Wednesday, October 15, 2014

Fermat's last theorem - III

You thought you had escaped a bad dream with the last post, didn't you? But the title of this  post would have revealed to you that, like George Bush, you were a bit hasty in arriving at the conclusion that the mission had been accomplished.

Where we last left him, Andrew Wiles had become a celebrity but the peer review process was only beginning. This is a process that all scientific works have to go through before being accepted as correct. Wiles had to submit a complete manuscript to a leading journal.whose editor will then choose a team of experts who will then examine the proof line by line. The proof was so complicated that 6 referees were appointed with each given responsibility for one section of the proof. The process took a few months.

At first all the problems were minor and Wiles sorted them out quickly but there was one 'little problem'. And you are right - it had  to do with the Kolyvagin-Flach method which he had used in the proof. No matter how hard he tried he couldn't fix the problem. If he fixed the problem in one place,another problem cropped up somewhere else. The issue dragged on for months. (But tell me how did you guess it was the Kolyvagin-Flach method?)

Rumours started flying in the mathematical community - perhaps this was another in the long line of failed proofs for Fermat's Last Theorem. Wiles wanted to work in isolation and concentrate completely on the problem but this was not possible since he had become a celebrity. There was pressure on him to publish the incomplete proof so that someone else could try to correct the flaw but this would have meant the end of a childhood dream.

In desperation he took on a collaborator and struggled on for a while but the problem seemed intractable. He was on the verge of giving up when  one day in Sept. 1994, over an year after his initial presentation, he had an inspiration - all he had to do  to make the Kolyvagin-Flach method work was to use it in conjunction with the Iwasawa theory! Aren't you amazed by the man's brilliance? Simon Singh gives the views of a mathematician on the final 130 page proof in Fermat's  Last Theorem :
'I think that if you were lost on  a desert island and you had only this manuscript then you would have a lot of food for thought. You would see all the current ideas of number theory. You turn to a page and there is a brief appearance of some fundamental theorem by Deligne and then you turn to another page and in some incidental way there is a theorem by Helleguarch - all of these things are just called into play and used for a moment before going on to the next idea.'
Oh....Ah.. That Sounds Very Interesting. (As Douglas Adams said in Dirk Gently's Holistic Detective Agency , "Capital letters were always the best way of dealing with things you didn't have a good answer to.") But... er, can I exchange it for a Wodehouse? Much obliged.

But the question is, was Wiles' solution the same as Fermat's? It couldn't have been because Wiles' proof involved 20th century mathematics which had been unavailable to Fermat so Fermat's proof, if it existed, must have been simpler.

Believe it or not, I am through with Fermat's Last Theorem. Now you can safely breathe a sigh of relief, grab a restorative  drink and check Facebook.

PS: As a reward for your patience, here is a bit of nonsense math.

Sunday, October 5, 2014

Fermat's last theorem - II

Computers had been used to check Fermat's Last  Theorem (Fermat's Last Theorem states that xn + yn =  zn    has no whole number solutions for x, y and z when n > 2) for the first trillion numbers and it had proved to be true. For most of us this would have been enough but mathematicians are a finicky lot. There still remained infinite numbers to be checked so the brute force of a computer could never be used to prove the theorem. Simon Singh relates a story in Fermat's Last Theorem to illustrate mathematicians'penchant for exactness:
An astronomer, a physicist and a mathematician (it is said) were holidaying in Scotland. Glancing from a train window, they observed a black sheep in the middle of a field. 'How interesting,' observed the astronomer, 'all Scottish sheep are black!' To which the physicist responded, 'No, no! Some Scottish sheep are black!' The mathematician gazed heavenward in supplication, and then intoned, 'In Scotland there exists at least one field, containing at least one sheep, at least one side of which is black.'
Enter Andrew Wiles. As a 10 year old he saw the theorem in a library book and was fascinated by it. (You would have thought that 10 year olds had other things to do but there it is.) That fascination became an obsession for the next 30 years before he finally cracked it. For the last 7 years, he shut himself off from the rest of the world to focus solely on the problem. Nobody in the world knew what he was doing. This was highly unusual in the field  of mathematics where people frequently checked new ideas with each other.

Wiles' proof involved deadly beasts like elliptic equations, modular forms, L-functions, Taniyama-Shimura conjecture etc. I know you want to know more about these things but since I know only their names and nothing else, I must respectfully avoid throwing any light on them. Browning said that one's reach should be beyond one's grasp or what is heaven for? I am sure he meant well but I have the unfortunate tendency of steering well clear of things that are way beyond my reach. What to do, I am like that only (sic). A thousand apologies!

