Performativizing Papyrocentricity #27

Papyrocentric Performativity Presents:

Sex/Dream Metaphors – Extreme Metaphors: Selected Interviews with J.G. Ballard, edited by Simon Sellars and Dan O’Hara (Fourth Estate 2014)

DNAncientNeanderthal Man: In Search of Lost Genomes, Svante Pääbo (Basic Books 2014)

The Cult of CthulhuH.P. Lovecraft: The Classic Horror Stories, edited by Roger Luckhurst (Oxford University Press 2013)

Rauc’ and RoleMortality, Christopher Hitchens (Atlantic Books 2012)

#BooksThatShouldNotBe — Tip-top Transgressive Texts…

Or Read a Review at Random: RaRaR

The World as Worm

In “Hymn to Herm”, I wrote about a religion based on √2, or the square root of two, the number that, multiplied by itself, equals 2. In the religion, neophytes learn the mystery and majesty of this momentous number when they try to calculate its exact value. The calculation involves adding and subtracting fractions based on powers of two. The first step is this: 1 x 1 = 1. So that’s too small. Add 1/2^1 = ½ and re-multiply: 1½ x 1½ = 2¼. Too big. So subtract 1/2^2 = ¼, and re-multiply. 1¼ x 1¼ = 1+9/16. Too small. Add 1/8 and re-multiply. 1+3/8 x 1+3/8 = 1+57/64. Too small again. Add 1/16 and re-multiply. And so on.

In effect, what the neophytes are doing is calculate the digits of √2 in binary, or base two. When the multiplication is too small, put a 1; when it’s too big, put a 0. Like this:

1 x 1 = 1 < 2, so √2 ≈ 1·…
1½ x 1½ = 2¼ > 2, so √2 ≈ 1·0…
1¼ x 1¼ = 1+9/16 < 2, so √2 ≈ 1·01…
(1+3/8) x (1+3/8) = 1+57/64 < 2, so √2 ≈ 1·011…
(1+7/16) x (1+7/16) = 2+17/256 > 2, so √2 ≈ 1·0110…
(1+13/32) x (1+13/32) = 1+1001/1024 < 2, so √2 ≈ 1.01101…
(1+27/64) x (1+27/64) = 2+89/4096 > 2, so √2 ≈ 1.011010…
(1+53/128) x (1+53/128) = 1+16377/16384 < 2, so √2 ≈ 1·0110101…
(1+107/256) x (1+107/256) = 2+697/65536 > 2, so √2 ≈ 1·01101010…
(1+213/512) x (1+213/512) = 2+1337/262144 > 2, so √2 ≈ 1·011010100…
(1+425/1024) x (1+425/1024) = 2+2449/1048576 > 2, so √2 ≈ 1·0110101000…
(1+849/2048) x (1+849/2048) = 2+4001/4194304 > 2, so √2 ≈ 1·01101010000…
(1+1697/4096) x (1+1697/4096) = 2+4417/16777216 > 2, so √2 ≈ 1·011010100000…
(1+3393/8192) x (1+3393/8192) = 1+67103361/67108864 < 2, so √2 ≈ 1·0110101000001…

Mathematically naïve neophytes, seeing the process miss 2 by smaller and smaller amounts on either side, might imagine that eventually the exact root will appear and the calculations end. But they would be wrong. They could work a year or a million years: they would never calculate the exact square root of two. There is no ratio of whole numbers, a/b, such that a^2/b^2 = 2. In other words, √2 is an irrational number, or number that can’t be represented as a ratio of integers (please see appendix for the proof).

This discovery, made by Greek mathematicians more than two millennia ago, is both mind-boggling and world-shattering. In fact, it’s mind-boggling in part because it’s world-shattering. √2 shatters the world because the world is too small to contain it: in the words of the Cult of Infinite Hermaphrodites, “Were the sky all parchment, the seas all ink, and gulls all plucked for quills”, the square root of two could not be recorded in full. This is far more certain than tomorrow’s sunrise, because predicting tomorrow’s sunrise depends on fallible scientific reasoning from incomplete knowledge. Proving the irrationality of √2 depends on infallible mathematical reasoning.

