EXPERIMENTS IN MASS EDUCATION 2000-2010

METAMATHEMATICS DEMYSTIFIED ver 2.0

A FAIRY TALE CONCERNING THE P=NP PROBLEM

(A Computational problem predating the Ages of Man)

(Computation in all Environments predating all Ages, all Revolutions, all Civilizations)

(Competing with Michelangelo)

[Note:- Being sporadically updated directly online on the server using pidgin-HTML and weak, unrealiable, sporadic internet connectivity.]

SCROLL DOWN A WEE BIT FOR THE STORY

SIX BLIND MEN AND THE ELEPHANT

SIX BLIND MEN AND THE ELEPHANT

SIX BLIND MEN AND THE ELEPHANT

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

O NE BLIND M AN LEADIN G ANOTHER

SIX BLIND MEN AND THE ELEPHANT

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

SCROLL DOWN FURTHER A WEE BIT FOR THE STORY

O F THE DUMM IES

FOR TH E DUMMIES

BY THE DUM MIES

METAMATHEMATICS DEMYSTIFIED

A FAIRY TALE CONCERNING THE P=NP PROBLEM

(Computation in all Environments predating all Ages, all Revolutions, all Civilizations)

"IT WAS THE BEST OF TIMES"

THE STATEMENT OF THE PROBLEM

THE EMPEROR INITIATES STEPS FOR A SOLUTION

THE PRIME MINISTER IS FRIGHTENED

THE STRATEGY FOR A SOLUTION

THE OTHER USES OF THE SOLUTION

STILL OTHER USES OF THE SOLUTION

THE COURTIERS SING A SONG

THE SEARCH FOR THE SOLVER

THE SOLVER ENTERS THE STAGE

THE STRATEGY OF THE SOLVER

THE METHODS OF THE SOLVER

THE NEW BED OF PROCRUSTES THE SOLVER

THE SOLVER IS EMPLOYED

THE EASY PART OF THE SOLUTION

THE STRATEGY FOR GEOMETRY

THE HARD PART OF THE SOLUTION

PROCRUSTES TACKLES THE VOID

POINTS DO NOT EXIST IN THE CONTINUUM

THE CREATION OF A POINT

ENTER ARISTOTLE

THE EXPULSION OF THE CRETEAN

PROCRUSTES AND THE SERPENT

PROCRUSTES CREATES POINTS IN THE CONTINUUM

PROCRUSTES AND HIS PIGGY BANK

PROCRUSTES INVENTS THE NATURAL NUMBERS - I

PROCRUSTES DREAMS OF THE OVERLOADED CAMEL

PROCRUSTES INVENTS THE NATURAL NUMBERS - II

PROCRUSTES CREATES THE INTEGERS AND OPERATIONS ON THEM

THE SUPERNATURAL PROPERTIES OF THE INTEGERS

OPERATION ON INTEGERS

PROCRUSTES STARTS COUNTING

PROCRUSTES AND THE TORTOISE

THE ARGUMENTS OF THE TORTOISE - I

THE ARGUMENTS OF THE TORTOISE - II

THE ARGUMENTS OF THE TORTOISE - III

THE ARGUMENTS OF THE TORTOISE - IV

THE ARGUMENTS OF THE TORTOISE - V

THE TORTOISE IS PUNISHED

PROCRUSTES BEGS THE QUESTION

PROCRUSTES VISITS THE BOTTOMLESS PIT

PROCRUSTES TAMES THE INFINITIES

PROCRUSTES REACHES INFINITY

PROCRUSTES IN THE CIRCUS

THE PROPERTIES OF SETS

THE ROYAL BARBER OBJECTS

PROCRUSTES TAMES MATHEMATICS

ANALYSIS IN THE BED OF PROCRUSTES

MATHEMATICS IN THE BED OF PROCRUSTES

THE SUPPRESSION OF DISSENT

THE CANNIBALS ARE EXCITED

THE CANNIBALS ARE CONFUSED

PROCRUSTES LAUNCHES THE WORLD WARS

PROCRUSTES DEFINES COMPUTATION

THE CREATION OF A MONSTER

THE MONSTER AS DR JEKYLL

THE MONSTER ENGAGES ITSELF

THE METHODS OF THE MONSTER

THE MONSTER AS MR HYDE

THE MONSTER AS A SUPERNATURAL

THE MULTI-HEADED MONSTER

ARCHIMEDES COUNTS THE HEADS OF THE MONSTER

THE BOTTOMLESS PIT OF THE MONSTER

THE FOOD OF THE MONSTER

THE MONSTER IS ETERNALLY ACTIVE

THE MONSTER IS A LINGUIST

THE MONSTER AND SYMBOLISM

THE MONSTER IS A SCHOLAR

THE MONSTER CHALLENGES ALL SEARCH ENGINES

THE MONSTER AND GOOGLE

COOKING FOOD FOR THE MONSTER

EDIBLE FOOD FOR THE MONSTER

THE MONSTER AS A CANNIBAL

THE MONSTER SOLVES THE PROBLEMS OF INDUSTRY

THE MONSTER AS THE IDEAL CUSTOMER

THE FINANCES OF INDUSTRY BECOME SOUND

THE COLLECTION AGENTS ARE OUT OF BUSINESS

PROCRUSTES DEFINES EFFECTIVE COMPUTATION

MODELS FOR EFFECTIVE COMPUTATION

THE MONSTERS LOVE THEMSELVES

THE MONSTER OF MONSTERS

THE MONSTER OF MONSTERS SETS AN EXAMPLE -- I

THE MONSTER OF MONSTERS SETS AN EXAMPLE -- II

THE MONSTER OF MONSTERS SETS AN EXAMPLE -- III

THE MONSTER OF MONSTERS SETS AN EXAMPLE -- IV

THE MONSTERS GET INTO A LOOP

THE MONSTERS ARE STUMPED -- I

THE MONSTERS ARE STUMPED -- II

THE MONSTERS ARE STUMPED -- III

THE MONSTER OF ALL MONSTERS LOVES ITSELF

THE MONSTERS CONTINUE TO LOVE THEMSELVES

THE MONSTERS KEEP ON LOVING THEMSELVES

PROCRUSTES CONSULTS ORACLE AFTER ORACLE

THE FINAL SOLUTION

PROCRUSTES WANTS A MEDAL

"IT WAS THE WORST OF TIMES"

PROCRUSTES IS GIVEN A PRIZE

PROCRUSTES MEETS MICHELANGELO

PROCRUSTES IMITATES MICHELANGELO

ENTER THE MANDARIN

PROCRUSTES CONSULTS THE MANDARIN-I

PROCRUSTES CONSULTS THE MANDARIN-II

PROCRUSTES FINDS THE CLOCK TICKING AWAY

PROCRUSTES LEARNS ABOUT THE DRUIDS

PROCRUSTES SEEKS OUT THE DRUIDS

PROCRUSTES AS A DOUBTING THOMAS-I

PROCRUSTES AS A DOUBTING THOMAS-II

PROCRUSTES AS A DOUBTING THOMAS - III

PROCRUSTES AS A DOUBTING THOMAS - IV

PROCRUSTES CONSULTS THE WISE MAN FROM THE WEST - I to VI

PROCRUSTES CONSULTS THE MATHEMATICIANS

PROCRUSTES PASSES THE BUCK - I

PROCRUSTES PASSES THE BUCK - II

WINNIE THE POOH, TIGGER AND THE CHILDREN DECAMP

THE CLOCK TICKS ON RELENTLESSLY

PROCRUSTES IS HECKLED

PROCRUSTES CONSULTS THE WISE MAN FROM THE WEST - VII to IX

PROCRUSTES TRIES TO ENSLAVE MICHELANGELO - I

PROCRUSTES TRIES TO ENSLAVE MICHELANGELO - II

PROCRUSTES TRIES TO ENSLAVE MICHELANGELO - III

THE CLOCK FURTHER TICKS ON RELENTLESSLY

PROCRUSTES IS WORRIED

PROCRUSTES PASSES THE BUCK - III

PROCRUSTES CONSULTS THE WISE MAN FROM THE WEST - X to XIV

THE TOM AND JERRY SHOW NEVER ENDS

PROCRUSTES CONSULTS THE WISE MAN FROM THE WEST - XIV to XV

THE CLOCK FURTHER TICKS ON MOST RELENTLESSLY

PROCRUSTES IS MOST WORRIED

PROCRUSTES PASSES THE BUCK - IV

PROCRUSTES IS FRIGHTENED

PROCRUSTES DECAMPS

PROCRUSTES RESURFACES

"THE WILDERNESS OF THE ETERNAL QUEST"