But I know that you will not let me live in peace till I give you some idea of how he went about his quest so here is my honest effort. Elliptic equations and modular forms are two widely separated areas of mathematics that didn't seem to have any obvious connection with each other like say, probability and calculus. Then the Taniyama-Shimura conjecture was proposed out of blue stating that thees two forms were actually two different manifestations of the same underlying property. Wherever it was checked, it proved to be true but it remained a conjecture. It faced the same problem that Fermat had: how to check for infinite possibilities?

It was then shown that if Taniyama-Shimura was right then Fermat was right i.e. either both were right or both were wrong. So Wiles decided to try to prove Taniyama-Shimura. If he could do that then Fermat was right by default. During the final stages of the proof he began to wonder if he was on the right right track. He decided to confide in another mathematician, Nick Katz. They decided they would design a lecture course for graduate students which Katz would also attend and they would check the calculations. Simon Singh writes:
'So Andrew announced this lecture course called "Calculations on Elliptic Curves",' recalls Katz with with a sly smile, 'which is a completely innocuous title -it could mean anything. He didn't mention Fermat, he didn't mention Taniyama-Shimura, he just started by diving right into doing technical calculations. There was no way in the world that anyone could have guessed what it was really about. It was done in such a way that unless you knew what this was for,then the calculations would just seem incredibly technical and tedious. And when you don't know what the mathematics is for, it's impossible to follow it. It's pretty hard to follow it even when you do know what it's for. Anyway, one by one the graduate students just drifted away and after a few weeks I was the only person left in the audience.'
After some more time and a few more steps (sounds simple, doesn't it?) Wiles was ready with the proof. He decided to announce it at the Isaac Newton Institute in Cambridge during a workshop called 'L-functions and Arithmetic' where he was slated to give 3 lectures called 'Modular Forms, Elliptic Curves and Galois Representations' (there are people who go to such conferences and attend such lectures) which was later called 'The Lecture of the Century'.  There were rumours circulating that some big result was going to be presented but Wiles didn't let on.

After the first 2 lectures the audience was still not sure whether he would actually come up with a big result. Finally towards end of the 3rd lecture he read out the proof, wrote up Fermat's Last theorem and said, "I think I'll stop here." And then the audience burst into applause - he had solved a 350 year old problem. E-mails were flying and soon newspapers, TV crews and science reporters descended upon the institute wanting to interview the 'greatest mathematician of the century'. He had become a celebrity. Simon Singh writes:
This was the first time that mathematics had hit the headlines since Yoichi Miyaoka announced his so-called proof in 1988: the only difference this time was that there was twice as much coverage and nobody expressed any doubt over the calculation. Overnight Wiles became the most famous , in fact the only famous, mathematician in the world, and People magazine even listed him among 'The 25 most intriguing people of the year' along with Princess Diana and Oprah Winfrey. The ultimate accolade came from an international clothing chain who asked the mild-mannered genius to endorse their new range of menswear.

Wednesday, September 24, 2014

Fermat's Last Theorem - I

Hang on. Don't give up so soon. Take heart from this feisty droplet. In a devastating book review, Peter Medawar wrote:
Just as compulsory primary education created a market catered for by cheap dailies and weeklies, so the spread of secondary and latterly tertiary education has created a large population of people, often with well-developed literary and scholarly tastes, who have been educated far beyond their capacity to undertake analytical thought.
I have been secretly fearing such a withering comment given my  penchant for indulging in sesquipedalianism. Given the title of this post, you have another reason for sending me such colourful remarks especially as there are some equations to follow. That gives me an excuse for a digression. It is just to tell you that God resides in equations so you better know something about them. As Richard Dawkins said in Unweaving the Rainbow, "What is this life if, full of stress, we have no freedom to digress.."

Apparently, once at the court of Catherine the Great, Euler met a French philosopher named Denis Diderot. Diderot was a convinced atheist, and was trying to convince the Russians into atheism also. Catherine was very annoyed by this and she asked for Euler's help. Euler thought about it and when he began a theological discussion with Diderot, he said: " (a+ bn)/n = x,   therefore God exists. Comment." Diderot was said to know almost nothing about algebra, and therefore returned to Paris.

End of digression. I would not have known much about Fermat's Last Theorem if Simon Singh had not been sued by the British Chiropractic Association. (Here is a talk by Simon Singh  where he gives some background about the problem. If for nothing else, watch the video for his hair style. Isn't it cool?) While checking out who Simon Singh is, I found out that he had written a book about Fermat's Last Theorem which I thought I will read. But as so often happens, it was a few years before I finally read it.

