26 November 2005

Under the Banner of Heaven and child abuse

I would like to write about this article in The New York Times and this book, which I'm currently reading. But I have little directly to say, beyond that I think y'all should read both of them.

The two pieces report on practices of polygamy and rape, the article in poverty-stricken northeast Africa, the book in Fundamentalist Mormon communities in North America. Both discuss the horrors of the systems where thirteen is not an uncommon age for girls (or are they now women?) to become second or third wives to men many decades older. Both mention the rape, physical abuse, and complete lack of freedom meted out to the young wives. Krakauer's book, Under the Banner of Heaven, is much longer, and he easily fills it, giving him time to also discuss the history and community that leads to and perpetuates such heinous behavior.

I would like popular reportage like these to mention modern psychological research in abuse, and perhaps Krakauer does later in the book (I'm only half way through), but he hasn't yet, whereas he mentioned most of his main themes early on. In fact, a complete understanding of rape, incest, child physical and sexual abuse, etc. needs more than an understanding of a "culture of obedience," for which Krakauer rightly condemns both the Fundamentalist Mormon and the larger Mormon communities. Discussions of "violent faith" skirt the issue too. These are important issues, no doubt — certainly a liberating theology would not make its adherents so prone to such victimization — but to understand how adherents can be prone to perpetration as well as victimization also requires understanding cultures of abuse. I have no doubt that child abuse rates are much higher among the depicted societies than among societies with healthier and less violent relations; Krakauer does not shy from mention of childhood beatings and incestuous sexual abuse. What needs to be connected better is how abuse begets abuse, how abusive societies can self-perpetuate.

There's another, related phenomenon about which I'm curious. Well-developed psychological theories to which I subscribe, and which have strong empirical backing, causally link a host of pathologies to a history of childhood abuse, especially abuse by a caregiver. In particular, syndromes in which forms of dissociation are central — Dissociative Identity Disorder (formerly known as Multiple Personality Disorder), for instance, is an extreme case, but also hearing voices, certain forms of Post Traumatic Stress Disorder, amnesia of abusive events, and possibly out-of-body experiences — are primarily caused by variously intense (and possibly fear-inducing or violent) betrayal traumas. So, what about receiving Divine inspiration, a la Mohammed or Joseph Smith, or the many modern prophets described in Under the Banner of Heaven?

There is current, modern research on the similarities and differences between the voices a prophet hears and the voices heard by someone with a hebephrenic-schizophrenia-style dissociative disorder. But, with a long-range global trend of decreasing rates of child abuse, will we (or our descendants) approach the end of the age of religion? Will there be a time, after we have eradicated child abuse, when God and Satan no longer possess people and teach them to write great books? I myself would rather that than one mighty and strong building an even stronger patriarchal oppressive World Religion.

23 November 2005

Courses to teach

Some day, I hope, I'll get the chance to be a mentor at Mathcamp. In the meantime, I've started accumulating classes I'd like to teach. All are, not entirely unsurprisingly given my interests, motivated by physics.

  1. Order-of-Magnitude Physics

    I'm taking a course this quarter for first-year physics grad students called "Back of the Envelop Physics". It comprises a ten-week review of undergraduate physics and an attempt to hone physical intuition and order-of-magnitude calculation skills. We're roughly working out of Sanjoy Mahajan, et. al.,'s online "notes" — really a textbook — titled Order-of-Magnitude Physics: Understanding the World with Dimensional Analysis, Educated Guesswork, and White Lies. These notes are based on the course taught yearly at Caltech.

    Versions of this course have been taught in past years at Mathcamp, sometimes by Sanjoy himself. In a year in which Sanjoy is unable to visit, I'd like to take a crack at teaching mathcampers some general physics.

    I could easily imagine this as a two-day course or one lasting all camp; probably one or two weeks is preferable. The course would probably be roughly two stars. I'd stick closely to a teaching style I've heard termed "Berkeley Standard" (although I might have misheard — someone actually from Berkeley, please correct me), which is very successful especially for problem-solving classes. And order-of-magnitude physics is indeed a problem-solving area. To wit: an hour class would consist mostly of time for the students to work (in small groups at the chalkboard) on problems presented either orally or on a handout, and time would be made for students to present their solutions. For new problem-solving methods, we might work a problem "lecture-style" first, to introduce the techniques. (For instance, I'd expect the students to be able to estimate how many trees there are in the U.S., but not to know a priori how to apply dimensional analysis to compute the size of a falling raindrop, or even to compute to drag-force on the raindrop.)

    After a few days of high-school level estimation and unit conversion, we would, however, need to go into some actual physics. So I would not be averse to spending half an hour occasionally giving lectures explaining the physics behind a phenomenon — surface tension, for instance, or sound or stars — so long as I can still spend part of class-time each day providing problems for students to solve. A good idea in order to get the right balance of lecture and work time would be to watch a few of Dorin's classes.

