Shirley Jackson og wikipedia.

Dette er et plot summary fra Wikipedia af bogen 'we have always lived in castles', tror jeg. Der er vist en diskusion, en akademisk diskusion (i den dobbelte betydning af både virkelig at være akademisk og ligegyldig.) der handler om wikipedias unøjagtighed og måske mere generelt om nettets tendenser til at udflade, bl.a. fakta. Som sagt er diskusionen ligegyldig. Der sker hvad der sker og hvis man vil have noget at sige der til, må man skabe noget. Man skaber noget ved at være bagrstræberisk, men det har visse tendser til at afmontere sig selv igen, det man har skabt.
Nå men der er i hvert fald et punkt hvor Wikipedia nogen gange er mange gange mere suverænt end et hvilket som helst opslagsværk, det være et leksikon, eller mere specefikke bøger om forskellige emner og det er i plot opsurmeringer. Wikipedia er den perfekte blanding af saglighed og lidenskab. Der er i nogle af dem en sjælden evne til at kondensere ikke bare handlingens ydre træk, men også handlingens kondeseren af sig selv. Denne opsumering(nedenfor) er et eksempel. Den formår på ret godt tid, at skrue de forskellige elementer ind i hinanden, på en måde så den faktisk også opsurmere historiens bevægelse og ikke bare dens elementer. Det lyder fint, er sikkert ret ligegyldigt, men en vældig nydelse at læse ikke desto mindre.

Plot summary

The people in the village have always hated us.

The novel, narrated in first-person by 18-year-old Mary Katherine "Merricat" Blackwood, tells the story of the Blackwood family. A careful reading of the opening paragraphs reveals that the majority of this novel is a flashback.

Merricat, her elder sister Constance, and their ailing uncle Julian live in isolation from the nearby village. Constance has not left their home in six years, going no farther than her large garden and seeing only a select few family friends. Uncle Julian, slightly demented and confined to a wheelchair, obsessively writes and re-writes notes for an autobiography, while Constance cares for him. Through Uncle Julian's ramblings the reader begins to understand what has happened to the remainder of the Blackwood family: six years ago, both the Blackwood parents, an aunt (Julian's wife), and a younger brother were murdered — poisoned with arsenic, mixed into the family sugar and sprinkled onto blackberries at dinner. Julian, though poisoned, survived; Merricat, having been sent to bed without dinner as a punishment for an unspecified misdeed, avoided the arsenic, and Constance, who did not put sugar on her berries, was arrested for and eventually acquitted of the crime. The people of the village believe that Constance has gotten away with murder (her first action on learning of the family's illnesses was to scrub the sugar bowl), and the family is ostracized, leading Constance to become something of an agoraphobe. Nevertheless, the three Blackwoods have grown accustomed to their isolation, and lead a quiet, happy existence. Merricat is the family's sole contact with the outside world, walking into the village twice a week and carrying home groceries and library books, often followed by groups of the village children, who taunt her with a singsong chant:

Merricat, said Connie, would you like a cup of tea?
Oh no, said Merricat, you'll poison me.
Merricat, said Connie, would you like to go to sleep?
Down in the boneyard ten feet deep!

Merricat is a strange young woman, fiercely protective of her sister, prone to daydreaming and a fierce believer in sympathetic magic. As the major action unfolds, she begins to feel that a dangerous change is approaching; her response is to reassure herself of the various magical safeguards she has placed around their home, including a box of silver dollars buried near the creek and a book nailed to a tree. After discovering that the book has fallen down, Merricat becomes convinced that danger is imminent. Before she can warn Constance, a long-absent cousin, Charles, appears for a visit.

It is immediately apparent to the reader that Charles is pursuing the Blackwood fortune, which is locked in a safe in the house. Charles quickly befriends the vulnerable Constance. Merricat perceives Charles as a demon, and tries various magical means to exorcise him from their lives. Tension grows as Charles is increasingly rude to Merricat and impatient of Julian's foibles, ignoring or dismissing the old man rather than treating him with the gentle courtesy Constance has always shown. In an angry outburst between Charles and Julian, the level of the old man's dementia is revealed when he claims he has only one living niece: Mary Katherine, he believes, "died in an orphanage, of neglect" during Constance's trial.

