Paul Klee

Thursday, July 30, 2009

Infinite Chess

The conventional game of chess provides more than enough challenges for even the strongest player, but many extensions to the game have been proposed. Among these are three-dimensional chess, of Star Trek fame, and designs using an infinite board. These latter attempts use the idea of potential infinity rather than absolute infinity. The designs presented below imply actually infinite domains as well as potentially infinite subspaces.

Extending chess to an infinite domain involves defining the required space and also redefining how the chessmen move within it. The conventional chessboard has 64 squares, which are indexed from a1 at the bottom left hand corner to h8 at the top right. This provides a convenient notation for recording the moves of a game. So, the rows are indexed by the first eight numbers and the files (columns) by the first eight letters, as shown below.

The board can be rendered unlimited simply by allowing the index to include all the finite numbers and all the finite combinations of letters. For example, a square such as g100 or ay39 would be legitimate. In this way the chessmen could move about in an infinite space, without changing the conventional rules by which they move.

A peculiarity of this extended index is that the bottom and left hand boundaries of the board are preserved. On such a board the white chessmen can move forward or to the right without limit but remain constrained to the left and behind. However, it is not clear where the opposing black pieces are to be located or whether they should be symmetrically constrained behind and to the left like the white pieces, which would clearly never do.

One solution is to confine the starting positions of the opposing armies to the dimensions of the standard board but relativise their position in the infinite plane. This can be done by extending the index to include negative values, analogous to the notation of the Cartesian plane. For example, ac42, -ac42,

ac-42 and -ac-42 would be legitimate and distinct locations, where ac represents 26 + 3 = 29 squares.

The extension to an infinite board would affect the powers of the chessmen differently. The queen, rook and bishop could make unlimited moves but the king, knight and pawn would be restricted to a single move, and so would gain limited freedom on the extended board. Their relative powers would be diminished accordingly.

An alternative scheme is to separate the white and black chessmen by an infinite space. The immediate consequence would be that no matter how far the queens, rooks or bishops moved according to their enhanced powers, they could never engage the enemy. To rescue the game from this impasse requires a further extension to the powers of the pieces and the pawns.

The solution is to allow all the chessmen to make infinite moves, from one domain to another, according to strict but familiar rules. These rules are as follows:

Rule 1: A man may either make a short (finite) move or a long (infinite) move but not both.

Rule 2: In making a long move, a man must move from one domain to another in the same manner as required by a short move.

Rule 3: A move from one domain to another preserves the finite position of the man.

The first rule is self-explanatory. The player may either make a move within the domain the man occupies or move the man to another domain, subject to rules 2 and 3.

The meaning and relation of the infinite domains needs to be explained before elaborating on rules 2 and 3. Each domain is a replication of the infinite space defined above. The domains are arranged in a square matrix, which must be sufficiently large to allow long moves as defined in rule 2. For example, a 5 x 5 matrix is necessary to allow the knights access to every domain. Any larger matrix could be adopted but, for aesthetic purposes, an 8 x 8 matrix of domains is ideal.

A system of notation can now be defined to locate the men within both domain (board) and its finite subspace. Each of the 64 infinite boards is indexed from A1 to H8, analogous to the indexing of the conventional chessboard. A double reference of the form XYxy locates an individual square within a domain. For example, the white king is located on the square E1e1 at the start of the game and the black king is on square E8e8.

The initial positions of the 32 men can now be described. The rule for setting up the board is simple. On the conventional board, the white queen sits on square d1: on the infinite board she sits on square D1d1. The trick is to duplicate the local reference in the board reference. The white queen’s pawn conventionally starts on d2, so it occupies D2d2 on the infinite matrix. An infinite bird’s eye view would show the initial set up to be identical to that of the conventional game.

Rules 2 and 3 can now be explained more fully. The white king’s knight begins on square G1g1. The knight is free to make a short move to either G1f3 or G1h3. In addition, the knight can make a short move to G1e2, because all the pawns start off in domain 2. The knight can make initial long moves to F3g1 or H3g1 but not to E2g1, because this square is occupied by the king’s bishop’s pawn.

A notable feature is that all the pieces can make unrestricted finite moves at the opening, because each one is alone in its domain. This allows the players to jump into a new domain from an unlimited number of positions. Like many art forms, it is the constraints rather than absolute freedom that leads to interesting works. No less so in the game of chess. For this reason the proposed game can be modified by restricting each domain to the usual finite 8 x 8 matrix of squares. The result is an extremely complex finite expansion of the traditional game of chess.

The diagram below shows some examples of long moves. A domain set of 3 x 3 boards has been used for compactness of presentation. It can be seen that knight, bishop and rook can reach across domains. The power of a pawn to take diagonally in a long move is also illustrated. The pawns power to move two squares on its first move allows it to make a double long move. The en passant rule is similarly preserved.


The extension of the game of chess to multiple domains generates a family of games, which may be either finite or infinite. This can be achieved by the addition of the three special rules for long moves and by adding a square or rectangular matrix of boards of one’s choice. The double notation allows the computerisation of the game. The implications for geometry and the theory of infinite number will not be considered here. Suffice to say that the examination of such models should provide useful insights in these areas of enquiry.

Sunday, July 19, 2009

Ich habe meinen Regenschirm vergessen

Who would have cared I owned an umbrella,
let alone that I had forgotten it,
had I not written that note to myself?
If it had rained outside Plato's cave, they would
not have known it, but I know, as I stare
into the sunlight, that it rains sometimes,
somewhere, at times and places that I may
chance to be, sans umbrella in Turin
or, maybe, mindless in Sils Maria.

To bear or not to bear an umbrella,
that is the unbearable question, since
we cannot know that it is going to rain
when we reach our final destination.
Posterity may never know why I
wrote, "I have forgotten my umbrella",
nor what the meaning of my act entailed,
unless some ingenious Frenchman can
explain it to future generations.

But the paradox remains; I could not
have truly written, "I have forgotten" it
at the time of writing the note because
that was the time when I remembered it.
Clearly, then, it was not the existence
of the umbrella that I forgot but
rather not remembering to take it
with me to the place where I had not predicted
that it would rain; a place I now forget.