THE MONSTER IS CONFUSED

THE FAMOUS PROBLEM

ATTEMPTS TO SOLVE THE PROBLEM

DENYING THE PROBLEM

USING PROCRUSTEAN MATHEMATICS FOR A SOLUTION

THE DANGERS OF SIMULATION

THE SOLVER DOES NOT EXIST

THE DIFFICULTY OF PROVING THE EXISTENCE OF A SOLVER

THE SOLVER OF PROBLEMS MAY EXIST AFTER ALL

THE VERIFIER IS SUPREME

RESTATEMENT OF A FAMOUS PROBLEM

THE SOLUTION TO THE PROBLEM

THE USE OF THE SOLUTION

SOLACE AND CONSOLATION

THE EMPEROR'S ROYAL PROCESSION

EPILOGUE

"COMPUTATION PREDATES THE AGES OF MAN"

(Computation in all Environments predating all Ages, all Revolutions, all Civilizations)

(Computation in all Environments predating all Ages, all Revolutions, all Civilizations)

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

O F THE DUMM IES

FOR TH E DUMMIES

BY THE DUM MIES

"IT WAS THE BEST OF TIMES"

When we were in our centuries of sleep an interesting thing happened in the world. In a far off land about two centuries ago a delegation of children led by Winnie the Pooh and Tigger met the Emperor.

They complained about the horrid mathematics subject they had to study and the birch of the mathematics teacher. The Emperor recalled having had similar problems when he was young.

The Emperor was pleased to order the Prime Minister to find a Solver for a permanent solution with the rider that he (i.e. the Emperor), the children and their leaders would be Verifiers of the solution.

The Prime Minister was frightened. The Emperor may cut off his head if he failed. He asked the courtiers for help.

A suave courtier suggested the strategy for a solution.

Like the alchemists of old had tried to do in their area mathematics had to be distilled and its essence found. This could be encapsulated and converted to a sugar coated pill. The pill could be swallowed with milk and honey. The mathematicians and mathematics teachers can be shown the door. A mathematics teacher can be left in the wilderness to use his birch on himself.

PRACTICE WHAT YOU PREACH

Another suave courtier convinced everyone that the mathematics pill will explain all the mysteries of the Universe and the High Priest and his minions can be bundled off. The High Priest could be left in the desert and can listen to his own sermons.

PHYSICIAN HEAL THYSELF

Another suave courtier convinced everyone that the mathematics pill will explain all the mysteries of sickness. The horrid Royal Physician and his minions can be expelled. The Royal Physician could be left in the Artic wilderness and can swallow his own bitter medicines, to heal himself.

The courtiers went on and on. They were all in the pay of a foreign scoundrel.

A high power Search Committee was formed and messengers were sent full speed in all directions to find the Solver.

Half a century later a legendary scoundrel named Procrustes descended from the legends of ancient Greece and entered Mathematics with an aim to reform it, distill it, get its essence and create the sugar coated pill.

Procrustes had a reputation for having a magic bed which he said would fit anyone.

Procrustes seems to have taken a leaf out of a course in Programming Languages. If a person was bigger than the bed he employed 'narrowing' and lopped of the limbs etc. till the person could fit his bed. If a person was too short he employed 'widening' and stretched the person's limbs so that he could fit into his bed.

Procrustes now had a new bed which he called Arithmetic and claimed all mathematics would fit into it.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

Procrustes was employed by the Emperor on the recommendation of the courtiers and charged with the task of taming mathematics. He was to be paid a penny a day for his efforts. He was given a Piggy Bank to store his money.

Procrustes left Algebra and Geometry alone. Much to his disgust a century of effort by the mathematicians he had employed showed that Algebra & Geometry could be put into various knots & contortions and actually fit into his bed.

Geometry did create some trouble, till one used the ancient proverb: "Once a goon always a goon".

In any Geometry there are invariants which do not basically result in any change to a goon when applied to the goon. Once these are isolated it seems possible to reduce that Geometry to Arithmetic.

Analysis was a troublesome affair. It depended on the continuum which seemed to be a model for the Universe.

Procrustes started with the Void as the Universe. By 'narrowing' he declared the Universe is the Void.

Procrustes widened the Universe to a massive lump where there was no such thing as a point.

By 'narrowing' Procrustes reduced the Universe to a mathematical point. By 'widening' he said this point was eternal, immortal but 'dumb'. It could not do anything by itself.

Procrustes 'narrowed' the concept of differing points of view allowed to the human race to two valued logic, putting the blame on Aristotle. He claimed that for every statement he was correct and the other person wrong.

By 'narrowing' he did not allow suo motto change by the immortal truth values for an eternal amount of time.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

The ancient ambassador from Crete had serious objections on this two valued logic business. He said he was a bonafide ambassador and all ambassadors were liars, who lie for the good of their country. He was promptly expelled to ancient Crete to eternally contemplate on the validity of his objections. He was excommunicated by the High Priest. He was by Royal Proclamation declared as nonexistent. He was declared to be a figment of one's imagination. All records pertaining to him were expunged from all archives.

Procrustes had a frightening dream. He saw a fearful spotted Snake in the continuum. He woke up with a Divine Inspiration and by 'narrowing' claimed there was something called a real number line/curve in the continuum.

Procrustes considered the Universe of the single eternal, immortal, dumb point. He 'widened' it to an infinite collection of distinct eternal, immortal, dumb points. By 'narrowing' he did not allow suo motto birth, death or change in any point.

Procrustes had a dream that he was taking stock of his piggy bank where he kept his pay.

Starting with a penny, he kept on adding another penny as a successor, and found out that he had progressively increasing small but tidy sum.

Procrustes woke up with a Divine Inspiration that God created the natural numbers and all else is the work of Procrustes.

Procrustes had a dream that he was in the desert. He saw a hungry, starved, old camel with a heap of straw on its back. Its master kept on adding straw after straw to the load on the back of the camel. Nothing seemed to happen to the camel.

Procrustes woke up with a Divine Inspiration that God created an infinite 'bunch' the natural numbers and all else is the work of Procrustes.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

Procrustes applied his dream.

Starting with a single immortal, eternal, dumb integer representing the Universe by 'widening ' he created an unlimited number of immortal integers.

He 'widened' the integers and declared they were immortals and eternal. By 'narrowing' he did not allow birth, death, suo motto changes.

He 'widened' many operations on integers and declared by 'narrowing' that they and their results could be defined. He used 'widening' to declare that they were immortal and eternal entities.

He 'narrowed' the collection of points in the continuum and said it consisted of distinct points that could be described by 'bunches' of integers or combinations of them. By 'narrowing' he did not allow suo motto change by the immortal points for an eternal amount of time.

Procrustes dreamt that he saw a person called Zeno traveling on a tortoise. The tortoise claimed that the points in the continuum cannot be assigned numbers, or 'bunches' of numbers.

The tortoise could never travel to any point from any other point, by traveling a fraction of the remaining distance in a stage of its journey.

The tortoise claimed that there was no such thing as length in the continuum.

The tortoise claimed that in an suitably overcrowded metropolis everyone's house extends and includes his neighbor’s houses in all directions.

This was the situation in the continuum. Every point includes all the points adjacent to it. Any mapping to integers will show that.

The continuum cannot be described by natural numbers or integers.

This was the stand of the tortoise.

There is only one house in a suitably overcrowded metropolis, and thus just one point in the continuum.

There is no such thing as infinity claimed the tortoise.

Only a finite number of natural numbers and integers exist.

Even if infinity exists the finite cannot comprehend or comment on the infinite.

This is as per the fable of the mother bullfrog takng deeper and deeper breaths to become as big as the Ox. It can at most burst.

A visitor to a house is a welcome guest. Longer he stays the happier we are. He cannot stay for ever. Then he is evicted. A guest is different from the member of a house.

