Archive for August 2012
Somehow I am saddened to hear of the death of William Thurston, even though I never knew him nor can I even understand his work.
For years I have been pointing people at this article of his, which I find illuminating and inspiring, even in ways that have nothing to do with mathematics: http://arxiv.org/abs/math.HO/9404236
5-paragraph version: http://mathoverflow.net/questions/43690/whats-a-mathematician-to-do/44213#44213
Riffs on his “derivative” exercise: https://profiles.google.com/114134834346472219368/buzz/hTVJiP5LoPb
User profile (http://mathoverflow.net/users/9062/bill-thurston):
“Mathematics is a process of staring hard enough with enough perseverance at at the fog of muddle and confusion to eventually break through to improved clarity. I’m happy when I can admit, at least to myself, that my thinking is muddled, and I try to overcome the embarrassment that I might reveal ignorance or confusion. Over the years, this has helped me develop clarity in some things, but I remain muddled in many others. I enjoy questions that seem honest, even when they admit or reveal confusion, in preference to questions that appear designed to project sophistication.”
“At that time, I prided myself in reading quickly. I was really amazed by my first encounters with serious mathematics textbooks. I was very interested and impressed by the quality of the reasoning, but it was quite hard to stay alert and focused. After a few experiences of reading a few pages only to discover that I really had no idea what I’d just read, I learned to drink lots of coffee, slow way down, and accept that I needed to read these books at 1/10th or 1/50th standard reading speed, pay attention to every single word and backtrack to look up all the obscure numbers of equations and theorems in order to follow the arguments.”
“When listening to a lecture, I can’t possibly attend to every word: so many words blank out my thoughts. My attention repeatedly dives inward to my own thoughts and my own mental models, asking ‘what are they really saying?’ or ‘where is this going?’. I try to shortcut through my own understanding, then emerge to see if I’m still with the lecture. It’s the only way for me, and it often works.”
“My first tenure-track job interview was at Cornell. During and after my job talk, most people were pretty quiet, but there was one guy who kept asking very penetrating and insightful questions. And it was very confusing, because I knew all the number theorists at Cornell, and I had no idea who this guy was, or how it was that he obviously understood my talk better than anybody else in the room, possibly including me.”
“When I grew up I was a voracious reader. But when I started studying serious mathematical textbooks I was surprised how slowly I had to read, at a much lower speed than reading non-mathematical books. I couldn’t just
gloss over the text. The phrasing and the symbols had been carefully chosen and every sign was important. Now I read differently. I rarely spend the time and effort to follow carefully every word and every equation in a mathematics article. I have come to appreciate the importance of the explanation that lies beneath the surface. […] I have little expectation for the words to be a good representation of the real ideas. I try to tunnel beneath the surface and to find shortcuts, checking in often enough to have a reasonable hope not to miss a major point. I have decided that daydreaming is not a bug but a feature. If I can drift away far enough to gain the perspective that allows me to see the big picture, noticing the details becomes both easier and less important.
I wish I had developed the skill of reading beneath the surface much earlier. As I read, I stop and ask, What’s the author trying to say? What is the author _really_ thinking (if I suppose it is different from what he put in the mathematical text)? What do I think of this? I talk to myself back and forth while reading somebody else’s writing. But the main thing is to give myself time, to close my eyes, to give myself space, to reflect and allow my thoughts to form on their own in order to shape my ideas.”
“A prominent mathematician once remarked to me that Thurston was the most underappreciated mathematician alive today. When I pointed out that Thurston had a Fields medal and innumerable other accolades, he replied that this was not incompatible with his thesis.”
He believed that this human understanding was what gave mathematics not only its utility but its beauty, and that mathematicians needed to improve their ability to communicate mathematical ideas rather than just the details of formal proofs.
Benson Farb, a mathematician at the University of Chicago and a student of Thurston, said in an email, “in my opinion Thurston is underrated: his influence goes far beyond the (enormous) content of his mathematics. He changed the way geometers/topologists think about mathematics. He changed our idea of what it means to ‘encounter’ and ‘interact with’ a geometric object. The geometry that came before almost looks like pure symbol pushing in comparison.”
Reactions like this are hard to explain: About the Proof&Progress article, http://golem.ph.utexas.edu/category/2006/10/wittgenstein_and_thurston_on_u.html#c005263 says:
“Reading Thurston’s response was one of the most stirring intellectual experiences of my life. It truly struck a cord with my own conception of mathematics. To me, it has the status that the Declaration of Independence has to many Americans, or the U.N. charter has to other global citizens.”
See these for a better summary:
Edit [2012-09-28]: OMG. Is this man human? He has also done work in computer science! A fundamental result in data structures (on the number of rotations needed to transform one binary tree into another) was proved in a paper by Sleator–Tarjan–Thurston (with 278 citations). According to DBLP, he has three STOC papers.
See also: http://arxiv.org/abs/math/0503081 (“Mathematical Education”)
Almost 8 months ago I left the first part of this post unfinished planning to complete it in the morning; seems I slept too long. (Or as this guy said after a 2-year hiatus: “Sorry for the pause guys. I was in the bathroom.”)