In the course of her efforts to drive Charles away, Merricat breaks things and fills his bed with dirt and dead leaves. When Charles insists she be punished, Merricat demands, "Punish me?... You mean, send me to bed without my dinner?" She flees to an abandoned summerhouse on the property and loses herself in a fantasy in which all her deceased family members obey her every whim. She returns for dinner, but when Constance sends her upstairs to wash her hands, Merricat pushes Charles' still-lit pipe into a wastebasket filled with newspapers. The pipe sets fire to the family home, destroying much of the upper portion of the house. The villagers arrive to put out the fire, but, in a wave of long-repressed hatred for the Blackwoods, break into the remaining rooms and destroy them, chanting their children's taunting rhyme. In the course of the fire, Julian dies of what is implied to be a heart attack, and Charles shows his true colors, attempting to take the family safe (unsuccessfully, as is revealed later). Merricat and Constance flee for safety into the woods. Constance confesses for the first time that she always knew Merricat poisoned the family; Merricat readily admits to the deed, saying that she put the poison in the sugar bowl because she knew Constance would not take sugar.

Upon returning to their ruined home, Constance and Merricat proceed to salvage what is left of their belongings, close off those rooms too damaged to use, and start their lives anew in the little space left to them: hardly more than the kitchen and cellar. The house, now without a roof, resembles a castle "turreted and open to the sky". Merricat tells Constance they are now living "on the moon." The villagers, awakening at last to a sense of guilt, begin to treat the two sisters as mysterious creatures to be placated with offerings of food left on their doorstep. The story ends with Merricat observing, "Oh, Constance...we are so happy."

Novalis.

Novalis and Philo-Sophie
Works now being published in English reveal the key role Novalis played in German culture
Jeremy Adler

The success of the French Encyclopédie and its place in the Enlightenment has tended to obscure the role of encyclopaedism in German culture. Yet the ideal of universal knowledge has been a potent force in Germany, shaping the way the nation defined itself ever since the seventeenth century. Novalis played a key part in this debate, not least in seeking to redefine what he called “total science” – his name for encyclopaedism – as a means to achieve cultural renewal. Yet he was sentimentalized after his early death as the dreamy poet of the Blue Flower, and while this ensured his posthumous appeal, it resulted in the comparative neglect of his philosophy. His contribution to German idealism was only fully revealed, a century and a half after his death, by the editors of the critical edition of his works, a literary monument forty years in the making (reviewed in the TLS, October 13, 2000). The edition showed the full extent of the unpublished journals and notebooks, including hundreds of jottings and aphorisms, often circling round a plan for a Romantic encyclopaedia. The new image of Novalis, not unlike that of Coleridge brought about by the editing of his Notebooks, led to a wider revaluation, in which the Romantic dreamer has given way to the incisive philosopher. Now, two centuries after his death, the new material is at long last becoming available in English in versions beginning with Margaret Mahony Stoljar’s Philosophical Writings (1997) and Jane Kneller’s Fichte Studies (2003), and continuing with the volumes under review.

The new Novalis more than confirms Thomas Carlyle’s view of him as “the German Pascal”. Both men had practical talents, yet they both evinced a radical purity that drove them to treat the infinite as the only measure, and hence to redefine the thinking of the age; moreover, they both pursued a trajectory from mathematics to theology and did so with such intensity that their precocious beginnings could perhaps only be fulfilled in an equally premature death; while the search for a higher, absolute truth ended in fragmentary utterance. Yet if Pascal’s Pensées were the anguished conscience of the neoclassical age, Novalis’s Fragmente were rather the electrifying consciousness of modernity.

With Friedrich Schlegel, Novalis regarded Germany’s task in modern Europe as a dialectical reversal of the French Revolution: the reflective German Geist should respond to and transcend the materialistic excesses of the Terror. Novalis’s speech “Christendom or Europe” (1799), though on Goethe’s advice excluded from the founding journal of German Romanticism, the Athenaeum, constitutes the most potent political manifesto of the first Romantic school. At its core lies an encyclopaedic vision of European diversity that goes back to the Middle Ages, when the opposing states were spiritually united under Catholic hegemony, a period Novalis treats as a golden age. The German tradition from Kant and Lessing to Goethe and Schiller regarded enlightenment as the means for humanity to prevail over strife, and Novalis explicitly invites the enlightened “encyclopaedists” to participate in the movement towards not just a German revival, but a new, spiritually self-aware Europe.

The ultimate reliance of the political visionary on the mystical poet of the Blue Flower is evident in later poems such as the Hymns to the Night and in “Wenn nicht mehr Zahlen und Figuren . . .”. The latter, as Ludwig Tieck recognized, distils Novalis’s belief into its most limpid form:

When no longer numbers and figures
Are the keys to all God’s creatures,
When those who sing or kiss
Know more than the greatest wits,
When the world is given back to life
And frees itself from earthly strife,
When light and shade in unity
Create a higher clarity,
And people see world-history
In fairy tales and poetry,
Then all confusion will fly away
At a single secret word.