Lollipops are stimulants for a child. More lollipops the better. All lollipops means the child collapses. Similarly the natural numbers are stimulants for mathematics. More numbers the better. All natural numbers means mathematics collapses.

The tortoise threatened to descend into the Empire and complain to the Emperor that Procrustes was all bogus.

The tortoise was banished to hold up the entire Earth on its back eternally.

It was condemned to be the eternal Beast of Burden in all ScreenSavers dedicated to children the world over.

Procrustes further 'narrowed' the points in the continuum to a class of immortal 'computable' real numbers. By 'narrowing' he did not allow suo motto change by the 'computable' real numbers for an eternal amount of time.

Procrustes had a dream that he was cast into the Bottomless Pit. Hurtling down the pit he met friendly Titans who were also hurtling down the pit. They would hurtle down for an infinite amount of time, throughout eternity. He found out that they were basically a friendly, simple lot who had been taken for a ride by some unscrupulous youngsters.

Procrustes woke up and claimed Divine Inspiration insofar as infinity was concerned.

He 'narrowed' infinity and reduced it to some kindly, harmless, naive, simple old persons next door. They were simple persons, perhaps a little foreign, but just like us. By 'narrowing' he did not allow suo motto changes by any of the infinities for an eternal amount of time.

Procrustes dreamt his pennies were adding up. It was just a matter of time till his Piggy Bank reached, overtook Croseus and then reached some infinity.

Procrustes dreamt that the Emperor had thrown him into the Arena. Herds & herds of herds, of wild animals were chasing him. They were rushing in from all the waterholes in the deep interiors of Africa. There were herds of bison chasing him. There were herds of tigers, elephants, pandas, penquins, kangaroos,kites, vultures, eagles, etc. etc. all after him.

He woke up with a sudden start and claimed by Divine Inspiration that there were some things called 'sets' which were fundamental to all mathematics.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

Procrustes naively 'narrowed' the concepts of sets and operations on them till they behaved themselves, and 'widened' their attributes by granting them immortality.

By 'narrowing' Procrustes did not allow suo motto change by sets at any time throughout eternity.

He pompously talked of a Set of All Sets and compared himself to it. He said any goon can determine if he belongs to a gang.

He wrote a voluminous treatise showing that if a child is given one lollipop for the left hand and one lollipop for the right hand the child ends up with two lollipops.

He proclaimed he had laid the rock foundations of mathematics.

The powerful royal Barber of all Barbers had some serious objections. He protested by putting up a board that he would shave anyone who does not shave himself. He was promptly banished to the Antartic wilderness to eternally contemplate on the dignity of shaving himself. He was formally excommunicated by the High Priest. He was by Royal Proclamation not to be mentioned or included in any set of decent persons.

OR

ATTEMPTS TO TACKLE THE ROYAL BARBER

Procrustes was shaken up by the behaviour of the barber. He further 'narrowed' the concepts of sets and operations on them to avoid all troublesome barbers.

He 'widened' their attributes by retaining their immortality.

By 'narrowing' Procrustes did not allow suo motto change by sets at any time throughout eternity.

The Problem of the Royal Barber of Barbers still persisted.

OR

ENTER HAMMURABI OF BABYLON

Procrustes was shaken up by the behaviour of the sets. He consulted the courtiers out of desperation.

A suave coutier advised him that the solution to the problem was to consider a set as a collection of goons. They have to civilised. They needed a Code of Conduct.

Hammurabi of Babylon had a code. This was preserved on ancient clay tablets. A sample commandement of the code was: If a child's lollipops are 'pinched' by its neighbour, then the child should 'pinch' the neighbour's lollipops.

All other codes were mere refinements and variations of the above code. He suggested that Procrustes should 'narrow' the concepts of sets and operations on them by defining a Code of Commandments to avoid all troublesome barbers.

Considering the Visibility of the Project the courtier volunteered to be Chairman of the Law Givers of the Code. His 'cronies' would be the Directors of the Board of Law Givers.

OR

THE CREATION OF THE CODE OF CONDUCT FOR SETS

Procrustes thought 'long and hard' about the problem and the suggested solution.

He recalled the advice of Bluto of The Sailors his childhood Mentor in Mathematics, that mathematics is best left to mathematicians.

He kept a legion of mathematicians in chains in GoonLand supervised by mercenary Sumo Wrestlers. For half a century they were to toil, slave and evolve a minimal sets of commandments using sets for all the mathematics they thought was mathematics. They were Mentored and Guided by The Beagle Boys in their efforts.

OR

THE PROCLAMATIONS OF THE CODE OF CONDUCT FOR SETS

His Proclamation of Commandments for Sets or Goons were issued from time to time. For a sample......

THE DEMOCRATIC LAND OF SETS AND GOONS

A. Everything is a set. Every goon is a set, every gang of goons is a set and one can determine membership in a set or a goon gang.

THE COMMANDMENT OF THE NULL SET OR EMPTY GANG

B. Procrustes allowed the Void to exist as the empty set of goons or empty set.

THE COMMANDMENT OF THE INFINITE SET OR INFINITE GANG OF GOONS

C. Procrustes allowed an infinite set of goons which starting with the empty set adds an element or a goon at a time, just like his Piggy Bank starting from being initally empty had a penny added to it every day.

The gang of goons can grow eternally like the Piggy Bank can grow eternally reaching Croseus, leaving him in tears, and rushing along to infinity.

THE COMMANDMENT OF PAIRS OR THE GANG OF PAIR OF GOONS

D. Procrustes banned the 'gestalt' concept when combining sets, but allowed the combination to result in a set.

If a goon helps himself to the music player from one neighbour's house and a compact disc from another then all that he has is a set of two items. His enjoyment of music is not counted.

ALTERNATIVELY

If a new gang of goons is formed by taking a distinct element from two sets of goons, then all we have is the set of two goons in the new set. They do not do anything extraspecial because they are together. There is no synergestic effect.

THE COMMANDMENT OF EXTENSIONABILITY OR EQUALITY OF GANGS OF GOONS

E. Procrustes banned the class concept and enforced Socialism in sets.

If the Royal Prince and the Plebian Pauper Fisherman went fishing, and had the same catch, then Royal basket of fish and the Plebian basket were equal sets.

ALTERNATIVELY

If the miserly Grand Mughal Emperor Aurangazeb weaves baskets to sell at fancy prices to his courtiers and the lowly weaver of baskets weaves some baskets to eke out a miserable living, and both weave the same number and type of baskets, the sets of baskets they weave are equivalent.

ALTERNATIVELY

Two goon gangs which have the same members are always equal, regardless of their individual experience and toil in life. A gang of goons learns nothing from experience. It is basically dumb.

THE LEGALITY OF NEW GOON GANGS FROM OLD ONES

F. Procrustes allowed new sets to be created by goons from existing sets.

THE GOON GANGS CAN LEGALLY GROW

1. A goon moves into an empty house in some locality, and finds the well furnished houses of his two neighbours have different sets of things with nothing in common. The goon can create a new set by helping himself to items from both his neighbours.

ALTERNATIVELY

A new goon gang can be formed by drawing goons from different gangs, which have no goon in common.

THE COMMANDMENT OF THE POWER SET OR THE POWER SET OF GOON GANGS

2. An instance of the creation of the power set consists of all the different ways the goon can help himself to the belongings of one of his neighbours.

ALTERNATIVELY

All the ways goons from a goon gang can gang up together is the power set of the gang of goons. It is also a set.

THE COMMANDMENT OF SEPARATION OR THE GOON HELPS HIMSELF

3. A function is defined as follows. The goon treats the items in his neighbour's house as the domain of the function. The goon's house is the range of the function. The range is restricted to the items the goon transfers to his house from his neighbour's house. Paradoxes are to be avoided.

THE COMMANDMENT OF THE BLACK SHEEP OR THE BANNED GOON

G. Procrustes banned a set from being a member of itself. The Barber was under permanent exile.

THE COMMANDMENT OF THE FOUNDATION OR THE ORIGINAL GOON

H. Procrustes demanded a nonempty set of goons to have a goon as a member who is distinct from the set itself. Only the empty set is brankrupt and a nonempty set has some assest.

ALTERNATIVELY

A empty mother Kangaroo has no babies in her pouch. A nonempty mother Kangaroo has at least one baby in her pouch.