I am in some street, and it starts to rain,
but I find my umbrella is missing.
For some reason this seems significant,
so, I write a note to remind myself
not to forget it again, when there is rain.
"I have forgotten to bring my umbrella"
would have made plain the act is intended,
not the little shadow I forgot to keep
by me, as defence against bad weather.

But I may have left my umbrella on
the train, in the café, at the brothel
or anywhere else where it could have been
set down carelessly and quite forgotten.
Would our Frenchman think of that, I wonder?
No, he would be too busy undermining
my reputation by analysing
my forgetfulness, on one occasion,
as if this were a judgement on my work.

What really happened I fear to tell you
because it links me to an awful crime.
An acquaintance of mine, Monsieur X, say,
borrowed my umbrella, but on the way
home he was attacked by ruffians who tried
to rob him. In self-defence he stabbed one
in the chest and the other ran away.
Needless to say, my umbrella is stuck
in a place that's best forgotten by all.


Thursday, July 16, 2009

The Old Man of the Woods

The great man sits alone within the hut,
his peasant face not much to look at, but
the frigid thoughts that flow within his mind
are much concerned with time, not humankind.

Outside the window change is measured by
the dripping icicles that signify
the end of winter's elongated clock,
days pounded out upon the chopping block.

Too far away the prison camps that cleanse
the nation of its multiethnic wens,
for myopic philosophers to see
the discarded piles of humanity.

The winter pools are deep and cold,
his tread is heavy, growing old.
Now wearied by the tasks ahead,
he scratches runes inside the shed.
His mind unpicks the threadbare lore,
a cloth worn thin from Grecian shore
to Konigsberg, where spiders spun
those webs abhorred by Englishmen.
I, Hugin know who is to blame.
Why should he claim a famous name?

Once more the wolves have eaten Sun and Moon.
Dread giants rule the Earth again and soon
the flow of blood becomes unstoppable.
Meanwhile the sage ponders at his table.

He thinks and scribbles, seeing none of this,
in his small shelter of domestic bliss.
What is the import of these mighty words
that fly above the heads of human herds?

Time and being crush down the suffering
but to him, unbearable lightness of being
is just reward for self inflicted toil
in the cultivated land of German soil.

He delves in mines for Rhenish gold,
and grasps the ring that was untold,
when time's old serpent's tail uncurled,
between its fangs to show our world.
We brothers say he has no name,
and fly to Odin with the same
dire message that his fate is sealed.
without our runes, his heart's not healed.
I, Munin shall forget his lore,
and shall forget it evermore.


Saturday, July 4, 2009

Going Home

The old man sat by the window overlooking the courtyard; his disciple, Lai Tan, sat at the big table, cutting bamboo slips.
“Master, shall I join yesterday’s slips or would you like to dictate today?”

The sage pulled his worn cloak tighter round his neck. The wind from the gorge was spinning the leaves into golden wreaths, sending gusts of cool air into the dimness of the room. The clack of footsteps sounded in the corridor and a house servant, head bowed, entered the room.

“What is it? My master is not to be disturbed,” Lai Tan said, looking over to the old man for guidance. The sage raised his eyebrows and held a finger to his lips, then looked towards the servant with a kindly smile.

“Well, tell the master what you want,” Lai Tan snapped.

“Sir, the Keeper of The Pass respectfully requests an audience, he is waiting outside.”

The old man rose unsteadily to his feet, nodded his assent and signalled with his hand to Lai Tan.

“Shall I leave you to talk with our jailor alone, Master?”

The old man nodded again and remained standing, in readiness to meet the gentleman outside.

“Tell your master he is welcome to come in but that we are unable to entertain him in the accustomed style,” Lai Tan said to the servant.

The servant left, followed by Lai Tan; His Master, Kuan Yin, swept into the room.

The keeper, a tall, spare man, was dressed in his red robe of office. Middle-aged with a neatly trimmed beard and bobbed hair, he bowed respectfully to the grey haired figure standing before the window.

“Master, forgive the intrusion. The King has asked me to enquire about your health and the progress of your book.”

The old man bowed slightly in return and motioned towards the unoccupied chair. The keeper sat down, after a slight hesitation, and looked enquiringly at the old man. The sage sat down again on the window bench and said, “A cold wind is blowing from the North West.”

The Keeper waited, expecting more but was disappointed.

“We are all waiting anxiously for your wisdom to be recorded – especially my lord the King.”

“Has he not wisdom enough from the mouth of Kung Fu Tzu?” the elder replied, smiling softly as he looked out of the window at the clouds, now piled up as if before a storm.

“He teaches the court of familial piety, of bending men to the rule of law, but what do you say?”

“The family is older than Dukes and Kings but the Way of Heaven is older than time.”

“But the Empire is in disorder. Isn’t that why you left the capital? The States war with each other and take no heed of the suffering of the people.”

“The wind is blowing the leaves about, a storm is coming,” the Old Man replied, looking out over the courtyard to the looming mountains beyond.

“I hope you will forgive me, I dared not let you leave Chou, at least not without some token of your wisdom to appease the King.”

Smiling meekly, the old man turned his head to look at the official.

“Does the King of Chou prefer wisdom to its source?”

The Keeper thought carefully, knowing that the question was a test of his sincerity.

“The water is sweeter at the source – to benefit the people it must flow down to the sea. Becoming muddied on the way it attains to saltiness.”

The sage sat quietly, looking down at the pattern on the tiles between his feet. The Keeper felt pleased with his reply: it spoke of the Yellow River flowing from the mountains to the sea, rushing through muddy canyons before nourishing the plain in the East. Listening carefully, one could hear its roar over the rising wind.

“Autumn has arrived, it will soon be spring,” the old man said, by way of acknowledgement.

The Keeper glanced at the table strewn with bamboo cuttings and a couple of completed rolls.

“How long will it be?” he said, worried by the directness of his question.

