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Quotes From Here And There

Albert Einstein
Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius - and a lot of courage - to move in the opposite direction.
Dylan Thomas
I fell in love – that is the only expression I can think of – at once, and am still at the mercy of words, though sometimes now, knowing a little of their behavior very well, I think I can influence them slightly and have even learned to beat them now and then, which they appear to enjoy.
Eddie Cantor
It takes twenty years to become an overnight success.
Edward Abbey
May your trails be crooked, winding, lonesome, dangerous, leading to the most amazing view. May your mountains rise into and above the clouds.
e e cummings
To be nobody but yourself in a world which is doing its best night and day to make you like everybody else means to fight the hardest battle any human being can fight and never stop fighting.
Eyler Coates
We've all heard that a million monkeys banging on a million typewriters will eventually produce a masterpiece. Now, thanks to the Internet, we know this is not true.
Friedrich Nietzsche
Without music, life would be a mistake.
Gustave Flaubert
The one way of tolerating existence is to lose oneself in literature as in a perpetual orgy.
Going to the Opera is like making love; we get bored but we come back.
Johann Wolfgang von Goethe
If I love you, what business is it of yours?
John Steinbeck
Only through imitation do we develop toward originality.
Lady Mary Wortley Montagu
There is no remedy so easy as books, which if they do not give cheerfulness, at least restore quiet to the most troubled mind.
Leonard Cohen
Ring the bells that still can ring;
Forget your perfect offering.
There is a crack in everything;
That's how the light gets in.
Montaigne
The greatest thing in the world is to know how to belong to oneself.
Paul Sweeney
You know you've read a good book when you turn the last page and feel a little as if you have lost a friend.
Peter Altenberg
I never dreamed of being Shakespeare or Goethe, and I never expected to hold the great mirror of truth up before the world; I dreamed only of being a little pocket mirror, the sort that a woman can carry in her purse; one that reflects small blemishes, and some great beauties, when held close enough to the heart.
Robert Frost
In three words, I can sum up everything I've learned about life: it goes on.
There's no money in poetry, but then there's no poetry in money.
Satchel Paige
Work like you don't need the money. Love like you've never been hurt. Dance like nobody's watching.
Thomas Mann
A writer is somebody for whom writing is more difficult than it is for other people.
V S Naipaul
The writer has only to listen very carefully and with a clear heart to what people say to him, and ask the next question, and the next.

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Showing posts with label Essay. Show all posts
Showing posts with label Essay. Show all posts

Indian Education: The Essential Questions and Problems

This blog is written in response to the turmoil and choices I have faced over the last year and over the last couple of weeks in particular. It serves to analyze the notions and meaning of education and the sheer difficulty of realizing it in the Indian context. Sometimes, the discussion here may be so generic, simplistic and undeserving of the people I quote that it may seem little thought has gone into shaping the arguments. Yet, in the consciousness of the average Indian, questions of this nature are very rudimentary and the preconceptions very rigid. I have come to believe that the arguments below are essential to how I understand certain things that I increasingly regard to be very important in life. Also, since I have written this piece primarily as an identification of various factors that affect Indian education, I realize that the treatment here is very informal and can do with a great deal of systematic elaboration.

I remember an American movie where there is this dialogue:
The purpose of college is to make one think.
I came across this line years ago, and have yet to find any other definition of education so comprehensive as to capture the essential theme of the process of education so succinctly. No matter what form education may take - from the informal brainstorming, workshops, seminars and apprenticeships that go by various names to the formal lecture series, theses and dissertations - the very crux of it all is to impart to the learner the ability to think. There is very little success in education unless all the effort is singularly directed at developing this ability for independent thought. A quote may serve to clarify the difficulty of realizing this ideal in the Indian context: "Education to the Indian middle class is largely a job ticket carried in a wallet, or a designer shirt flaunted at a party. Education is acquired and owned, it does not permeate and transform people."

In much of our educational process, the background or the reasons are often eclipsed by the actual topic of study. There is little realization that the purpose of studying the former may not just be plain curiosity, but also so that we can better anticipate and live through the changes in our times. The result is a highly misplaced emphasis on the statement of facts, without having to deal with the accompanying motivations. For example, a science textbook may talk at great length about a discovery or invention without adequately considering the background of necessity and the reasons that led to the said discovery or invention. Or a history textbook may talk about events like wars, peace treaties, etc. without happening to put forth and analyze any motivations that propelled the events. Or a work of fiction might place great emphasis on the plot and its twists and turns without having much of the turmoil that a character faces or the philosophical musings that characterize and capture for the reader a bygone time and place and that lend it the greatness that story-telling alone cannot. Though the past may seem very obvious to us, its changes were lived as a novelty by the people of those times. History shows how ill-prepared we always are for change, fumbling through our inventions and revolutions. Education that turns its back on the future while satiating its preoccupation with the past does not fulfill its purpose. We cannot view the future through the prism of the past, but it would be foolish to know the history of one's existence without knowing the reasons and the path of our greatness, the mistakes committed and the lessons learned therefrom. Yet this closed-door-closed-mind policy seems to be increasingly a problem with our nations, religions, and 'civilizations'.