This was a case of Kanga without Roo on one hand, and Kanga with Roo peeping out on the other.

THE COMMANDMENT OF UNIQUE UNION OR THE UNION OF ALL GANGS

I. Procrustes demanded that given a set of goons a set of a goon gang can be formed that included all the goons. Like in Nursery & KG every goon is included in the party,everybody wins the race and everybody gets a prize.

THE COMMANDMENT OF REGULARITY OR THE LEAST IMPORTANT GOON

J. Procrustes demanded that from a set of goons one should be able to choose any subset of them as a subgang. A strict linear hierarchy can be imposed on this subgang and the least important goon isolated.

ALTERNATIVELY

Ginen a nonempty set of goons. A drill master asks some of the goons to 'fall in line' by ascending height. The shortest goon in the line can be isolated.

THE COMMANDMENT OF CHOICE OR THE 'MAMOOL' OF THE GOON

K. Procrustes demanded that a goon moving into a neighbourhood with nonempty neighbours can "wet his beak". He can help himself to exactly one item from each of his nonempty neighbours' house.

L. The above Axioms for Acceptable Goons were Verified formally by various demonstrations. Read Aztec and Inca civilisatons for 'neighbours', and the set of friendly visitors they had as 'goon' or 'gang of goons' for a real life Example of Verification.

Procrustes now claimed by narrowing that the sets following his commandments were all that mattered. They were his rock foundation for mathematics. Any set found violating his commandments could join the Royal Barber in the wilderness of Antartica and have the penquins for company.

Coming back to the continuum, Procrustes now said by 'narrowing' that what was left of the butchered universe, the 'bunch' of immortal points described by his contortions of integers and sets of integers was all that mattered.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

He had thus by 'narrowing' reduced the Universe and hence the continuum to computable real numbers, and hence to integers, and hence Analysis was reduced to arithmetic.

So now he had put the lid on mathematicians & mathematics and reduced everything to Arithmetic.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

There were some complaints on the way the continuum was butchered, the use of his Proclamation of Commandments for Sets seemed farfetched, and the cavalier treatment of infinity did not go down well in some quarters. However, dissent was put down with a strong hand. The damp, leaking dungeons and the dusty, rusted torture chambers were renovated. They were painted with all the unpleasant colors of the Rainbow. This for housing dissenters of the commandments relating to sets. Multiple instances of straw filled Pits and Pendulums of bamboo shoots were installed and commissioned.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

The cannibals at the edge of the Empire were excited when they saw the butchered continuum. They asked it to guess correctly as to whether it would be roasted alive or boiled alive.

The Cannibals had a strong lobby, but Procrustes managed to put them into an eternal loop. The army was called out to drive them into the deepest interiors of the Island Continent. They were left to eternally contemplate on the confusion created in their confused minds.

One thing leads to another, perhaps unintended consequences. A hypothetical view can be taken, that this Denial of the continuum and 'narrowing' of infinity, resulted in the human race declaring itself as 'gangs' of supermen out to control gangs of 'superdunces'. The human race merrily rolled downhill for half a century.

Procrustes 'narrowed' the purpose of the Universe to something indulging eternally in 'computation' describable by arithmetic and hence encompassing mathematics.

Procrustes had a frightening dream. He was being chased by the monsters Grendel and her family. He ran and ran and ran with the monsters behind him. He fled through the Pampas of South America into the deep Jungles of the mighty Amazon with the monsters right behind him. He sought out Beowulf the Hero who destroyed the monsters and saved him.

Procrustes woke up with a start and created an unspeakable horror called the Turing machine to frighten future generations of naughty children.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

A first acquaintance with the Dr Jekyll like machine gave it naive, user friendly toy like characteristics. It seemingly had a kindergarten type vocabulary, literacy and manipulative ability.

OR

THE CHURNING OF THE MILK OCEAN

The monster indulged in simple activities like simulating picking up the Pacific Ocean and transferring it to the Atlantic Ocean.

OR

THE POT OF NECTAR IS FOUND

The monster used a thimble to simulate the transfer of water: a thimbleful of water at a time.

It carried on the process for eons of time.

Procrustes exhibited its true Mr. Hyde character.

OR

THE MONSTERS SCRAMBLE FOR THE POT OF NECTAR

Procrustes 'widened' the capabilities of his machine by endowing it with immortality, infinite reliability, and the ability to compute eternally, much beyond Odin's feast or Shiva's dance of Death.

It could by metamorphosis convert itself into a multi-headed hydra or a ferocious multi-headed demon descending directly the from myths and legends, of many cultures the world over.

OR

THE SERPENTS WITH FORKED TOUNGES

The number of heads and their associated tapes could be more than the number of grains of sand Archimedes estimated to be in all the beaches of the world.

The monster had a digestive system that was a bottomless pit for a never ending, voracious consumption of food.

The monster had a very simple staple diet of ink and tape. Take away its ink and the monster would become a simple, well-behaved, regular tamed houshold toy or dumb pet.

THE PERPETUAL MOTION MACHINE

The monster did not require any energy for its operation, unlike the voracious demands for power of other computing devices like a Google server farm.

The monster was comfortable with all the symbols of all languages in the world, past, present and the foreseeable future.

The monster was comfortable with all the symbols, of all stereotypes, of all the cultures, past, present and the foreseeable future.

The monster had a finite control which could store the entire library of Alexandria, all the libraries of the world, all the information, knowledge and associations of the human race in the past, present and possible future. It could compete with Odin in its thirst for knowledge.

The monster could organize, store and retrieve this information from its finite control with a speed surpassing that of light, in fact instantaneously.

The monster should make any Google search engine, past, present or future end up with a strong inferiority complex.

Procrustes used 'widening' to allow any entity to be encoded as a symbol or sequence of symbols even if the entity had immortal, eternal or infinite characteristics.

By 'narrowing' Procrustes demanded the sequence represent a sequence computable by his monsters. He now called these monsters his machines.

In particular his machines could be encoded as a sequence of symbols.

This sequence could be fed to his machine.

Procrustes 'widened' the capabilities of all industries & businesses, of all civilizations and cultures the world over, from cottage industries to MNCs, from supermarkets to grocery stores, by granting them their dreams. By 'widening' he granted them infinite reliability.

Industry would find in the Turing machines an eternal captive market of satisfied customers. They can supply tape and ink eternally to these immortal, eternal customers who will never complain.

Procrustes 'widened' the capabilities of the environment of his machines to reliably settle the invoices raised by industry & business the world over.

Industry could do without the use of horrid collection agents who would like to auction the machines for a fat commission.

Procrustes 'narrowed' the concept of computation to 'effective computation'.

Procrustes took a leaf out of Ovid’s Metamorphoses and demonstrated, for about half a century that his multi-headed hydras of machines, can effortlessly change into many different models of effective computation.

Procrustes declared by 'narrowing' that the concept of 'effective computation' was identical to the capability of his machines.

Procrustes found his machines ended up contemplating their own descriptions, and some of them got into trouble as a result of the same.

Procrustes found that some of his machines could reproduce themselves.

Procrustes found that among his machines there was one that could simulate the behavior of any other machine. He called it a Universal Machine.

SETS AN EXAMPLE-I

Procrustes announced by 'narrowing' that if any one had pencil, paper and the ability to simulate the Universal machine one could do all 'mathematics'.

SETS AN EXAMPLE-II

Procrustes announced by 'narrowing' that if any one had pencil, paper and the ability to simulate the Universal machine one could do all 'computation'.

SETS AN EXAMPLE-III

Procrustes announced by 'narrowing' that if any one had pencil, paper and the ability to simulate the Universal machine one could do all 'computation'.

By 'narrowing' one can simulate whatever mattered in the Universe.

SETS AN EXAMPLE-IV

Procrustes announced by 'narrowing' that if any one had pencil, paper and the ability to simulate the Universal machine one could do all 'mathematics'.

The mathematicians and mathematics teachers can seemingly be shown the door.

OR

THE FIRST ENCOUNTER WITH INCOMPLETENESS

Procrustes found that when asked to solve some problems in mathematics his machines ended up in a loop, and an eternal beating around the bush sometimes. They did not know when to stop and announce failure.

