Specular Reflections
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Thoughts on Life, Science, Writing and the Universe at Large

Saturday, March 03, 2007
A note on the frequency of this Weblog.

Iíve decided after some pondering to post a longish message every month or so, rather like a mini-zine, instead of attempting and failing to post more frequently. At least during the semester, writing of a non-work nature is nearly impossible on a daily or weekly basis. So here goes.

In this installment:
* Cantor and the mathematics of the Infinite
*Younguncle Comes to a Boston public school

Cantor and the Mathematics of the Infinite

March 3, 2007. Today is Cantor's birthday.

Cantor was born in St. Petersburg on this date in 1845. He lived most of his life in Germany as a professor of mathematics in Halle. Among his remarkable discoveries were set theory and the mathematics of infinity.

The reason why I am going to go out and get a little cake (and put an infinity symbol on it in marzipan) and sing happy birthday to this man is because he was one of the most remarkable mathematicians ever, both in terms of his mathematical discoveries and his personality. And because he took on that formidable thing, the notion of infinity. The ancient Greeks apparently abhorred the idea of the infinite, other than Archimedes, who had some speculations about it, and the immortal Zeno. The ancient Indians were familiar with it, and indeed were obsessed with unbelievably large numbers. But to formalize the mathematics of infinity in a rigorous manner was a monumental task left to Cantor.

Cantor developed set theory on the way to formalizing the mathematics of infinity. I remember well my first encounter with set theory, when I was in 5th grade (I think) at Notre Dame Academy, a school run by Christian nuns in Patna, India. I was poor at mathematics then, and would remain so for many years. (If you had told me then that I would end up getting a physics PhD in a highly mathematical sub-field, I would not have believed you.) Anyway I remember set theory being introduced to us, and a test given, and to my utter and complete surprise I got a 100%. It was a heady moment.

Cantor began by considering the set of natural numbers, which is obviously an infinite set. He then realized some very strange things. The set of even numbers, which is also infinite, has the same number of elements as the set of natural numbers. You might think it should be a smaller kind of infinity, but because you can make a one-to-one correspondence between the elements of the two sets, you realize that the sets have the same number of elements (or the same cardinality). By a similar logic, a line has the same cardinality as a plane. But a line (which contains all real numbers) has a greater cardinality Ė that is, more elements --- than the set of natural numbers.

So even though you cannot count the members of an infinite set, you can tell that one infinity is bigger or smaller than another, using Cantorís techniques.

The arithmetic of infinity as developed by Cantor is surprising. Infinity + infinity = Infinity. Infinity X Infinity = Infinity. It reminds me of the line from the Vedas, 1500 B.C. or thereabouts:

Take the infinite from the infinite and lo! Infinity remains.

It is no surprise that infinity has been associated with a higher power in more than one tradition. Christian and Jewish mystics also believed that God was infinite. In fact Cantor came to believe that in working on the mathematics of infinity he was doing Godís work. He organized different orders of infinity into a sequence: his famous Aleph-null, Aleph-one, etc. but spent the last years of his life wondering whether there were other orders of infinity in between the ones he had enumerated. This last problem, the continuum hypothesis, was one he failed to prove and it broke him mentally and physically. He died in a mental institution.

The poor man didnít know what another great mathematician, Kurt Godel, would discover after his death: that the continuum hypothesis is one of those mathematical conjectures that cannot be proved or disproved within the current framework of mathematics. Godel himself was fascinated by infinity and, by a strange coincidence, lost his mind to it just as Cantor did.

The other mathematicians Iíve been reading about recently include Riemann and his great unsolved problem regarding prime numbers (the Riemann Hypothesis, you can win a million dollars if you prove it), the brilliant and unfortunate 18th century mathematician Sophie Germain, who was denied recognition due to her gender, and the remarkable Sofia Kovaleskaya (19th century) who ended up with a professorship in Sweden and was to write: to be a mathematician one must have poetry in the soul. (Quoting from memory).