You must be wondering why I wanted to read about this of all things. The mathematician E.C. Titchmarsh once said, 'It can be of no practical use to know that pi is irrational, but if we can know, it surely would be intolerable not to know.'It is just a matter of curiosity and challenge. It is like the guy who was asked why he wanted to climb Everest. He replied, "Because it is there." The book is written for a lay audience so I thought maybe I can follow it.

Enough of my ramblings and on to Fermat's Last Theorem which is what I know you have been waiting so patiently for.  But first I have to tell you a fundamental idea about equations which I heard in a talk by Lawrence Krauss. I urge you all to remember it come what may:

L.H.S. = R.H.S.

Now that we are all clear about this idea, we will move on to an equation which you may have heard about somewhere: Pythagoras' Theorem. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In algebraic terms, a² + b² = c² where c is the hypotenuse while a and b are the legs of the triangle. This equation has an infinite number of solutions eg  3² + 4² = 5² or 5² + 12² = 13²

Fermat's Last Theorem states that in an equation of similar form, if the power is more than 2, the equation will have no whole number solutions i.e. If n is a whole number which is higher than 2 (like 3, 4, 5, 6.....), then the equation  xn + yn =  zn     has no whole number solutions when x, y and z are natural numbers (positive whole numbers (integers) or 'counting numbers' such as 1, 2, 3....). This means that there are no natural numbers x, y and z for which this equation is true i.e., the values on both sides can never be the same. What caught mathematicians' attention was that in the margin of the book where he wrote the theorem, Fermat wrote
I have discovered a truly remarkable proof which this margin is too small to contain.
That kept mathematicians busy for 300 years after his death in 1665. The brightest minds around the world struggled over the problem and some breakthroughs were achieved but a complete solution remained elusive. (As Simon Singh says in this TED talk, it even appeared in The Simpsons.)  Despite large prizes being offered for a solution, Fermat's Last Theorem remained unsolved. It has the dubious distinction of being the theorem with the largest number of published false proofs. (Of course there is always the possibility that some Hindu zealot might claim that it had already been proved in the Vedic period.)

Tuesday, September 16, 2014

Startle reflex

Startle reflex is a defensive response to sudden or threatening stimuli, Sometimes when I am concentrating on a book or deep in thought about something (one of the iconic scientific images is Darwin's words 'I think' scribbled beside a tree-diagram;  my thoughts won't be so deep), if someone suddenly calls out to me, I will give a start as if a bomb has just gone off beside me. The sound won't be close to these sounds in intensity. (According to this interesting Radiolab podcast, hearing is our fastest sense.) In Laughing Gas by P.G. Wodehouse, the protagonist says:
I remember once, when a kid - from what motive I cannot recall, but no doubt in a spirit of clean fun - hiding in a sort of alcove on the main staircase at Biddleford Castle and saying 'Boo!'to a butler who was coming up with a tray containing a decanter , a syphon, and glasses. Biddleford is popularly supposed to be haunted by a Wailing lady,and the first time the butler touched the ground was when he came up against a tiger-skin rug in the hall two flights down. 
My reaction to an unexpected sound would also be as undignified as that of the unfortunate butler. My heart would jump up higher than Wordsworth's did when he saw a rainbow. It  will almost 'killofy my heart' as Epifina the bad-tempered great-grand mother of Salman Rushdie's The Moor's Last Sigh would have said. Sujit used do this often when he was smaller but he would then overdo it so the element of surprise is lost and my reactions will become normal.

A similar thing happens when slightly cold water falls on me. Sponging or any cleaning on my body is always done with lukewarm water - water that is in the Goldilocks zone: neither too hot nor too cold. I have got used to this so when normal tap water falls on me, I give a start as if ice-cold water has fallen on me.

Tuesday, September 9, 2014


In some situations, small changes stop having small effects and result in sudden qualitative changes called phase transitions. For eg., the temperature of a solid keeps increasing as you keep heating it but after a point, if you supply it with a little more heat, the crystalline structure of the solid collapses and the molecules start slipping and flowing around each other i.e. it starts melting.

Phase transitions need not occur only in chemistry. They can occur in social systems too like spreading of fads and fashions, speculative bubbles, stock market crashes, etc. The occurrence of my stroke can also be described as a phase transition. One moment I was like millions of others preparing to go  to office and from a moment later, I was unable to scratch my nose on my own. Over time, I have developed a healthy respect for itching like Ogden Nash who said:
I’m greatly attached to Barbara Fritchy; 
I bet she scratched when she was itchy.
When my nose itches, I twitch my nose and surrounding areas (part of it is involuntary) which is the signal to Jaya about what the problem is. Over time this has  become the signal for any kind of itching in any place. Through trial and error, Jaya will find out the exact spot. I am usually given head bath about once a week. By the end of that period, my head will start itching which is signal for my next head bath. At this time if my head is scratched, it feels divine. There is actually a word for the part of the body where one cannot reach to scratch.