This is the doctrine of a world history founded on inwardness that the late Penelope Fitzgerald so admired. In an essay on Yeats, she rehearses the credo almost verbatim: “the world will not be right till poetry is pronounced to be life itself, our own lives but shadows and poor imitations”. The Birth of Novalis, edited by Bruce Donehower, the title of which recalls the outworn image, actually dismantles the Novalis legend. This invaluable biographical collection concentrates on the engagement to Sophie von Kühn, from the poet’s meeting with the twelve-year-old to her excruciating death at just fifteen. It includes letters by Novalis, his brother Erasmus and Schlegel, and Sophie’s pathetic journal with its jottings such as “today was like yesterday nothing at all happened” – four days before her engagement to Novalis, which didn’t even rate an entry. The texts culminate in Novalis’s Journal of 1797, and conclude with the most important sources: the life by his brother Karl (1802), that by his mentor, August Cölestin Just (1805) and, still the best essay, that by Ludwig Tieck (1815). As Donehower aptly comments: “contrary to the stereotypical image of the otherworldly, solitary romantic”, Novalis is rarely alone. The diaries are filled with references to social events, to conversations, meals, walks, and so on. There are also some fairly frank notes on his sexual activity, what Novalis calls “the satisfaction of my fantastical desires”. Apart from occasional solecisms (“the father” for “father”, for example) the translation reads well.

Donehower follows recent scholarship in teasing out the poet’s changing identities, from the philosophy student, aspiring lawyer and gallant (“Fritz the flirt”), to Sophie’s admirer, her grief-stricken fiancé, the committed student at the Freiberg Mining Academy and the conscientious mining engineer. Sophie’s forbearance in her suffering became a cult – even Goethe visited her sickbed. She suffered three operations, but her liver tumour was incurable. Yet it was less the by all accounts remarkable living Sophie than the experience at her grave, the stimulus for the Hymns to the Night, which proved the defining factor in the poet’s life. The journal – as translated by Donehower – narrates:

"In the evening I went to Sophie. There I was indescribably joyful – lightning-like moments of enthusiasm – I blew the grave away from me like dust – centuries were as moments – her presence was palpable – I believed she would appear at any moment – "

Novalis anatomizes his unio mystica with Sophie in quasi-scientific detail, dissecting his actions and emotions to disclose the physical basis for the transcendental:

"As the mortal pain subsides, the spiritual sorrow grows stronger, along with a certain calm despair. The world becomes ever stranger – I feel increasing indifference towards the things around me and inside me. The brighter it gets around me and inside me – "

The narrative recalls the spiritual exercises practised by the Pietists to encourage the “inner light” to emerge. In following this goal, Novalis unites the mental with the affective sides of his personality to establish what he calls his “Philo-Sophie”. In his elevation of her into his ideal, Sophie becomes a mythical cult-figure, sharing aspects of the Virgin Mary and Christ, and personifying knowledge and wisdom. Human identity in general becomes a complex phenomenon for Novalis:

"A truly synthetic person is one who resembles many persons at once – a genius. Each person is the germinal point of an infinite genius. He is able to be divided into many persons, yet still remain one. The true analysis of person as such brings forth many persons – the person can only be individualized as persons, dissolution and dispersion. A person is a harmony – no admixture no movement – no substance such as “soul”. Spirit and person are one. (Energy is origin)"

It remained for Proust to realize Novalis’s starry dream, and to complete a novel as memory (“Er-Innerung”), a fiction that recreates the plural self by manifesting society as an inner cosmos.

Collectivism, on this Romantic view, was the social correlative of the plural self, and with Schlegel Novalis pursued what they called “symphilosophical” collaboration, a central axis of Jena Romanticism. The chief impetus for Novalis’s intellectual development, however, came from his encounter with Fichte at the University of Jena, and his breakthrough as a thinker is documented in the notebooks now known as the Fichte Studies. These form the philosophical counterpoint to his relations with Sophie. Jane Kneller’s translation of these is now followed by David W. Wood’s excellent version of the fragments from the next major phase in Novalis’s thought, generally known as The Universal Brouillon, to which Wood gives the more plausible and attractive title, Notes for a Romantic Encyclopaedia. It is to be hoped that this dextrous change will help establish the notebooks as a central text of early German Romanticism. Like other recent translators, Wood follows the historico-critical edition, and thereby confirms that the apparently intuitive thinker presented in the Athenaeum aphorisms (1798) was in fact a systematic seeker after truth. Wood’s volume also includes a short selection from the Freiberg Studies in Natural Science (1798–9). With its lucid introduction and notes, this essential volume enables the English-speaking reader to approach the Notes for a Romantic Encyclopaedia (1798–9) for the first time as a coherent text, part of a wider search in Germany for a new scientific method, a plan only later realized in modern physics. It should now take its rightful place alongside the “Oldest System-Programme of German Idealism” (1796) by Hegel, Hölderlin and Schelling, the first Romantic work to herald a poetically orientated physics; and Goethe’s exemplary fusion of science and poetry, On Morphology – which partly prompted, partly responded to the younger men’s theses. From anti-Newtonian musings such as these, the German scientific revolution associated with Planck, Einstein and Heisenberg was to draw a significant cultural inspiration.