I also have a lovely four volume set entitled ďThe World of MathematicsĒ edited by John Newman, which has original articles and essays by mathematicians through the ages. I just read a piece by Bertrand Russell and a marvelously lucid exposition of Cantorian infinity by Hahn.

There is something deeply satisfying and addictive about mathematics. I wish I had the time to probe deeper into its mysteries but it is hard to do in one life, and a busy one at that. Still, I think about it as I wash the dishes, or grade papers.

When I hear my students express fear and dislike of mathematics I want to beat them over the head (metaphorically speaking) with this four-volume set.

Younguncle Comes to a Boston Public School

Tuesday, February 13, 2007. Today I had my first public book reading in the U.S.

It wasnít a fancy affair at some snooty bookstore with hundreds of people, arranged by an agent or publicist. No, it was much better than that.

The venue was a school in Boston and the arranger was the parent-volunteer who also works at the library in the school. It happened because she sent me a personal email and I replied. She met me very kindly some ten minutes from the school, at a place which represented the boundary of the Known for me --- beyond that perimeter lay the unknown navigational dangers of one of the most confusing street systems Iíve ever had the terror of getting lost in, if you know what I mean. We drove together in her car, uphill and about, past narrow little lanes and broad, busy roads until we got to this large, very institutional looking building. On the way she told me that over 50% of the students were from blue-collar families who qualified for government-subsidized meals. The school, however, was a pilot school, which meant that they had a lot of freedom with curriculum and books, and she thought it was a wonderful enough place that both her daughters went there. Iím very interested in off-mainstream approaches to education, so I was more than a little curious.

Upon entering the building I was struck by how the institutional air --- the high ceilings and barracks-like architecture --- was lightened by art-work all over the walls, including mobiles suspended from the ceilings. A lot of the art-work was rather remarkable. I peeked in a couple of classrooms and in one of them a multi-age group of little ones was sitting on a carpet, drumming with their teacher. I saw that most of the children were African-American. That is not a common sight in most places in and around Boston that Iíve visited. The kids seemed happy and engaged. When I glanced in the Art room I found the same thing. I was struck by some kind of ornate robe that was hung in the middle of the room, heavy with embroidery. It looked like a king might have worn it, in Egypt or thereabouts. Along the corridors were posters made by the children on slavery.

My reading was in the library. The first group to come in consisted of some forty or fifty kids from kindergarten to grade 3. They were cheerful, squirmy, restless, eager, curious and couldnít stop asking questions even before the whole show began. They wanted to see my copy of the book, which had been read to them in their classes. (This is, of course, Younguncle Comes to Town that Iím talking about). One microscopic little girl in a pink turtle-neck and very curly hair charmed me by saying that her favorite story was ďthe one about the picky-pockets!Ē So I talked about the main character, Younguncle, and told them one of the stories (much condensed due to lack of time). In the last few minutes I had them close their eyes and do an imagination exercise in which I took them (by means of words) to another planet. Then they had questions. And more questions. Things like: did I really make the book? I explained the process of how a book gets published, which is mysterious even to me, but they accepted my explanation generously. They wanted to know all kinds of whys and wherefores ---- and they were so courteous and serious and giggly and utterly charming that I was happy to answer them. In retrospect I wish I had read aloud to them as well, since they were obviously used to it --- their dedicated teachers seemed to have made sure of that. In fact my guide, the librarian, told me that most of the kids in the school are readers. This is not a school of privileged rich kids, so I suspect that a lot of love and effort went into making it into such a special place.

