LaTex Happiness

Whoa. Thanks to Lee (and the folks at watchmath.com), I can now put stuff like this

$\int_{a}^{b}f(x)dx$

on my blog. Aren't you glad? (Note that if you view this in an RSS reader it likely won't come up right. Sorry!)

Today's Accomplishments

Life is hard around here lately. School kind of ate me, and I'm waiting to see if I will agree with its digestion. (I'm not sure which outcome to prefer, frankly.) I'm not technically "behind" at anything right now, but I still feel I'm being crushed. At any rate, here are my accomplishments for today:

• Wrote up the last lecture of analysis notes. (I rewrite, with additions/modifications/clarifications, the notes from each analysis lecture. This is critical to my understanding and ability to reference the material later. Lately I was several lectures behing, but I am now caught up.)
• Talked to Ed about a couple of things I didn't understand from the analysis notes. We were able to resolve them together.
• Looked extensively at the two problems on this week's analysis homework. I solved (I think) the first problem, which I typed up in LaTex. The second problem was less tractable but I identified some of the difficulties I have with it, and thought about those for a while.
• Typed up the Logic homework problems. I haven't started working on this homework yet, and I think we haven't covered almost anything that is on the homework (due 11 days from now), but at least the problems are typed up, so I can modify this document when I'm ready to start working. Also, by typing up the problems, I now have a much clearer idea of what the homework entails. ("Entails," ha ha ha.) These homeworks have typically been 15-20 pages of dense handwriting on college-ruled paper, so I've decided to experiment with typing this one up instead of killing my hand. (My right middle finger has developed a chronic bruisey ache when I write by hand for more than an hour or so at a time.)
  
• Tested the draft lecture notes I wrote for the 10-minute mini-lecture on completing the square that I might have to deliver on Wednesday (but probably won't deliver until the next week). My first draft took 12 minutes to deliver, which is pretty good, so I revised it downward. The second draft took 14 minutes. Oops. I have done a third draft but I didn't have the heart to deliver this (complete with writing on the board, of course) to an empty room for the third time, so that will have to wait.
• Brought my Probability book home from school so I can finish up the homework due Wednesday (which was posted to our course website yesterday, when I did about 3/4 of it when I saw that it had shown up).
  

Despite the above, it hasn't felt like a very productive day. I can't tell what is or is not productive. I'm pretty sure I didn't work anywhere near 8 hours, which is a standard work day. My program doesn't seem, so far, to require that amount of work. Yet the amount of work that it does require feels like it might be beyond me. And yet, I'm basically crushing (with enormous effort) my classes so far, and, as I said earlier, I don't think I'm actually behind on anything. So I don't know what to think. I feel like a person driving a car that has lost control of steering and brakes but who happens to be going down a straight freeway with no traffic...for the moment.

At least this isn't as bad as Tuesday night, when, as I stayed up until 4AM to finish my logic homework, I kept thinking about how grad school was pitting the intolerable (finishing the homework) against the unthinkable (not finishing it).

Fiasco Week

The past couple of weeks, before this one, have been pretty easy. I knew that they were going to be pretty easy because I didn't have any logic homework due until this week and nothing big was happening in my pedagogy class, leaving only the normal weekly analysis and probability homework. And the first of these two weeks, I worked hard to make sure I was doing enough not to make this week hellacious. But last week I didn't do very well, and in fact I am not sure I accomplished anything at all Thursday through Saturday aside from attending classes.

This morning, our third mammoth logic homework was due. It had 10 problems. We usually have two weeks for these homeworks, but our professor was out of the country for a week, so we had three weeks for this. Last night I had finished the 8th problem by 8:30, so I had two more problems to do.

I finished (mostly) at 4:19 AM. And...ugh. That is just way too late to be up doing homework. Part of the reason it took so long is that sometime after 8:30 I just really broke down. I had a bad headache, I felt hopeless about the derivations that I had to do, and I just...I don't know. A friend from our program invited me over to her house, and I went, and working with her was great, but didn't prevent me from falling apart.

I skipped logic class this morning (got Ed to turn in my homework for me) and slept in until 12:30, then barely made it to my probability class on time at 2. (Thank goodness I had finished my probability homework, also due today, some days earlier.)

Tomorrow I have analysis homework due. We get this homework once a week and it's always one problem. Sometimes the problem is fairly tractable and other times it fills me with despair, but so far I have always gotten them done on time, correctly, for full credit, so that's promising. This is my little mountain to climb each week, and doing them, and doing them well, fills me with a lot of joy every time.