It is not surprising that strange itching problems catch my eye. There is an article by Atul Gawande where he writes about a phantom itch:                                                                                                            
“Scratching is one of the sweetest gratifications of nature, and as ready at hand as any,” Montaigne wrote. “But repentance follows too annoyingly close at its heels.” For M., certainly, it did: the itching was so torturous, and the area so numb, that her scratching began to go through the skin. At a later office visit, her doctor found a silver-dollar-size patch of scalp where skin had been replaced by scab. M. tried bandaging her head, wearing caps to bed. But her fingernails would always find a way to her flesh, especially while she slept.
One morning, after she was awakened by her bedside alarm, she sat up and, she recalled, “this fluid came down my face, this greenish liquid.” She pressed a square of gauze to her head and went to see her doctor again. M. showed the doctor the fluid on the dressing. The doctor looked closely at the wound. She shined a light on it and in M.’s eyes. Then she walked out of the room and called an ambulance. Only in the Emergency Department at Massachusetts General Hospital, after the doctors started swarming, and one told her she needed surgery now, did M. learn what had happened. She had scratched through her skull during the night—and all the way into her brain.
I didn't know you could scratch past your skull  into your brain! Then there is an itch which  occurs when you run.  And what about Morgellons syndrome?

Saturday, August 30, 2014

Tolerance of dissenting opinions- II

In India the choice could never be between chaos and stability, but between manageable and unmanageable  chaos, between humane and inhuman anarchy, and between tolerable and intolerable disorder. - Ashis Nandy, sociologist 

In the years after Independence, the civil service  was shielded from politics so promotions, transfers and the like were not dependant on whether you please your political masters. Post retirement sinecures were not dangled before them as inducements to toe the line. These days, almost  the first action of any  government is to transfer bureaucrats perceived to be loyal to the previous government and appoint their own favorites. If all top decision makers think similarly, there is a problem. (The same thing happens in corporates where a new CEO surrounds himself with yes-men and refers to them as 'my team'.) Ramachandra Guha says in India after Gandhi:
As P.S. Appu points out, the founders of the Indian nation-state respected the autonomy and integrity of the civil services. Vallabhai Patel insisted that his secretaries should feel free to correct or criticize his views,so that the minister, and his government, could arrive at a decision that was the best in the circumstances. However, when Indira Gandhi started choosing chief ministers purely on the basis of their loyalty to her, these individuals would pick their subordinates by similar criteria. Thus, over time, the secretary of a government department has willingly become an extension of his minister's voice and will. 
Following Indira Gandhi's massive victory in the 1971 General Elections, Kushwant Singh commented ,"...if power is voluntarily surrendered by a predominant section of the people to one person and at the same time opposition is reduced to insignificance, the temptation to ride roughshod over legitimate criticism can become irresistible." Ambedkar had warned against the dangers of bhakti or hero-worship, of placing individual leaders on a high pedestal and treating them as immune from criticism. Ramachandra Guha writes:
...most political parties have become extensions of the will and whim of a single leader. Political sycophancy may have been pioneered by the Congress Party under Indira Gandhi, but it is by no means restricted to it. Regional leaders such as Mulayam, Lalu and Jayalalithaa revel in a veritable cult of personality, encouraging and expecting craven submission from their party colleagues,and their civil servants and the public at large. Tragically, even Ambedkar has not been exempted from this hero worship. Although no longer alive, and not associated with any particular party, the reverence for his memory is so utter and extreme that it is no longer possible to have a dispassionate discussion about his work and legacy.
Witness the furor over an innocuous cartoon that both Ambedkar and Nehru would have laughed over. Many people seem to take themselves too seriously and lack a sense of humour. Arguing with people who lack a sense of humour is an impossible task. As is arguing with people who are proud of their ignorance, as Christopher Hitchens says while discussing the fatwa on Salman Rushdie.

As soon as some senior person raises his or her voice against the ruling party, CBI, Income tax dept. etc seem to find cases against them. The CBI is a useful tool to harass your opponents so no government will grant it autonomy. They will all speak in self righteous tones when in opposition but will sing a different tune when in power. It is like the Women's Reservation Bill - everybody seems to be for it but it never gets through parliament.