The next set of kids that filed in consisted of about 25-30 kids from grades 4-5 with a couple of 6th graders as well. I was a little nervous about this group as I wasnít sure if they were at the fashionably jaded stage yet, but these kids were also intelligent, curious and attentive. I took one of my shorter stories from the book and did a reading aloud plus narration that took only about twenty minutes, and they were completely engaged. After that we had a Q&A in which they asked all kinds of interesting questions that gave me a chance to talk about the writing process, and how much practice it takes, like anything else one wants to do well, and how I donít believe in dumbing down the vocabulary because I know kids can find out the meanings of words they donít know. Some of the teachers had some pretty interesting questions as well. At the end of the whole thing I felt truly privileged to have been there.

posted by Vandana 4:54 PM
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Sunday, February 11, 2007
Miscellaneous Ramblings on a Sunday Night

I really should be sleeping, but the moon is on my mind. My little house backs on to some woods and every once in a while we hear coyotes howling, which sets off our dog as well. I don't know if coyotes howl only on full moon nights or whether that is a canard invented by ignorant humans --- one of these days I should do a mini-study of my own. But in any case NASA announced recently that the moon has gone metric by international agreement, so the only places on Earth and the Moon using the old English system are the United States, Liberia and Burma. Future missions to the moon will be in terms of seconds and kilometers and kilograms --- yay!

The moon is both prosaic --- because we take it so much for granted --- and poetic. Poetic not only because of its luminous beauty but because it reminds us that we are part of a much larger universe. The stars are too remote, the sun in the daytime too bright. But the moon is so clearly another world, its battered, cratered, rounded profile so familiar and yet alien. And there are mysteries there still --- for instance the notion that the moon is geologically dead is now being challenged by new evidence of gaseous upwellings below the surface (yes, a flatulence problem). There are stragely beautiful things like dust fountains that hang over the airless surface. Every rock that hits the moon leaves an impact that cannot erode away because there are no such agents as wind or water. Imagine being scarred each time, never healing. Only another impact can obliterate a crater. How tragic, somehow!

The moon is a source of much mystery, myth and metaphor. I wonder what it will be like to stand there, to live there, to walk its low-g surface, to have static-cling-moondust messing up one's electronics. As it loses some of the old mystery, will it gain new meaning? Will future generations see it as a giant step toward other worlds?

Will our grandchildren stand upon it and look beyond Earth to Mars?

Will they bring pet coyotes?

posted by Vandana 8:37 PM
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Saturday, December 02, 2006
Some Good Books I have Recently Read

Yes, I know, two posts in one day, when my posts are usually separated by weeks or months. Well, it is one of those days. Procrastination in action. All those chores calling seductively to me like sirens, and me girding loins and turning away from the temptation of it all to dutifully write a blog...

One of the things I've been wanting to do with this blog is to note books that I've read that have affected me in a profound way, but I haven't had a moment, and I doubt it is something I could be consistent with anyway. But just as a note to myself I want to mention two books, one, an anthology from India's top feminist publisher, Zubaan (an imprint of Kali for Women) called The Inner Line, and the other, a collection of stories by Jeffrey Ford called The Empire of Ice Cream and Other Stories.

The anthology just came out and it has one of my stories in it, but I don't want to comment on that. What it has are a bunch of stories by India's top women writers, many of them translated from Indian languages other than English, including some writers I utterly admire, like Mahashweta Devi (Bengali) and Ambai (Tamil). The stories travel the world from village to city to foreign country, and there isn't a single poor story in it (although some stories translate better than others). What stands out for me is how this anthology showcases the different ways in which women voice their pain and their triumphs. Some of these voices are subtle, others strident. We need them all.

The second book I mentioned, Jeffrey Ford's collection, contains some very fine stories, including two I'd come across before. I've been priviliged to be part of a jury that gave an award to one of these stories, but I had no idea of the breadth (or depth) of Ford's work until I read this anthology. Ford uses language like a true artist, and some of the stories may even be called clever in the way that they play with ideas and tropes, including things that have been around for a while, like Greek mythology --- and to all these he manages to give something fresh and uniquely Fordian --- but what stands out for me, what makes this collection not only outstanding but moving, is the compassion he has for his characters. The Boatman is a case in point. After reading this collection I realized that this quality --- compassion --- developed, I imagine, from a life fully and thoughtfully lived, with responsibilities and hard times and all --- is not something that surfaces often in the fiction that I've read, particularly in SF and Fantasy (notable exceptions including Ursula K. Le Guin). The other interesting thing about this collection is that the author has posted end notes after each story. These are interesting little gems in their own right, illuminating Ford's creative process and (because they are post- rather than pre-) avoiding the problem of spoilers.