Recall: To get a power of 2 that starts with a prefix (like ), we want such that the fractional part of lies between those of and (all logarithms here to base 10), and similarly to get a Fibonacci number starting with , we want (with some hand-waving) such that the fractional part of lies between those of and . The more general problem is:
Problem: Given an irrational number and an interval , find such that lies in the interval .
Here is one method, based on Edward Burger’s book Exploring the Number Jungle. Let be the midpoint of the interval . Then we are trying to find such that is close to .
- Find a fraction approximating , such that . (These are the convergents of the continued fraction of , but in practice it seems you can also get away with taking semi-convergents that may not satisfy this property.)
- Let be the closest integer to . Note that this automatically means
- Write with . This you can do quite easily with the Euclidean algorithm.
- Then for and , we have (it is a simple exercise to prove this)
- This means that the distance between and is small, modulo 1. If this distance turns out to be still too large, start with a bigger convergent .
The thing we’re doing here is called inhomogeneous Diophantine approximation.
[A bit too much on a stupid pop song. Move along. :-)]
The Day Before You Came is the last song that ABBA recorded. It is interesting for more than being their swansong: it is also highly atypical of ABBA.
Here is the song (Link is to Dailymotion because Youtube has videos only of live perfomances):
The music and the video are non-ABBAish too (gone are the exuberance and the outlandish clothes, the video is almost entirely Agnetha with the others getting only a few seconds of screen time and no action), but confining ourselves to the lyrics:
Must have left my house at eight, because I always do
My train, I’m certain, left the station just when it was due
I must have read the morning paper going into town
And having gotten through the editorial, no doubt I must have frowned
I must have made my desk around a quarter after nine
With letters to be read, and heaps of papers waiting to be signed
I must have gone to lunch at half past twelve or so
The usual place, the usual bunch
And still on top of this I’m pretty sure it must have rained
The day before you came
I must have lit my seventh cigarette at half past two
And at the time I never even noticed I was blue
I must have kept on dragging through the business of the day
Without really knowing anything, I hid a part of me away
At five I must have left, there’s no exception to the rule
A matter of routine, I’ve done it ever since I finished school
The train back home again
Undoubtedly I must have read the evening paper then
Oh yes, I’m sure my life was well within its usual frame
The day before you came
Must have opened my front door at eight o’clock or so
And stopped along the way to buy some Chinese food to go
I’m sure I had my dinner watching something on TV
There’s not, I think, a single episode of Dallas that I didn’t see
I must have gone to bed around a quarter after ten
I need a lot of sleep, and so I like to be in bed by then
I must have read a while
The latest one by Marilyn French or something in that style
It’s funny, but I had no sense of living without aim
The day before you came
And turning out the light
I must have yawned and cuddled up for yet another night
And rattling on the roof I must have heard the sound of rain
The day before you came
It is putatively a love song, but it makes no explicit declaration of love. The entire lyrics of the song are merely a catalogue of an average day’s events. The song is a timetable. You’ve got to admire the sheer cheek of this, if nothing else. :-) (a la Peter Cushing lives in Whitstable.)
Yet it is interesting. I think this could be argued to be a first-class example of svabhāvokti, the achievement of a poetic effect by simply and ably describing things as they are. The author of the song describes her usual boring routine, presumably to contrast against her much-changed life after meeting the “you” of the song. This is an inversion of the much more common poetic convention, that of recalling time spent in love, time spent together, etc. (Bilhana’s चौरपंचाशिका Chaura-panchashika comes to mind.)
It is linguistically interesting as well: with few exceptions, the verbs are all accompanied by “must have”, “I’m certain”, “undoubtedly”, “I’m sure”, or “I think”. [I’m not sure what this characteristic of the verb is called. In the old days I think we’d just call it verb tense, but now “tense” is reserved for verb forms that indicate time, and “tense/aspect/mood” is used instead. I don’t know what this particular “must have”-type of construction is called: a Google search throws up terms like near-certainty mode, deductive, non-factual, evidential, presumptive, etc.] It’s as if the author isn’t sure. (It is remarkable that, in general, adding “I’m certain” or “undoubtedly” makes a statement less certain.) One interpretation is that the narrator after meeting her lover no longer remembers her life from before; so different it has become. But considering the level of detail, is this really possible? Besides, how much can a daily routine really change? The trains will still run at the same time, at any rate. A different possibility suggests itself: that the narrator is simply unreliable.
Suddenly a lot of things make sense: she is not describing her life “before you came”. She hasn’t met anyone at all, but is instead hoping to meet someone that will turn her colourless life exciting. The song is not reminiscences about a dull past (who would want to do that?), but she is instead imagining how different she will feel in the future after finding someone, yet so sad is her case that all she can imagine and describe is her present. The video lends credence to this idea: it starts with a scene of her daily commute, a guy appears and the video goes into a (presumably) imagined mode, before the guy disappears and she’s back at the same train station.
This would finally explain why the prevailing mood of the song, as experienced by the listener, is not one of love (it does not evoke śṛṅgāra शृङ्गार in other words): instead it is one of melancholy and weariness.
See also: Guardian article.