Much greater is the problem when even the actual topic of study is not considered as important enough to be discussed as the fame and the aura surrounding it. An example of this inanity can be found here. The common man must rely on the media for understanding the many events that befall our little, crazy world. Unfortunately, the mediapersons are themselves so uninformed and quite often so full of mediocrity and the accompanying arrogance that little light is shed upon the essential aspects and the peripheral (mis)information is all that one can get. So, one could have watched a prominent journalist talk patronizingly to lawyer-minister Mr. Kapil Sibal, without having done his homework on the nuclear liability bill or the foreign universities' legislation or the right to education act. Or you could have seen the failure behind the coverage of the parliament proceedings in consideration of the finance bill (including the communist and other parties staging protests outside the parliament, the destruction of public property, the 'reform' of the Jharkhand government, and the murkiness of CBI waters) without the media taking a moment to put forth what exact change is proposed by the legislation and how is it supposed to make things better or worse for the aam aadmi.

Education is singularly about recognizing the questions as being equally important as the answers, if not more. When it is reduced to its basic form, instruction, it becomes a one-way street. Slowly, the teacher-student relationship is transformed into a pedagogue-follower one, and the interaction is poisoned with a lack of questions, atleast the 'whys'. At this point, there is very little to be had from further pursuit of such study except familiarity with the subject being studied. It is perhaps the worst form of education one can receive even when it is delivered in a highly refined manner. In India, we seem to be largely familiar with this and only this form, completely disregarding the two other pillars of education viz. research and facilitation/mentorship. The fact that we are saddled with shoddy instructors only makes matters worse. We do not seem to be alarmed at the complete lack in our educational system of the ability to ask essential questions, while we are concerned with exasperating and often unimportant details like saffronization of history textbooks and reservations. And then we wonder why we don't have internationally acclaimed institutions (commensurate with our populace) that produce world-class thinkers and innovators.

Another thing that we never seem to consider is how functional the entire educational system of our country is. We have a great heritage of arts, literature, mathematics, natural and social sciences from which we might claim whole new schools of thought, extremely rich, robust and comparable alternatives to the current lines of thought as existing in the West. Yet, the achievements of our educational institutions lie merely in understanding, replicating and propagating systems of thought and conduct of affairs borrowed from the West, and amongst these only those that have any immediate practical value. I am not in favour of lambasting the West and its educational heritage; my whole education (as that of most Indians who are considered literate/educated) has been in the pedagogic traditions laid down by the British. However, a generally widespread ignorance of the existence of Indian pre-colonial traditions of thought and an unwillingness to develop and incorporate these side by side with the Western legacy of education in the hallowed portals of even the best of our educational institutions is proof of how little we understand of our past. This functional approach to education is crippling and takes away from it many of the ideals that education tries to inculcate. In the essay 'Indian Traditions and the Western Imagination' from 'The Argumentative Indian', Amartya Sen argues:
While it is true that any useful knowledge gives its possessor some power in one form or another, this may not be the most remarkable aspect of that knowledge, nor the primary reason for which this knowledge is sought. Indeed, the process of learning can accommodate considerable motivational variations without becoming a functionalist enterprise of some grosser kind. An epistemic methodology that sees the pursuit of knowledge as entirely congruent with the search for power is a great deal more cunning than wise. It can needlessly undermine the value of knowledge in satisfying curiosity and interest; it significantly weakens one of the profound characteristics of human beings.

Our best institutions have sprung from the need for imparting professional education. So we have the IITs to deal with technology, IIMs to deal with management, and a recently strengthening string of national schools of law to deal with imparting legal education. These institutions lay claim to arguably the best minds in the country as far as 'talent' is concerned, as also the biggest share of state patronage in the form of funds allotted by the government to further the cause of education. Also, due to the tendency of good researchers to cluster at the best institutions, these schools have arguably the best faculty. However, all these institutions mainly impart education in the professional (read functional) spheres! We have yet to see comparable institutions that serve as centers of higher learning in liberal arts, performing arts, music, cinema (yes there is the overarching FTII, but what else), literature, philosophy (Every once in a while, one hears eminent colleges in Bombay shutting down this department due to a lack of students wanting to study philosophy.), political strategy and ethics, change in favour of social equity, etc. There are institutions that serve to understand these causes but they are fragmented islands of thought, often afflicted with dogma, corruption in ideals and practical affairs, and without the ability to connect to other schools of thought around the country and to bring about a constructive change in the society. Quite often, they might simply be 'educating' people with neither the academic rigor nor the inclination to question, change or innovate in their various streams of study, because a 'sharp' mind becomes accustomed in its very impressionable stage to the exclusive idea of the necessity of attending a professional school. Nothing can be sadder than this extremely functional approach to education, leading to the very slow but nevertheless persistent process of decay in many important areas of study and research.

The chemist Hofmann (who taught Perkins, father of the dyestuff industry and inventor of aniline mauve and mordants) in one of his addresses said,
Whenever one of your chemical friends, full of enthusiasm, exhibits and explains to you his newly discovered compound, you will not cool his noble ardour by asking him that most terrible of all questions: 'What is its use? Will your compound bleach or dye? Will it shave? May it be used as a substitute for leather?' Let him quietly go on with his work. The dye, the leather, will make their appearance in due time. Let him, I repeat, perform his task. Let him indulge in the pursuit of truth - of truth pure and simple - of truth not for the sake of Mauve - let him pursue truth for the sake of truth.
What Hofmann is speaking against is an overt emphasis on the functionalist approach to education.