posted by Vandana 3:20 PM
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Finding the Balance: Notes on a Saturday Morning

For some years now, I've been walking a thin line, falling off, clambering up again, stagerring a few centimeters, falling off... and so on. The balance I am talking about is one familiar to most people --- and women in particular, and my specific version is this: the balance between the competing demands of job, family life, writing, math/physics passions, friendship and community, and social responsibility. I've spent far more time falling off and attempting to climb back on than I have spent actually walking that thin line. How to apportion that elusive thing, time, so that one does justice to all these things? Because I need them all, I should not have to choose between them. That would be like choosing between breathing and eating. The way our modern urban societies are set up, however, make this balancing act difficult in the extreme.

Part of it is to become super-organized. I'm trying, and it is difficult, but I make progress the way a snail on a sidewalk makes progress toward the green verge, the promised land. Perhaps one day...

One of the ways I've thought of becoming more efficient is in organizing the information I receive or acquire about things in which I am interested. Take my current obsession, prime numbers. I've been reading about them off and on for the past couple of years, having to put aside my reading frequently due to other pressing things, and by the time I go back to the subject what I'd learned earlier is half-forgotten. Last year I set down all the main results I'd read about until that point on a piece of paper and put it up in my kitchen. (How many kitchens do you know that have the Prime Number Theorem as part of the wall decor?) But that, I find, is not enough. So my latest idea (probably impractical, like most of my ideas about organizing) is to write up a little amateur's monograph on the subject, for myself alone --- not simply a compilation of facts but an account, a narrative, as well.

I'm interested in primes because they are interesting --- the best of all possible reasons --- but also because they play a major role in a story I'm working on, about a mathematician. When I am obsessed like this, with, say, Subject A (could be primes, or the nesting habits of mole rats, or the possibility of hyperspace) and I let my mind become slack and receptive, then I see aspects and analogs of Subject A everywhere, even when I wash the dishes. It is a very strange feeling. And I'm digressing, which is not a very organized thing to do. Sigh...

Talking of balance, I've got to go do some chores. Perhaps if I keep my mind open, like the sail of a boat turned to hold the wind, prime numbers will pop out everywhere, wiggling their tails among the fruit at the grocery store, doing cartwheels across the student papers I have to grade...

posted by Vandana 12:31 PM
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Wednesday, November 22, 2006
Things of note in my little life, from the past couple of months, in reverse chronological order:

--- The Times of India ran a feature called Almost Famous last Sunday (nov. 19) highlighting emerging Indian writers. "Reaching for the stars, but not yet there, these are promising writers, who've got what it takes to make it big." Two were picked from a number of languages, including English, Hindi and Tamil, and yours truly was one of the writers in English picked. The story ran in all 11 editions of the Times. My parents in Delhi said that one uncle called them in the morning with the news, while another came by, excitedly brandishing the paper. I've also gotten a note from a friend in Bangalore. The eminent writer Githa Hariharan picked my name and that of Rana Dasgupta for English. Although I write for the world, the Indian audience is crucial to my work, so recognition from one's compatriots is particularly welcome. Among the Hindi writers is the much lauded Neelakshi Singh, who seems to be a fellow Bihari, and I'm going to track down her work...

--- two astronomical events turned out to be damp squibs, literally and figuratively. The transit of Mercury and a particularly productive Leonid meteor shower were both not visible to Massachusetts residents due to bad weather. For the first one I'd gotten my students all worked up, given them a special assignment on Mercury, found a room in the Humanities building where we could set up a solar telescope... all to no avail. Ah well.

--- right now I'm obsessed with prime numbers and also intrigued by recent reports on the existence of dark matter as established by pictures of two colliding galaxies...

