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Archive for October, 2008

Today morning I just happened to have a look at and admire my collection of books. And I picked up Fermat’s Last Theorem by Simon Singh, a book that I had read almost 5 years ago for the first time. It is a brilliant book, especially for somebody like me to whom the mathematical details of the very long proof featuring the taniyama-shimura conjecture, modular forms and many many other things would have made no sense and I don’t pretend to understand the details. Ofcourse I know in a broad outline how the proof progressed and what it was about. Simon Singh’s book gives a gripping, almost electric history of the famous 358 year old problem and also chronicles Andrew Wiles’s love affair with it. I finished the book in one sitting.

[“I think I’ll stop here”: Andrew Wiles making history]

The Times gives a fitting one liner on the book:

To read is it to realize that there is a world of beauty and intellectual challenge that is denied to 99.9 percent of us who are not high-level mathematicians.

The book is divided into seven chapters and the epilogue. Each begins with a beautiful quote. I intend to write them down here:

1. Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean.

– G. H. Hardy

2. “Do you know,” the Devil confided, “not even the best mathematicians on other planets – all far ahead of yours – have solved it? Why, there is a chap on Saturn – he looks something like a mushroom on stilts – who solves partial differential equations mentally; and even he’s given up.”

– Arthur Porges, “The Devil and Simon Flagg”

3. Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.

– W. S. Anglin

4. Proof is an idol before which the mathematician tortures himself.

– Sir Arthur Eddington

5. The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.

– G. H. Hardy

6.  An expert problem solver must be enodowed with two incompatible qualities – a restless imagination and a patient pertinacity.

– Howard W. Eves

7. A problem worthy of attack

proves its worth by fighting back.

– Piet Hein

Posting your favorite quote on mathematics in comments is encouraged! Thanks!

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Richard Feynman has always been one of my role models. I have many role models but not that I like everything about them, just some particular traits. However for Feynman I was never very sure what i liked but I really like him. I hardly discussed Feynman with anyone but I gradually noticed that he was very popular, with a popularity amongst people who had heard of him rivaling Einstein.

[Richard Feynman: Image Source, Wikipedia Commons]

I never thought about it seriously on why he became so popular as he did, I mean there have been many physicists who did fundamental work but people have hardly heard of them. Take for example Poincaré and Einstein, Poincaré worked on the same things as Einstein and did very fundamental work, but people today have hardly heard of him but everybody knows Einstein. However the reasons for Einstein becoming popular are not very difficult to understand.

I had been provoked to think about it a few times after some discussions on a forum on Feynman that I own, a brief discussion in comments on Reasonable Deviations and once with a professor of mine. However I never thought about it beyond a point.

I have not read anything related to Feynman over the past year or so, but last week I just took out Perfectly Reasonable Deviations from the Beaten Track from my own personal library and just read some letters that I had marked in my first reading a couple of years back as very incisive and insightful. I came across the foreword to the book by Timothy Ferris again and also a couple of reviews by Freeman Dyson on books on Feynman in Scientist as Rebel. I entirely agree with the analysis these two gentlemen give. And I would like to share it and add my own ideas and thoughts.

Being a great scientist and being famous are two separate things, like Ferris rightly points out that for every great scientist who became a public figure like Albert Einstein, Marie Curie and Werner Heisenberg there are others who have done fantastic work like Subramanyam Chandrashekar, Linus Pauling etc who did not.

Let’s take a case from the above: Werner Heisenberg.