More recently, in response to a question on why India is unable to produce a global management school, Jagdish N Sheth associated with the Goizueta Business School opined,
The main issue with Indian institutes is that they only concentrate on teaching and not research. We still have a system of knowledge dissemination and not knowledge creation. Our excellence lies in our admission process...
He may as well have been speaking for the educational structure that mars the entire country.

The problem of a functionalist approach is something the West deals with very well, by not merely responding to the need of the industry and by recognizing that the unconditioned consciousness does not recognize the divisions between the various streams of study that we have created for the convenience of organization and specialization. So the universities in the West have schools that study political thought and change, classics (including their mythological heritage in Greek and Latin), socio-economic change and its repercussions, etc. in addition to the professional streams. These schools are highly vibrant and respected centers of learning, have splendid infrastructure and generous funding, and are vied for by students. This competitive excellence in streams of non-professional study must come from a deep recognition of the fact that political philosophy and economic ingenuity play as important a role in the vision of a nation or civilization as technological progress. Indeed, studies that have immediate tangible value cannot be advanced without progress in the underlying 'pure' fields such as physics, economics, linguistics, etc. A healthy stance towards learning in all spheres is important if we are to protect the wholistic nature of education from decay.

Another issue that is related to the functional aspect of Indian education is the positive versus normative divide (terms of perspectives used mainly in economic study). A nation that has left the reins of rule and thought to outsiders for centuries cannot wake up all of a sudden and start asking stirring questions. So, as far as it concerns positive studies, the study of true and false, we are good at it. This might perhaps be the reason why Indians do so well in the sphere of sciences and why the IITs (despite their lack of extravagant infrastructure) are recognized the world over for their contribution in sciences and technology. However, when it comes to the normative questions, the questions of good versus bad and right versus wrong, we seem to get uncomfortable and squeamish and prefer to keep it beneath veneers. In a land which not only has rampant corruption, prejudice, and a general lack of awareness or questioning about ethics but also recognizes these as a part of life to be given no greater attention than the very act of breathing, it can be very difficult for a common next-door conditioned mind to comprehend the fallacy of acts such as propagating venality, not being passionate about one's work, paying "donations for a seat in an educational institute", prejudice against sexual minorities, etc. Such an attitude transforms itself into an educational failure in social sciences which rely heavily on the normative line of thought, because prejudice cripples the mind of any ability for independent, constructive thought.

The question of divisions between the various streams of study brings us to another important issue that is hardly ever considered in the Indian milieu. I have become aware of this flaw after having read of it in an unlikely but appropriate place, and have become increasingly observant to it cropping up in everyday questions pertinent to my education and of those around me. While it focuses much on literature, it is equally applicable to understanding the pitfalls of education in general. In the preface to The Golden Notebook, Doris Lessing says:

It starts when the child is as young as five or six, when he arrives at school. It starts with marks, rewards, 'places', 'streams', stars - and still in many places, stripes. This horse-race mentality, the victor and loser way of thinking, leads to 'Writer X is, is not, a few paces ahead of Writer Y. Writer Y has fallen behind. In his last book, Writer Z has shown himself as better then Writer A.' From the very beginning, the child is trained to think in this way: always in terms of comparison, of success, and of failure. It is a weeding-out system: the weaker get discouraged and fall out; a system designed to produce a few winners who are always in competition with each other. It is my belief - though this is not the place to develop this - that the talents every child has, regardless of his official 'IQ', could stay with him through life, to enrich him and everybody else, if these talents were not regarded as commodities with a value in the success-stakes.

The other thing taught from the start is to distrust one's own judgement. Children are taught submission to authority, how to search for other people's opinions and decisions, and how to quote and comply.

As in the political sphere, the child is always taught that he is free, a democrat, with a free will and a free mind, lives in a free country, makes his own decisions. At the same time, he is a prisoner of the assumptions and dogmas of his time, which he does not question, because he has never been told they exist. By the time a young person has reached the age when he has to choose (we still take for granted that a choice is inevitable) between the arts and science, he often chooses the arts because he feels that here is humanity, freedom, choice. He does not know that he is already moulded by a system: he does not know that the choice itself is the result of a false dichotomy rooted in the heart of our culture. Those who do sense this, and who don't wish to subject themselves to further moulding by a system, tend to leave, in a half unconscious, instinctive attempt to find work where they won't be divided against themselves. With all our institutions, from the police force to academia, from medicine to politics, we give little attention to people who leave - that process of elimination that goes on all the time and which excludes, very early, those likely to be original and reforming, leaving those attracted to a thing because that is what they are already like. A young policeman leaves the Force saying he doesn't like what he has to do. A young teacher leaves teaching, her idealism snubbed. This social mechanism goes almost unnoticed - yet it is as powerful as any in keeping our institution rigid and oppressive.

These children who have spent years inside the training system become critics and reviewers, and cannot give what the author, the artist, so foolishly asks for - imaginative and original judgement. What they can do, and what they do very well, is to tell the writer how the book or play accords with current patterns of feeling and thinking - the climate of opinion. They are like litmus paper. They are wind guages - invaluable. They are the most sensitive of barometers of public opinion. You can see changes of mood and opinion here sooner than anywhere except in the political field - it is because these are people whose whole education has been just that - to look outside themselves for their opinions, to adapt themselves to authority figures, to 'received opinion' - a marvelously revealing phrase.