--- We have been going to concerts and other art events as a family. It is good for all of us, and even more so when we take friends along. Highlights from October include an early music concert in which my husband's lute teacher, the inimitable Olaf Chris Henriksen performed, a visit to a special exhibit at the Museum of Fine Arts, featuring Indian miniatures, an Indo-Persian fusion concert featuring the group Niyaz, and a mini-film festival at the aforementioned MFA. The film festival featured the movies of one of Bollywood's most sensitive filmmakers, Muzzafar Ali. We made it to two of the movies. First was Anjuman, which is about a young Muslim woman in Lucknow and her emancipation in the backdrop of the struggle for embroidery workers rights --- one of Shabana Azmi's early films, and a fine rendition of the notion that the personal is political and vice versa (with some lovely music too). The second was the classic Umrao Jaan, about the Lucknow Courtesan that the Urdu writer Mirza Ruswa immortalized in his novel. A lush visual and musical treat, the film has been described as a feminist tragedy. What made both occasions special for us was that the director, Muzaffar Ali, was present to answer questions. He turned out to be a tall, striking-looking man with a mane of white hair, unexpectedly humble and not given to (or incapable of) blowing his own trumpet. He spoke at length about Rekha's eyes and his love for textiles and period clothing. The film made Rekha's name as an actress. He talked about the language of the eyes, and that Indian women are adept at that language, although actresses of Rekha's calibre are few and far between. My husband and daughter enjoyed both events, even asking the great director some questions. Their intelligent appreciation meant a lot to me, especially as we don't make it to many Indian cultural events.

posted by Vandana 8:43 AM
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Sunday, September 17, 2006
An autumnal view of things...

The weather continues to be strange. Apart from death and taxes, it seems that global warming is also inevitable. Be that as it may, the leaves are slowly turning color, despite days like today that remind one of summer. It has been getting cooler on the average, and my dog has been trying to eat acorns. He sniffs one, takes it in his mouth, gives it an experimental, tentative sort of crunch, and then spits it out. Then, being eternally optimistic, he tries another one. I think he likes the smell of acorns and can't see why he shouldn't be able to eat them.

Autumn brings death to some things, sleep to others. Death is necessary to make room for new life and renewal. This autumn I, too, have tasted of death, and something in me, finding room at last, has begun to grow. I carry, in a manner of speaking, a sapling in my chest.

I predicted this (the thoracic arboretum part anyway) through a story I wrote many months ago, well before the events that precipitated this dying and renewing occurred. (Sometimes I am beset with apprehension that my stories are a foretelling of what will come in my own life). The story began as an image of a man lying in a grassy sort of place, an alpine meadow, perhaps, watching the sky. As he watches, shoots of grass begin to grow through his chest. I won't say more because I still have to work on the next draft, and it involves a city and a woman as well...

Autumn brings other things as well. My parents, who are visiting from Delhi, are able to behold the grandiosity of American pumpkins at country-side farm-stands. They will get to see the colors turn, a sight not available in countries like India, where seasonal changes are generally gradual for most of the country (except when the monsoons come, of course). In California, where my in-laws live, there are no such dramatic fall colors either.

One of the things I am trying this fall that is new for me is singing Western music in a choir. Currently renewing my classes in Indian vocal music is difficult for me, and since my tastes in music are fairly eclectic (I love Bach, for instance) I have decided to take the plunge, so that I will have *some* kind of music in my life. Today was the first time I joined this choir, and it was interesting and challenging, because they put me in the alto section since they had too many sopranos. I come from a highly melodic tradition where there is very little harmony, so I found that singing alto parts that don't make melodic sense (independent of the whole, which may be very beautiful) rather difficult. Still, I like a challenge, and I am determined to tap out the alto section for next week on the piano at home, so that it becomes familiar enough that I don't automatically lock on to what the sopranos are singing.

There's got to be a metaphor in there somewhere!