After the first world war, the dominant mood in Germany and in most of Europe was of doom. Dyson mentions in a review of the book “Weimar Culture, Causality, and Quantum Theory, 1918-1927: Adaptation by German Physicists and Mathematicians to a Hostile Intellectual Environment” that a theme song that represented this mood was Der Untergang des Abendlandes or Decline of the West by Oswald Spengler, after the German defeat on the eastern front the book took Germany by storm and within some years almost everybody had read it and everybody talked about it. Even people who strongly thought that Spengler was indulging in false rhetoric were highly influenced by his work.  He said that the decay of the western civilization must bring with it a destruction of the rigid ideas in Classical Physics and Mathematics. Quoting him:

Each culture has its own new possibilities of self expression which arise, ripen, decay and never return. There is not one sculpture, one painting, one mathematics, one physics, but many, each in its deepest essence different from the other, each limited in duration and self contained…Western European physics let no-one deceive himself has reached the limit of its possibilities. This is the origin of the sudden and annihilating doubt that has arisen about things that even yesterday were the unchallenged foundation of physical theory, about the meaning of the energy principle, the concepts of mass, space, absolute time, and causal laws generally.

There were many similar works to follow up by other authors that built upon this environment. At about this time Hermann Weyl and Schroedinger were highly influenced by Spengler’s work and the mood in the country and the rest of Europe that was of revolutionary expectation. So, when Heisenberg actually came up with his theory it at that time was seen to challenge the primacy of causality in Physics. It was revolutionary.

[Werner Heisenberg]

The point being that Heisenberg became famous for reasons that largely were extraneous to his actual work. His work came in a period of great intellectual and philosophical turmoil and expectation. And hence he became as famous as he did.

Feynman worked with the Manhattan project and gained some notoriety from it but seldom made any headlines otherwise, and his work was not “revolutionary” in the broad sense above so again it is not clear what made him famous.

Actually that way Feynman was not a “revolutionary” at all. Quoting from Scientist as Rebel:

Great scientists come in two varieties, which Isaiah Berlin, quoting the seventh-century-BC poet Archilochus, called foxes and Hedgehogs. Foxes know many tricks, hedgehogs only one. Foxes are interested in everything, and move easily from one problem to another. Hedgehogs are interested only in a few problems which they consider fundamental and stick with them for years or decades. Most of the great discoveries are made by hedgehogs. most of the little discoveries by foxes. Science needs both hedgehogs and foxes for its healthy growth, hedgehogs to dig deep into the nature of things and foxes to explore the complicated details of our marvelous universe. Albert Einstein was a hedgehog, Richard Feynman was a fox

Feynman was a great storyteller as is apparent from “Surely You are Joking..” and “What do you care What other people think“. People of all ages always like storytellers. And his stories were very very spicy, very funny and very interesting. And through this his personality came to be known. Feynman’s appeal as Timothy Ferris rightly points out was more in his core conduct as a working scientist. His enthusiasm, freedom and integrity, reflected the spirit of science in action.

Feynman loved his freedom. He wrote home while on the Rogers Commission probing the Challenger Space Shuttle crash:

“I am completely free, and there are no lovers that can be used to influence me”

He always advocated in his own style freedom of choice for his students. Something that resonates with almost all of us when we look around at the rigid ideas about what is right and wrong and loads of bureaucracy. Most of us sometime or the other are harried by the “politically correct” ideas that infest social structure and academia. Feynman embodies a welcome change that finds favor with most people. As Dave Brooks wrote about him:

Feynman is the person that every geek wants to be: very smart, honored by the establishment even as he won’t play by its rules, admired by people of both the sexes, arrogant without being envied and humble without being pitied. In other words he is young Elvis, with earth shaking talent transferred from the larynx to the brain cells and enough sense to have avoided the fat Vegas phase. Is such celebritification of such scientists good? I think so, even if people do have a tendency to go overboard. Anything that gets us thinking about science is something to be admired, whether it comes in the form of an algorithm or an anecdote.

Another thing about Feynman was his integrity and humility. As Ferris rightly puts it and I agree with him from my own personal experience, once someone gets in a position of power he or she starts wielding that to defend their own views. As Einstein himself once remarked:

To punish me for my contempt for authority, Fate made me an authority myself.

