January 2007 Archives

MZ-F: "Why is everyone at this school so screwed up?"

Me: "I suppose that includes me."

MZ-F: "Well... yeah."

I'm starting to wonder whether people in the humanities and social sciences (to a much lesser extent) actually do work. Seriously.

This was classic:

"But as habits go, Mr. Obama's smoking is less annoying than John Kerry's poetry-writing and less odd than George Bush's obsessive brush-clearing. American will have to resign themselves to the fact that no one is perfect, not even Mr. Obama. It has also emerged that his middle name is Hussein, and that his ears stick out. If this is the worst that can be said, so much the better for him."

--The Economist, January 20th-26th, 2007

Richard Feynman published a collection of his essays and interviews which he appropriately entitled, "The Pleasure of Finding Things Out." It's a great book, and everyone interested in science and math should read it. But the best part is perhaps the title itself. There really is some kind of addictive and strangely unique pleasure that comes from suffering through a problem and all of its brambles, suddenly rising above the thicket, and seeing the expanse of the journey from problem to solution laid out neatly. If there was anyone who understood this, it was certainly Feynman.

A couple of nights ago, I had been struggling through a tricky physics problem. For various reasons, I rarely work with other people on problem sets: I like to start working on them sooner in the week, I like be able to completely exhaust my creativity before resorting to that of others, and I think I learn better that way. An end result is that I'm horribly inefficient compared to most people. So I had probably spent two or three hours thinking about this problem, and was at an impasse. Walking back from the library around 2 AM, I had this sudden epiphany. Lo and behold, when I sat down in my room and reformulated my approach, the correct answer popped out.

As I put down my pencil, my clock hit 3:02 AM. A box enclosed my equations of motion, and the pleasure of finding things out finished washing over me. Anyone used to doing heavily analytic activities, like math or physics or writing, knows that the brain does not quite function normally at this time of night, after it has been subjected to many hours of work. Things seem strangely different.

I dropped into my armchair and stared at the wall for a moment. Now, I realized that the reason that physicists do physics instead of philosophy is mainly because they're not very good at the latter. I don't think the philosophers are very good at philosophy either, but they're certainly a bit better than the people who study physics. There is a lot of evidence for this claim, but a good example occurs when physicists learn Hamilton's principle. It basically states that particles always travel in a manner that minimizes a certain quantity. If you listen carefully, you can hear aspiring physicists arguing before and after lecture about how on earth these particles could possibly "know" the exactly correct path to take. The knee-jerk reaction is to slap these anthropomorphizing dummies in the face and demand to know why they suddenly think that particles are sentient beings. They've already learned mechanics once, but when we teach them the exact same thing over again using different words, they start thinking that dirt clods have brains.

I was thinking about this, and for the sake of argument, I supposed these people were right. Suppose there is some reason for the particle's motion. God would be one possible reason. I don't like thinking about God very much, because most of the relevant arguments are extraordinarily banal, but I suspended my moratorium on God for a moment.

Most major religions associate an afterlife with God. These religions usually privilige adherents with some sort of eternal bliss following death. Accordingly, nonadherents and sinners are punished to endless misery in some variant of hell. There are all sorts of terrifying analogies that Catholic schoolchildren can tell you to relate the meaning of spending "eternity" in hell. The Koran remarks:

"Know they not that whoever opposes and shows hostility to Allah and His
Messenger, certainly for him will be the fire of Hell to abide therein? That
is extreme disgrace". (9:63)

Now I think that all rational people agree that we cannot be certain of the existence or nonexistence of all this stuff. In the most simple terms, a logical person can choose to believe or disbelieve without being technically wrong. So we can attribute a utility stream to each outcome:

Case I: God exists. Nonbelievers face a very, very large negative utility after death. I hesistate to call it infinite (who knows, there might be some sort of diminishing marginal (negative?) utility of misery here...), but it is certainly so large as to render all mortal utilities vanishingly small. From what I hear about heaven, adherents receive some equally large positive utility upon death.

Case II: God does not exist. There is no afterlife, and the adherent has the same fate as the nonadherent. Post mortem utility is zero.

Now there are certainly costs associated with belief and disbelief in religion, which we must consider in our decision calculus. I argue that they are negligible for two reasons. First, the costs/benefits of belief and disbelief probably average out to be about the same for both groups. Disbelievers don't have pay for the gas to go to their place of worship, and they don't have to bother with all of these arbitrary moral injuctions. Believers feel good about god being "on their side," and they have the smug satisfaction of feeling more enlightened than all the unwashed heathen. Secondly, these mortal costs are trivial compared to the grand, cosmic costs in the afterlife. This is true unless the probability of the existence of god is vanishingly small, which we already asserted to be unknowable anyway.

The upshot, you ask? A simple glance at the figures show that it is clearly irrational to not believe in God. I'm disobeying the lessons of elementary economics.

At this point, I decided that it was time to go to bed. So I did.