I am almost always either completely finished by Monday or I basically know what I'm doing and just need to clean up the execution a tiny bit during the week. But even though this one is due tomorrow, I still don't know how to do it. I did work on it a little bit (read: four pages worth of notes' worth of work) on Saturday, but I didn't get anywhere with it. I do have things I can try next, so I don't feel hopeless quite yet, but I'm not in a great position.

Another thing I do every week is neatly rewrite my analysis class notes, filling in the missing details and making sure that I understand them. I'm three lectures behind on doing that (there are two lectures each week), so that's not great either.

I think one thing that I need to do is regularize my sleep schedule. I have morning classes M/W/F but only afternoon classes T/Th so it's always very tempting to sleep in on those days, especially if I've stayed up late the night before working on something, but really in any case. But I don't think that's doing any favors for my productivity overall, because it means there are more days on which I feel disoriented due to getting up at a strange time.

I am also thinking of giving up caffeine (for the umpteenth time). It's getting to the point where I feel mentally dull all day until I have my tea, and that's not good, and last night's headache may have been caffeine-imbalance-related as well.

Now it's time for me to go tackle the analysis homework for real. What's unfortunate is that I am much more willing to work on something that isn't due yet than on something that is due soon. I don't like the feeling that I have to figure this out in the next, say, eight hours in order to have a legitimate shot of being able to turn in something decent tomorrow, and it makes me not want to look at it at all (or, you know, not yet).

Looking forward, next week should be a bit easier. We don't have a new logic homework yet, possibly because we have a (small) paper due in 2.5 weeks, and the only big thing I need to do other than next week's analysis is prepare and be ready to deliver a 10-minute mini-lecture on a college algebra/pre-calc topic for my pedagogy class. That means I'd better work hard on that paper next week.

The Axiom of Choice

You can't get too far in analysis without running into the Axiom of Choice (AC), which is an easy idea to explain but deceptively tricky to grasp, I think. (Analysis, for those who aren't aware, is basically the study of functions - it is what calculus is called when it gets theoretical.) I've wanted to write about AC for a while.

What the Axiom of Choice says is that if you have an infinite collection of nonempty sets, it is possible to choose an item from each set. So if you had, for instance, an infinite set of sock drawers, you could choose a sock from each drawer.

There are two "choice" types of situations where you don't need AC. If you have a finite number of sets, no matter how many, then you don't need AC. You can use the principle of mathematical induction instead. That is, you can say, basically, OK, I can choose something from the first bin because, duh, it's not empty. Then, if I've chosen something from some number of bins up to this point, I can always choose something from the next bin, because again, it's not empty. But even though this works for any finite number of bins (even one billion bins), it doesn't cover an infinite number of bins.

You also don't need AC if you have a specific method of choosing from the sets (bins). For instance, if you have an infinite collection of pairs of shoes, you could say, "From each pair, choose the left shoe." That's basically creating a function from the pairs to the chosen objects, which is what we want. (AC says there is such a function whether we can define it explicitly or not.) People often contrast shoes with socks to explain this difference, because shoes have a right and left and so there is an explicit function for choosing, but socks are undifferentiated.

Of course, AC is usually used with sets of numbers, not sets of socks, because there are not actually an infinite number of socks even in the entire universe, as best I'm aware.

Now, if we were talking about sets of natural numbers (subsets, that is, of {1, 2, 3, ...}) we could just say, "Always choose the smallest one." Every set of natural numbers has a smallest element. This property is called being "well-ordered."

The real numbers, though, in their normal order, don't have this property. There isn't a smallest one of all, and there are a lot of sets of them, even bounded sets, that don't have a least element. For instance, "Every real number larger than 2" doesn't have a least element. (2 isn't in the set, so that's not it, and no matter how close you to get to 2, even if you pick, say, 2.000000001, there is always a smaller one still in there, say 2.0000000000000000000001.)

The Axiom of Choice is equivalent to saying that the real numbers are well-ordered. It's not true in their normal order, but AC says that there is some order you could put them in such that every subset of them would have a least element. (It's sort of a crazy idea - don't try it at home. AC doesn't provide such an order, it just claims that it exists. In fact, if we could define the order, we wouldn't need AC at all!)

To see the equivalence, let's say you had an infinite collection of sets of numbers, and you wanted to choose a number from each set. If you have well-ordering then you can use the rule "always choose the smallest number."

Similarly, if we have Choice, and we want to well-order the reals, we can first choose one to be the lowest one, then choose another one to be the next lowest, and so on ad infinitum.

So why this is interesting? First, AC is an Axiom. That means you can't prove (or disprove) it from anything else in the normal theories we use about numbers. It's just an assertion from the heavens. And while most axioms that you commonly encounter (such as that two points determine a unique line, or that a*b = b*a) are what we might call "obvious," AC is...well, is it obvious to you?