At least, it’s as close to infallible as human beings can get. But that’s another part of what is mind-boggling about √2. A finite, feeble human being, with a speck of soon-decaying brain, can prove the existence of things larger than the universe. A few binary digits of √2 are shown above. Here are a few more:


The distribution of 1’s and 0’s seems effectively random, as though the God of Mathematics were endlessly tossing a coin, putting 1 for heads, 0 for tails. Yet √2 is the opposite of a random number. Change a single digit anywhere and it ceases to be √2. Every 1 and every 0 is rigidly determined by “unalterable law”. So are the position and magnitude of the digits of √2 in every other base. Here, for example, is √2 in base 4:


Another word for base-4 is DNA: genes are in fact written in a base-4 code based on the chemicals guanine, adenine, thymine and cytosine, or G, A, T, C for short. If the digits of √2 are truly random, in the statistical sense, then all genomes, actual and potential, occur somewhere along its length: yours, mine, the Emperor Heliogabalus’s, Bilbo Baggins’, the sabre-toothed tiger’s, the dodo’s, and so on. But almost all the “DNA” of √2 in base-4 will be meaningless: although √2 is the opposite of random, it is effectively a typing chimpanzee. Or a typing worm – a type-worm. √2 is like an endless worm that types out its own segments on a typewriter with two keys (for binary numbers) or four keys (for quaternary numbers) or ten keys (for decimal numbers) and so on.

But √2 doesn’t just encode the genomes of individual people, animals and plants: it encodes everything they do throughout their lives. In fact, it encodes the entire universe. And perhaps the universe is √2 or some number like it. Perhaps, in some sense, everything exists within the digits of an irrational number, or a sufficiently large rational number. If so, then √2 has become aware of itself through human beings: the World as Worm has bitten its own tail.

Appendix: Proof of the irrationality of √2

1. Suppose that there is some ratio, a/b, such that

2. a and b have no factors in common and

3. a^2/b^2 = 2.

4. It follows that a^2 = 2b^2.

5. Therefore a is even and there is some number, c, such that 2c = a.

6. Substituting c in #4, we derive (2c)^2 = 4c^2 = 2b^2.

7. Therefore 2c^2 = b^2 and b is also even.

8. But #7 contradicts #2 and the supposition that a and b have no factors in common.

9. Therefore, by reductio ad absurdum, there is no ratio, a/b, such that a^2/b^2 = 2. Q.E.D.

Ave Aves!

Front cover of Collins Bird Guide by Lars SvenssonCollins Bird Guide: The Most Complete Guide to the Birds of Britain and Europe (second edition), text and maps by Lars Svensson, illustrations and captions by Killian Mullarney and Dan Zetterström, with a significant contribution by Peter J. Grant, translated by David Christie and Lars Svensson (HarperCollins, 2009)

A literate musician can read a score and hear a symphony in his head. I wonder whether the mega-minds of the future will be able to do something similar with genomes: read a DNA recipe and see the animal or plant cooked from it. The mega-minds will need to know about the oven, that is, the womb, egg or seed, but then musicians need to know about instruments, not just notes. The code can’t exist in isolation: it needs a world to be realized in and a musician’s mind can mimic that world.

But mega-minds aren’t here yet for genetics, so we have to use books like this to see the product of DNA-recipes. Collins Bird Guide is effectively a genetic cook-book or genomic score, but we don’t see the naked genes, just the dish or symphony cooked or played from them. Lars Svensson describes thousands of birds of all shapes, sizes, colours, diets and habitats, from the huge golden eagle, Aquila chrysaetos, which can carry off a lamb, to the tiny goldcrest, Regulus regulus, which isn’t much bigger than a bumblebee. But these two, like all other birds, have a common ancestor: when you see a bird sitting in a tree, it is also, metaphorically speaking, sitting in a genetic tree whose twigs, branches and boughs spring from a single trunk. One DNA-recipe has turned into many under the influence of natural and sexual selection.

