Thoughts on MaBloWriMo

At some point when Thanksgiving break began, I felt exhausted and didn’t want to any blogging. Forcing myself to blog was beneficial for the first two weeks, but after some point, it seemed counterproductive. Taking an excess amount of time to think and develop ideas reaps no benefits, as the long hiatus of this blog demonstrates. But forcing out every newly-thought idea out before it’s full conceived results in some poor writing. I would like to go back one day to revise some of the blog posts I made. Even so, attempting to blog everyday has taught me valuable lessons about what I want to do with this blog. So hopefully this project will produce some fruit in terms of new posts. Thanks for reading!

Socrates, A Personal Role Model

I’ve been listening to the History of Philosophy Without Any Gaps podcast, and arrived at Socrates. Hearing about Socrates and Plato’s dialogues reminded me of reading those same dialogues when I was younger. In particular, reading Plato’s Apology made a large impression on me. The idea of knowing that one knows nothing had an appeal, especially as a child who felt that the world was vast and largely unknown. When I grew a bit older, I felt that it’s possible to have a systematic structure to everything one wants to know. But now, I’ve returned to original position of Socratic ignorance, albeit with perhaps a bit more wisdom. I find myself watching for underlying assumptions that people make when giving arguments, especially ethical or political, and given my mathematical training this becomes easier. Since I myself have various biases and assumptions of which I am largely unaware, I try to point these out, if I point at all, as modestly as I can. But it surprises me when people declare the certainity and obviousness of ideas and theories that seem anything but. Maybe they’re correct that certain fields of knowledge are complete, but I find such declarations to be hasty. Perhaps, it’s from mathematical training or even some postmodern thought slipping in, but the idea of being wholly ignorant, at least in philosophical questions, still shapes my thought.

The Concept of Statistical Independence

In probability theory, the notion of independent events is a purely computational notion. The probability of both A and B is the probability of A times the probability of B. Mark Kac, in his amazing monograph Statistical Independence in Probability, Analysis and Number Theory notes near the end of the first chapter that the intuitive notion of events being independent (as unrelated events) served to hinder considering probability as strictly mathematical since all the examples (until Borel) had some empirical baggage so to speak. I am reminded here this assumption of the relation between this intuitive concept and the computational concept of independence is not deducible, but an empirical observation. I have no conclusion to draw, just that I find this association strange.

The Trouble with Thanksgiving

The Thanksgiving holiday, taking place on the fourth Thursday of November, has a troublesome position in the academic calendar for quarter-based schools. It occurs too close to the end of the quarter and destroys the sense of rhythm in the lectures at a crucial position near the end. The week is nearly cut in half, and given that regardless of when your class takes place, most everyone will want to take the Wednesday off. By the time everyone gets back from their overeating, no one wants to learn anything and are in anticipation of the winter holidays (and living in dread of finals). So the rest of the quarter becomes difficult to try to get any last minute material. So in other quarters, late-quarter fatigue does set in anyway, but the holiday is still rather disorientating.

Worse than the lectures being all shot, we have the difficulty of what to do about homework. Do you become the villainous, cruel teacher who would dare assign homework to be due after Thanksgiving? But again, next week is the last week of class, so the most reasonable time would be after Thanksgiving to make homework due.

People suspect that if instructors and professors were to remember their own student years, then they’d have more sympathy and act kindly. But in this case, it backfires. I had to study and do homework over the break and go to class on Wednesday. So the recollection only makes me angry that when the roles are reversed, I am still worse for it.

Okay. Rant over.

Constructions involving Straightedge, Compass and A Conic

A paper “On Points Constructible from Conics” by Carlos R. Videla, published in Math. Intelligencer came to my attention recently that answered a question that a few of my colleagues had asked back in my undergraduate algebra course and later in graduate school. We know from Galois theory that some ancient Greek open problems of the duplicating the cube, trisecting an angle, constructing a 7-gon, and squaring the circle are impossible when limited to a straightedge and compass.

It turns out that if one extends the notion of constructibility to include any type of conics whose foci are constructible points and whose directrix is a constructible line, then these problems can be answered in the affirmative (except of course squaring the circle, since pi is transcendental). It turns out that this extended notion allows not just for square roots but cube roots as well. Therefore a number is constructible if and only if the degree of the field extension over the rationals is 2^n3^m. So we see that as a natural generalization of the theorem about constructible polygons, the n-gon will be constructible provided that n=2^r3^s p_1\dotsb p_t, where p_i-1 factors into powers of 2 times powers of 3. It seemed strange at first that this would involve cube roots, since conics are zeroes of degree-2 polynomials. But the method is to exploit the asymmetry of the non-circles to get a linear equation is equal to a quartic, providing cube roots.

In addition to this paper, there was another paper recently posted to arXiv (arXiv:1210.8046) that proves that only one conic (any nondegenerate, noncircle will do) is enough to obtain the constructibility with all conics, and hence the classical Greek problems are solved in the affirmative with the addition of a single conic (say an ellipse).

Trying to do Too Much

A recent graduate of the University of Oregon visited not long ago, and as a fellow go (game) enthusiast, I had the pleasure of playing a small (13 x 13 board) game with him. At some point a crucial body of stones was split in half. I had the option of trying to hold on to the side, or connecting my stones and keeping something in the middle. I had tried to keep both and ended losing both. Thus the game ended.

Today my advisor explained to me that the approach I had been taking with a problem was mathematically impossible to actually accomplish. I wanted too many conditions to be satisfied to ever be true. If I had let go of something and used an approximate version of what I was doing, it would have been fine.

Someone told me once that the a game of go reflects the character of the players. It is easy in life, as in go, to say that if things had gone ideally, if I had thought the right thoughts and time was conserved correctly, everything would have gone well. But circumstances will also be less than ideal, there is never enough moves, nor enough time. The problem is with “everything”.

Chairs Keeping Breaking on Me

Today during class, a chair broke underneath me. The cushion was loose and I fell on my rear. Incidentally this reminded me that this has happened at least twice before. Once in elementary school, a bench broke while some friends and I were sitting on it. Then in my first year of undergraduate study, when I was getting to Spanish class, my seat broke in what I thought was going to be a rather embarrassing moment. It turns out the seat was already broken and propped up to look like it was normal. One of my colleagues jokingly called out the teacher for letting me sit there. Is this a normal phenomenon? Are chairs more fragile than I assumed? At this point, it would be a bit characteristic of me to some navel-gazing and suggest an analogy of how chairs breaking is sort of like the insecurity of our foundational assumptions, or something.

In a bit more serious news, Qiaochu Yuan, the original inspiration for me to do MaBloWriMo is calling the November project quits. Of course, his actually mathematical blog requires much more work than my ramblings about chairs, so his decision is perfectly reasonable. My much less ambitious project of writing posts of this¬†quality has me exhausted and tapped. So when I read that he called it in, I felt tempted to do the same. Of course the difference in quality alone makes me feel like I’m just being a whiny baby. But there is something to be said of not posting poor quality posts for the sake of posting every day. But I will continue, because I’ve used “quality” as an excuse not to update this blog for nearly a year. So I’m afraid the internet will have to tolerate a bit more of my rambling for at least half a month more.