  2. Differential Forms and Hamiltonian Mechanics

    This four- or five-week four-star course with homework is based largely on a course Yasha Eliashberg taught Winter and Spring of 2004 at Stanford, helpfully titled "Topics in Analysis and Differential Equations with Applications". The goal would be to arrive at the punch-line which I would give in the opening-assembly advertisement: say you have a box with a slider in the middle dividing the box, so that one one side of the box is one flavor of gas and on the other side of the box is another flavor, and say that you raise the slider at time t=0; the gasses will diffuse together under the second law of thermodynamics, but provided that the universe is static, eternal, and Newtonian, there will be some (bounded, and without any probabilistic argument at all) integral number of years from now in which the two gasses are completely sorted.

    The proof of this rather surprising assertion is straightforward. Say there are N total gas molecules; then the current configuration of the gasses in the box is some point in 6N-dimensional phase-space (3N for momentum, 3N for position). And the time-evolution of the system is given by some (in general very complicated) Hamiltonian, and its interactions with the canonical symplectic form \omega=dp\wedge dq. But the box is compact, and provided there are no outside forces on the box, total energy is conserved and finite, so the path of the system through phase-space is confined to some compact region. However, the vector-field (aka differential equation) determined by H and \omega (exactly \dot{p} = \d H / \d q; \dot{q} = - \d H / \d p) preserves the volume element \Omega = \omega^{3N}. Thus by pigeon-hole, the infinitely many images at time t = n years of a ball of size \epsilon around the initial state (\epsilon chosen so that all near-by states are also "sorted" states) cannot be all disjoint; running time backwards gives us a time many years from now in which the gasses are within epsilon of the current state. (There are, of course, some subtleties in the last step of this argument; you actually need a much more detailed description of volume-preserving flows in compact space, in which you, say, take the path through your starting position and consider its closure, and then argue that the flow restricted to that closure is still volume-preserving. But that's not the interesting physics.)

    I'd hope to present this proof at the end-of-camp "Thirty Proofs in Thirty Minutes", but to get there I'd need to develop quite a lot of mathematics, and it would, admittedly, be a challenge to fit it all in in four or five weeks. I would assume that students have seen high-school one-variable AP calculus, and are reasonably comfortable taking on faith that words like "vector", "volume", etc., all make sense in high dimensions. I'd probably begin by "reminding" students about tangent spaces and vector fields, and I'd do so rather intuitively. (I remember one class in which the teacher began with "review" and actually defined what that word meant: these are things that I realize you don't understand, but will pretend that you do anyway. I will probably begin my class that way too.) I'd talk about coordinate maps, fields, and partial derivatives, probably, and then I'd take some time to discuss dual spaces. Because what I want to be able to describe are cotangent bundles on (smooth) manifolds. I need concepts like the canonically-defined "d" operator, and I need to have differential n-forms at my disposal. My hope, of course, is to never evaluate an integral, or even define an integral of a differential form. I care that differential equations (which for my purposes are exactly the same as vector fields) have solutions, but I most likely will not prove this. No, this would be a lightning course based largely on pictures of line fields and anti-symetric bilinear functionals.

    Will we solve any actual physics problems? I'm not sure yet, since, as you can see, I have a lot of work before I'll understand the curriculum for the class. I expect that we'll work a few examples; say small oscillations of the coupled pendulum. And perhaps, if we're ambitious, we might derive Kepler's laws. It depends on how much time it takes to just develop the formalism.

    (Incidentally, I've only really worked this formalism in the Newtonian regime. How much carries over into the non-relativistic Quantum universe? I'd expect that quite a lot does, since there we also use words like "Hamiltonian", but now position and momentum are no longer simply coordinates. Indeed, all the previous scalar fields on phase space (and coordinates are (local) scalar fields that happen to intersect transversally and uniquely) are replaced by (Hermetian) operators which now no longer need commute. So even if many of the formulas hold, the argument would need to be completely reworked. And I can basically visualize stuff in 6N-dimensional phase space; time-perameterized paths in complex Banach space don't yet make sense to me.)

  3. Linear Algebra and Quantum Mechanics

    Of my three courses, this is the one I'm least sure of, and the one that would be hardest to teach, especially in four weeks to high schoolers. It would also be four-star.

    The goal of the course would be to develop both linear algebra and quantum mechanics, based on a claim I made last summer: "When we mortals look at Quantum Mechanics, we see differential equations. When God looks at Quantum Mechanics, He sees linear algebra." And it's true: the differential equations in QM are all linear, and since we're working over complex space, all diagonalizable, etc. And the mathematics used in QM is all just linear algebra. I'd go so far as to claim that you don't understand linear algebra until you've used it in QM, in the same sense that Newtonian physics is essential for actually understanding high-school calculus, and Maxwell's electromagnetism for understanding a first course in multi-v.

