Why keep blogging? - I

In Fermat's Last Theorem, Simon Singh quotes from a book by the mathematician, G.H. Hardy:

I will only say that if a chess problem is, in the crude sense, 'useless', then that is equally true of most of the best mathematics... I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.  Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow.  I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating.  And that I have created something is undeniable: the question is about its value.

After my stroke, I was also similarly engaged in useless activities, reading about evolution, human irrationality, etc. (although I was not creating anything). I gradually found that I was better at doing these useless activities than I had been in doing any useful activities earlier. Then someone suggested that I start a blog. I started writing tentatively and then with more confidence. The blogging went on for longer than I had expected and I also started writing about the books that I read, including about topics not directly connected to my stroke.

Dan Dennett has written a book called Intuition Pumps And Other Tools for Thinking. As you would have guessed, it is about Intuition Pumps And Other tools for thinking. It is a heavy book. Very heavy - and I am not referring to its  bulk. (But maybe I am underestimating you and  you may find it suitable for casual reading.) I will quote from a relatively easy section of  the book:

In his excellent book on Indian street magic, Net of Magic: Wonders and Deceptions in India, Lee Siegel writes,  

"I'm writing a book on magic," I explain, and I'm asked, "Real magic?" By real magic people mean miracles, thaumaturgical acts, and supernatural powers."No," I answer: "Conjuring tricks, not real magic." Real magic, in other words, refers to the magic that is not real, while the magic that is real, that can actually be done, is not real magic.

A couple of years back, I got something that would have been considered "real magic" a few decades back - a neuro headset with which I could type on my own. As Arthur C. Clarke said, 'Any sufficiently advanced technology is indistinguishable from magic.' This eased the process of typing a bit and my posts got longer. I will read something and think, 'How can I not tell you about this? 'Lately, I have been blogging more about other things than about my stroke. (You can write only so much about a guy who can't eat, walk or talk without it becoming an outstanding bore.) But every so often, the typing will feel tedious and I will feel like making a final post titled 'So long and thanks for all the fish'  and calling it a day.

At such times, I will remember the ending of this splendid speech by Robert Sapolsky to Stanford students. He tells the story of a nun who spends all her time ministering to prisoners on the death row of a particular prison. These are some of the most horrible people on earth so naturally she is always asked how she is able to  do such a thing. She always replies, 'The more unforgivable the act, the more you must try to forgive it; the more unlovable the person, the more you must find the means of loving him.' He tells the students to adopt a similar attitude (I have deleted some words from the speech to make it read better in print):

You guys, as of tomorrow around noon, are officially educated. And as part of your education, what has happened is that, you have learnt something about the ways of the world, how things work, you have learnt the word 'realpolitik', you have your eyes opened up, you have wised up and one of things that happens when you have wised up enough is, you reach a very clear conclusion that, at the end of the day, it is really impossible for one person to make a difference. The more clear it is that it is impossible for you to make a difference and  make the world  better, the more you must. You guys are educated, you are privileged, you are well connected, you are enormously lucky if you are sitting here at this juncture and thus what that means is that there is nobody out there in better position to be able to sustain a  contradiction like this for your entire life and use it as a more moral imperative. So do it and good luck and have good lives in the process.

And what happens?  I begin to think that a post on Fermat's Last Theorem is just what the doctor ordered for you to feel better about the world, to help take your mind off those nasty sales targets or that stressful presentation to your boss about how to make your product move up the value chain i.e how to charge more for it.

And so it goes.

Some interesting properties of numbers

In  Fermat's Last Theorem, Simon Singh writes about some interesting patterns among numbers:

1. Perfect numbers - Numbers whose divisors add up to the number itself. Eg.6 has divisors 1, 2 and 3 which add up to 6. The next perfect number is 28. If the sum of the divisors is more than the number, Pythagoras called it an 'excessive' number and if the divisors added up to less than the number, he called it 'defecttive'.
2. Friendly numbers or amicable numbers are closely related to perfect numbers.They are pairs of numbers such that each number is the sum of the divisors of the other number. For eg. 220 and 284 are friendly numbers. 220 is the sum of the  divisors of 284 (1, 2 4 71 and 142). 284 is the sum of the divisors of 220 (1, 2, 4, 5,10, 11, 20, 22, 44, 55and 110). Fermat discovered 17,296 and 18,416. Descartes discovered a third pair (9,363,584 and 9,437,056). Leonhard Euler discovered 62 pairs. Strangely,they all had missed a much smaller pair which was discovered by a 16 year old Italian - 1184 and 1210.
3. Sociable numbers are 3 or more numbers which  form a closed loop. Consider the loop of five numbers: 12,496; 14,288; 15,472; 14,536; 14,264. The divisors of the first number add up to the second, the divisors of the second add up to the third, the divisors of the third add up to the fourth, the divisors of the fourth add up to the fifth, and the divisors of the fifth add up to the first.
4. Fermat proved that 26 is the only number sandwiched between a square and a cube (between 25=5²  and 27= 3³⁾
5. All prime numbers (except 2) can be placed in two categories: those  which can be written as 4n + 1 and those which can be written as 4n - 1 where n equals some number.Thus 13 is in the former group (4*3 + 1) while 19 is in the latter  group (4*5 - 1). Fermat's prime theorem claimed that the first type of primes were always the sum of two squares while the second type could never be written in this way. The theorem was proved by Euler almost a century after Fermat's death.

PS: Since I will rarely do any maths related posts (I know that you are dreadfully disappointed but as a wise philosopher of yore said, you can't always get what you want) I am generally giving a couple of links about maths that I had saved:

1. An excellant series in NYT by Steven Strogatz on The Elements of Math
2. A talk by Simon Singh about "The Simpsons and Their Mathematical Secrets" 

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.

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.

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.)

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.

Itching

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?