OR

THE MONSTERS ARE SHY

Procrustes found that when asked to solve some problems in mathematics his machines declined to volunteer.

OR

THE SECOND ENCOUNTER WITH INCOMPLETENESS

Procrustes found that when asked to solve some problems in mathematics, the machines one by one declined to solve the same. The problem had no takers.

Procrustes wanted to know why him machines were stumped.

Why when asked to solve some problems in mathematics his machines declined to volunteer.

OR

THE FAMILY ASTROLOGER IS CONSULTED

Procrustes found his machines ended up contemplating their own descriptions, and some of them got into trouble as a result of the same. He used 'widening' and equipped the machines with an oracle.

OR

THE ASTROLOGER IS STUMPED

Procrustes found his machines with an an oracle ended up contemplating their own descriptions, and some of them got into trouble as a result of the same. He used 'widening' and equipped the machines with another oracle.

Procrustes found his new machines with two oracles still ended up contemplating their own descriptions, and some of them got into trouble as a result of the same. He used 'widening' and equipped the machines with yet another oracle.

OR

HIERARCHIES OF ASTROLOGERS ARE STUMPED

Procrustes found his machines with three oracles still ended up contemplating their own descriptions, and some of them got into trouble as a result of the same. He used 'widening' and equipped the machines..........

HIERARCHIES OF UNDECIDABILITY

Procrustes then used 'widening' and equipped the machines with another oracle. He then used 'widening' and equipped the machines with yet another oracle. He then used 'widening' and equipped machines .........

Procrustes announced that now he had his tool of mathematics to explain the Universe by simulation.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

Procrustes declared his contraption to be the greatest intellectual achievement of not only the century, but for all time throughout eternity.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

"IT WAS THE WORST OF TIMES"

As a prize Procrusteswas allowed to personally carve a "PERFECT" statue of himself.

He was commanded to exhibit the result along with his "PERFECT" statue in the Court in a short span of a few decades of time.

PROCRUSTES now wanted to follow the method of Michelangelo. He was running out of time. He was desperate.

Procrustes kept waiting and hoping.

P=NP is valid

P=NP is not valid

P=NP STUMPS THE MONSTERS

OR

ENTER THE FIRST VISITOR

Procrustes consulted the Visitor and complained about the confused behaviour of his Monsters.

He wanted to know if he was as clever as Michelangelo.

OR

THE MONSTERS AS DEVAS AND BHEESHMAS

OR

THE MONSTERS AS DEVAS AND BHEESHMAS

He had considered the capabilities of different classes of Monsters.

He considered the Superior Monsters who followed Procedures. These Monsters could choose when to to stop 'computing'.

There was no independent party who with Ctrl+Alt+Del could preempt the computation. There was no Bouncer to throw them out of the Bar. Whenever like Oliver Twist they asked for more of anything they were given more. The catch was that some of the Monsters could compute 'for ever'.

He had predicted in ancient times that the immortal Monsters that Procrustes had created, or their equivalents by Metamorphoses would be intellectually 'stunted'. They would be 'stumped' by some problems. The Monsters were intellectually Incomplete.

There were problems that no Monster would volunteer to solve. There were problems that would lead some monsters to confusion and they will not be able to announce failure. They will keep on trying for ever.

He had determined that mathematics based on the famous Proclamation for Commandments of Sets was Incomplete. Just like a totally drunken person trying to proclaim he is sober it could not prove its own consistency.

These predictions along with their proofs were recorded by Fate in Plates of Bronze.

OR

THE MONSTERS AS BHEESHMAS

He now had Inferior Monsters who followed Algorithms. These Monsters however could choose when to to stop 'computing'.

There was no independent party who with Ctrl+Alt+Del could preempt the computation. There was no Bouncer to throw them out of the Bar. Whenever like Oliver Twist they asked for more of anything they were given more. The catch was that all the Monsters had to 'end' their computation.

OR

THE MONSTERS AS BHEESHMAS

The bulk of the human race had no Vision to help them and had to try all possibilities for a solution.

A Monster simulating the bulk of the human race and trying to solve the combinatorial problem for large inputs the long way took eons of time.

If however an oracle existed which could give a yes/no answer to a possible solution for an instance of the problem, then life becomes easy.

The oracle can dictate step by step the solution to the problem. Alternatively the person/agent determining the solution could query the oracle for the correctness of each step. The problem is 'easily solved'.

Does a monster exist which can 'easily' do the job of the oracle?

OR

THE MONSTERS AS EKALAVYAS?

Does a creative person exist for all sizes of a problem? Can the infinite union of creative persons for a problem be simulated by some creative person? The problem is to determine if creativity can be acquired by the hoi polloi, in all cases, in all walks of life. Can a dummy be trained in any problem by a creative person who can then be sacked? Can a creative person be replaced by an Expert System, which can serve as an effective MEMEX for a dummy to use? Can an effective (Software Engineering?) process be found for all problems without those troublesome creative visionaries who cannot be managed easily? Like the concept of Programmer Scope in Software Engineering, is there a size limit for a combinatorial problem beyond which no creative person can have a Vision? Can a superprogrammer be replaced by ordinary programmers? Do creative persons have a problem size limit like the last straw on the camel's back? Can the monsters with a finite lifetime comment on the infinite?

However in this case the Vision of the Grand Master of the Incomplete failed him. He had suffered from the Joan of Arc syndrome. He could not resolve the problem.

He suspected it to be a 'finite analog' of the Incompleteness syndrome of the immortal monsters. We can suspect he smelt Incompletness lurking in this problem also, but we cannot be sure. His views were recorded by Fate in Plates of Bronze.

These Bronze Plates have been preserved by Fate in places dedicated to Advanced Study for future generations.

OR

THE GRAND SLAVE OF THE CELTIC MONSTERS

In the Land of Celts there was a famous mathematician. He was the Grand Slave of the Celtic Monsters. For eons of time, as a result of an ancient curse he had to continuously 'visit' and 'kowtow' to all the Celtic Monsters. His activity was subject to some constraints.

The Visitor suspected that a study of the curse and the associated activity would help resolve the problem Procrustes faced.

OR

THE GRAND SLAVE IS INVITED

OR

THE HAMILTONIAN CYCLE PROBLEM

In the ancient times as a result of a misspent youth he had in an inebriated state damaged the nose of the Sphinx and illegally entered the Pyramids of Egypt. This was 'taboo' and an eternal Curse was upon him by Isis and her son Horus whose repose he had disturbed.

Whenever the Sun was eclipsed, he had to make a circuit starting from his house, visit all the Celtic Monsters, without visiting the same monster twice, 'kowtow' to each monster, and finally return to his house. He had to undertake the journey provided the circuit existed.

The vast mass of Celtic Monsters, who were subject to birth and death, were always on the move, but when the Sun was eclipsed they froze in their places.

Just like Joan of Arc had her Visions, in this case when the sun was eclipsed and the Monsters froze in their places the Grand Slave had a Vision of the circuit, if it existed, and could excecute his journey satisfying the Curse.

Just as in the case of Joan of Arc, sometimes the Grand Slave's Vision failed him. Then he had to spend eons of time to determine the circuit by trying all possibilities. Many times he had given up in vain, and he was severely punished by the Furies as per the terms of the Curse.

He had tried in vain to find a simple way to determine the circuit when his Vision failed. The futile tears and sweat he had generated in the process had sunk the great continent of Atlantis and the Atlantic Ocean had formed.

He was unable to determine whether a 'simple way' to determine the circuit existed or not, or whether perhaps he will never know.

An eclipse was due and the Grand Slave hastily departed to serve the Celtic Monsters.

OR

THE DIRECTIONS OF THE VISITOR FOR A SOLUTION

He promised Procrustes that he would convince the Wise Man from the Land of the Redwood Trees to 'drop in' and elaborate on similar problems.

He opined that the Mathematical tools to resolve the problems have not been found as yet. The foundations of Mathematics must be examined all over again with a fresh clear mind.

The Visitor from the Land of Man Made Mountains departed. He had learnt all that was to be learned.

OR

ENTER THE THIRD VISITOR

Procrustes consulted the Visitor and complained about the confused behaviour of his Monsters, and the raw deal the children and their leaders had given him.

He wanted to know if he was as clever as Michelangelo.