“About five thousand characters,” the sage replied.

No stranger to the art of writing, the Keeper calculated from the rolls on the table and the shelves that the work was about half complete. He rose and bowed to the old man.

“I must not disturb your thoughts any longer. I will tell your scribe he can return.”The old man rose and bowed his head as the official crossed the room and went out through the doorway.


As autumn turned to winter the old man and his scribe worked on to complete the great book. When the first plum blossom opened in the capital and the snows had melted, turning the great river into a raging flood, Lai Tan had drawn the last character of the five thousand and twenty five and bound up the final roll.

“Master, what will happen now that you have completed Kuan Yin’s demands?”

The old man looked at the brightening and fading sunlight in the courtyard.

“The moon will be waxing soon,” he replied.

“Shall I tell the Keeper that we have finished?”

The sage turned to look at the young man and nodded.

Lai Tan went to the Keeper of the Pass and knelt before the official.

“Please rise and sit here,” Kuan Yin said, pointing to a low stool set before his chair.

“Have you come to tell me that it’s finished?”

“Yes Master Yin, the great work is now complete. I have come to ask you whether we are free to leave. I was wondering what would happen to the book now. It would be unwise for us to risk it on a journey. Will you keep it safe here for wandering scholars to read?”

“Wandering scholars would soon bring great repute to your master but I must deliver it to the King in Luoyang. When your master has rested from his great task, my servants will make provisions so that you can continue on your way. Return to your master now and give him the good news.”

Lai Tan rose, bowed low and returned to the room where he had worked for almost two years.

“Master Lao, I fear the keeper wishes to steal your book. He intends to present it to the King. What if he claims to have written it himself?”

“Why would a wise man claim to have written so much foolishness?” the sage replied.

Having drawn every character of the book, Lai Tan had often asked himself the same question.

“The Keeper expects us to leave soon. He says he will provide for our journey. When do you intend to leave?”

“When the moon is half full.”

“But where will we go?”

The old man sat with his hands in his sleeves and thought for a while.

“I shall return to the place where I was born.”

“To Hu Hsien?” the disciple asked.

Lao Tzu nodded.

“There too,” he said before nodding off to sleep.


Three days later the Keeper presented Master Lao with gifts from the King that he had withheld until the book was finished. There was a fine yellow robe and a silk hat for the Master, a blue silk robe and a sash for his scribe and a goodly weight in silver. A mule was loaded with provisions for the journey and with the King’s gifts.

The old man was helped onto the back of the ox, the same one that had delivered him to the Keepers lodge, now grown fat with good feed and little work. Kuan Yin bade them farewell and watched the little party until it disappeared behind the rocks on the road to the western pass. At the bottom of the winding road, the Master pointed south and Lai Tan led the mule away from the river, into the hills where the village of Hu Sien lay hidden in the mists.

The next day Kuan Yin rode out with a retinue of guards and servants towards Luoyang, taking the precious rolls with him. The journey was long and dangerous with the constant threat of bandits on the road to the capital. The journey proved uneventful until they came within a few leagues of the city. The captain wheeled his horse round and came up to the Keeper.

“My Lord, we must find shelter, a storm is brewing in the west.”

“Very well, we will improvise a camp behind those trees by the river,” Kuan Yin shouted back over the rising wind.

They left the road and pitched the official’s tent, complete with banner, tethering the horses and mules beneath the trees.

The storm broke just after dark, tearing rotten branches from the trees and flattening the tent.

“Typhoon!” the captain screamed, over the roaring wind, “lie down and cling to the ropes.”

Concerned about the book, the keeper fought his way to the tethered mule and tried to carry the bundle of rolls into the shelter of the trees. The wind was so strong that the big bundle, tied up with hides, was torn from his grasp and rolled down the bank into the foaming waters of the river. Heedless of the danger, Kuan Yin plunged down the muddy bank and tried to drag the bundle back to shore. The raging torrent carried him away with the bundle, towards the distant capital.


One-foot dragged himself back through the mud towards the river. A storm always meant good pickings. He had already come across the body of a rich man, clothed in red silk all smeared with mud. He had stripped away the finery and pocketed the ornaments and rings. He had pitched the naked body back into the water for the fishes to pick clean before hiding the booty in his hut. There had been a big bundle too, probably full of valuables, but it had been too heavy for him to shift alone. He cursed the gods. There was nothing to be done, he would have to get help from his partner in crime before dawn or someone else would take the prize.

He stumbled down the riverbank towards the lights of the tavern. Poking his head through the door he spied One-tooth slumped over a table strewn with food scraps and empty wine bowls. He lay between a fishwife and some other clod; drool hanging from his gaping mouth, snoring in his stupor.

One-foot shook him roughly.

“Get up, quick,” he shouted into a battered ear.

One-tooth woke with a snort and recognised his partner.

“What now” another of your wonder finds I suppose; pots of gold, strings of pearls, precious silks fit for a king.”

One-foot decided to say nothing of the body.

“Get up, it’s a big bundle stuck in the mud bound up with hides; it could contain all kinds of valuables. I can’t move it by myself, it’s too heavy.”

“Curse you! How far is it?”

“It’s close by. I’ll find someone else if you won’t come.”

One-tooth stumbled to his feet, pushed the woman aside and spat on the dirt floor.

“Lead on then.”

With much cursing and muffled shouts the pair dragged the bundle to the hut and cut the hides open.

“It’s nothing but a bundle of firewood tied up with red string, curse it,” One-tooth shouted angrily.

”No, there must be something inside; unroll it.”

Holding the lantern over the rolls of bamboo, the scavengers beheld the rows of characters painfully drawn by Lai Tan.

“It’s just some stupid official’s tax rolls. It’s worth nothing,” spat One-tooth. I’m taking the hides, they’re good quality at least, you can keep the firewood.”

With a final curse, he rolled up the hides and went on his way.


The Chief Minister of Chou finished making his report to the King about the storm damage to the city.

“There was one other matter, my Lord.”

“Speak then.”

“A beggar was captured in possession of a mysterious book.”