[Source: American Physical Society]

Such use of position though in a psychological way understandable, can be extremely irritating for the newbie, which everyone is at some point right? Feynman never got into that business. Again quoting Ferris:

He remained the instinctive rebel who sympathized with the students in the hall than the sage on the stage

He was a great authority himself. However he always preferred clarity of thought than anything else. He extremely disliked authority and honors. He thought they had no point and it was a rotten system in which a group of individuals would decide who is “good enough” to get an honor. He nearly declined the Nobel prize but later decided to take it at the insistence of his wife Gweneth. He said this when asked if it was worth winning the Nobel:

I don’t know anything about the Nobel prize. I don’t understand what it is all about and what’s worth what. And if the people in the Swedish academy decide that X,Y or Z should win a Nobel prize then so be it. I won’t have anything to do with it. It’s a pain in the neck. I don’t like honors, I appreciate it for the work I did and for people who appreciate it. I notice that other physicists use my work. I don’t need anything else. I don’t think there is any sense to anything else. I don’t see any point that some one in the Swedish academy decides that this is work is noble enough to receive a prize. I have already got my prize. The prize is the pleasure of finding things out, the kick in the discovery, the observation that other people use it. Those are the real things. The honors are unreal to me. I don’t believe in honors. It bothers me, honors bother me, honors as epaulets, honors as uniforms. My pappa brought me up this way, I can’t stand it, it hurts me.

Feynman was always willing to admit his ignorance. Most of the times people around us talk in a way that is “clearer than they ACTUALLY think”, he never got into the trap. If he did not know anything then be it. He was never afraid of being uncertain and admitting that he did not know something. Look at the video below and let him talk about it himself (05:00 onwards)

A lot of people have read “Surely You are Joking..” but few have read the great Feynman Lectures in Physics. He was a great teacher, always taught in a racy non-linear style which was as if he was thinking out aloud instead of reading from notes prepared in advance. I still read some chapters from the Feynman lectures whenever there is the time. If you have such a teacher in your lifetime, it would be one of your greatest achievements. We are only lucky that we can have access to such books. Also one thing to note is that Feynman never really wrote a book, all the books that bear his name are actually compilations edited by somebody else, mostly from his audio-tapes.

In fact his seminal paper on the famous Feynman diagrams would have never been published had it not been for coaxing by friends. There is a funny anecdote regarding that, but let’s not get into that. For about a year after his work on Feynman diagrams he refused to publish it. He said he was just too lazy to do it, he could talk to anybody who wanted to listen about it. But he would not publish it. He frequently said he was a fool and extremely lazy. People avoid saying that, but he was just reflecting on human condition. Again something that strikes a chord equally amongst the less gifted and the well gifted.

The world has known him as a great scientist, a great teacher and a great clown. But in Perfectly Reasonable Deviations from the Beaten Track we see another side of him. That of a wise counselor. He is not trying to be smart in any of the letters, just trying to be clear. He never spoke of his research or what he wanted to do in those letters, but they were only meant to help those who wanted to learn. The letters are a pleasure to read. Do read them if you have not.

And to think that people around us have SOME work and they start cribbing that they are just too busy to reply to a letter or even a text message, and here you had a great scientist, a Nobel laureate, a great teacher writing personally to the letters he used to get from all parts of the world, doesn’t it sound too good to be true? Every single letter in the collection is personal.

As Dyson writes:

I described him in a a letter to my parents as “half genius and half buffon”. Here in the letters he is neither a genius nor a buffon, but a wise counselor, interested in all kinds of people, answering their questions, and trying to help them the best he can.

He wrote letters to Kings, scientists, politicians, students, fans and just about anybody. Amongst these letters are some letters to his first wife Arline. Which describe day to day difficulty they had between their marriage and her death from TB. For most of these years Feynman was at the Manhattan project and Arline was at a nursing home some sixty miles away.

His letters to his second wife, Gweneth are full of anecdote about his travels. Some writing about the stupidity and snobbery of kings and some writing about the wonderful things in life.

He is famous as a great joker who played to the crowd. The prankster who found it was cool to break safes at Los Alamos or when it comes to trying to decode the Mayan Hieroglyphics or talking about adventures in topless bars. Feynman admired people with practical skill and said philosophers had no use. He controversially maintained that it was only through science that one could admire the true beauty of nature. He was a person of strong opinions.