OR

THE DESIDERETA IS CONSULTED FOR FINDING A SOLUTION

IP=PSPACE

IP=PSPACE

He can use the method suggested by the Grand Master of the Incomplete:-

1. He can take the role of an oracle and dictate the solution step by step.

2. He can alternatively guide a 'dumb' sculptor by answering it the 'next step' contemplated is correct.

Another strategy is to use the Interactive Verifier=Solver syndrome for a large set of problems:- He can be used as a Solver by a 'dumb' Interactive Verifier. They can carry on a conversation, and even if Michelangelo tries to be evasive or mislead the 'solution' can easily be found.

Procrustes was the boss, the Verifier. He only has to Verify the result.

A better solution was for Procrustes to be the nagging boss. He could continuously audit Michelangelo, harass him, test him by misleading him and so on.

Procrustes could various the titles for himself, like "Interactive Verifier", "Auditor-General","Director", "Manager" or just plain "Boss".

To Manage creative persons like Michelangelo & Co was easy. Any Allosaurus with a brain of a few grams in its tail can do the job of Management in such a situation.

IP=PSPACE

IP=PSPACE

Procrustes took sleeping pill after sleeping pill. He slept and slept. He had dream after dream. No Michelangelo was sighted.

Procrustes became desperate.

THE MEMEX AND IP=PSPACE

In a dream within a dream, on a Christma Eve, the Kindly Spirit of an Ancient Sage from the Land of Great Lakes advised Procrustes that he cannot capture Michelangelo in a dream. Perhaps consulting the Memex and outsourcing will help him if Michelangelo exists in "flesh and blood".

Either he could 'isolate & buy' Michelangelo's services or if that fails he should use the fable of The Fox and the Crow effectively.

Procrustes became desperate.

The remaining Monsters were still computing the answer. There was no sign of the children or their leaders. The Mandarin had gone home to the Land of Sleeping Giants. The Visitor from the Land of the Man Made Mountains had departed. The Visitor from the Land of Wanderers had departed. The Mathematicians slyly reminded the Courtiers, who in their turn slyly reminded the Prime Minister of the delayed Mathematics Pill.

OR

THE THIRD VISITOR ARRIVES

Procrustes consulted the Visitor and complained about the confused behaviour of his Monsters, the raw deal the children and their leaders had given him, the elusive Michelangelo, the petty politics of the petty mathematicians etc. etc.

He wanted to know if he was as clever as Michelangelo.

He wanted to know how to become as clever as Michelangelo.

OR

THE GIFT OF THE VISITOR

When Thor went to the Land of the Giants he quenched his thirst by drinking from a Horn. He could not see the bottom of the Horn and ended up almost emptying the oceans in a draught or two. The Can was created from that bottomless Horn.

The original purpose of the Can was to house the progeny of the Midgard Serpent.

OR

THE CAN OF WORMS

Each worm in the can had a unique search string associated with it. One could use any decent Search Engine, the search string and the MEMEX for the details of the problem associated with a worm.

OR

THE EXCLUSIVE CLUB OF WORMS

Initially the Can was empty and various worms claim to have started the club.

The Visitor however said he prefered to consider the Adam of all the worms was the one representing the Curse of the Grand Slave of the Celtic Monsters.

This started the club and the membership exploded.

Archimedes and his assistants had counted and catalouged 3000 worms so far, but the end was not in sight.

His problem also corresponded to one of the worms.

The worms represented nontrivial problems in practically all activities of the human race, and perhaps Nature itself.

OR

THE ATTRIBUTES OF A WORM IN THE CAN

If the term 'easily' is varied liberally a different club of worms may result. This was now standardised to represent only the Can of worms he had presented.

Each worm represented a COMPLETE-problem which a creative person like Michelangelo can guess the answer in a jiffy and Verify it 'easily'.

A COMPLETE-problem beyond a small size, takes a normal person trying all possibilities, and all errors, eons of time to Solve.

To claim the 'COMPLETE-' status a problem has some restrictions. Like the Jinn of the desert or Rakshasas of the Subcontinent, the normal person is allowed to make many copies of himself. One of thes copies closely simulates a creative person solving the problem 'easily'. However one does not know beforehand which copy is the 'simulator'.

OR

THE HIGH STATUS OF A WORM IN THE CAN

In the latter case the obnoxious creative persons can be shown the door.

In such a case the statue in question can be carved in a very short span of time, by any ordinary person, simply by following the steps of a procedure. Michelangelo & Co can join the lists of unemployable, unemployed persons.

OR

THE ETERNAL TOM AND JERRY SHOW

OR

HOW TO BECOME A WORM IN THE CAN

OR

THE DIFFICULTY OF MEMBERSHIP IN THE CAN

A worm which had been trying for centuries, employing all devious methods and tactics, to obtain membership was declared bogus in the Land of the Five Rivers and its membership banned. It represented the problem of primality.

A worm created in ancient times, and studied in ancient Babylonian, ancient Indian, ancient Moorish civilisations was attributed to Diophantus of Alexandria. It membership in the can could not be proved from time immemorial. Half a century ago it was exposed in The Land of Leo Tolstoy. It was shown that it was not a worm at all. It was just as in the fable the Ass in the Lion's skin. Its membership application was turned down.

OR

THE DIRECTION FOR A SOLUTION

Procrustes 'wept on the Stranger's shoulder' and told him the whole history as how he had been taken for a ride. He felt everybody excluding himself was at fault and upto tricks.

For countless tens of thousands of years, from time immemorial, his long line of ancestors starting from Adam, had tried to solve a problem.

A human being (or any being), like himself, feels like a 'wonky' Pot. Other human beings (or any beings) are like beautiful, carved Pots. This is particularly applicable to 'beautiful', 'gifted' and 'creative' persons. The questions were whether the 'Hand of the Potter shook' in making him, whether he is 'worse off' than the beautiful Pots, how can he become a beautiful Pot, anf finally why some beautiful Pots end up becoming 'wonky'.

His problem was identical to the problem Procrustes was facing.

Omar Khayyam had sold off his property, and invested in Baskets of Grapes. These were then placed in a large Caravan.

The Caravan was heading for a distant Oasis of Lotus Eaters, in a far off desert, deep inside the Continent of Circle.

He planned to relax, "take the cup in hand...." and concentrate on drinking "Iced Grape Juice" for the rest of his life.

He invited Procrustes to join him in the Party and bring his overflowing Kiddy Bank with him..

Procrustes quietly absconded like the weavers of the Emperor's magic clothes. He left no address but took his Piggy Bank with him.

It is understood that Procrustes has surfaced again of late and is now trying new types of 'narrowing' like quantum computation.

"THE WILDERNESS OF THE ETERNAL QUEST"

Needless to say using Procrustean mathematics to model and explain reality resulted in coils of undecidability springing up. Even asking Procrustean mathematics to speak up for itself resulted in coils of undecidability. This was just as in the Tower of Babel.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

Oracle after oracle was desperately consulted but enigmas like the generalizations of the P=NP problem surfaced. Very few seem to understand their nature and implications clearly.

Toil, tears and sweat for over a century were of no avail in solving the problem. This applies to numskulls like me, the hoi polloi, the glitterati and the geniuses.

Attempts to deny the problem as in the fable of the Fox and the Grapes does not seem to solve the problem. It keeps on surfacing everywhere.

FOR SIMULATION

The problem is that simulation is a very dangerous game. Procrustean mathematics can be used to specify computations to simulate the Universe.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

In simulation we end up with a situation like the multiple choice examination and know all the wrong answers.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

There is no such thing as a Solver of problems. We end up as Verifiers of solutions, and experts in knowing what is not the Universe. We will know all the wrong answers.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

THE EXISTENCE OF THE SOLVER

We end up working very hard and come up with something trivial. We end up in learnedly knowing everything about nothing. We end up beating around the bush and making never ending excursions to the periphery, but never to the center. Whatever Procrustes may try his brand of mathematics can only simulate something not create anything. Simulating eating is different from eating.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

There seems to be a catch. Dreaming up something is different from reality. Or is it???? In case Procrustes succeeds to find a suitable model of mathematics, then the time immemorial 'eternal quest' of the environment will end.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

So it is not clear if Verifiers of solutions are as good as Solvers of problems.

