On the end of the world


A running theme in the past few decades has been something we used to read only on the screen at the conclusion of movies: The End.

In recent years, we have had books and articles on the End of the World, the End of Physics, the End of History, the End of Communism, and the End of Civilization. There seems to be no end to publications of this kind, fueled by the internet free-for-all and the flourishing market for any kind of sensationalist proclamation.

A couple of generations ago, monster movies and murder mysteries used to sell very well too. Now frightful finalities seem to be the fad. I am almost inclined to say, à la Hamlet, “America, thy name is fad!”

Unfortunately, the ends predicted by current Nostradamuses are not the fantasy-calculations of star-mongers and soothsayers, but the serious prognostications of well-read scholars and well-informed scientists. But they do remind us of the Biblical Book of Revelations and the Puranic Kalki Avatara: both of which were inconvenient truths of by-gone ages, but those projected unhappy endings were to be followed by new happy beginnings, for in the end Almighty God would establish harmony and bliss (according to one interpretation, at least for Christians).

But the ends foreseen by current diviners are dismal beyond compare, and indiscriminate in sweep.

True, in medieval times, astrologers used to foresee disasters now and again because comets showed up in the sky or planets were coming into conjunction. But these days, we are heading towards a kind of thermodynamic disaster which, in its entropic fury, is leading us down a path of no return, with no chance of a cyclic reinstatement of an Arthurian Utopia.

Like so many other people who are counting on spending more years of planetary residence for themselves and their progeny, I am somewhat concerned about, perhaps I should say, quite disturbed by all this. Our only hope seems to lie in these scientists being dead wrong, and that doesn’t seem very likely at this point. I am frantically looking for some loophole in arguments that prophesy inevitable catastrophes, but like peace in the Middle East, that doesn’t seem to be within my grasp either.

But then, when these pundits talk of civilization, they are primarily thinking of the creature-comfort-abundant hyper-technological voraciously-energy-devouring modes which are the goals of billions all over the world who are, not so slowly, and surely, catching up.

Many thinkers and most people in the Non-West, indifferent to or unaware of the dire warnings of Western wise men, are eager for mid-century when, per economic forecasters, India and China would have equaled, perhaps surpassed, the U.S. in GNP and what not. They are dreaming, not of a white Christmas or about the end of the world, but the beginning of a new era when they will be the lead economic players.

Then again, there are billions now living in countless villages in Africa and Asia and South America where people live fairly peaceful lives as all our ancestors did centuries ago, making do with whatever is available, hunting and in harmony, sitting around for chats and songs and folktales, listening to ancient stories narrated by elders, laughing and merry-making, unaware of asteroids, and above all, being quite content with what is available, even if the average life-span may not be considerable. I wonder if their civilizations will also come to an end because of the unintended catastrophic forces that scientific knowledge, industrial pollution, and ecological rampage have unleashed.

Sometimes I tell myself that like individual death, the end of civilization will also be but a sad and passing phase of history. When it is all over, the surviving few will manage somehow or other, perhaps sitting in groups and recalling how their more energy-gobbling ancestors lived and enjoyed and ruined the whole damn thing. Or perhaps, it will all be over for Homo sapiens, and birds and beasts will once again fly and roam following the dictates of evolutionary instincts, with no wars or religions or wanton intrusions that would upset the balance in the biosphere. But the world will also then be deprived of poets and singers, astronomers and pure mathematicians and humorists and that wold be the greatest tragedy of all!

On Fermat’s Last Theorem


We live in an age in which the depressing dimensions of the human condition in their myriad manifestations are reflected day in and day our in the media. Newspapers and magazines are filled with reports of crime and war, of cheating and hatred, of drug and disease. The TV screen, whether for entertainment or as news, often offers us violent, tragic, or obscene images. Such are the impressions of the human race formed in the impressionable minds of our young. In their professed commitment to presenting the news and inspired by blatant greed, media manipulators blow up the gruesome and the gory, the sleazy and the sordid, for these seem to sell more. They are indifferent the impact of parading such truths on the emerging generation.