Have you heard one word from the BJP about CBI autonomy even though they had made a lot of noise about it earlier? Don't tell me you are surprised.Saying one thing when in the Opposition and doing something  else when in Government is nothing new. One is reminded of the conclusion of George Orwell's Animal Farm: 'The creatures outside looked from pig to man, and from man to pig, and from pig to man again; but already it was impossible to say which was which.'

Whenever I hear comments in news channels like 'people are wise', 'people know the truth', 'people can't be fooled', etc., I can't help smirking. Really? Winston Churchill's most famous comment is that 'democracy is the worst form of government if it were not for the rest' but he also said that 'the best argument against democracy is a two minute conversation with a vvoter'. Talk about 'informed voters' reminds me of a nurse who asked me, "What is this BJP? Is it Congress?" Kejriwal will say ,"I told  you so."

Democracy often works because of the idea of emergence - a lot of units that are individually stupid giving rise to group intelligence - but there are some assumptions in it which could cause problems. Even the wisest and most educated among us have only a partial idea of what is really going on and we reach our own conclusions based on our own biases. (You don't have theses biases of course. I mean other people.) Like the protagonist of Joseph Heller's Something Happened, you never really know what happens behind closed doors.Contrary to what this song says, the public doesn't know many things.

PS : Democracy of Our Times, a talk by Prof. AndrĂ© BĂ©teille

Thursday, August 21, 2014

Tolerance of dissenting opinions- I

It is not the function of our government to keep the citizens from falling into error; it is the function of the citizen to keep the government from falling into error. - US Supreme Court Justice Robert R. Jackson

I came across an interesting comment by J.B.S.Haldane in Ramachandra Guha's India after Gandhi. Haldane was a famous British biologist who moved from London in 1956 to reside in Calcutta. He joined the Indian Statistical Institute and became an Indian citizen. He once described India as 'the closest approximation to the Free world'. When an American friend protested at this surprising statement, he said:
Perhaps one is freer to be a scoundrel in India than elsewhere. So one was in the USA in the days of people like Jay Gould, when (in my opinion) there was more internal freedom in the USA than there is today. The 'disgusting subservience' of the others has its limits. The people of Calcutta riot, upset trams, and refuse to obey police regulations, in a manner which would have delighted Jefferson. I don't think their activities are very efficient, but that is not the question at issue.
The reference to Jefforson is because he believed that one of the chief duties of a citizen is to be a nuisance to the government of his state. I saw this comment at around the time when there was news about an IB report about Greenpeace. The report sounded silly stating that Greenpeace reduced India's GDP by 2-3%. Greenpeace is an advocacy group that puts forth its point of view and there are others who convey the opposite point of view. If there is anything illegal, prosecute them otherwise what is the problem? Magnifying the effect of a contrary position is a good strategy before clamping down on it.

In some talk show, a BJP spokesperson said they have nothing against NGOs who do "good work" but will act against NGOs that "create mischief". Who defines these terms? What is "good work"  for me may be "creating mischief" for you. One BJP spokesperson implied that the IB should  not be criticised. No institution, individual or idea can be beyond criticism otherwise it becomes the refuge of choice for scoundrels. A prime example of this is religion.

In a talk show about something else, about 70% of the studio audience was in favour of a proposition. A BJP spokesperson said that if you ask the same question in a year's time, 100% of the audience will support it. I would be uncomfortable living in a society where 100% of the people are for something. That level of conformity is a ready recipe for an unscrupulous leader to '"create mischief". We are not talking of philosopher kings here. If we know only our side of the argument, there is a problem. In his celebrated  treatise, On Liberty, John Stuart Mill says:
If all mankind minus one, were of one opinion, and only one person were of the contrary opinion, mankind would be no more justified in silencing that one person, than he, if he had the power, would be justified in silencing mankind..... If the opinion is right, they are deprived of the opportunity of exchanging error for truth: if wrong, they lose, what is almost as great a benefit, the clearer perception and livelier impression of truth, produced by its collision with error.
In The Demon-Haunted World, Carl Sagan writes:
Even a casual scrutiny of history reveals that we humans have a sad tendency to make the same mistakes again and again. We are afraid of strangers or anybody who's a little different from us.When we get scared, we start pursing people around. We have readily accessible buttons that release powerful emotions when pressed. We can be manipulated into utter senselessness by clever politicians. Give us the right kind of leader and, like the most suggestible subjects of the hypnotherapists, we'll gladly do anything he wants - even things we know to be wrong.
Most of us are for freedom of expression when there is a danger that our own views will be suppressed. We are not upset though when views we despise encounter a little censorship here and there.