In the limiting case of using a computing device for the simulation of anything in the Universe the Solver vanishes. Solvers just degenerate to Verifiers when Procrustean mathematics is used as a tool.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

Provers=Verifiers??? simplifies to (a procrustean) P=NP???? simplifies to are all equally clever in Verifying solutions???.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

THE SOLUTION TO THE PROBLEM

At the present moment it has been left as an exercise for future generations of children led by Winnie the Pooh and Tigger to solve the problem.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

When the children and their leaders report back to us we will Verify the solution so that it can be presented to the Emperor of that far off land.

We can take solace in the consolation given in a canonical book on computation by the legendary Prof. Marvin Minsky.

When describing the computable real numbers, to numskulls like me, he has consolingly remarked in his book: "Nobody understands it terribly well anyway."

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

The show must go on, and the Emperor dressed as a swineherd must keep on parading in his new magic clothes, with the mathematics people bringing up the rear with their birches.

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

So it is a case of --"God is in heaven and all is well on earth!". Just let us "carry on with it" and keep on doing whatever we are doing!!!!!

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

METAMATH EMATICS

(Computation in all Environments predating all Ages, all Revolutions, all Civilizations)

During his Visit to India in the mid-seventies Prof, C A R hoare,FRS had suggested and directed that the concepts in Computer Science were basically very simple. The had been made unnecessarily complex and complicated. Common Sense explanations had to be found. The Goedel Turing Society of India has been attempting the implementation of the directive.

Mass IT literacy for a couple of million school children has been attempted for two decades by ECIL, India. This particular study is an updation & adaptaion of studies made during 1996-2006.

Earlier 'hobbyist' studies were from 1975-1995, at ECIL, India. This was the classical period in the environment when industrial strength software had to be developed without a reliable computer. Barring the isolated islands of a few higher tier institutions, education and training in Computer Science and Software in the vast mass of the environment had to be done without a computer, with very limited literature, with at times only traces of the industrial and information technology revolutions. At times the environment could be pre-agrarian and feudal.

The main reliable computational tool for education, training and development of software was the RASP, a variant of the Monsters considered above. The RASP (along with other models of universal computation) was the tool used successfully for mass instruction during the Y2K period. This shows perhaps the 'power' of metamathematics.

The dry subject of Metamathematics is considered. For decades in the environment it has been found that all attempts to teach, study or force feed the subject have been avoided and/or evaded by most students.

To make it palatable it is converted to an attempted user friendly fairy tale.

The treatment is however superficial. Today a motivated person has the MEMEX at his/her disposal. The tale is written so that it can be adapted and translated easily to any culture and language. This localisation is essential to cater to local cultural sensitivities, prejudices and beliefs.

Common sense depends on the environment, language, history, geography, economic situation, way of life and culture of the audience. So examples have to be tailored by localisation. Someimes it is Universal like animal stories, provided the animals are acceptable. Some cultures ban animal stories. Some consider astrologers amd oracles aa farce. This goes on and on.

The core of the present exposition is however based on the accepted standard fare in the curriculum of Metamathematics and The Theory of Computation. The spice can be dispensed with. Search engines and pointers to the MEMEX can be used for formal studies.

The yardstick used to measure the success of the exposition matters. Here it is a simple requirement that the students should 'clear' their standard examinations and get over the long lasting 'terror' and 'hate' of the areas involved.

They should be able to appreciate the humourous side of the concepts, which results from the limitations of the human race in putting comprehensible order and purpose into what Lincoln in one of his famous letters termed the 'eternal mystery'.

The blind vociferous extrapolations from the finite and bounded to the infinte and unbounded always lead to humourous situations. This is like the naive software programmer assuming his methods can be indefinitely scaled up to colossal software and beyond.

It is like a Software Engineering Environment dedicated to small Organic software or just lower level Software Service activities extrapolating itself to all the activities of industrial strength Semi-detached and Embedded Software Engineering Environments.

It is like a villager or tribal from pre-agrarian societies extrapolating his environment to that of a modern advanced Megapolis.

It is like a child extrapolating the power of the contents of its Piggy Bank with the power of the 'kiddy banks' of Croseus or Kubera.

It is very, very difficult to comprehend meanng and effects of the explosive power of the exponential or Ackermanns function. One normally deals with a small limited size environment.

For most people it is very, very difficult to appreciate the complexity that can be there behind a simple very user friendly naive user interface. This refers to the first acuaintance one makes with a Turing Machine, any model capable of Universal Computaton, a Google-type searh engine, or any very userfriendly naive looking GUI of a software package.

If one wants to be 'scientific' one can remove most of the references to epics, myths and legends from this study.

The mapping to various myths, legends and cultures shows some of the crudeness of metamathematics. The 'beliefs' of the former and the 'axioms' of the latter are equally 'humourous'. Both have resulted in all the 'good' activities and 'bad' activities of the human race. Both 'miraculously' explain a lot of things. What they cannot express or explain are either 'open problems' which are conveniently left for future study or just declared or made 'taboo'.

It appears now that as expected both have suspiciously met each other at the same dead end.

The yardstick used to measure the success of the exposition and related instruction has shown that quite a few students like to study the areas further. Clearing the standard examinations is easy. The irrational fear reduces or vanishes. It is a case where the end justifies the means.

Samuel Johnson had commented on the dog standing on its hind legs. "It is not done well; but you are surprised to find it done at all." This applies to the present exposition and in general to most solutions to problems using Interactive Verfiers to replicate the solution of a Solver.

This attempted Common Sense exposition is for a target group of K-12 schools and beginning undergraduates in any area.

The target groups do not easily accept two valued logic, rationalism and decidability as the norm. They are at times hostile or not able to comprehend the treatment of the infinite in the metamathematics of the current times. For some cultural backgrounds the treatment of infinities is naive and common place.

The concept of proof does not exist in some environments. The reliance on intuition is common. One guesses the result snd verifies it. Alternatively one reuses the solution 'someone' has found. One waits patiently like a Turing Machine till the 'guesser' arrives.

The normal traditional tendency is to wait for decades for the gifted person to turn up to do the job. The Interactive Verifier is used to replace the gifted person for a low cost 'if possible'. The Verifier competes with the gifted using procedures for finding faults, and normally wins. The prized secure government jobs cater to Verifiers.

The person with Vision or any normal person suddenly getting inspired and finding a solution to a problem parallels the ancient legend of Prometheus bringing Fire. Thus we have a hypothetical model of Prometheus making never ending but asynchronous, unpredictable trips bringing "fire" and training Interactive Verifiers. The Solver vanishes and only Interactive Verifiers and Verifiers exist.

Some persons have a Vision that cannot be reduced to prcedures. They cannot articulate the steps involved.In this case we do not consider them to be Solvers. They are possessed. Either we put them on a pedestal or consign them to the stake. The 'pedestal' and the 'stake' have equivalents in all cultures and environments. Actually there are 'pedestals' of varying heights and a large variety of 'stakes'.

Every human being has both types of Visions many times during their life. Some have it more often and more intensively.

In the environment undecidability and hierarchies of undecidability are a usual accepted way of life.

However the target groups are very much interested in the opportunities the Global Village creates. Thus they are amenable to experimnents in "Leapfrogging Technologies and Cultures".

The discussion here tries to use only traces of the agrarian, industrial and information technology revolutions. This is the common situation.

Outsourcing in the Global Village demands a broad knowledge and tolerance of multifarious cultures and appreciation of their buried past.

The discussion here is to be supplemented by adding a base of (slightly Procrustean) general knowledge from History, Geography, Mathematics, comic characters, Folk wisdom, folklore, epics,myths and legends as required. This can be in the form of stories or can be obtained by surfing the MEMEX (i.e. Net).

The story starts from 1800 AD. This was the time Gauss posed the problem of primality, and articulated his views on the same. He also talked about Geometries with a curvature in space.

The great contributions of Germany from 1800-1900 AD are just not known in India. They has been effectively blanked out in educational systems in India for over a century, even after the end of Imperialism. This explains why metamathematics as it is defined today finds a vacumn in the local environment.