Novalis called the philosophy of his Encyclopaedia “magical idealism”. Perhaps the nearest he came to explaining it was in a jotting of July 1798:

"To be an empiricist means to see thinking as conditioned by the influence of the outer world and things – empiricists are passive thinkers. Voltaire is a pure empiricist and so are many of the French philosophes . . . . the transcendental empiricists . . . make the transition to the dogmatists – from there to the visionaries – or transcendental dogmatists – then to Kant – from there to Fichte – and finally to magical idealism."

The magical idealist “wonderfully refracts the higher light”, and poetically transforms nature by “the magical, powerful faculty of thought”. This involves reinstating the Renaissance concept of the magus and applying it systematically to modern science.

On one occasion, Novalis also compares his project to a voyage of discovery:

"I have been on my journey of discovery, or on my pursuit, since I saw you last, and have chanced upon extremely promising coastlines – which perhaps circumscribe a new scientific continent. – This ocean is teeming with fledgeling islands.

The Athenaeum aphorisms, Blüthenstaub, only intimated the greater project:

"We are connected to every part of the universe, as with future and prehistory. It only depends on the direction and length of our concentration which relation we particularly wish to develop, which will become the most important for us, which will take effect. A true method of this procedure is probably nothing less than the long-sought art of invention; in fact it is probably more . . . ."

The newly translated notes go much further in exploring the new science Novalis calls “encylopaedistics”. The name for this “science of the sciences” may echo Diderot’s Encyclopédie, but Novalis seeks to outdo the French model by introducing dynamism to the idea of an Encyclopaedia, to study the “relationships – similarities – equalities – effects of the sciences on each other” to create “a scientific Bible”. His procedure instances the root meaning of the word “encyclo-paedia”, that is, a “circle of learning”. The approach entails turning scientific method on its head, as when Novalis claims to transform Bacon’s inductivism into a deductive method for “generating truths and ideas writ large – of generating inspired thoughts – of producing a living scientific organon”. As the Freiberg notebook records: “The combinatorial analysis of physics might be the indirect art of invention that was sought by Francis Bacon”.

The “circles” Novalis envisages in his “combinatorial analysis” are inspired by the medieval ars combinatoria, whose ideas retained an attraction for German thinkers down to Leibniz and Kant. The concentric wheels that Ramon Lull devised as a tool for inventing new ideas also serve Novalis as a model, and provide him with a motor for recombining existing ideas to create new ones. This method is, incidentally, related to the ones which the late Mary Douglas traces with such passion in Thinking in Circles (2007). As Novalis writes:

"There exists a sphere in which every proof is a circle – or an error – where nothing can be demonstrated – that is the sphere of the developed Golden Age. This and the polar sphere also harmonize. I realize the Golden Age – by developing the polar sphere. I am unconsciously in [the Golden Age], insofar as I am unconsciously in the polar sphere – and consciously, insofar as I am consciously in both."

The encyclopaedic Bible inducts the reader into the Golden Age: man returns to the prelapsarian state by rearranging the totality of all knowledge, thereby achieving a higher, paradisal consciousness. Man’s intellectual versatility reflects the universality of his creator. Yet the construct, like the self, remains unstable:

"Philosophy disengages everything – relativizes the universe – And like the Copernican system, eliminates the fixed points – creating a revolving system out of one at rest."

Novalis musters a dazzling array of disciplines to constitute his Romantic Encyclopaedia including mathematics, mineralogy, medicine, law, economics and music. Everything he touches he illuminates. Yet the totalizing aesthetic has its risks, both in precipitate insights, and in aspects of his theory of the State, understood as a “spiritual being” comparable to God. To combat absolutism, however, the Romantic Encyclopaedia looks for Kantian limitations: “Resolution of the main political problem . . . . Are combinations of opposed political elements possible a priori?”.

The philosophy of magical idealism led inevitably to the practice of literature. When Novalis abandoned the Romantic Encyclopaedia, it was to write the poetry it preaches, the “art of transforming everything into Sophie – or vice versa”. The Hymns to the Night brilliantly exemplify the turn: the poem’s success stems in no small part from the way it illumines the poet’s grief and mystically resolves his problems by an exegesis of world history. It is the first modern panoptic lyric, unmatched in visionary compass before Eliot’s Waste Land and Rilke’s Duino Elegies. In The Novices at Sais, his fragmentary Bildungsroman, Novalis develops the conceit of encyclopaedic circles to educate its main character, thereby also showing how world history advances by the combinatorial progress of humanity. The novel stems from an infatuation and later disappointment with Wilhelm Meister’s Apprenticeship. Goethe may not have approved, but he listened. In Wilhelm Meister’s Travels, he likewise favours the scientific path for his central character, and adapts Novalis’s method to represent the circles (“Kreise”) that compose society. In so doing, he replaces the abstract ars combinatoria used by Novalis with a sociological principle, more in tune with his own novel’s social theory, which offers a peaceful alternative to the route later proposed by Marx: a revaluation of labour, to remove the alienation that might lead to revolution, and a new respect for collectivism as a value.