Birds, which often come in very distinct male and female forms, offer lots of good examples of sexual selection. One of the most spectacular examples isn’t native to the region covered by the book, but it has been introduced here. And so there are pleasant surprises in store for some European ornithophiles. I once came across a wild-living golden pheasant, Chrysolophus pictus, early one morning in a park in northern England. I thought for a moment that I was hallucinating: the bird has a crest of spun gold, a scarlet breast and belly, and an orange/black “nuchal cape”, or neck-feathers, that “can be raised like a fan when displaying” (“Partridges & Pheasants”, pg. 59). It also has yellow legs, blue wings and a long, attractively patterned tail. “Unmistakable!” notes the book.

That’s true of the ♂, at least. The ♀, whose eyes and brain are responsible for the spectacular appearance of the ♂, is undistinguished and similar to the ♀ of Lady Amherst’s pheasant, Chrysolophus amherstiae, whose ♂ is again “Unmistakable!”, thanks to the sexual selection of its ♀. These closely related species are native to eastern Asia and “occasionally hybridize” in Britain (pg. 59). In other words, their common ancestor was fairly recent and their DNA recipes can still work together. But these hybridizations may also be a function of small populations and restricted habitat in Britain. “Function” is the operative word: birds, like all other forms of life, are mechanisms with inputs, throughputs and outputs. For a pheasant, some of the input is sense-data. The throughput is the processing of sense-data in the brain. The output is behaviour: for example, mating with a less-than-ideal partner under the restricted conditions of Britain.

All this can be modelled mathematically, but in the widest and deepest sense it already is mathematical: the human invention of mathematics, with a small “m”, is a symbolic representation of Mathematics with a big “M”. Mathematical symbols represent entities and operations and are manipulated according to logical rules. This mimics the inter-play of entities in the real world, which are subject to the rules of logic implicit in physics and chemistry. Human mathematics is fallible, albeit self-correcting. The mathematics underlying reality realizes the pipe-dreams of the papacy and is infallible, in the sense that it never disobeys the rules by which it is governed.

But this infallible mathematics can fail the entities for whom it operates: birds can die young and fail to reproduce or have fewer offspring than their competitors. But this is the fuel of a larger mechanism: evolution, which is a mathematical process. Genes mutate and vary in frequency under the influence of natural and sexual selection, inter alia. Birds offer more good examples of the effects, because they have wings, beaks and feet. These are mathematical mechanisms, shaped by and for the physics of a particular environment: wings have input from the air and provide the output of flight. Or the output of swimming: some wings are adapted for movement underwater, as in the cormorants, or Phalacrocoracidae, whose beaks are adapted for seizing fish and feet for paddling.

Sample page from Collins Bird Guide by Lars Svensson

You can look through this book and survey the varying geometry of wings, beaks and feet, from gliding gulls to hovering warblers, from seed-cracking finches to flesh-tearing owls, from tiny-toed swifts to wading egrets. The tool-kit of the common ancestor has become many tool-kits and evolution has been morally neutral as it has worked its multiplicative magic. The feet of the odd and endearing wallcreeper, Tichodroma muraria, are adapted to clinging onto vertical rock; the feet of eagles and owls are adapted to puncturing nerve-filled flesh. And presumably each species enjoys using its adaptation. A distinct psychology will accompany each distinct wing, beak and foot, because no organ can change in isolation: it is evolving within the environment of the body, influencing and influenced by other organs, in particular the brain.