It may be that there is no other way of educating people. Possibly, but I don't believe that. In the meantime, it would be a help at least to describe things properly, to call things by their right names. Ideally, what should be said to every child, repeatedly, throughout his or her school life is something like this: 'You are in the process of being indoctrinated. We have not yet evolved a system of education that is not a system of doctrination. We are sorry, but it is the best we can do. What you are being taught here is an amalgam of current prejudice and the choices of this particular culture. The slightest look at history will show you how impermanent these must be. You are being taught by people who have been able to accommodate themselves to a regime of thought laid down by their predecessors. It is a self-perpetuating system. Those of you who are more robust and individual than others, will be encouraged to leave and find ways of educating yourself - educating your own judgement. Those that stay must remember, always and all the time, that they are being moulded and patterned to fit into the narrow and particular needs of this particular society.'

Lessing talks about the imaginary division between arts and science. In India, we seem to have kept the division but altered its banks. We have an unwritten but nevertheless quite rigid and oppressive hierarchy: science, commerce, arts. Mental ability (often construed as talent) dictates what one pursues, often with little or no justification in terms of talent, temperament, or inclination. However, most Indians concerned with science have some liking for the Arts and do dabble (sometimes quite successfully) in artistic activities they like. The same cannot be said for the people in arts dabbling in the Sciences, not because of lack of inclination or ability but because it is difficult to succeed in science without sustained training inputs. The bottomline is that the divisions between arts and science have been a lot more fuzzy than the ones between unseemly commerce and science/arts. Indeed, arts and sciences share topics such as logic and certain esoteric parts of philosophy; and attitudes are often found on the wrong sides of the divide - it is not uncommon to find scientists and technologists having a highly emotional, attached attitude to work, something identified easily in artists; likewise, it is not hard to find people associated with literature and philosophy utilize scientific methods and detached attitudes in analyzing the connections and repercussions of work in their domain. On the other hand, commerce is considered to be separate from science/arts. It is rare to find a person with a good degree of knowledge in both commerce and science/arts, unless the person belongs or has belonged to both sides.

In order to clearly understand the nature of graduate studies in CSE and management and the sort of career to be expected later, I recently met a few of my friends who have already handled such studies before. I was surprised (though I shouldn't have been) to learn that people invariably thought a dichotomy was staring me in the face, that a choice was inevitable. Now, quite a few areas of study in economics and finance can be understood easily by engineers who are usually trained in optimization; broadly speaking, economics and finance are essentially studies in optimization themselves. Surprisingly, a huge share of the managers you find in Indian B-schools are engineers who don't want to be engineers anymore, and seem to see management as a kind of escape route and not as an applicatory augmentation of their already acquired analytical skills. It surprised me to find that even people with a CSE background who had chosen to study management were consciously clueless that a lot of topics studied in management intersected with CSE, viz. financial engineering, decision sciences, operations and logistics, etc. Indeed, they believed that they had made a complete 'switch'. Also, there is very little inclination in an MBA student to do academic research, though I must blame this on the utilitarian nature of the curricula followed. On the other hand, the people who were pursuing graduate degrees in CSE and were involved in heavy-duty research seemed to think of the MBA as a completely different way of life (which it is) and were again clueless that any academic research happened in the sphere of management. I believe that CSE is perhaps the only engineering stream that generally intersects almost all other spheres of scientific study, and that it shares more than just a few subjects of interest with management. And yet, graduates of this stream with immense possibility, whether they pursued CSE or management, came to believe that they had made a choice between the two which Lessing calls "the result of a false dichotomy rooted in the heart of our culture."

The Lessing identified division between arts and science has been transformed in the Indian context into a commerce versus arts/science divide. And both are equally convenient, imaginary and ridiculous. For example, consider a painting, obviously a work of art. Yet its valuation and its tenability as an investment option is a question in commerce and its trends. Likewise, its preservation is a question that chemical scientists and practitioners must answer. Also, with a flower, you may write a poem about it, dissect it for its various morphological parts, or go sell it in the market. The various streams simply serve as perspectives to look at a single thing. In everyday life, it is extremely easy to observe, if one is aware and alert, as to how these seemingly different principal streams of study run into each other, feed off each other, and are dependent on each other. And how a perspective allows us to think of a thing without creating conflict, and is best used for this purpose alone and not to limit the horizons of thought. In my belief, a truly educated mind can change perspectives, can wear various hats, and in this must never be constrained by its wish or its dexterity in accommodating these perspectives, although ability may be a straitjacket, easily gotten rid of with some effort.

Since this is just a blogpost and not a formal essay, here's more from the highly interesting and educative preface to The Golden Notebook:

Meanwhile there is a country where...

Thirty or forty years ago, a critic made a private list of writers and poets which he, personally, considered made up what was invaluable in literature, dismissing all others. This list he defended lengthily in print, for The List instantly became a subject of much debate. Millions of words were written for and against - schools and sects, for and against, came into being. The argument, all these years later, still continues... no one finds this state of affairs sad or ridiculous...