For a fragile moment around 1800, then, there was a balance between individualism and collectivity in German culture, recalling the “Symphilosophie” envisaged by Schlegel: “Perhaps a whole new era will begin in the arts and sciences if Symphilosophie and Sympoesie become so general . . . that . . . complementary natures produce collective works”. Goethe paid homage to this ideal, when he called his Faust an “être collectif”. Novalis’s Romantic Encyclopaedia translates this joint activity to the political sphere, as in an entry on “Theory of a Nation. Pedagogy of a Nation”, concerning the interdependence of individual and collective. The protean method of his Romantic Encyclopaedia underpins much of his writing, where the disarming negations, reversals and pirouettes dissolve the rigidities of linear thought into a supple, lyrical dialectic. Thus Novalis the advocate of the State can also conclude: “In many places States should not be established at all . . .”. Such provocations retain a startling topicality.

Bruce Donehower, editor
THE BIRTH OF NOVALIS
Friedrich von Hardenberg’s Journal of 1797, with selected letters and documents
159pp. State University of New York Press. $25.
978 0 7914 6969 9

Novalis
NOTES FOR A ROMANTIC ENCYCLOPAEDIA
Das Allgemeine Brouillon
Translated, edited, and with an Introduction by David W. Wood
290pp. State University of New York Press. $35.
978 0 7914 6973 6


Jeremy Adler's translation of Hoelderlin's philosophical essays will be published next year.

Chaitin - Chaitin! Chaitin!

The Omega Man
Photo: Kevin Knight
Photo: Kevin Knight
He shattered mathematics with a single number. And that was just for starters, says Marcus Chown
TWO plus two equals four: nobody would argue with that. Mathematicians can rigorously prove sums like this, and many other things besides. The language of maths allows them to provide neatly ordered ways to describe everything that happens in the world around us.
Or so they once thought. Gregory Chaitin, a mathematics researcher at IBM's T. J. Watson Research Center in Yorktown Heights, New York, has shown that mathematicians can't actually prove very much at all. Doing maths, he says, is just a process of discovery like every other branch of science: it's an experimental field where mathematicians stumble upon facts in the same way that zoologists might come across a new species of primate.
Mathematics has always been considered free of uncertainty and able to provide a pure foundation for other, messier fields of science. But maths is just as messy, Chaitin says: mathematicians are simply acting on intuition and experimenting with ideas, just like everyone else. Zoologists think there might be something new swinging from branch to branch in the unexplored forests of Madagascar, and mathematicians have hunches about which part of the mathematical landscape to explore. The subject is no more profound than that.

The reason for Chaitin's provocative statements is that he has found that the core of mathematics is riddled with holes. Chaitin has shown that there are an infinite number of mathematical facts but, for the most part, they are unrelated to each other and impossible to tie together with unifying theorems. If mathematicians find any connections between these facts, they do so by luck. "Most of mathematics is true for no particular reason," Chaitin says. "Maths is true by accident."

This is particularly bad news for physicists on a quest for a complete and concise description of the Universe. Maths is the language of physics, so Chaitin's discovery implies there can never be a reliable "theory of everything", neatly summarising all the basic features of reality in one set of equations. It's a bitter pill to swallow, but even Steven Weinberg, a Nobel prizewinning physicist and author of Dreams of a Final Theory, has swallowed it. "We will never be sure that our final theory is mathematically consistent," he admits.

Chaitin's mathematical curse is not an abstract theorem or an impenetrable equation: it is simply a number. This number, which Chaitin calls Omega, is real, just as pi is real. But Omega is infinitely long and utterly incalculable. Chaitin has found that Omega infects the whole of mathematics, placing fundamental limits on what we can know. And Omega is just the beginning. There are even more disturbing numbers--Chaitin calls them Super-Omegas--that would defy calculation even if we ever managed to work Omega out. The Omega strain of incalculable numbers reveals that mathematics is not simply moth-eaten, it is mostly made of gaping holes. Anarchy, not order, is at the heart of the Universe.

Chaitin discovered Omega and its astonishing properties while wrestling with two of the most influential mathematical discoveries of the 20th century. In 1931, the Austrian mathematician Kurt Gödel blew a gaping hole in mathematics: his Incompleteness Theorem showed there are some mathematical theorems that you just can't prove. Then, five years later, British mathematician Alan Turing built on Gödel's work.