“A scholarly beggar?”

“No my Lord, a river scavenger and a criminal called One-foot.”

“He still lives?”

“I don’t know my lord. The local magistrate ordered the lopping of his other foot.

“Oh dear, soon there will be nobody left standing.”

“A jade disc and a muddied red robe were found in the scavenger’s hut.”

“Ah, I see. We are in need of a new Keeper it seems. But what of the book?”

“It consists of a great many rolls, Lord. I’ve read some of it. It appears to be some kind of philosophy, so I wondered if it would interest your majesty. Its rather obscure stuff, I’m afraid.”

“How fortunate that Kung Fu Tzu is still within the palace walls. Be so good as to request his presence so that he can explain this mystery to us.”

“Yes my Lord, I will summon him at once.”

A low table and a stool were arranged before the King and a couple of the damp rolls placed on the table. A short while later the stately figure of the sage appeared. Bowing at the waist only, he sat at the table and opened one of the rolls.

“What does it say?” the king asked, a little impatiently.

“It’s full of aphorisms and rhymes about the old ways,” Kung Tzu replied without looking up from his reading.

“Read some out loud,” the King commanded.

“There is a thing confusedly formed,

Born before Heaven and Earth.

It stands alone and does not change,

Goes round and does not weary.

It is capable of being the Mother of the World.

I know not its name

So I style it ‘The Way’.”

“Mysticism,” the King said. “You’ve studied the Book of Changes for years, haven’t you?”

“That is true,” the sage replied, “but this book speaks of that which lies behind the changes.”

“Is it worth anything?” the King asked.

“The Dragon’s ascent into Heaven on the wind and the clouds is something which is beyond my knowledge,” Kung Tzu replied, quoting his own words after his meeting with the old man.

“Make copies for the library right away,” the King commanded, remembering the famous words quoted by the sage.


On the fourth night of their journey, the Sage and the scribe camped under an old chestnut tree overhanging a lake. The evening was calm and the bright moon reflected in the limpid waters.

“How far is it now Master? Lai Tan enquired.

“In another day the moon will be full,” the old man replied.

The next day was a fine spring morning. The old man had already bathed by the time Lai Tan woke up and was sitting under the tree watching the sunrise over the water.

“Master, why are you all dressed up in your new robe and hat? Surely this is not fitting wear for a humble village.”

“See how the blossoms have opened on this tree”, the sage replied.

Lai Tan washed quickly, ate a millet cake and put on the blue gown and sash.

It was noon when they rode down into the clearing where the houses lay nestled among the trees. The sound of the lyre and flute rose up to meet them, where young boys and girls in bright costumes were dancing on the green. As they entered the village, the dancers ran to meet them, followed by the headman and the rest of the villagers.

“My Lord, it is years since a magistrate passed through our village; you are most welcome. Let me help you down.”

The sage dismounted and stood unsteadily among the giggling village girls.

“Master, is it seemly to be among these women? We must go to the headman’s house at once.”

The girls liked the look of the young scribe and rushed over to where he stood with the mule. They touched his blue robe and tried to drag him away to the green. The music began again as the young people continued with the spring festival of the new moon.

“Master, what of your teaching about curbing desire?” Lai Tan shouted over the heads of the dancers.

“Time enough for that later,” the sage shouted back gaily as he entered the headman’s house.

“Will you take a cup of herb tea with me, brother?”

“Is the moon not full?” the sage replied. “Let’s talk over old times.”


The Combination Room

The Combination Room

Dramatis Personae:
Gordon Handley Professor of Mathematics } Trinity College
Sanjeev Ramangita Handley’s student and protégé }
Bernard Rushwell Professor of Philosophy }
Ludolph Wittgemein Rushwell’s student and protégé }
James Canard-Means Professor of Economics } Kings College