In fact, its use is rather contested.

If you don't use AC, then you can't prove a lot of the important theorems of calculus. And that's not just a matter of theoretical concern - we use calculus all the time to solve all sorts of problems, and it demonstrably works. Calculus is important, and it would be nice to think that it has a sound theoretical basis and isn't just a bunch of malarkey that works by chance, or for reasons beyond human comprehension.

On the other hand, if you do use AC, then you get some crazy results like the Banach-Tarski paradox. Those guys proved that, using AC, you can cut a sphere into a finite number of pieces and then reassemble the pieces into two spheres the same size as the original, which is more or less obviously not true. (The way the cuts are done is not something we can actually replicate, even though it is a small number of cuts, so this isn't an empirical question.)


So, there you have it: the Axiom of Choice.

An Idea

I never write stories or even try to write novels, but I was thinking about this today. It would be funny if you had a story involving a person (like an agent) time-traveling back to Nazi Germany to complete some mission, and the mission was put into peril when they had to wait for a late train. Imagine the annoyance at learning that the canonical one good thing about life under fascism was not true.

(Come to think of it, that is sort of how I feel when Republicans are not fiscally conservative.*)

(*No comparison of Republicans to fascists is intended or should be inferred.)

The Working-Procrastination Continuum

My work habits have definitely changed a little bit since I started grad school, shifting towards the better end of what I see as a continuum between working and procrastinating that goes something like this:

Flow: You're working and not even thinking about not working. You might not notice that you're getting hungry or stiff, and when it's time to stop, you wish you could go on. If you do take a break, you spend it wanting to get back to work.

Work is Work: You're working pretty steadily, but it's rough going. You take breaks when you can, and think a lot about how much longer you have to go, or how much more you need to do.

Pretending to Work: You're sort of doing some work, but you stop every few minutes to check email or play solitaire or stare into space. You're trying to get settled down and do some work, but not much is being accomplished.

Trying to Get to Work: You have a definite plan to start working, but you're trying to pry yourself out of bed/away from the TV/off the Internet. There might be a couple of things you need to do first, like clean off your desk or get a glass of water, but you're not quite doing those things yet. But you will soon - honest!

Procrastinating: There's something you could, maybe should, be working on, but you figure you can work on it later, maybe tomorrow, maybe next week. You definitely plan to do it, there's no doubt about that, but not right now.

Pretending to Procrastinate: You claim that you're going to do something, but if you look into yourself, there is no plan at all for getting it done. You might be in a sort of passive rebellion against doing it. There is no time that it could occur to you to work on it that you would actually then go and actually work on it. It is not possible that the conditions under which you would do the work could occur. Some change in attitude (perhaps partly unconscious) would be required in order for it to happen.

Refusal/Blowing Off: You consciously have no intention of doing a particular thing, though you realize that in some sense you should. Perhaps you've given up because there is no longer enough time to get it done before it's due, and it won't be accepted late, or maybe you've just decided it's not a priority for you.

I used to spend the bulk of my working hours in the range from "Pretending to procrastinate" to "Pretending to work" range. I find that, now that I'm in school, I'm never (so far) pretending to procrastinate, and most of my work times are in the "Trying to get to work" to "Flow" range. It's hard to distinguish between procrastinating and just not working right now in my current life, since I always have work that I could be doing, and yet I don't need to work 12 or 16 hours a day either. But cutting out that "pretending to procrastinate" stage is a big deal for me, and spending more time in the various working stages is great.

I still spend a vast amount of time in the "trying to work to work" and "pretending to work" phases. I'm not sure how to get better at that.

A Knock at the Door

Around 4:15 this morning, I was dreaming of something with the feel of fractions, or nested intervals, or cups of varying sizes. Suddenly, something happened whose translation into the world of my dream was alarming, necessitating some sort of action. A couple of minutes later it happened again, and I put words to it: someone was knocking on the door.

I bolted upright, eyes open, heart racing. What did this sign mean? Surely it required a response, but what kind? "Someone's knocking on the door. What is - why?!" I asked out loud to Ed, who was still asleep. I patted my bedside lamp to turn it on.

Once I figured out what door-knocking means in our world, I crept to the door and peeped through the fish-eye lens set therein. I saw, I thought, two women in their early 20s.

Should I open the door? I should not, I thought. My door has (I verified) no chain or little bar to allow it to be opened partway. Perhaps these women were the harmless front of some attack. Why were they knocking at such a late hour? I crept back to my room.