In order to save myself from criticisms, I want to preface the entry below. I talk about this question of abstractness in education at the end. It sounds a little pretentious. So I want to clarify that I'm not suggesting that theoretical endeavors are any better than the concrete, applied, or computational. The point is simply to propose an alternate educational model, to question whether our current model is really the best. I talk a lot about "abstraction" because it seems to be the most complete and general way of teaching people about ideas and how to solve problems (in my opinion). This I take to be the general goal of education.

My blogging has been a bit sparse lately. It's ironic, though, because I've been thinking about a lot of blog-worthy topics lately. I think that my problem is that I've been arriving at more questions than conclusions. So this is one question.

I'm taking a course on classical mechanics this quarter. The main focus is on Lagrangian and Hamiltonian formalisms in physics, as well as some related topics like the calculus of variations, chaos, and rigid-body motion. The relavant point is that the alternative formulations of mechanics are entirely equivalent to Newton's laws. In fact, Newton's laws can be derived from the one simple postulate and series of transformations of Lagrangian mechanics. If these formulations are indeed equivalent, it raises the interesting question: how do we decide the order in which to teach them? The current state of affairs is to preach the Newtonian formulation as a mantra until the second year of the undergraduate physics curriculum, at the very earliest (less demanding physics programs may even postpone this until the third or fourth year).

Answering this requires that we define differences between the formulations and evaluate their pedagogical effectiveness. I will restrict myself to Newtonian and Lagrangian formulations, because that is what I know best. Newtonian mechanics is centered around the notion of forces causing acceleration on bodies (F=ma), from which the equations of motion may be calculated. I find two important characteristics in this view. Firstly, the form of the equations conceptually suggests that motion is determined by some sort of external agent on the system; the second law implies a causal mechanism. Secondly, Newtonian mechanics is coordinate-dependent. Consequently, it is invalid in noninertial reference frames, and has an explicit reference to the coordinates imposed on the system. The primary pedagogical implication of these two observations is that Newton's laws lend themselves to being intuitively obvious but rather tedious and awkward to use for all but the simplest situations. Even someone with an undeveloped physical intuition can understand the meaning and application of the Newtonian formulation. But this same simplicity, I think, can ultimately be limiting. The notion of what a force actually "is," I think is somewhat lacking. True, force is something proportional to acceleration, and I apply a force when I push you, but that doesn't really say much. What makes force important? It seems aesthetically undesirable to define our mechanics in terms of some quantity that we simply define for the job. The result is that students think of mechanics in terms of these superficial causal mechanisms that just happen to work for some reason, rather than being a result of the structure of space.

Lagrangian mechanics, on the other hand, is much more abstract and useful than Newton's laws. It has one postulate: the path that any particle takes is such that the action is minimized (where action is an integral of kinetic minus potential energies over a path). At first, this may seem more arbitrary than Newton's simple laws, but it seems that this postulate ultimately says something about the nature of space, whereas F=ma does not. Considering the transformations of a particular space, conservation laws follow immediately, and are not introduced in a sort of deus ex machina fashion. Furthermore, there is the advantage that it is, in some sense, independent of coordinates and reference frames: it works in noninertial systems and arbitrary coordinates, without much trouble. It also leads to tremendous simplification of calculations. From a pedagogical point of view, the downside is its analytical complexity: although fairly straightforward, it requires more advanced mathematics than the laws of Newton.

It seems that the teaching of physics follows a primarily historical development. We first teach Newton's laws, then E&M, then Lagrangian mechanics, then more advanced E&M, then quantum mechanics, then some miscellaneous modern topics, then after a review of basic physics in the first year of grad school, quantum field theory, string theory, and the latest developments. On some level, this makes sense. If man discovered physics roughly in this order, then the student will be able to learn everything in a perfect and logical progression from the historical approach, if you throw out a few of the logical inconsistencies. Everything will follow naturally from everything else. But the real question--the biggest question of all: is the historical approach the best? Strictly speaking, I see no reason why this answer should be yes. If our goal is to train physicists, shouldn't we be trying to force them to think the way physicists actually think? Doing this requires that physical intuition about symmetries, fields, and other important things be stressed from the very beginning, rather than slavish devotion to calculation of uninteresting and unmeaningful problems. There is this problem in physics education, where each class tends to teach something that contradicts the previous class. Special relativity contradicts classical mechanics, quantum mechanics contradicts both, QED supercedes quantum mechanics and electrodyanmics, and so forth. The historical way theories were developed requires that this always be the case to some extent. But it shows how the physical intuition developed at each level leaves one unprepared for the next level.

Considering these observations, it seems theoretically more desirable and useful to teach Lagrangian mechanics first, or at least treat Newtonian mechanics as secondary to it. But there is the practical problem of mathematical complexity. It is unreasonable to expect any more than a handful of high school students to comprehend the calculus of variations. Since most people take their first physics courses in high school, our program appears doomed to failure. But most people only get serious about physics in college, so we will limit ourselves to that stage. I see no reason why there could not be a small class for highly motivated students of physics that starts immediately with Lagrangian mechanics. I am effectively in that situation right now. I was able to place out of the first-year honors physics course, and I started with a course on elementary quantum mechanics (which incidentally was extremely easy) and this current classical mechanics course. There are many first-year students at good colleges and universities possessing the mathematical machinery needed for such concepts. The rest of the students in physics could continue with their current Newtonian-based, historical program, which seems to currently work okay, and which might be more approachable.