Soft April sunlight filtered through the narrow windowpanes of the Combination Room, gilding the tousled hair of a lone man not yet of middle age. His garb was unconventional; grey herringbone tweed trousers, a cricket sweater and a knitted scarf. He had draped a second sweater, of a drab colour, over a wooden framed mirror, opposite to where he sat in a high backed, wicker chair.
He was of slight build and rather short, his legs barely touching the floor. A cigarette protruded at an angle from his compressed lips, as he concentrated on aligning a walnut exactly between the jaws of a nutcracker. When the necessary precision had been achieved, he squeezed the nutcracker hard, using both hands. The nut skittered across the wooden floor, disappearing under the oak table.
“Blast,” he said, snatching the cigarette from his lips, parking it in the ashtray on the table. He contemplated taking a new nut from the bowl, but his sense of duty, not to mention tidiness, forced him to jump up and search for the nut beneath the table.
The errant nut had lodged in a crack between the uneven boards. He was just reaching for it when the heavy door to the room creaked open. Jerking upright, his head struck the underside of the table.
“Drat it all,” he shouted, backing out without the nut.
From a kneeling position, he turned his head to see the round, fleshy features of his student peeping round the door.
“Handley,” the face hissed, through stained teeth and fleshy lips, “I must speak with you at once.”
“What is it Sanjeev? You know the senior rooms are reserved for Fellows only.”
“I know Handley. You are topmost professor and I am still very low, but I have damaged the WC in the staircase, and I don’t know what to do.”
“What do you mean damaged?” Handley asked, rising to his feet.
“It’s the pump-action. I pull the chain, many, many times, but always there are many unhappy returns of the bowel motion.”
“You should report it to the porter. Well, come in, for God’s sake, you’re creating a draft.”
The door opened to reveal a rather uncouth figure in his late twenties, stout and not recently shaven.
“Now you’re here, you’d better sit down. I’ll cover for you if anyone comes.”
“Oh, thank you Handley. But I must explain, the motion was not mine, but some other dirty fellows’. I am still in need of relief, you see.”
“I don’t think I need to know the details, thank you, Sanjeev. As long as you do use the WC and not the garden bed, I will be satisfied. I couldn’t open my window for a week after your last escapade.”
“Most sorry, Handley, but it takes a lot of getting used to this English custom of sitting down to do business.”
“Yes, yes, I’m sure it does, but you must persevere if you are going to fit in with our quaint little ways.”
“I will persevere, Handley, I will be most deciduous.”
“Assiduous, Sanjeev, from the Latin assidere.”
Sanjeev Ramangita sat down on the floor with crossed legs, his large eyes rolled, looking round the room. His gaze lighted on the bowl of nuts, lips moving in the act of silent counting.
“Seventeen nuts, Handley and one on the floor. What do you think it means? Shall I calculate the Goldbach ratio?”
“It means that I accidentally dropped one of the nuts on the floor.”
“Butterfingers. Why do they say that Handley, do the English butter their fingers? ”
“No they don’t; it means dropping a ball in the game of cricket.”
“But what if your ball has already dropped, and what about buttery boards?”
“If you keep asking silly question you will become a butt yourself. Now, if you don’t mind, I would rather like to look at the cricket scores now,” Handley said. He sat down and picked up a crumpled copy of the times from the table.
“I could crack nuts for you.”
“No thank you, I prefer to crack my own nuts.”
Sanjeev fell silent. The ticking of the black clock on the mantelpiece, interspersed with the occasional rustle of Handley’s paper marked the passage of time.
The sound of footsteps and voices echoed in the passage outside the door.
“Shall I hide, Handley?” Sanjeev whispered.
“Well, you might try the wardrobe but I don’t think the smell of naphtha and vegetable curry is an ideal combination. Just stand by the window, and gaze intelligently into the distance.”
“Like Rabindranath Tagore?”
“Yes, something like that.”
While Sanjeev moved to the window, Handley quickly smoothed down his hair and lit another cigarette. He just had time to arrange The Times on the table, with the completed crossword prominently displayed, before striking a pose.
A slim man of medium height entered, talking in fluting tones over his shoulder.
He looked like an animated turtle, snapping out his words with exaggerated clarity. His companion, a decade and a half younger, was very tall and of athletic build. The tall man’s face was gloomy, with dark circles under the eyes. Unusually, he wore no tie. He listened intently as the older man spoke.
“It’s all up to you, now that my Magnum Opus has been published. You must take over the torch and build on what I have achieved. You can see more clearly than I what must come next in the great story of philosophy. It’s a great burden, I know, but I believe you are the only one who can carry the work forward.”
The tall man closed the door behind them and then stopped, transfixed in front of the mirror that Handley had covered with his pullover.
“Isn’t that a bit of a mixed metaphor?” Handley said, “Unless he’s going to burn down the old building first.”
“I thought I might find you here,” Bernard Rushwell said, advancing towards the table where Handley sat. “Perusing the cricket scores, I bet. I wanted to tell you that the prodigal son has returned from Norway, but only on a flying visit. He has some important results to communicate. I was sceptical at first - we had a terrible row - but he has almost won me over.”
Handley wondered why he ought to care about Wittgemein’s return. He knew the Austrian by sight, but had hardly spoken to this new Apostle. He was an Angel himself, but disapproved of some of the newer members of the society, particularly Canard-Means’ Bloomsbury friends.
When Ludolph Wittgemein came over to shake his hand, Handley thought of Mary Shelley’s monster. The Austrian’s grip was surprisingly limp and brief for such a muscular man, but Handley had no desire to hold hands with the chap. He would leave that sort of thing to Canard-Means and company.
With a pang of guilt, he turned to the window, where Sanjeev was casting his broad shadow into the room.
“I would like to introduce my pupil, Sanjeev Ramangita. Bernard, this is Mr Ramangita. Sanjeev, this is the renowned philosopher Professor Bernard Rushwell.”
“I am most honoured to meet you, Sir. I have only just arrived in England, and have yet to conquer the plumbing, but I hope soon to appreciate the greatness of your work.”
“It’s already out of date, I’m afraid,” Rushwell said, “so it may not be worth your while. Ludolph, you should meet Professor Handley’s protégé, Handley expects great things from him when he has learned the ropes. A future Apostle, eh Handley?”
Like the contact between Adam and God on the ceiling of the Sistine Chapel, their hands barely touched; the worlds of philosophy and mathematics repelling each other like oil and water, despite Rushwell’s struggle to make them mix.
“Well, we might as well sit down,” Rushwell said. “Sherry is in order, I think. Ludolphus. Would you do the honours, please. It’s really rather important, you see. Whitehorn and I have laboured in the vineyard all these years and we might have produced a barren crop.”
“Er, no sherry, for Sanjeev, Wittgemein, he’s a Hindu,” Handley said.
“Save it for Canard-Means,” Rushwell interjected, “I asked him to pop in later so we could get his views on Ludolph’s new ideas. Right, sit down Ludolph, we might as well begin.”
Wittgemein moved a wooden chair a little way back from the group, as if delivering a tutorial, and rummaged in a voluminous jacket pocket. He pulled at a battered spiral bound notebook, whose wire had become entangled in the lining. After a brief struggle and the tearing of cloth, he got it out and located the starting point of his notes.
“The world is all that is the case,” he began, in a hoarse voice. “It is the totality of facts, not of things. The world is determined by the facts, and by their being all the facts.”
“Right, this is very compressed,” Rushwell interrupted, “I’d like to say why I think this approach is so important. In the Principia, we tried to forge a link between the most primitive logical ideas to the objects and relations of mathematics. The underlying assumption was that logic was the proper place to start. But, unless we know what we mean by logic we can’t know that it is fundamental to our enquiries.”
“Well, where else could you start?” Handley said, “You could have left well alone. Most mathematicians of sufficient calibre can get on with their business without worrying too much about the philosophical underpinnings.”
“There is the question of rigour. While few could match your standards, Gordon, they still fall a long way short of axiomatic proofs as we have defined them,” Rushwell said.
“Yes, but there is quite a difference between doing real mathematics and merely laying down the law about how it should be done. You may be forgetting that mathematical beauty often determines the direction of an enquiry into fundamental problems rather than a philosophical roadmap. Where we aspire to go there are no maps.”