What else? Classes have begun at the college where I teach, and I have a keen lot this time. I keep them at the edge, keep them thinking --- the only way to teach a two-hour class without watching the kids fall asleep. Teaching motion and the difference between uniform and non-uniform motion, I have them get up and walk, and mark their positions every second with chalk. We work out some pretty basic relationships on the blackboard, then I throw a problem at them where they have to figure out the right length of a runway so the plane can take off. It is fun to see their minds working. Eventually I'll subject them to my imaginative science fictional problems (my earlier engineering physics students know I love coming up with these) which I hope they'll enjoy. Too many people think science is a boring compedium of facts, where imagination has no place.

A scientist must order.
One builds science with facts as a house with stones
But an accumulation of facts is no more science
Than a pile of stones is a house.

Jean-Henri Poincare, quoting from memory.

posted by Vandana 9:47 PM
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Tuesday, September 05, 2006
Do not go gentle into the good night...
Rage, rage against the dying of the light...

Dylan Thomas's famous lines can be applied as much to the current tragic situation in the Middle East as to one's personal life. Since this blog is neither political commentary nor personal memoir (being mainly reflections of a general nature) I will not say more precisely where and how those lines apply. But I've been thinking about how writing, at least, is simultaneously an act of hope and of defiance in times of political and personal trouble. So here's to the Muse that compels me to write, that saves me from extinction, that reminds me through my own words who I am in this sorry, troubled world.

Last week I was in New York city, visiting relatives for a couple of days. There's something about the rush and flow of a huge metropolis, the streams of people and traffic, that results in a comforting anonymity. There I was, standing at a subway station in the bowels of the earth, watching the trains flash by, watching the murals on the opposite walls through the windows of the trains, hearing live music from a black man with a guitar and a voice like gold. Times like these it is enough to simply be. To try loneliness on for size and to discover that it fits like an old shirt. To let the world blow through your senses as though you were nothing but an open window. To ponder stray facts like this one: that any city large enough is filled with metaphors ripe for the picking.

In the train there was a young man singing, his voice husky and occasionally rasping, but he did a good job. Coins rained into his battered hat. Then a woman came along, strident, her eyes bright and tragic, asking people for coins to help raise her children. I am raising them alone, she said. I have nobody. Her diction was polished, her address direct, but nobody stirred. People looked away. Embarrassed? I don't know. I don't know if she was just another youngish black woman looking for drug money, but her words hit me hard. I reached for my purse; she was some distance away, however, and before I knew it she was gone, leaving behind a muttered curse that seemed to hang in the air for quite a while later.

One another random note, I've been pondering the nature of pain, and how opening oneself up to it --- whether it is one's own pain or that of another --- opens the way to empathy. To suppress pain is to deny that opportunity. They go together like Siamese twins. A good friend of mine who is (apart from being a wonderful writer) a wonderful person, tells me of a trip to Poland, where among other things she visited Holocaust sites. She's Jewish. It was traumatic for her, but I can see the necessity of opening oneself up to that experience, the necessity of honoring and acknowledging what her people went through. Her eyes filled with tears as she talked (as did mine). But her tears were not only for the Jews who were killed; they were also for the other 6 million that Hitler had murdered, including the Romanys. This is a woman who, despite being Jewish (or perhaps because she is true to the precepts of her faith) opposes Israel's current actions against Lebanon (as much as she opposes the terrorists). This is a woman who has that attribute that so many lack: true empathy, divested of political or religious labels.

Sometimes it is almost too much to write about. Sometimes you wake up knowing that every day is a battle, not only in far-off lands where bombs may casually eliminate whole families, but here and now, in your own soul, your own little life. Every morning you lie there, persuading youself to rise and don your armor despite the pain that is like iron in your chest. You rage against the loss of hope and trust, the betrayals and hurts that the world inflicts, the infinite possibilities of damage and deceit that may still lie in wait for you in the course of the day. And then you get up. You get up, and you do what you have to do, and you don your armor and strengthen it in weak places, and you go off into the battlefield of the day.

So here I am, raging strong.

posted by Vandana 6:57 AM
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