Using a hypothetical computer that could mimic the operation of any machine, Turing showed that there is something that can never be computed. There are no instructions you can give a computer that will enable it to decide in advance whether a given program will ever finish its task and halt. To find out whether a program will eventually halt--after a day, a week or a trillion years--you just have to run it and wait. He called this the halting problem.

Decades later, in the 1960s, Chaitin took up where Turing left off. Fascinated by Turing's work, he began to investigate the halting problem. He considered all the possible programs that Turing's hypothetical computer could run, and then looked for the probability that a program, chosen at random from among all the possible programs, will halt. The work took him nearly 20 years, but he eventually showed that this "halting probability" turns Turing's question of whether a program halts into a real number, somewhere between 0 and 1.

Chaitin named this number Omega. And he showed that, just as there are no computable instructions for determining in advance whether a computer will halt, there are also no instructions for determining the digits of Omega. Omega is uncomputable.

Some numbers, like pi, can be generated by a relatively short program which calculates its infinite number of digits one by one--how far you go is just a matter of time and resources. Another example of a computable number might be one that comprises 200 repeats of the sequence 0101. The number is long, but a program for generating it only need say: "repeat '01' 400 times".

There is no such program for Omega: in binary, it consists of an unending, random string of 0s and 1s. "My Omega number has no pattern or structure to it whatsoever," says Chaitin. "It's a string of 0s and 1s in which each digit is as unrelated to its predecessor as one coin toss is from the next."

The same process that led Turing to conclude that the halting problem is undecidable also led Chaitin to the discovery of an unknowable number. "It's the outstanding example of something which is unknowable in mathematics," Chaitin says.

An unknowable number wouldn't be a problem if it never reared its head. But once Chaitin had discovered Omega, he began to wonder whether it might have implications in the real world. So he decided to search mathematics for places where Omega might crop up. So far, he has only looked properly in one place: number theory.

Number theory is the foundation of pure mathematics. It describes how to deal with concepts such as counting, adding, and multiplying. Chaitin's search for Omega in number theory started with "Diophantine equations"--which involve only the simple concepts of addition, multiplication and exponentiation (raising one number to the power of another) of whole numbers.

Chaitin formulated a Diophantine equation that was 200 pages long and had 17,000 variables. Given an equation like this, mathematicians would normally search for its solutions. There could be any number of answers: perhaps 10, 20, or even an infinite number of them. But Chaitin didn't look for specific solutions, he simply looked to see whether there was a finite or an infinite number of them.

He did this because he knew it was the key to unearthing Omega. Mathematicians James Jones of the University of Calgary and Yuri Matijasevic of the Steklov Institute of Mathematics in St Petersburg had shown how to translate the operation of Turing's computer into a Diophantine equation. They found that there is a relationship between the solutions to the equation and the halting problem for the machine's program. Specifically, if a particular program doesn't ever halt, a particular Diophantine equation will have no solution. In effect, the equations provide a bridge linking Turing's halting problem--and thus Chaitin's halting probability--with simple mathematical operations, such as the addition and multiplication of whole numbers.

Chaitin had arranged his equation so that there was one particular variable, a parameter which he called N, that provided the key to finding Omega. When he substituted numbers for N, analysis of the equation would provide the digits of Omega in binary. When he put 1 in place of N, he would ask whether there was a finite or infinite number of whole number solutions to the resulting equation. The answer gives the first digit of Omega: a finite number of solutions would make this digit 0, an infinite number of solutions would make it 1. Substituting 2 for N and asking the same question about the equation's solutions would give the second digit of Omega. Chaitin could, in theory, continue forever. "My equation is constructed so that asking whether it has finitely or infinitely many solutions as you vary the parameter is the same as determining the bits of Omega," he says.

But Chaitin already knew that each digit of Omega is random and independent. This could only mean one thing. Because finding out whether a Diophantine equation has a finite or infinite number of solutions generates these digits, each answer to the equation must therefore be unknowable and independent of every other answer. In other words, the randomness of the digits of Omega imposes limits on what can be known from number theory--the most elementary of mathematical fields. "If randomness is even in something as basic as number theory, where else is it?" asks Chaitin. He thinks he knows the answer. "My hunch is it's everywhere," he says. "Randomness is the true foundation of mathematics."