Where there are critical books of immense complexity and learning, dealing, but often at second or third hand, with original work - novels, plays, stories. The people who write these books form a stratum in universities across the world - they are an international phenomenon, the top layer of literary academia. Their lives are spent in criticizing, and in criticizing each other's criticism. They at least regard this activity as more important than the original work. It is possible for literary students to spend more time reading criticism and criticism of criticism than they spend reading poetry, novels, biography, stories. A great many people regard this state of affairs as quite normal, and not sad and ridiculous...

When I recently read an essay about Antony and Cleopatra by a boy shortly to take A levels. It was full of originality and excitement about the play, the feeling that any real teaching about literature aims to produce. The essay was returned by the teacher like this: I cannot mark this essay, you haven't quoted from the authorities. Few teachers would regard this as sad and ridiculous...

Where people who consider themselves educated, and indeed as superior to and more refined than ordinary non-reading people, will come up to a writer and congratulate him or her on getting a good review somewhere - but will not consider it necessary to read the book in question, or ever to think that what they are interested in is success...

Where when a book comes out on a certain subject, let's say star-gazing, instantly a dozen colleges, societies, television programmes write to the author asking him to come and speak about star-gazing. The last thing it occurs to them to do is to read the book. This behaviour is considered quite normal, and not ridiculous at all...

Where a young man or woman, reviewer or critic, who has not read more of a writer's work than the book in front of him, will write patronizingly, or as if rather bored with the whole business, or as if considering how many marks to give an essay, about the writer in question - who might have written fifteen books, and have been writing for twenty or thirty years - giving the said writer instruction on what to write next, and how. No one thinks this is absurd, certainly not the young person, critic or reviewer, who has been taught to patronize and itemize everyone for years, from Shakespeare downwards.

Where a Professor of Archaeology can write of a South American tribe which has advanced knowledge of plants, and of medicine and of psychological methods: 'The astonishing thing is that these people have no written language...' And no one thinks him absurd.

Where, on the occasion of a centenary of Shelley, in the same week and in three different literary periodicals, three young men, of identical education, from our identical universities, can write critical pieces about Shelley, damning him with the faintest possible praise, and in identically the same tone, as if they were doing Shelley a great favour to mention him at all - and no one seems to think that such a thing can indicate that there is something seriously wrong with our literary system.

Now That There's Time To Read And Introspect

Received via email from Peter Griffin:-

Suketu Mehta in NYT - http://www.nytimes.com/2008/11/29/opinion/29mehta.html

Dilip D'souza in the Washinton Post - http://www.washingtonpost.com/wp-dyn/content/article/2008/11/28/AR2008112802247.html

Naresh Fernandes in The New Republic - http://www.tnr.com/politics/story.html?id=4ef869a4-0c91-4a83-8d3e-2b7ca1501996 See also http://www.tnr.com/politics/story.html?id=8e4fc4e9-5298-4f0c-bf66-980c253c43e0 - his piece on Jews in Bombay.

And these pieces, on their blogs, by Amit Varma, Sonia Faleiro and Rahul Bhatia.

And these by Prem Panicker: http://www.prempanicker.com/index.php?/site/respiro_ergo_sum/ & http://www.prempanicker.com/index.php?/site/an_officer_and_gentleman_and_a_moron/ & http://www.prempanicker.com/index.php?/site/end_game/ (the latter two link to some excellent stuff as well)

And this, by Ingrid Srinath - http://citizensforpeace.in/blog/2008/11/29/this-is-not-indias-911 (Read also Priyanka Joseph's comment to that post)

Godel's Theorem

An absolutely fantastic essay found here:

John von Neumann, the legendary mathematician who mastered calculus by the age of eight, who devised the familiar set-theoretic definition of the ordinal numbers at twenty, whose powers of calculation surpassed those of at least one early electronic computer, and who was described by Polya as "the only student I was ever afraid of", had the following to say regarding a certain episode in mathematical history:

This happened in our lifetime, and I know myself how humiliatingly easily my own values regarding the absolute mathematical truth changed during this episode, and how they changed three times in succession!
The cause of such awe was a short paper published in 1931 by the 25-year-old logician Kurt Godel, entitled Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems"). The revolutionary (and, to von Neumann and many others, disturbing) implication of the results therein was that any logical system comprehensive enough to describe elementary arithmetic necessarily contains propositions which can neither be proven nor disproven. Moreover, Godel proved that the internal consistency of such a system can never be proven except by employing reasoning which is not expressible within the system itself.


Background

To better understand the impact which Godel's findings must have had on his peers, we should first describe the mathematical climate of the time.

In the nineteenth century it had been discovered, through the work of Riemann, Lobachevsky and others, that coherent models of geometry could be constructed in which Euclid's parallel postulate (that, given a line L and a point P in the plane, exactly one line exists which contains P and is parallel to L) did not hold. This, in itself, was a shock to many mathematicians: for millenia it had been assumed that Euclid's description of geometry, founded as it was on a "self-evident" and minimal set of axioms, was one of the firmest, most trustworthy branches of mathematical knowledge. The existence of non-Euclidean geometries not only challenged mathematicians' geometrical intuition, but also the Platonist view that mathematics consisted of discoveries about eternal, pure forms whose existence was objective and unquestionable. More "monstrosities" such as continuous functions which were nowhere differentiable soon appeared, further fueling the general loss of faith in geometry.