But changes in the brain aren’t easily visible. If they were, some parts of evolution would be much less controversial: racial differences in human intelligence, for example. But races differ in other ways: in their attitudes to animals, for example. One generalization is that northern Europeans like listening to songbirds and southern Europeans like shooting them. So it’s not surprising that this book was originally published in Swedish as Fågelguiden, Europas och Medelhavsområdets fåglar i fält (1999). It would also be interesting to see the statistics of ornithological publishing in Europe. Those statistics will reflect genetic differences in the white European race, and so will readers’ reactions to the book.

My interest is partly aesthetic and mathematical, for example, and I quail at the thought of learning the differences between what bird-watchers call “little brown jobs”: the various kinds of warbler are hard enough to tell apart in pictures, let alone in the wild. But things can get even worse at night: Lars Svensson notes of Savi’s warbler, Locustella luscinioides, that “A possible confusion risk at distance and at night in S and C Europe is the mole-cricket” (“Warblers”, pg. 318). Birdsong and bird-cries are another aspect of ornitho-mathematics, but it’s hard to represent them in print: “kru-kih karra-kru-kih chivi trü chivi chih” (clamorous reed warbler, Acrocephalus stentoreus, pg. 322), “glipp-glipp-glipp” (common crossbill, Loxia curvirostra, pg. 386), “trrsh, trre-trre-trre-rrerrerre” (sand martin, Riparia riparia, pg. 258), “pyük…popopo…” (pygmy owl, Glaucidium passerinum, pg. 226), “brrreep, bip bip bip” (red phalarope, Phalaropus fulicarius, pg. 162), and so on.

In an electronic manual of ornithology, you’d be able to hear the songs, rather than imagine them, but electronic manuals, by offering more, in some ways offer less. Because the book has so many species to cover, it can’t describe any species in detail. So there are occasional fleeting comments like this:

Asian Desert Warbler, Sylvia nana V*** [= rare vagrant in northern Europe]… has the peculiar habit of sometimes “tailing” the Desert Wheatear [Oenanthe deserti] (“Warblers”, pg. 310-1)

The accompanying illustration shows a desert warbler standing under a small bush and peering out at a nearby wheatear. It’s anthropomorphic and anthropocentric to be amused by the behaviour, but ornithology is a human invention and humans don’t have to be purely scientific. I get a boy-racer thrill from another “V***” bird, the white-throated needletail, Hirundapus caudacutus:

Big, with heavy compact body, neckless, stub-tailed (shape somewhere between fat cigar and “flying barrel”). Flight impressively fast, the bird seems to draw easily away from other swifts (though these are still fast flyers!). (“Vagrants”, pg. 415)

That I would like to see. In the meantime, I have this book and the multiplex mutational mathematics it captures in pictures and words.

Flesh and Binary

It’s odd that probability theory is so counter-intuitive to human beings and so late-flowering in mathematics. Men have been gambling for thousands of years, but didn’t develop a good understanding of what happens when dice are rolled or coins are tossed until a few centuries ago. And an intuitive grasp of probability would have been useful long before gambling was invented. Our genes automatically equip us to speak, to walk and to throw, but they don’t equip us to understand by instinct why five-tails-in-a-row makes heads no more likely on the sixth coin-toss than it was on the first.

Dice from ancient Rome

Dice and gambling tokens from ancient Rome

Or to understand why five-boys-in-a-row makes the birth of a girl next time no more likely than it was during the first pregnancy (at least in theory). Boy/girl, like heads/tails, is a binary choice, so binary numbers are useful for understanding the probabilities of birth or coin-tossing. Questions like these are often asked to test knowledge of elementary probability:

1. Suppose a family have two children and the elder is a boy. What is the probability that both are boys?

2. Suppose a family have two children and at least one is a boy. What is the probability that both are boys?

People sometimes assume that the two questions are equivalent, but binary makes it clear that they’re not. If 1 represents a boy, 0 represents a girl and digit-order represents birth-order, the first question covers these possibilities: 10, 11. So the chance of both children being boys is 1/2 or 50%. The second question covers these possibilities: 10, 01, 11. So the chance of both children being boys is 1/3 = 33·3%. But now examine this question:

3. Suppose a family have two children and only one is called John. What is the probability that both children are boys?

That might seem the equivalent of question 2, but it isn’t. The name “John” doesn’t just identify the child as a boy, it identifies him as a unique boy, distinct from any brother he happens to have. Binary isn’t sufficient any more. So, while boy = 1, John = 2. The possibilities are: 20, 21, 02, 12. The chance of both children being boys is then 1/2 = 50%.