The fact that randomness is everywhere has deep consequences, says John Casti, a mathematician at the Santa Fe Institute in New Mexico and the Vienna University of Technology. It means that a few bits of maths may follow from each other, but for most mathematical situations those connections won't exist. And if you can't make connections, you can't solve or prove things. All a mathematician can do is aim to find the little bits of maths that do tie together. "Chaitin's work shows that solvable problems are like a small island in a vast sea of undecidable propositions," Casti says.
Photo: Kevin Knight
Photo: Kevin Knight

Take the problem of perfect odd numbers. A perfect number has divisors whose sum makes the number. For example, 6 is perfect because its divisors are 1, 2 and 3, and their sum is 6. There are plenty of even perfect numbers, but no one has ever found an odd number that is perfect. And yet, no one has been able to prove that an odd number can't be perfect. Unproved hypotheses like this and the Riemann hypothesis, which has become the unsure foundation of many other theorems (New Scientist, 11 November 2000, p 32) are examples of things that should be accepted as unprovable but nonetheless true, Chaitin suggests. In other words, there are some things that scientists will always have to take on trust.

Unsurprisingly, mathematicians had a difficult time coming to terms with Omega. But there is worse to come. "We can go beyond Omega," Chaitin says. In his new book, Exploring Randomness (New Scientist, 10 January, p 46), Chaitin has now unleashed the "Super-Omegas".

Like Omega, the Super-Omegas also owe their genesis to Turing. He imagined a God-like computer, much more powerful than any real computer, which could know the unknowable: whether a real computer would halt when running a particular program, or carry on forever. He called this fantastical machine an "oracle". And as soon as Chaitin discovered Omega--the probability that a random computer program would eventually halt--he realised he could also imagine an oracle that would know Omega. This machine would have its own unknowable halting probability, Omega'.

But if one oracle knows Omega, it's easy to imagine a second-order oracle that knows Omega'. This machine, in turn, has its own halting probability, Omega'', which is known only by a third-order oracle, and so on. According to Chaitin, there exists an infinite sequence of increasingly random Omegas. "There is even an all-seeing infinitely high-order oracle which knows all other Omegas," he says.

He kept these numbers to himself for decades, thinking they were too bizarre to be relevant to the real world. Just as Turing looked upon his God-like computer as a flight of fancy, Chaitin thought these Super-Omegas were fantasy numbers emerging from fantasy machines. But Veronica Becher of the University of Buenos Aires has shown that Chaitin was wrong: the Super-Omegas are both real and important. Chaitin is genuinely surprised by this discovery. "Incredibly, they actually have a real meaning for real computers," he says.

Becher has been collaborating with Chaitin for just over a year, and is helping to drag Super-Omegas into the real world. As a computer scientist, she wondered whether there were links between Omega, the higher-order Omegas and real computers.

Real computers don't just perform finite computations, doing one or a few things, and then halt. They can also carry out infinite computations, producing an infinite series of results. "Many computer applications are designed to produce an infinite amount of output," Becher says. Examples include Web browsers such as Netscape and operating systems such as Windows 2000.

This example gave Becher her first avenue to explore: the probability that, over the course of an infinite computation, a machine would produce only a finite amount of output. To do this, Becher and her student Sergio Daicz used a technique developed by Chaitin. They took a real computer and turned it into an approximation of an oracle. The "fake oracle" decides that a program halts if--and only if--it halts within time T. A real computer can handle this weakened version of the halting problem. "Then you let T go to infinity," Chaitin says. This allows the shortcomings of the fake to diminish as it runs for longer and longer.

Using variations on this technique, Becher and Daicz found that the probability that an infinite computation produces only a finite amount of output is the same as Omega', the halting probability of the oracle. Going further, they showed that Omega'' is equivalent to the probability that, during an infinite computation, a computer will fail to produce an output--for example, get no result from a computation and move on to the next one--and that it will do this only a finite number of times.

These might seem like odd things to bother with, but Chaitin believes this is an important step. "Becher's work makes the whole hierarchy of Omega numbers seem much more believable," he says. Things that Turing--and Chaitin--imagined were pure fantasy are actually very real.

Now that the Super-Omegas are being unearthed in the real world, Chaitin is sure they will crop up all over mathematics, just like Omega. The Super-Omegas are even more random than Omega: if mathematicians were to get over Omega's obstacles, they would face an ever-elevated barrier as they confronted Becher's results.

And that has knock-on effects elsewhere. Becher and Chaitin admit that the full implications of their new discoveries have yet to become clear, but mathematics is central to many aspects of science. Certainly any theory of everything, as it attempts to tie together all the facts about the Universe, would need to jump an infinite number of hurdles to prove its worth.