An item like the discovery of a new elementary particle or the proof of a mathematical theorem cannot have the same claim for headline on the first page as deception in a fund-raising campaign, a verdict of guilt in a murder trial, or the career-transition in the life of a midnight buffoon. So it was good to see in bold-faced heading -even if it was only on the second page- a report on a major ripple in the world of mathematics. We need to be reminded now and again that while the world goes wild with its atrocities, sick with its fanaticism, and helpless with its countless problems, there are men and women who still compose music and write poetry, explore the universe and do mathematics for the sheer joy of it all.

Seeing the prominence given to the report that Fermat’s Last theorem had finally been proved, an economist friend of mine called me up to ask what the fuss was all about. After listening to my spirited discourse on the significance of the achievement in the world of mathematics, he sighed plaintively, unconvinced that an abstruse proof about some puzzling property of numbers deserved all that attention.

I asked him if he thought that the split between Charles and Di, or the honeymoon of Naruhito and Masako should demand our attention more than the conquest by the human mind of a challenge that has taunted the greatest of mathematicians for more than three and a half centuries. “But this is not as interesting,” he replied. Ruskin talked about books of the hour and books of the ages. Likewise, there are also news of the hour and news of the ages. While the former is what we are inundated with, the latter is seldom given much prominence.

Yes, scientific discoveries, artistic achievements, literary creations, and selfless acts do find a place in Sunday magazines. But I have often wondered how we would perceive the world if these were the headlines in our papers, while bank robberies and briberies, political scandals and wasteful wars were relegated to small prints in later pages. Would the public be any less informed as a result? Perhaps not, and it is even possible that the thoughts and interests of people, young and old, will be more attuned to the nobler elements of the human potential.

This is not the place to discuss Fermat’s Last Theorem, but the mere story of how it arose could fascinate and inspire the young. Suffice it to say that Pierre de Fermat (1601-1665) was not even a professional mathematician. He wrote poetry in French and Spanish, was a scholar in Greek and Latin, and served as councilor to the King of France. Yet, he is remembered as one of the creators of the mathematical theory of probability and the proponent of a fundamental principle governing the physical universe, and as a curious contributor to the theory of numbers. It was in this last context that he scribbled in the margins of a book that he had discovered a truly marvelous demonstration of an apparently simple property of numbers (integers). But generations of the most prolific creators in mathematics have been baffled by the result, unable to prove it with rigor. [The interested reader with only minimum familiarity with arithmetic can find out about the theorem from a book in the local library.]

And now, if the report be true, the theorem has been proved to be correct. Not that anybody doubted its veracity, but proof in mathematics is like tasting in kitchen-creations: by looking at an elegantly served platter, we may be convinced that it is delicious, but we need to taste it before we can be absolutely certain.

So, to recall my friend’s question, “why all this fuss about the proof?”, I say that proving Fermat’s Last Theorem is the equivalent of the hoisting a flag on the peak of a mountain defiant thus far. Not all of us may reach the mountain top, but we can all share in the excitement. It is a triumph of the human spirit, and we can all rejoice in the achievement. Problems and politicians come and go, fights and frustrations arise and abate, but the positive landmarks left by human minds and efforts, from the Vedas and the Pyramids to sublime symphonies and the unveiling of the secrets of the physical world and of the magic of numbers: these and the like will remain as our lofty legacies for as long as our species treads this planet. These too deserve frequent and prominent mention in the media.

Reactions to news of an imminent apocalyptic end.


There is enough real bad news gruesomely floating around the various news channels of every TV set. Depressing news has become the order of the say: riots, suicide bombings, ice-caps melting, AIDS spreading, tornados and floods, and more.

Even while the helpless Tibetans are beaten and silenced by the largest nation on earth whom we have helped become the richest nation soon by consuming everything they produce and paying for it all, while every move to formulate a peace settlement for forming a sovereign Palestinian State is torpedoed by chronic hate-mongers who wish to obliterate a tiny nation, and the ruthless and indiscriminate retaliaton by that nation, and while there is genocide and starvation in many parts of the world, and the self-righteous fanatics call for the head of those who dare publish what they think, I don’t have the appetite for Evangelicals rubbing it all in with their fantasies and rot-in-hell threats in the guise of Biblical prophesies. Enough is enough!