“Yes, Handley, you are right,” cried Sanjeev, jumping up from his chair, “if you cannot follow the beautiful things in your head, you will never reach the topmost heights.”
“Thank you, Sanjeev. But of course, you do need a very considerable technique to climb the highest mountains, and I suppose that is where you logicians can give us a leg up. Anyway, if you don’t know what you mean by logic by now, you may be in the wrong game, Bernard.”
“But do you know what you mean by mathematics?” Rushwell retorted, adopting his frozen, defensive smile.
“Probably not, but I expect you’re going to tell me,” Handley replied, brushing fallen ash from his trousers.
Rushwell paused for a moment, drew breath and said, “mathematics is the science in which we do not know what we are talking about and do not care whether what we say is true.”
Handley lit a cigarette and paused for reflection. “By the first part, I understand you to mean that we do not know what the objects of mathematics are exactly, since they clearly are not among the things of this world. I’m not too keen on the use of the word science though in this context. For me, a science is not just a systematic enquiry but also one that has empirical connotations. This sort of science has nothing to do with pure mathematics, which is quite unrelated to worldly things. The second part of your reply is more complex. Mathematicians do care a great deal about whether their theorems are true or not, but I suppose you mean true in some absolute, ontological sense. I’m not much of a philosopher, so I can’t instruct you about whether mathematical truth is fatally confined to its own domain or has some mysterious relation to what happens in the real world.”
“Perhaps we should ask Ludolph what he thinks, Rushwell said, looking expectantly at his protégé.”
The Austrian had gone pale and was leaning forward slightly, as if in pain.
“It’s the words, the language, you see, it’s just no good.”
“I’m not sure I understand you, Ludolph, could you explain more clearly?”
“I’m sorry, Bernard, I’m rather tired from the journey. So much is about to happen in the world, all this seems so remote now, even though I know it is the most important thing for people like us. I’m an Austrian remember. If the Balkan war continues much longer, Austria will have to intervene. If She does, She could be at war with Russia, and that will be the end of the world, as we know it. Of course, I would have to return home and fight for my country.”
“Nonsense, it doesn’t mean that at all. I hope Asquith would have the good sense to keep Britain out of it. There is no reason why you should leave England, just to satisfy some chauvinistic instinct.”
“I don’t think you would say that if our country were threatened and you were abroad somewhere,” Handley said. “We all hate war, but we can’t turn our backs on our homeland.”
Rushwell made an impatient gesture with his hand. “If you are able, Mr Wittgemein, we would be interested to hear your latest views on the matter in question.”
Ludolph put his hand to his temple and massaged it a bit before replying. “Very well, what I really would like to say is that you’ve got it all wrong. I know how important you think it is to pin down exactly what logic is, Bernard, but I have come to believe this is a hopeless task. Like Sanjeev said, you see some wonderful truth in your head, but you can’t express it clearly without a great deal of analysis, maybe years of work.“
“Exactly,” Rushwell interjected.
“No, not exactly,” Ludolph said, his eyes lighting up for the first time. “That’s what I’m trying to tell you. Not only is this kind of discussion a waste of time, at least for the purpose of arriving at the truth about the world, it entirely misses the point.”
“And exactly what is the point, Ludolph,” Rushwell asked icily.
“If you keep that frozen smile up much longer, I think I shall go quite mad,” Ludolph said, getting up from his chair to pace up and down parallel to the wall. What is that bally pullover doing over the mirror, anyway?”
“Need to confirm your existence, do you?” Rushwell snapped, his mouth finally hardening into a thin line.
“For Christ’s sake, Bernard, not now. No wonder Othalia has chucked you over. Don’t you realise how cold and cruel you can be sometimes?”
“Perhaps we should continue this another time, when we have all calmed down a bit,” Handley said, stubbing out his cigarette without looking at the antagonists.
”The pullover belongs to Gordon,” Rushwell said, jumping up. “If he accidentally catches sight of himself he will realise the futility of his existence and have to do away with himself. That’s it, isn’t it Gordon. It’s just one of those Trinity things we all have to get used to. We’re all mad in one way or another but we have to learn to get along. Being a cry-baby does nobody any good.”
“But who is this ‘nobody’?” Ludolph asked, turning to smile at Bernard.
“Very funny,” the older philosopher replied. “What now?”
“I do feel rather unwell. I haven’t eaten since dinner last night at High Table.”
“It could be your last, if you don’t pull yourself together. You know damn well how much faith I’ve invested in you. You can’t suddenly walk out now and throw everything away. You could be a Fellow in a few years; we would all support you, wouldn’t we Gordon?”
Ludolph stood up, clutching his belly. “I have a frightful cramp in my stomach.”
“Pie,” observed Handley. “
“Yes,” cried Sanjeev eagerly, “ pi is most important. I have devised many new ways to calculate this wonderful number.”
“No, mutton pie; for dinner; last night at High Table, I’ve been feeling a bit off colour myself,” Handley said.
“That’s why the WC is broken,” Sanjeev said, excitedly. All those dirty fellows have been ridding themselves of impure food and wearing out the pump.”
“I had the mutton pie too, it had no effect on me,” Bernard said, “but then I was weaned on Pembroke pies.”
“A little lamb enclosed within a wheaten shell,” Handley mused. “Sanjeev, would you be so kind as to escort Mr Wittgemein to the staircase, so that he can relieve himself. Meanwhile we will await the appearance of Apostle number 243.”
“243, Handley that is a nice number. It is three to the power of five.”
“I was aware of that,” Handley said, “but it is also the membership number of Professor Canard-Means.”
“But what about the broken WC, Handley?”
“I’m sure a man of Mr Wittgemein’s intellect will find a way round any local difficulties,” Handley replied.
When they had left, Rushwell said, “I hope you will forget what I said in the heat of the moment. I too have been under considerable strain lately.”
“I think we have known each other long enough not to attach too much importance to such little spats,” Handley replied. “You ought to take up something a bit more relaxing than logic.”
“Fortunately for you, you never married. Domestic bliss can end up being an unforeseen torment.”
“And your diversions?”
“Even worse,” Rushwell replied, “the very Devil. Speaking of whom, I think he has arrived.”
A soft-featured man in his thirties with a large moustache entered the room and strolled over to the seated pair.
“Your sherry’s gone cold, James,” Rushwell observed, pointing to the full glass on the table.
“Many thanks,” canard-Means replied. “I just saw your acolyte, accompanied by his Indian bearer, going into the male lavatory. I hope Lindon has not been leading him astray.”
“You’re a little out of touch, Ludolph found the Apostles were not to his taste after all. He’s resigned.”
“Pity, he became so much more animated among his peers.”
“He has important work to do. I think it best if he isn’t distracted by too much empty prattle,” Rushwell replied
“And what is this important work, pray? Some pet scheme of yours in disguise, perhaps.”
“On the contrary, he is working on finding the fundamental object that underlies all propositional forms. Without it, the primacy of logic remains in doubt ”
“And what do you think of these endeavours? I mean, doesn’t this cast doubt upon your theory of types?”
“Exactly. I had to invent that theory to obviate the pernicious antinomies of sets. These infect the basic propositional form, as you know, so a new, primitive notion of the proposition is essential if the whole enterprise is not to collapse like a house of cards.”
“Hark, I think I hear genius approaching now,” Canard-Means said.
Sanjeev entered the room, beaming, followed by Ludolph.
“I trust your expedition has met with more success than Captain Scott’s,” Handley said.
“Oh, yes Handley, much more. Ludolph is truly a great engineer. He pulled the chain many times and listened to the harmonics of the machine. Without even looking, he knew that there was a blockage in the cistern, by the way it sang to him.”
“And what was this blockage?” Rushwell asked.
“I am very ashamed to say it was Carr's Synopsis, Handley. I know you told me to get rid of it, but I still love it very much.”
“What was it doing in the cistern?” Handley asked.
“I need something to read in the WC when your British food causes a blockage. I wrapped it in an oilskin to keep it dry. See, I have it here.”
Sanjeev held up the dripping package, which began to form a pool of water on the floor.
“I think this meeting is adjourned,” Rushwell said, taking Ludolph’s arm and leading him back to the door. “I’ll see you in my rooms, James, should you wish to learn more about the future of philosophy.”