The discovery of Omega has exposed gaping holes in mathematics, making research in the field look like playing a lottery, and it has demolished hopes of a theory of everything. Who knows what the Super-Omegas are capable of? "This," Chaitin warns, "is just the beginning."
Further reading:

* Exploring Randomness by G. J. Chaitin, Springer-Verlag (2001)
* "A Century of Controversy Over the Foundations of Mathematics" by G. J. Chaitin, Complexity, vol 5, p 12 (2000)
* The Unknowable by G. J. Chaitin, Springer-Verlag (1999)
* "Randomness everywhere" by C. S. Calude and G. J. Chaitin, Nature, vol 400, p 319 (1999)
* http://www.cs.umaine.edu/~chaitin/

I believe that what most people want and need is a unified system for all their work and records, not a bunch of separate "applications." This means the management of versions and copies and their variations, built of tracked small pieces with their origins known-- not big lumps with their origins lost (files as we know them today). This means juggling and managing vast amounts of fine-grained information-- thousands of items-- in multiple contexts. An item should only need to be entered once, and become available in all its contexts at any time. And all the contexts of an item should be simultaneously visible.

Øhm jeg har ikke rettet det nedenfor stående... og jeg gider fanme heller ikke.
Det her er en note til ted nelson ... jeg tror hans ambition skal omformes noget, for jeg tror ganske enkelt det er et intrinsisk problem, at computeren bliver rodet. At danne platformen helt fra bunden, eller udefra ind, i en eller anden samlende process - hvor der opnås total gennemsigtighed - er for det første ikke muligt af praktiske grunde. Det ville kræve at man kunne tage hensyn til fremtiden allerede, have foregrebet den, mens virkeligheden er, at den udvikle sig med forskellige ryk og ganske enkelt ikke er opdaget før den er det. For det andet er det ikke muligt, fordi det er en systematisk sandhed, at funktioner fungere som et lukket system, i hvert fald altid på visse punkter. Dette er et formelt problem.
Dette er ikke døden for Nelsons tanker. At det er svært at tænke fremtiden behøver ikke, at vi ikke skal det og computeren har vitterlig ikke opfyldt sit litterære potentiale, for den står vitterlig i en anden situation for os, end noget der er kommet før den, den folder en masse problemer som fx har behersket filosofien ind i sig. Forestillingen om at tænkningen kunne blive funktionel via skriften har altid behersket filosofien. Forestillingen var, at man igennem skriften kunne transportere sandheder, tænkningen ud blandt mennesker. Skriften var her i centrum som kommunikationsmiddel, det var den som fungerede som hovedmediator, både fra hovedet til skriften og fra skriften ud til de andre hoveder. Som kommunikations middel har skriften især en fejl, den er ikke selv en funktion, den skrives og læses, men bogstaverne ligger stille. På den måde etableres der et misforhold mellem skriftens mål og det man gør når man læser. Lad os forestille os en filosofibog der handler om virkelighedens stat, dvs om den rigtige statsindretning, men læsningen er ikke skabelsen eller visionen om staten. Læsningen bliver en mediator og det at blive god til at læse en bog og forstå dens skrift er ikke det samme som at blive god til at lave stater, ensige det rent faktisk at skabe stater. Skriften kan ikke fungere, den sætter tanken i et alt for bestemt forhold, i forhold til en bestemt art og der ligger et enormt problem og venter, hvordan skal man bruge det læste, dette er virkelig en anden lærdom end det at lærer at læse.
Computeren er et svar på dette skriftens problem, det at tænke, skabe noget kan nu blive funktionelt og kodet. Men i forhold til Ted Nelson må vi påpege, at computeren ikke er det totale skrive apparat som han forventede. Skabelsens problem forbliver der, forholdene mellem forskellige funktioner eksistere stadig. Det computeren udpeger er, skriftens, kodens forhold når den fungere, sættes materielt i spil og dermed kan udspille reele forhold iblandt os, hvor den ikke kun siger noget meningsfuldt, men agere meningsfuldt. Et program der kan øge demokratiet siger ikke nødvendigvis eksplicit hvad demokrati er, den er forskellige forhold, strukturer der samler forskellige funktioner, ord og kategorier og sætte dem i formelle spil med hinanden. I romantikken var der en forløber for videnskaben, som lød på, at gud havde skrevet verden i matematik, at enhver del af naturen var guds skrift, at dens love var guds måde at virke på osv. Gud var altså istand til at skrive verden. Videnskaben kan i nogen forstand ses i et metaforisk forhold til denne forestilling. Den har overtaget, ikke guds ideen, men ideen om, at finde hans skrift, den skrift som naturen er skrevet med. Meningen bag er ikke problemet, det er droppet, men søgende efter skriften er blevet intensiveret og systematiseret. Det ultimative mål er altså at komme til at skrive verden eller hvad, som gud, som en, som enhed. Men der når vi aldrig til, sådan fungere ingen skrift, sådan er der ingen skrift der kan virke. Computeren illustere dette for os, den fungere i høj grad decentralt, de enkelte programmer er ikke gennemsigtige for hinanden, de fungere ind ad til og de programmer som skal systematisere dem har store kategoriseringens problemer, ganske enkelt fordi de er uden for.