If I did not have a lingering sense of humor, I’d be wailing and pulling the little hair left on my head, and jump up and down the closest staircase, before taking a leap into Lake Ontario to rid myself of all these terrible things that afflict humanity and threaten our species.

The other strong force keeping me away from such adopting that drastic personal solution to planetary problems (aside from the fact that the lake water is really, really cold now) is the aroma emanating from the kitchen which promises me yet another delightful dinner in the company of my dear wife. The least I can do is to do the dishes, however sloppily, instead of leaving her alone with an uneaten meal and some dishes to clean.

Is Mathematics an invention or a discovery?


Mathematics: In Nature or in the Human mind?

This question has been discussed and debated by mathematicians as well as by lay-thinkers for a long time. My own view is that in this context it is useful to distinguish between mathematics as a system, and particular instances of mathematics.

As a system mathematics is a framework which includes the following elements:

( Pure logical deduction. Pure logical deduction corresponds to strict causality (If this, then that) in an abstract realm. This abstract realm has its mapping in the human brain (this is a mystery). Thus, all normally functioning human brains are conditioned to the basic rules of logic.

( Integers. Any world of multiplicity incorporates integers which are subject to certain rules of combination, like three objects with four objects making seven objects. The integers and rules of combination are given names by the human mind.

(c Symmetries. There are any countless symmetries in the physical world: brightness and darkness, up and down, an object and its reflection in a mirror, are examples. These symmetries are recognized and explored by the human mind.

(d Patterns. There are many patterns in natural phenomena. For example, orbits of planets, and the behavior of aspects of physical world which we call laws of nature.

(e Relationships. Natural phenomena are interconnected in complex ways, often in precise and tractable modes. These interconnections exist as recognizable mathematical (functional) connections.

Thus, (assuming the existence of an objective reality which the human mind grasps and interprets) the basic ingredients of mathematical thought permeate that world of reality. These are reflected in the human mind which casts them in its linguistic modes.

In this sense, the mathematics we use is not an invention but a discovery that is formulated in a human-cerebral language. Whether there is a human mind or not, planets and electrons will move in elliptical orbits, the inverse-square law will govern gravitation and electromagnetic, and the symmetries of group theory will be implicit in elementary particle physics, etc.

One may compare the situaion to colors in the universe. etc. Does the human brain invent or discover colors? What the brain does is to interpret electromagnetic waves wihin a certain range of frequencies as colors. It discovered colors in this sense.

Having said this, it should be added that there are branches of mathematics (such as matrices, differential geometry, number theory, complex analysis, and quaternion) which are largely human inventions. On the other hand most mathematicians would agree that the Mandelbrot Set (whose boundary is a fractal) was a discovery.

On a Criticism of Science


An oft-heard comment: <Since science keeps changing its mind, we can’t put much stock in what it says>

One of the devastating consequences of Kuhn’s paradigm-shift thesis is the misinterpretation and caricaturization of science as an unreliable source of knowledge because some of its theories keep changing or have the potential for change.

This is like a child saying that it can’t rely on its parents for food and shelter because eventually they will get old and die.

The point is, in any phase of history, the goal of science is not to give ultimate and unalterable truths about the universe, but to provide the most consistent, coherent, and reliable understanding and information about natural phenomena relative to current knowledge.

There is not one critic of science, whether individual or institutional, that has added an iota to human knowledge, though all of them have much benefited from the applications of <unreliable> scientific knowledge.

None of this is to say that there is nothing else that is significant or valuable in human life than scientific knowledge and methodology. Many other matters, such as music and poetry and religion, let alone sports and gourmet food and humor, are no less meaningful, ejoyable, uplifting, and relevant than science.

But to decry science because its theories and interpretations change reflect more a misunderstanding on the part of its critics as to what science is all about than anything seriously wrong with science itself.