numbers begins the sequence (6, 3) – see the asterisks in the sequence above.
This happens again at the 3105th entry, which begins the sequence (3, 1, 0, 5). After that the gaps get bigger, which is where machine code comes in. An ordinary computer-language takes a long time to reach the 89,012,345,679th entry in the almost natural numbers. Machine code is much quicker, which is why I know that the 89,012,345,679th entry begins the sequence (8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 9):
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 63, 3105, 43108, 77781, 367573, 13859021, 77911127, 911360799, 35924813703, 74075186297, 89012345679…
And an ordinary computer-language might give you the impression that base 9 doesn’t have numbers like these (apart from the trivial 1, 2, 3, 4, 5, 6, 7, 8, 10…). But it does. 63 in base 10 is a low-hanging fruit: you could find it working by hand. In base 9, the fruit are much higher-hanging. But machine code plucks them with almost ridiculous ease:
1, 2, 3, 4, 5, 6, 7, 8, 10, 570086565, 655267526, 2615038272, 4581347024, 5307541865, 7273850617, 7801234568…
Some of the most beautiful patterns in nature arise from the interaction of three very simple things: sand and water, sand and air. Sculptrix Sabulorum, a side-project of the Exeter band Slow Exploding Gulls, are an attempt to do with sound what nature does with sand: turn simple ingredients into beautiful patterns. Here are extracts from an interview and review in the Plymouth fanzine EarHax:
Hector Anderton: OK. The obvious first. Sculptrix Sabulorum. What does it mean and why did you choose it?
Joe Corvin: It’s Latin and literally means “Sculptress of the Sands”. We chose it, well, because we thought it looked and sounded good. Good but mysterious.
Hector Anderton: And who is the sculptress? The sea?
Joe Corvin: Well, the sculptress is Mother Nature, in the fullest sense, but she uses the sea. The wind. Gravity. Simple things, but put them together with sand and interesting things happen.
Cath Orne: Which we wanted to explore, but we didn’t think S.E.G. [Slow Exploding Gulls] was the way to explore them.
Hector Anderton: But hadn’t you done that in Silica?
Joe Corvin: We’d started to, but Silica hadn’t exhausted the theme. Of sand, I mean. It’s something I’d always been interested in, but with S.E.G. we tend to go with the organic side of the sea, with sea life.
Hector Anderton: Whereas sand is inorganic?
Joe Corvin: Exactly. Silica was a bit of a departure for us, in that respect. It was as though we were walking down a corridor and we opened a door in passing and thought, yeah, that room looks interesting.
Cath Orne: So we’ll come back and have a proper look later.
Joe Corvin: Yeah. Under a new name. Which we’ve done. Hence, Sculptrix Sabulorum.
Extract © EarHax (1992)
Skulsonik, Sculptrix Sabulorum (Umbra Mundi 1995)
Macca to Madonna: “Listen to the music playing in your head.” In fact, we never do anything else. We don’t experience the world: we experience a sensory simulacrum of the world. Light or sound-waves or chemicals floating in the air stimulate the nerves in our eyes or ears or nose and the brain turns the resultant stream of electrical pulses into sight or sound or smell.
But it does more than that: it covers up the cracks. Raw nerve-stuff is not smooth and polished sensation. We have blind-spots, but the brain edits them out. Only a small part of our visual field is actually in clear focus, but we think otherwise. If we could see raw nerve-stuff, it would be a blurry, fuzzy mess.
The same is true of hearing. And Skulsonik is an attempt to record raw nerve-stuff: to capture not sound out there, but sound in here – the music playing in your head. Sculptrix Sabulorum have set out to answer a simple question: “What does music really sound like?” Or rather: what does music cerebrally sound like? What does it sound like in your head?
Extract © EarHax (1995)
Previously pre-posted (please peruse):
• Mental Marine Music – Slow Exploding Gulls
How many blows does it take to demolish a wall with a hammer? It depends on the wall and the hammer, of course. If the wall is reality and the hammer is mathematics, you can do it in three blows, like this:
α’. Σημεῖόν ἐστιν, οὗ μέρος οὐθέν.
β’. Γραμμὴ δὲ μῆκος ἀπλατές.
γ’. Γραμμῆς δὲ πέρατα σημεῖα.
1. A point is that of which there is no part.
2. A line is a length without breadth.
3. The extremities of a line are points.
That is the astonishing, world-shattering opening in one of the strangest – and sanest – books ever written. It’s twenty-three centuries old, was written by an Alexandrian mathematician called Euclid (fl. 300 B.C.), and has been pored over by everyone from Abraham Lincoln to Bertrand Russell by way of Edna St. Vincent Millay. Its title is highly appropriate: Στοιχεῖα, or Elements. Physical reality is composed of chemical elements; mathematical reality is composed of logical elements. The second reality is much bigger – infinitely bigger, in fact. In his Elements, Euclid slipped the bonds of time, space and matter by demolishing the walls of reality with a mathematical hammer and escaping into a world of pure abstraction.
• Continue reading Neuclid on the Block