The three questions above are very simple, but I don’t think Archimedes or Euclid ever addressed the mathematics behind them. Perhaps they would have made mistakes if they had. I hope I haven’t, more than two millennia later. Perhaps the difficulty of understanding probability relates to the fact that it involves movement and change. The Greeks developed a highly sophisticated mathematics of static geometry, but did not understand projectiles or falling objects. When mathematicians began understood those in Renaissance Italy, they also began to understand the behaviour of dice, coins and cards. Ideas were on the move then and this new mathematics was obviously related to the rise of science: Galileo (1564-1642) is an important figure in both fields. But the maths and science can be linked with apparently distinct phenomena like Protestantism and classical music. All of these things began to develop in a “band of genius” identified by the American researcher Charles Murray. It runs roughly from Italy through France and Germany to Scotland: from Galileo through Beethoven and Descartes to David Hume.

Map of Europe from Mercator's Atlas Cosmographicae (1596)

Map of Europe from Mercator’s Atlas Cosmographicae (1596)

But how far is geography also biology? Having children is a form of gambling: the dice of DNA, shaken in testicle- and ovary-cups, are rolled in a casino run by Mother Nature. Or rather, in a series of casinos where different rules apply: the genetic bets placed in Africa or Europe or Asia haven’t paid off in the same way. In other words, what wins in one place may lose in another. Different environments have favoured different sets of genes with different effects on both bodies and brains. All human beings have many things in common, but saying that we all belong to the same race, the human race, is like saying that we all speak the same language, the human language. It’s a ludicrous and anti-scientific idea, however widely it may be accepted (and enforced) in the modern West.

Languages have fuzzy boundaries. So do races. Languages have dialects and accents, and so, in a sense, do races. The genius that unites Galileo, Beethoven and Hume may have been a particular genetic dialect spoken, as it were, in a particular area of Europe. Or perhaps it’s better to see European genius as a series of overlapping dialects. Testing that idea will involve mathematics and probability theory, and the computers that crunch the data about flesh will run on binary. Apparently disparate things are united by mathematics, but maths unites everything partly because it is everything. Understanding the behaviour of dice in the sixteenth century leads to understanding the behaviour of DNA in the twenty-first.

The next step will be to control the DNA-dice as they roll. China has already begun trying to do that using science first developed in the West. But the West itself is still in the thrall of crypto-religious ideas about equality and environment. These differences have biological causes: the way different races think about genetics, or persuade other races to think about genetics, is related to their genetics. You can’t escape genes any more than you can escape maths. But the latter is a ladder that allows us to see over the old genetic wall and glimpse the possibilities beyond it. The Chinese are trying to climb over the wall using super-computers; the West is still insisting that there’s nothing on the other side. Interesting times are ahead for both flesh and binary.


1. Suppose a family have three children and the eldest is a girl. What is the probability that all three are girls?

2. Suppose a family have three children and at least one is a girl. What is the probability that all three are girls?

3. Suppose a family have three children and only one is called Joan. What is the probability that all three are girls?

The possibilities in the first case are: 000, 001, 010, 011. So the chance of three girls is 1/4 = 25%.

The possibilities in the second case are: 000, 001, 010, 011, 100, 101, 110. So the chance of three girls is 1/7 = 14·28%.

The possibilities in the third case are: 200, 201, 210, 211, 020, 021, 120, 121, 002, 012, 102, 112. So the chance of three girls is 3/12 = 1/4 = 25%.