The comic characters used are Tom and Jerry, Popeye & Co., Winne the Pooh, Tigger & Co, Asterix & Co., and in due course others like Bluto. The Beagle Boys are refered to so the the concept of 'goon' becomes user-friendly.They really are not essential and can be replaced by local cultural equivalents, as they are used as stereotypes. The block of Marble is an adaptation from the Menhirs of Obelix in the comics Asterix the Gaul. The Druds and magic potion is motivated by the same source.

The aim is to fit the fairy(?) tale in a setting of general knowledge. This is to stimulate the use of the MEMEX for the same. The MEMEX is seen to take the role of the Solver for Interactive Verifiers.

An attempt has been made to refer to all the continents including the mythical lost continent of Atlantis, the Island Continents of Antartica, Australia and Madagascar. Some other facts from Geography is thrown in.

Use has been made of Greek Mythology as per ancient renderings for children like Charles Kingsley, Bullfinch's Mythology, or the (Procrusten?) movie The Clash of the Titans. From Norse mythology Odin has been used both as a warrior and as a seeker of knowledge. He also models an oracle machines as per mythology. Thor from Norse mythology is just used using the story of his visit to the Giants. The Midgard Serpent is used for spice. The references to Hindu mythology or the High Priest and his activities are only for spice and not essential. The Christmas Carol is accomodated. The Vigin Godess Isia and her son Horus are from Egyptian mythology. The Jinn are from the mythology of the Middlae East and the Rakshsas from Hindu mythology. They are used for the concept of 'nondeterminism' by multiple copies.

Nostrodamus is refered to by using the prediction relating to the man made mountains. He is used as the living example of an oracle. Joan of Arc and Michelangelo are used as living examples of The Gifted Ones.The Atheist is both accomodated and not accomodated.

The concept of immortality is mapped to a machine computing for ever, or an entity living for ever. The concept of mortality is mapped to two types. The machine chooses when to terminate the computation, the Bheeshma concept. It is not preempted by an external agency. The other type is that it is preempted like a normal living thing. It appears only the first type leads to the current open problems, and it is only this which had been considered.

A novel of interest is Dr. Jekyll and Mr. Hyde to show that appearances are deceptive, and that two valued logic is limited. The other novel of interest is Oliver Twist adapted so that Oliver gets more so long as he asks for more. This models Growing Automata. The Pit and the Pendulum is from Poe. It has been diluted to make it palatable for small children.

Formalisation of mathematics is restricted to stating two formulas: P=NP & IP=PSPACE. No 'learned' explanations of them are attempted. The overuse of symbolism and abstraction in all methamathematics has led many persons with the feeling of encountering undpeakable horrors.

To interestingly state a NP-complete problem without mathematical notation the Hamiltonian circuit problem is recast with Celtic Monsters. The vandalism of the Sphinx and pyramids during the French invasion of Egypt is historically recast and pushed into antiquity.

The legend of Beowulf is added for just spice, to illustrate in a light way how motivation comes to a gifted person. The pre-agrarian, agrarian, industrial, information technology revolutions are bypassed to leapfrog from "the mythical times of Beowulf" to the invention of the Turing machine. The actual history can always be read up on the "MEMEX" in all the gory details of the environment of the World Wars.

The surface knowledge about ancient Rome only assumes one has seen some classical popular Hollywood movies. Rome is used for spice. Napoleon is used for his reference to China as the sleeping giant.

The Mandarin is used as the stereotype of a high IQ Verifier who only follows procedures. He represents the steel frame of effective high IQ beauracracy.

The Verifers form a pyramid and the upper end Verifiers are expected to be of the highest IQ. This was pointed out by Prof. C A R Hoare, FRS as being essential for the development of high quality large industrial strength software.

The Grand Master of the Incomplete is Goedel. The reference to Redwood Trees is an allegorical reference to the collective wisdom of Computer Scientists on Intractability from Western USA. The reference to Man Made Mountains is an allegorical reference to the collective wisdom of Computer Scientists on Intractability from Eastern USA. The reference to the Wanderers is an allegorical reference to the collective wisdom of Computer Scientists on Intractability from the International Community. The reference to Great Lakes is an allegorical reference to the collective wisdom of Computer Scientists on the creation and use of the MEMEX. The Land of The Five Rivers is the Indian subcontinent. The Land of Leo Tolstoy is Russia.

The references to Software Engineering are an adaptaion from Software Engineering Economics, by Capers Jones. The references to Google are an adaptation from The Google Story by Vise.

The Universal fables used are those of The Fox and the Crow, The Fox and the Grapes,The Ass in the Lion's skin, The Frog and the Ox & The Emperor's New clothes.

In an unkind cut, perhaps because they have made their subject unnecessarily dufficult, the metamatheticians have been held responsible for the bloodshed of the French Revolution, the Communist Revolution and the horrors of the World Wars, and perhaps curent wars also.

The Liar's Paradox, the Barber's paradox, Zeno's parodox, the Cannibals parodox are standard fare along with Archimedes and finite numbers.

The Proclamation of Commandments for Sets are the ZF, ZFC and similar finite collections of axioms.

Relating them to example involving the Aztecs and Incas is to kindle a historical study of the Voyagers, the discovery of the New World, the Pirates of the Carribean, etc. etc., to the current outsourcing scenario. The tortoise holding up the Earth is from Hindu mythology and using it as a Beast of Burden is from the Windows 7 desktop.

Decades ago the poor allosaurus was accused of having a brain of a few grams in its tail. This was later refuted. However it seems to be an interesting concept and is used here as a model for most Managers, Administrators, feudal Aristocrats, Bosses and hence most (Interactive) Verifiers.

Informal words which speak volumes are 'dumb', 'jiffy', 'soften', 'easily','goon', 'gang' and 'stumped'. They are freely used for communicating concepts succintly and effectively.

The concept of reducibility has been illustrated by using Ovid's Metamorphoses. This can be culturally mapped to the local folklore or legends.

The Piggy Bank mentioned can be replaced by Kiddy Bank or the local equivalent time immemorial pot. The use of the word 'Pig' is offensive in some cultures.

To cater to the sensibilities of some cultures the words 'High Priest' and 'excommunicated' will have to be replaced by terms like 'medicine man'.

THE GLOBAL VILL AGE HAS TAKE N OFF!!!

THE GLOBAL VILL AGE HAS TAKE N OFF!!!

THE GLOBAL VILL AGE HAS TAKE N OFF!!!

THE GLOBAL VILL AGE HAS TAKE N OFF!!!

THE GLOBAL VILL AGE HAS TAKE N OFF!!!

THE GLOBAL VILL AGE HAS TAKE N OFF!!!

THE GLOBAL VILL AGE HAS TAKE N OFF!!!

COMPUTATIONAL COMPLEXITY AND CHAOS

COMPUTATIONAL COMPLEXITY AND CHAOS

COMPUTATIONAL COMPLEXITY AND CHAOS

COMPUTATIONAL COMPLEXITY AND CHAOS

COMPUTATIONAL COMPLEXITY AND CHAOS

COMPUTATIONAL COMPLEXITY AND CHAOS

THE MEMEX + THE DUMM Y AND IP=PSP ACE! OOPS !

THE MEMEX + THE DUMM Y AND IP=PSPACE! OOPS !

THE MEMEX + THE DUMM Y AND IP=PSPACE! OOPS !

THE MEMEX + THE DUMM Y AND IP=PSPACE! OOPS !

THE MEMEX + THE DUMM Y AND IP=PSPACE! OOPS !

THE MEMEX + THE DUMM Y AND IP=PSPACE! OOPS !

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

COMPUTATIONAL COMPLEXITY AND CHAOS

COMPUTATIONAL COMPLEXITY AND CHAOS

COMPUTATIONAL COMPLEXITY AND CHAOS

COMPUTATIONAL COMPLEXITY AND CHAOS

COMPUTATIONAL COMPLEXITY AND CHAOS

COMPUTATIONAL COMPLEXITY AND CHAOS

COMPUTATIONAL COMPLEXITY AND CHAOS

COMPUTATIONAL COMPLEXITY AND CHAOS

COMPUTATIONAL COMPLEXITY AND CHAOS

COMPUTATIONAL COMPLEXITY AND CHAOS

COMPUTATIONAL COMPLEXITY AND CHAOS

COMPUTATIONAL COMPLEXITY AND CHAOS

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!

THE GLO B AL V ILLA G E ==> VERIFIERS RU LE!!