Conversations With Myself

A sparsely furnished room; white walls, black furniture, a little morning sunlight falling on the uneven boards of the floor from a window. A rigid figure sits in an upright, leather backed chair, clutching the wooden armrests. Parchment skin drawn tightly over broad cheekbones, haunted eyes staring into nothingness beneath craggy brows, burning within twin caves of forgetfulness. A veined and mighty forehead looms above, crowned with a shock of dark hair, still unsalted by half a century of unremitting toil. No mouth is visible beneath the shaggy growth, blooming from nostrils to jutting chin. The whole fearful aspect weakened, somehow, by small ears, as if to deny the soaring music within.

An old woman dressed in black enters, followed by another, a middle-aged simulacrum of the first.

“Elisabeth is here to see you, Friedrich; you know who she is, don’t you?”

The older woman reaches out with obvious concern, touching the veined hand clasped tightly round the arm of the chair. In a while, the leonine head turns in her direction, as if hearing rather than seeing the stooping woman.

”Is your name Franziska, by any chance?”

The voice is gentle, cultured, kind.

The hausfrau puts her hand to her mouth, stifling a spontaneous reply.

“Of course it is mother, who else could it be?” the younger woman says, with ill concealed annoyance.

“Ah! Die Mutter, Die Mutter, ha-ha, yes..”

“Friedrich, please try and pull yourself together.”

“And you, who are you?” the seated man replies, with the authority of a king.

“You know very well who I am. Elisabeth, your sister.”

“What do you want with me now, is it time for my walk?”

“We have received a letter from Oehler, your work is to be published in England; isn’t that wonderful?”

“Ha, pearls for the other swine!”

“I felt I should let you know. Mama has agreed, but I thought you should be told.”

“More light, more light, more..”

The old woman moves to the window and pulls on the already fully open curtains.

“Dead, summarily dead. More light, soon it will be dawn, more light..” the man continues to mutter, almost inaudibly now.

“It’s no good Mother, I shall have to proceed alone, I’m the only one who can act for him now.”


The women have left the room. Alone now, the invalid turns his head from side to side, as if scenting the sunlight from the window. Faint sounds drift up from the street, the shaft of sunlight moves slowly to illuminate the wall. He stares at the light reflected from the dark wood of a polished table, standing along the wall. Several leather-bound but unread volumes are arranged on the table, in a row. Above them hangs a wooden crucifix, its doleful figure looking down at the books.

The eyes of the author, too, come to rest for a while on his works, then look up at the crucifix. An inner voice reverberates from within.

‘Am I understood? Were you understood? Your time is past, your will is done; mine is yet to come. Let my song go forth to greet the new day and the new man who will be born, fit for a new beginning, for a new dawn.’

He glowers at the figure on the wall.

“What say you, son of man?” he says out loud.

The crucifix looms large in his vision; he strains to hear the voice rising from the abyss.

‘Yima, son of Vivanghat, why do you persecute me?’

The sage leans forward in his chair, smiling maliciously.

“Without persecution, where would you be? Just one of countless men crucified by your betters; just another body rotting on a tree. Was your pain worth the pain of all those who suffered in your name? Was it greater than the pain of even one of those millions, whose lives were twisted and tormented by your priests?”

He strains to hear the faint reply.

‘And were you not born out of yourself, as I was born, son to the father and father to the son?’

“Heh, heh. Why ask what you already know? From the Earth I was born and of the Earth I shall remain. No imaginary spirit I, but a real spirit, a spirit of the future born out of a dying past.”

‘None shall enter the Kingdom of Heaven, except through me.’

“Ha, then empty it shall remain, for my kingdom is here on Earth, a kingdom freed from the sickness of ideals and maundering ghosts.”

‘And what of suffering humanity?’