Attempts to re-establish the comfortable certainty of the past, by turning from geometry to set theory as the new foundation of mathematics, also ran aground. Set theory, when pushed too hard, soon yielded such abominations as Russell's "set of all sets which do not include themselves". It proved difficult to construct a theory of sets which outruled such objects without sacrificing one's principles in the process. Logicism, as espoused by Frege, Dedekind and Russell, gave birth to structures so complicated and unwieldy that the stated intention to formalize the intuitive laws of reasoning was obscured. Constructivism, which rejected even the law of trichotomy (that every real number is either greater than, equal to, or less than zero) was deservedly perceived as fanatical.

To sidestep the embarrassing possibility that multiple, equally defensible versions of mathematical truth might exist, mathematicians soon claimed to have never been searching for truth in the first place. The formalists, led by Hilbert, redefined mathematics as consisting of allegedly meaningless symbols which were not "about" anything in particular. The mathematician was recast as a practitioner who merely manipulated these empty signs, attempting to derive theorems (sentences consisting of the aforementioned meaning-free symbols) from axioms without concerning himself with the "truth" of his findings.

Hilbert hoped thus to outmanoeuvre intuition, and, more importantly, to make possible a proof of the consistency of mathematics. The logicists before him had already laid the foundation by developing a formal language in which mathematical statements could be expressed, along with symbolic transformation rules representing steps which could legally be followed in the progression from the beginning to the end of a valid proof. (The climactic, exhaustive chronicle of this endeavour is Russell and Whitehead's Principia Mathematica, page 362 of which finally yields the demonstration that 1 + 1 = 2.) With this framework in place, it should (Hilbert thought) be possible to study the combinatorial properties of the set of all sentences which could legally be derived from the system's axioms, and to prove that no two of them were logical opposites. This would obviously be an assurance that mathematics (or at least the portion modelled by this formal system) was free from internal contradiction: that is, that the axioms could not be used to prove both a theorem and its negation.

As well as proving the impossibility of internal contradiction, it was hoped that the set of "true" sentences (those which could be constructed by applying legal transformations to the axioms) could be proved complete in the sense that, given a sentence, one could be assured that either this sentence or its negation was a member of the set of true sentences. A formal system with this property is said to exhibit "decidability", since one need never be unsure of the truth of a given sentence.


Godel's theorem

Hilbert's dreams of reformulating classical mathematics as a formal axiomatic system equipped with absolute proofs of consistency and completeness were dealt a cruel blow by Godel's findings in 1931.

In his famous paper, Godel proved that it was impossible to find a metamathematical proof of such a system's consistency without employing rules of inference inexpressible within the formal system under consideration. (More precisely, Godel proved his results of any axiomatic system comprehensive enough to contain the whole of arithmetic. Henceforth, when the term "formal system" is used, it should be assumed that we are speaking of a system satisfying the aforementioned requirement. Less powerful systems, such as arithmetic equipped with addition alone or multiplication alone, can in fact be proved decidable and complete, as was shown by Presburger and Skolem in 1930.)

Godel's other main conclusion was that any such formal system is incomplete, and hence that "truth" within the system is undecidable. Specifically, he showed that it possible to construct a sentence such that neither the constructed sentence nor its negation is provable within the system. What is more, even if one were to arbitrarily decide that such a sentence was true and should therefore be added to the system's axioms, there would still exist other equally undecidable sentences within this new system; and no matter how far this process of augmentation is taken, there will always be further truths which elude proof.


Godel Numbering

The proofs of Godel's results hinge on the fact that the set of formulas expressible within a symbolic system is countable, and hence each formula may be mapped to a natural number. Therefore, metamathematical statements about these sentences may be construed as statements about natural numbers: meaning that these metamathematical statements are expressible in the system itself. As we will see, this power of the system to codify statements about itself turns out to be an Achille's heel of sorts, allowing Godel's ingenious construction of an undecidable sentence.

Godel considered a formal system containing only seven constant symbols: the left and right parentheses, as well as signs representing "not", "or", "for all", "zero", and "the successor of" (an operator which adds one to an integer, and can therefore be used to express all natural numbers via its repeated application to "zero"). Recall that Godel's aim was to assign a unique integer (usually called the "Godel number") to each sentence expressible within this system; to begin with, the constant symbols described above were allocated distinct natural numbers. Similarly, other primitive signs (such as letters representing sentential variables) are each assigned integers. Since the number of variables which might be needed in a sentence is potentially infinite, Godel was compelled to employ some simple number theory to avoid overlap between the integers associated with different types of variables. As such, a certain class of variables was assigned prime Godel numbers, while another class was allocated from the set of squares of primes, and so on.

A similar trick was used by Godel to calculate a unique integer associated with each sentence. A sentence is just a string of primitive symbols, each of which already has a natural number assigned to it. Obviously, a simple addition of the Godel numbers of symbols in the sentence is inadequate, since it does not guarantee uniqueness over the set of all sentences. Similarly, a weighted sum is out of the question since we do not have an upper bound on the Godel numbers of primitive symbols. (If such a bound existed, say N, then we could simply multiply successive symbols by 1, N+1, (N+1)2, etc., to obtain a unique Godel number for the sentence.) Instead, the Godel number of a sentence containing n symbols with respective Godel numbers G1,...,Gn is defined as the product p1G1* ... *pnGn where pi denotes the ith prime number. This representation allows us to unambiguously (as guaranteed by the fundamental theorem of arithmetic) retrieve a sentence from its Godel number via factorisation. Similarly, a sequence of sentences may be assigned a single Godel number by multiplying successive prime powers, the exponents being the Godel numbers of successive sentences in the sequence.


