A note on the frequency of this Weblog.I’ve 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 InfiniteMarch 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 – 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 Cantor’s 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 God’s 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 didn’t 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 I’ve 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 “The World of Mathematics” 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 wasn’t 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 I’ve 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. I’m 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 I’ve 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 couldn’t 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 I’m talking about). One microscopic little girl in a pink turtle-neck and very curly hair charmed me by saying that her favorite story was “the one about the picky-pockets!” 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 wasn’t 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 don’t believe in dumbing down the vocabulary because I know kids can find out the meanings of words they don’t 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 . . .