“Bah! How can there be joy without pain? Without pain man would be no better than some mindless sheep, grazing thoughtless in your father’s pastures. Great suffering, great agony, such as I have known, breeds great men, men fit to serve the one yet to be born – the Superman.”

‘Will this Superman choose the path of righteousness or the path of sin?’

“He will be beyond good and evil, he will follow his own creative path, unbound by the chains of history.”

‘In my father’s house, there are many mansions, fit for such as these, but none may set himself above the Almighty.’

“Have you not heard, then? Your God is dead.”

‘I am the way, the truth and the light; none shall be redeemed except through me.’

The light from the window fades, and the room darkens. The man closes his eyes and waits.


The servant, Alwine, enters the room, places a candle on the table and closes the curtains. She lights a taper from the candle flame, switches on the gaslight over the bed and ignites it. She adjusts the pressure until the mantel glows white-hot. The light hisses and illuminates the cavernous features of the seated man, casting a stark shadow on the wall, and on the crucifix.

Franziska enters the room, extinguishes the soft candle flame and turns down the covers on a narrow, wooden bed. As in a ritual, the two women haul the tall man to his feet. He stands patiently while they help him to undress. They put a white nightgown over his head and seat him at the table. Alwine leaves the room and returns with a meal of bread and soup. The man grunts with satisfaction and eats with gusto, washing down the food with a glass of water.

When the meal is over, the man is led to the bed. While Alwine clears the table, Franziska waits until he has finished with the bedpan before pulling up the covers. She kisses the domed forehead and turns off the gaslight, leaving only the flickering candle to illuminate the room.

“Alwine,” she calls.

In a while, the servant returns with a candle and removes the bedpan. Franziska follows her out of the room and closes the door.

The man stares at the ceiling for a while, and then at the candle, which casts distorted shadows of the crucifix on the wall. His eyes follow the flickering shapes until they finally close. His breathing assumes the regular pattern of sleep.


An old Hermit emerges from a cave and stands staring into the East at the brightening sky. A few grey clouds still hang over the horizon. A quickly moving speck catches his attention. He watches until it assumes the shape of a large bird, flying towards the mountain where he stands. He half raises a hand in greeting as the bird flies overhead. He sees that it is clutching a snake in its talons.

“My animals,” he calls out, in a raspy voice, long unused to speech, “you have returned.”

The eyes of the man in the bed move rapidly beneath the eyelids.

“A sign! I too must return to my people and give them the glad tidings.”

The sleeping eyes see a youthful figure walking up the slope towards the cave, his features hidden by the long shadow of the morning sun.

“You have come at last,” the hermit cries, running eagerly towards the approaching youth.

“Are you not afraid, old man, to finally meet your God?”

“I fear nothing, certainly not a figment of my imagination!”

“Have you not heard about the fate of Pentheus and of all those who oppose my will?”

“Ah! I was mistaken; you are not the one. Just an old Arcadian spirit whom I loved when I was young.”

“Beware, old man, lest you dance to my tune. You shall know my madness before Morpheus claims you.”

“I fear nothing, I have seen all that was and that will be. No one before me has seen so much.”

“I do not speak of your crazy dreams or of the fear of death but of that unknown terror, that eternal anguish of the self torn to pieces and scattered to the winds. It is I who was rent asunder, until my father retrieved my still beating heart and sewed it in his thigh.”

“Another bleeding heart; why not join your successor on his cross?”

“I too am merciful and accord the foolish a second chance, but you have dared too much.”

The old man cries out as the venom of the God enters his mind. A pit opens up beneath his feet, seething with corruption. Clawing hands and ravening mouths suck and tear at his body. In the distance, he hears the laughter of the young God as he is drawn down into the bowels of the earth. A tangle of vines and ivy boil up into the air, marking the place where the hermit stood.

The man jerks upright in his bed. A strangled cry escapes his gaping jaw. His hands feel his face convulsively in the flickering candlelight. He stares, wide eyed, at the dark shadows cast by the crucifix.

“Ask and you shall receive,” a gentle voice says from the shadow on the wall.

“Never, never! The man replies.

Elisabeth enters the room, followed by Franziska.

“Friedrich, whatever is the matter?” the old woman asks.

Elisabeth opens a drawer in the table and takes out a brown bottle and a metal spoon. She goes to the bed and forcefully administers the opiate. The man resists feebly, seemingly unaware of the identity of his carers.

“Light, more light!” he cries out, droplets of elixir running down his chin from beneath the luxurious moustache. He soon slumps back on the pillows, his head moving from side to side. The women wait until his fitful movements subside into a drugged sleep.


His body is suffused with a blissful warmth. A soft but ample breast is pressed against his cheek. He turns his head and suckles blissfully on the proffered teat. He feels a powerful erection. Soft hands caress his tightening scrotum and he feels his penis enveloped in the firm softness of a woman’s clasp. Sweet lips kiss his face, crooning his name softly, a heavy body moves rhythmically over his.

“Lou?” he mutters, revelling in waves of perfumed delight.

Voices whisper softly about him: many hands reach for him and many arms and legs entwine his body. His virility increases but without climax, like a Wagnerian aria. The women begin to cry out in ecstasy, and then in anger, fighting with each other, clawing at his flesh in a frenzy. They scratch his skin, pinch and bite, until the blood runs from the welts. Spurred on by the taste of blood, their teeth chew and rip strips of flesh from his neck, sharp nails open up holes in the softness of his side and pull at the organs within.

Like vultures, they probe into every orifice and finally pick out his eyes.


Alwine enters the room and draws the curtains. She looks at the dishevelled figure on the bed. The bedclothes lie on the floor and the nightdress is pulled up to reveal the still strong body, scarred and scratched from self-inflicted wounds. Elisabeth comes in.

“Alwine, quickly; fetch hot water and towels.”

“We should clip his nails too, Madam.

“Yes, but don’t tell Frau Nietzsche about this, it upsets her so.”

When the servant has left, the woman kneels at the bedside and begins to pray.


Tony Thomas

28 August 2003