But inspite of being a joker, a regular guy the general public could connect to and a genius he was a wise man.  When people came to him for help or wrote to him about problems, he spoke truth. His answers to most problems made a lot of sense and they still do. Be it concerning freedom, life, government etc. He mostly made great sense. I liked this part by Dyson most,

Like Einstein and Hawking he had come through times of great suffering, nursing Arline through her illness and watching her die, and emerged stronger. Behind his enormous zest and enjoyment of life was an awareness of tragedy, a knowledge that our time on Earth is short and precarious. The public made him into an icon because he was not only a great scientist and a great clown but also a great human being and a guide in time of trouble.

Recommended Reads and References:

1. Perfectly Reasonable Deviations from the Beaten Track

2. Surely you’re joking, Mr Feynman!

3. What You care what other people think

4. No Ordinary Genius

5. The Scientist as Rebel

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The Camera Lucida

While reading something by Leonel Moura a few months ago which described in some sense the evolution of art. I came across the Camera Lucida, I had not heard or read about it before and reading about it was fun. Not only is it elegant but is something that can be used with good effect even today and is available quite easily with art suppliers.

The Camera Lucida is a very elegant device, it works to the effect as if the the object we have to draw is reflected on the paper or the canvas we are drawing on. So one would only have to trace the object without having to worry about the perspective. Camera Lucida is Latin for light chamber which is the exact opposite of Camera Obscura which means dark chamber. That ancestral thread lead to the modern photography and as an illustration of that fact we still call our photographic devices as cameras. However there is no optical similarity between these two devices.

It was used as a handy drawing and painting tool by artists and even microbiologists till a few decades ago and was invented by  William Hyde Wollaston in 1807. There is some evidence that the device was first described by Johannes Kepler but over time his contribution it seems was forgotten and now the invention is largely attributed to Wollaston. Wollaston’s original design is given below

[Image Source: Wikipedia Commons]

In such an arrangement an artist looks down at the fabric or paper (labeled as P) through a half silvered mirror which is placed at 45 degrees. The mirror is adjusted so that the source or the object to be drawn (Label S) is in the field of view. Given the arrangement, a virtual image of the source is formed on the paper, this superimposition appears as if the object or person of who you are making a painting of is reflected on the sheet and thus the job is reduced to simply making the outline and coloring it aptly.

Note: The light coming from S is totally internally reflected at the surfaces of the four sided glass prism allowing all of the light from the source to the eye.

[Images Courtesy of The Camera Lucida Company]

The image on the left is simply of an artist using a camera lucida to paint a subject on paper, and the one on the right is simply a photograph taken with a camera lens in place of the artist’s eye and it shows how the image of the subject appears on the paper with the hand of the artist. The Camera Lucida as I mentioned earlier was for obvious reasons used by microbiologists as till recently photomicrographs were expensive!

A sample sketch of a Camera Lucida used for this purpose is shown below:

Click to Enlarge

[A Camera Lucida: Image Courtesy, The Botanic Gardens Trust, Sydney ]

Controversy

According to a controversial art history theory called the Hockney-Falco hypothesis advanced by a British-American artist David Hockney, quite a few of the great artists of the past whose works lead to advances in Realism were using optical aids and that their creations were not entirely due to their skill as is held. The evidence for this proposal is based solely on the characteristics of the paintings. Hockney’s collaborator Charles Falco who is a condensed matter physicist and an expert on optics calculated the amount of distortions that would result with the use of certain optical aids. Such distortions have been found in the works of quite a few artists such as Ingres, Carvaggio etc. Their controversial idea is summarized in Hockney’s book: Secret Knowledge: Rediscovering the Lost Techniques of the Old Masters.

[Image Source: Amazon]

The Camera Lucida is also very easy to make and I going to make one of these soon!

PS: Also check out the David Hockney page at Artsy.

Quick Links:

1. Buy a Camera Lucida

2. The Camera Obscura

3. Secret Knowledge: Rediscovering the Lost Techniques of the Old Masters.

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