Outline Of Godel's Proof

Since every symbol, sentence, and sequence of sentences in the formal system has now been assigned a Godel number, and since the system under discussion is capable of expressing statements about natural numbers, we now have a way of expressing metamathematical statements in the language of the system. For example, the claim that one sentence implies another can be interpreted as asserting a certain numeric relation between the Godel numbers of the two sentences. This relation will obviously be very complex, since it will need to express, in the domain of Godel numbers, all possible legal transformations which may be applied to a sentence in the system. However, since in the end it is merely a statement about integers, it is certainly expressible in the language of the system itself. Similarly, a yet more complex relation between natural numbers m and n exists which expresses the claim "The sequence of sentences with Godel number m is a proof for the sentence with Godel number n".

To prove that an undecidable sentence existed, Godel needed to find a formula G which, somewhat like Epimenides (the Cretan who claimed "All Cretans are liars"), expressed the assertion that no proof of G exists. More precisely, this claim could be expressed in the language of the system as

There does not exist a natural number m such that m is the Godel number of a sequence of sentences forming a proof for the sentence with Godel number g.
where g is actually the Godel number of the sentence just quoted. The sentence can therefore be construed as making a claim about itself, namely that it is unprovable.

A little thought should show that constructing such a sentence is somewhat difficult. To calculate the Godel number of the above sentence, one follows the process described above of splitting it into primitive symbols, whose Godel numbers are encoded as exponents of successive primes. However, the result of this calculation, g, appears in the sentence itself, and therefore affects the calculation! It would appear at first that we need to be "lucky" by stumbling upon a number g with the property that, when substituted literally into this sentence, brings about the coincidence that the Godel number of the resultant sentence is also g.

Luck, of course, plays no part. Godel conceived of a complex but elegant construction which, through a process of iteration, shows how to find such a number in a finite number of steps. The details of this process, while readily understandable, are somewhat tedious and will not be described here. The end result is the important point: for a very general class of formal sysems, we have an explicit method for constructing a sentence, G, which asserts its own unprovability. Further, Godel showed that if the axioms of the system are consistent (meaning that it is impossible to derive two contradictory sentences from them) then G is indeed unprovable: since if a proof for G existed, then it would also be possible to prove its negation, making the system inconsistent. The converse also holds: discovery of a proof for G's negation would imply the existence of a proof for G. In other words, if the axioms are consistent, then G is formally undecidable.

Godel further noted that, although unprovable within the formal system itself, the sentence G can in fact be proved true via metamathematical reasoning. In fact, the immediately preceding discussion shows this: since we have established that no proof for G can exist, and since this is exactly the assertion made by G about itself, G is a true statement. Thus the system not only contains an undecidable sentence, but: since it contains a true, unprovable sentence: the system is also incomplete. (The term "completeness", applied to a formal system, implies that all true statements in the system are derivable from its axioms.) What is more, simply adding G to the axioms would not suffice to make the system complete, since exactly the same process could be applied to this augmented system to obtain another, similarly undecidable, sentence. Godel thus shattered all hope of ever constructing a consistent, complete formal system.

The final blow landed by Godel's paper was a demonstration of the impossibility of proving a formal system's consistency via a proof expressible within the system itself. A brief description of how he obtained this result follows. Above we saw how, from the assumption that the system's axioms were consistent, Godel proved that it contained a true, undecidable sentence and was thus incomplete. It turns out that the proof of this fact:

If this system is consistent, then it is incomplete.
is achievable within the system itself. To see how, note that the sentence G, which asserts its own unprovability, is equivalent to the statement "This system is incomplete", since it gives an explicit example of a true, undecidable sentence. Thus the statement above is equivalent to:
(This system is consistent) implies that G is true.
Next, let A be the statement "There exists a sentence which is unprovable". This claim is in fact equivalent to asserting the system's consistency, since if the system were inconsistent, then every sentence would be provable. (This is closely related to the fact that, if we have a false statement p in any logical system, then the sentence "p implies q" is true for any sentence q.) Hence the above statement may be expressed within the formal system as simply "A implies G". Godel showed that this latter sentence was formally provable within the system. Now, assume that a proof for A, i.e., a proof of the system's consistency, also existed. Then since we have proofs for both A and "A implies G", we have a proof of G. But G was previously proven unprovable. Therefore no proof of A can exist: the system cannot prove its own consistency.

Consequences of Godel's proof

Godel's findings were the catalyst for many philosophical controversies which continue even to the present day. The Oxford philosopher J.R. Lucas has made the claim that Godel's theorem precludes the existence of artificial intelligence, since any calculating machine is isomorphic to a formal system to which Godel's theorem applies. Others, notably Douglas Hofstadter, dismiss this view as "a transient moment of anthropocentric glory" and claim that Godel's proof may even offer insights about the workings of human intelligence which will be useful in the creation of AI.