I am sure they saw that I had turned on a light. They knocked again, louder, and again a minute later. They were knocking quite violently. Did they need help? Had they been attacked, raped, left abandoned at my complex? Did they hope for me to call the police, a taxi, their mom? Was I prolonging their plight by ignoring them? Were they our downstairs neighbors, dealing with a water leak?

Ed sat up in bed, dazed, Frankensteinian in his sleeping mask and earplugs. He thought it was morning. What was happening?

"Someone is knocking on the door," I said. "I don't know what to do. I think I'm going to call the police."

Yes, someone is knocking on our door, and they won't stop, I imagined saying. I don't know who they are or what they want, but they won't go away. Maybe they're in some kind of trouble.

"Unless you want to answer it," I said. I told him what I had seen of the knockers. He crept to the door and back. He hadn't seen anyone and thought they had given up - he heard them knocking next door.

Holding my phone, and knowing they were no longer at our door, I opened our door and stepped partly out onto the walkway. A woman stood alone outside of the next door down. She saw me but said nothing.

"Did you need something?" I asked.

"Yeah," she said casually. "My friend lives here." She was pointing at the door. "Do you know Jared?"

"No," I said.

"At all?"

"No," I said.

She was silent.

I went back in and to bed. It took me a while to fall asleep again.

Library Excitement

Today I went to the math & science library at school to get some books I had identified as possibly useful for the 1000-word paper I need to write about a 19th century logician. I visited that library when I was here in April, but hadn't been since school started.

The main library here seems very nice and spacious, but the science library is byzantine, cramped, low-ceilinged, and noticeably fluorescently-lit. (Of course, everything on campus is fluorescently lit, but it's not usually objectionable.) However, there were multiple shelves of books about logic and logicians and I wanted to collect them all! It was very exciting.

Also, since I am a doctoral student, my books are not due until the end of the freakin' semester, which kind of blew my mind. Overall I am pretty psyched about the library situation, thus further proving, were it necessary, that I am a nerd.

Indian Cookery

I've heard many times that vegetarian Indian food can be very easy and cheap to make. I happen to love Indian food, and would be really excited to be able to make it, especially easily and cheaply. (I mean, what's not to like?) So this weekend, I googled around to try to find some easy recipes for daal (lentils) and aloo saag (potatoes & spinach). I read several of these recipes, and then suddenly they all kind of gelled together and I realized I didn't need a recipe. Or at least it felt that way.

So I got some things at the grocery store (Walmart, actually; I can't bring myself to pay grocery store prices these days) and tonight I made my food, roughly as follows:

Daal
1 lb. lentils
1/2 large white onion, diced fine
1/2 clove garlic, chopped
1 can tomato paste (the usual small size)
vegetable broth (about 4 cups, from a box)
butter
peanut oil
spices including chili powder, cinnamon, cumin, cardamom pods, cloves, etc.

I put butter and olive oil in the pan, cooked the onions and garlic at high heat, then put in the spices and stirred everything around in the spice paste until it seemed like going any longer would burn things. Then I put in the broth and tomato paste, and the lentils. I just cooked those forever (they took way longer than I expected!), adding more water as necessary, until they were done

Aloo Saag
4 small red potatoes, cut into bite-size pieces
1 large bag of frozen cut leaf spinach
1/2 large white onion, diced
1/2 clove of garlic, chopped
butter
peanut oil
spices including garam masala (2T), chili powder, and crushed red pepper
salt

I again started with butter and oil, and cooked the onion and garlic in that, and then added the spices, making a paste. I pre-boiled the potatoes (before starting with the skillet part, of course), let them air dry pretty well, then tossed them into the hot skillet with the spice paste. That mixture was a little bit dry, so I kept adding little bits of water to keep everything going. Once I thought the potatoes might have a nice crisp on them (they didn't, really, but whatever), I put in the frozen spinach and a bit more water, and just let that cook down, and then I salted the whole thing.

The lentils tasted amazing all along, but the potatoes & spinach scared me because they smelled extremely much like pumpkin pie, and I didn't want to taste it. I don't usually like it when savory foods go in too much of a sweet direction. But when I did finally taste a potato, my GOD! They were fantastic! Now maybe you just can't screw up potatoes, but the spinach in there was wonderful and...wow, it was just a great dish.

For dinner I had everything, with some brown rice under the lentils. It was really amazingly good, satisfying. I'm afraid of how much leftovers I have (a really enormous amount), despite the fact that Ed also dined on my stuff. The lentils were were well spiced, but very mild (of course), and the aloo saag actually succeeded at being slightly spicy. It wasn't Indian food like you'd have in a restaurant, and it probably would have been more Indian-tasting if I'd put in some cream, but it was recognizably Indian in its general flavor profile. So I have to agree with others: vegetarian Indian food is easy and cheap to make.