Lastly, I will anticipate one major criticism of my plan: abstractness. All you boring people say: this is too abstract! As we teach children arithmetic, then fake algebra, then calculus, then analysis, then abstract algebra, etc., so it is with physics: we must start with the most concrete, then work our way up. You say: analysis is harder than doing those stupid integrals from calculus, so it must be taught later. But is it really harder? Or is it just harder because your mind has been indoctrinated from day 1 to think of all math in terms of 1st grade arithmetic: if I follow a set of rules, I will calculate the correct result--the end? This is the general question I have been thinking about lately: why do educational programs move from concrete to abstract? Can people learn something by starting with abstract concepts, and developing the concrete applications later on? For a lot of aesthetic and practical reasons, I think they should if they can, but I don't know if it's possible. I really have no idea what the answer to this question is, nor how to answer it.

To finish off this discussion, here is one piece of anecdotal evidence that I have thought about a lot. It turns out that the University of Chicago's mathematics department is a fantastic place to be an undergraduate, for many reasons. One of these reasons, I think, relates to abstractness. Each year, all first-years take this mathematics placement exam that covers everything from arithmetic to analysis. The students who are very good at math and have at least some experience with basic calculus are placed into this year-long course colloquially called "the 160s" (i.e. honors calculus). From the very start, they only study the theoretical aspects of one-variable analysis: sorts of things like sequences, compactness, continuity, integration, Lebesgue measure, etc. Everything beyond the 160s is similarly abstract. In other words, abstractness is taught from day 1. People do quite well in such an environment; they tend to love it. It is a revealing fact that the U of C has more undergraduates who major in math than in english. This is opposed to most places, which have this battery of computation-centric courses on differential equations, multivariate calculus, and linear algebra, which are sometimes fun but not very abstract. They're useful for things like physics and engineering, but its the kind of thing that any competent physicist or engineer should be able to pick up "on the street," so to speak.

I'm not sure if that says anything definitive about my final question, but it suggests that abstractness isn't always unnatural or counterintuitive to the unintiated mind.

"If you'll permit me to use one of those formulas which come to me as I write my notes, human life could be defined as a calculus in which zero was irrational. This formula is just an image, a mathematical metaphor. When I say "irrational," I'm referring not to some unfathomable emotional state but precisely to what is called an imaginary number. The square root of minus one doesn't correspond to anything that is subject to our intuition, anything real--in the mathematical sense of the term--and yet, it must be conserved, along with its full function."

Lacan, Jacques. 1977. Desire and the interpretation of desire in Hamlet. Translated by James Hulbert. Yale French Studies 55/56: 28-29.

See note 47 in:
Sokal, Alan. "Transgressing the Boundaries: Toward a Transformative Hermeneutics of Quantum Gravity." Social Text (Spring-Summer 1996).

People typically consider significant anniversaries to be multiples of five. School reunions inevitably happen in multiples of five, and for the fiftieth anniversary of a wedding--the mother of all multiples of five that can occur in one's lifetime--one gives the best gift of all: gold. Yet in this sea of fives and tens and twenty-fives, the number four stays afloat. The dominance of fives makes a lot of sense: we have five fingers on each hand, fives toes on each foot, and a base ten number system. But where to these fours come from? I have no idea, but fours crop up everywhere. High school and college are both four years long, presidential terms are four years, and even congressional elections happen every two years--a factor of four. Twos and fours... it's a computer programmer's delight.

The point of all this is that four years somehow strikes me as a very significant number, and two days ago "überfluss" (formerly "Adams Blog" (the lack of the apostrophe was considered a feature back then)) turned that age. Its average of one entry every 1.81 days hasn't quite kept pace with the blazing pace of one entry every 1.24 days, which prevailed in the first year of its existence. Yet I think it has nevertheless been remarkably consistent. If you don't believe me, just consider the fact that poor readers like you have wasted their time posting an average of one comment every 21.4 hours for the past 4 years and two days. That is a lot of comments.

So it has been a success. There is no reason why the next four years should be any less of a success. Although the posting volume will inevitably drop due to a decrease in leisure time, I hope that the consistency will still remain.

I once proposed that our high school class celebrate its reunions at prime numbers instead of these boring intervals of fives. As you can imagine, the proposal met with stiff--unanimous, actually--resistance. So maybe powers of two are a little sacrilegious, but I think it makes a lot of sense. After four years high school was done, after four more years college will be done, eight years after that I hope to god that graduate school will be well finished--if I go down that route. The real point of all this is that multiples of five make absolutely no sense. It's a relic of an attitude of numerical unsophistication and rigidity. I see no good reason for its unearned dominance of our celebratory schedules, so I toast the new year and the next four years of writing, typing, and arguing.