Whilst the dream of establishing secure foundations for mathematics has never recovered from Godel's attack, his findings have not been construed as a reason to abandon all hope of extracting meaning from mathematical inquiry. Godel himself seemed to hold the view that Platonic realism provided the clearest definition of mathematical truth: of mathematical concepts, he said "It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence". According to Davis and Hersh, most modern mathematicians also secretly subscribe to Platonism: "like an underground religion, it is observed in private and rarely mentioned in public".

Godel's methods also sparked various fruitful lines of investigation which had far-reaching consequences. Since the publication of his paper, the first naturally-arising example of an undecidable set-theoretic statement has been found. Known as the continuum hypothesis, it is the statement that no set has a cardinality greater than that of the natural numbers but less than that of the reals. Godel himself showed in 1937 that this hypothesis cannot be proved from the axioms of set theory; Paul J. Cohen demonstrated in 1964 that neither can it be disproved.

A fascinating variant of Godel's theorem was discovered in 1970, when it was proved that no general algorithm for solving all Diophantine equations (polynomial equations with integer coefficients and roots) can be formulated. Loosely, it can be shown that in any formal number theory, a Diophantine equation exists which is in some sense equivalent to Godel's self-denying sentence G. Such an equation can be interpreted as stating of itself that it has no solutions; in fact, if a solution were found, one could construct from it the Godel number of a proof that the equation had no solutions. It seems unlikely that we have come close to exhausting the list of surprises derived from Godel's work. Perhaps von Neumann may be allowed the last word on Godel's significance:

Kurt Godel's achievement in modern logic is singular and monumental: indeed it is more than a monument, it is a landmark which will remain visible far in space and time... The subject of logic has certainly completely changed its nature and possibilities with Godel's achievement.

Rengetsu And The Cold Night



The story I am going to narrate today is a fable that has grown to be a leitmotif in my life. It is so important to me that I have spent days gathering the courage to (re)write it (though it is very unlikely that I'll be able to narrate it with the exalted erudition it deserves), and searching for an appropriate image to place it with. For me, it justifies much more than my absence from this blog for a long time - it justifies in ample measure much of my attitude and many of my thoughts, writings and actions. In a sense, it defines the difference between the me I was before I read this story ten years ago and after it.

It is about Rengetsu née Otagaki Nobu (1791-1875), a Buddhist nun who was well-versed with the schools of learning of the Pure Land sect, Zen and Esoteric Buddhism.


Born of a high-ranking Samurai and a Geisha in the city of Kyoto (famous for its long, famous tradition of comely Geishas that remains unbroken to the present day), she was raised by Otagaki Teruhisa, a priest serving at Chionji, head temple of the Pure Land sect of Buddhism. She suffered great tragedies earlier in her life, but later went on to become a successful potter, calligrapher and poet. She is one of the few women who have attained Enlightenment under the Zen school of Buddhism.


One day tired from her travel, Rengetsu reached a village and asked for a place to rest for the night. The village was home to orthodox Buddhists. As soon as they found out that she was following the path of Zen preachings, they turned her out of the village.


Rengetsu was now faced with the twin prospects of spending the night under a tree in the forest (for the village was surrounded by a dense forest on all sides), or continuing her journey in the dead of the night (which was certainly the better and more tiring option, for it did not subject one to the scrutiny of wild animals and the danger of being eaten up by them without the benefit of a struggle). She chose the former and lay down to rest under a tree that bordered the village. She soon fell asleep...


In the middle of the night, she woke up from her sleep due to a strong breeze blowing through the land and howling in her ears. When she looked up at the sky, she was awestruck by the majesty of the tree which was in full bloom and showered her with flowers possessing an ethereal fragrance.


At this act of Providence, she turned towards the village and said an oath of thanks for turning her away.

Due to your utter sensitivity
In turning me away without shelter

On this dark night of a pallid moon,

I find myself showered with flowers from heaven.
She thanked the very people who took offense to the idea of sharing her company for a single night. If it had not been for their act of denial, she would have been sleeping in someone's home, oblivious of the beauty that this night was to offer her - the showering of blossoms, the cold lonely whispering night, and the silent conversation with the moon.

She was not angry with the villagers, and through her acceptance of the tribulations that they had wished upon her, she found a reason to celebrate in the bone-chilling lonely night. She had thus proved herself to be Buddha - The Enlightened One!!!


Why I Need To Write?

‘The one way of tolerating existence is to lose oneself in literature as in a perpetual orgy,’ wrote Gustave Flaubert in a letter in 1858. Though I am not as great a skeptic as Flaubert, I can vouch for the fact that he knew what he was saying.

Well-chosen words can have an almost chimerical quality. They can bring to your face that dimpled smile, or have you in splits with their daggers of wit. They can make you wax nostalgic with ineffable emotions, or send you searching for the tissue paper with which to wipe your tears.

Early in life did I learn the magic that words can work, and I have been fortunate to have not forgotten its power through adolescence into adulthood.


I wish to write because no other thing excites me as much as words strung together beautifully like pearls on the thread of meaning. I wish to write so well that my reader cannot put me down even if he has his seat on a burning stove or a juggernaut hurtling towards him or a million other chores to do. Nothing less than this will do.


Often while reading a remarkably excellent novel or short-story or poetry, I have thanked the author for penning it and wondered when I will put my wits together to write as well as that.
I hope that someday I break my shackles of indolence finally and get down to doing what I have always wanted to do and always known I’ll end up doing!

 

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