    The course would probably start with Schrodinger's pet cat, in an effort to understand what physicists might mean by "the cat is half-alive in the box". We'd be uninterested in the philosophical underpinnings of the thought experiment, or even in what it means to "look at the cat". Then we'd probably ask what "spin" is, and I'd borrow discussion from the beginning of Feynman's third Lectures on Physics. This would lead nicely into comparison with geometric vectors: I can plot various spin-states of an electron as points in (complex) two-space. Pictures, intuition, and unjustified (and simplified) assertions taken on faith can lead us to a description of the classic "two-slit" experiment, based on the discussion in Feynman's QED.

    Then we'd move into the more interesting world of particles in free space and in a box. I'd like to compare the infinite-dimensional (Banach) space of states of a free particle (perhaps in a box) with the finite-dimensional (normed) \R^n (and also the finite-dimensional complex space of spins). Most concepts appear in both regimes: bases, change-of-bases, linear operators, "matrix" representation, non-commutativity and Lie brackets, inner products, orthonormal bases, eigenvalues and eigenvectors, diagonalization, and more. A ten-week course that meets every day, or a five-week ambitious course that assumes a basic introduction to differential equations, would solve the harmonic oscillator, and hope to solve the equations of light propagation in a vacuum. I would most likely avoid any discussion of angular momentum and of relativistic effects like the fine structure constant, because those would take the course too far astray from the central goal of teaching both QM and LinAlg. Besides, in a group that includes students who've never seen LinAlg, such discussions will not be particularly understandable.

    This course (in fact, all these courses) needs some rethinking before it can be presented to Mathcampers. I would give problem sets daily, and expect students to do some homework, because I'd hope to cover more material than is actually doable in twenty or so lecture hours. And there's no way in hell that I'd offer both this course and the one on differential forms in the same summer.

20 November 2005

Love the Globe

Congress reduces its oversight role: Since Clinton, a change in focus

This is what the media should be doing. Keeping the government honest. Acting as gadfly. Pointing out when politicians fail to do their jobs.

Also in the news, The New York Times a few days ago reported that Some Judges Criticize Court Nominee on Civil Rights.

Understanding very different world-views

I've started reading Speaking Natalie, a blog by a friend of mine, fairly regularly. Eric is an amazing writer; what I enjoy most about his blog is that he presents an amazingly different world-view from mine. Eric understands his life in terms of God — he writes, for instance, that "God called me to Stanford" and that "I was not born into the warrior caste either, but Jesus treats me that way. By grace my past is wiped away and I am given a new name, a new identity, a new heritage." His understanding of morality is also strongly based on his understanding of Biblical meaning (he was a classics major as an undergrad (now he is in law school), and can base his understanding both on his profound Christianity and on historical knowledge). His moral and ethical system is consistent and evolving; it's different from mine, and extremely interesting, because he asks questions that are not distant from the ones I am interested in (and will eventually ask here, when I get time to write).

What has fascinated me also is Eric's understanding of gender. In a recent entry on chivalry, Eric writes about when he "first discovered that girls are different – and wonderful", and goes on to discuss how men ought to behave towards women. He writes that "there is something … well, magnificent about [women]" (emphasis in original). And this is so different from my understanding of gender, in which the (wonderful) variation between people is not largely based on gender.

It surprised me to realize this dimension of Eric's world-view. On the dance floor — I know him as one of my favorite people to dance with — his behavior is relatively gender-neutral. He has a number of friends, male and female, with whom he regularly dances, and he treats me and my dancer partner with equal politeness and friendliness. He knows I'm queer, and never batted an eye. He follows and leads. A simplistic understanding of sexism and gender opinions tends to pair Eric's style of "benign" sexism, in which women are put on pedestals, with "negative" misogynist sexism. These do in fact pair (c.f. Catholic worship of Marry), but Eric seems to have high rates of the former and essentially none of the latter. Sure, he most enjoys combat in his roll-playing games; he also lets a generic medical student take the pronoun "she." But the dance-floor is well-understood as a microcosm of society — one class this quarter in the dance department is premised on the idea that a good way to understand gender relations in broader society is to understand how they are reflected within social dance — and on the dance-floor Eric does not even seem to show benign sexism.

I have greatly enjoyed every dance I've had with Eric; I have not spent much time talking with him, and I expect that before we can get to enjoyable discussions of topics like those he and I blog about we will first have to work through our very different understandings of the world. In the meantime I will continue to devour Speaking Natalie, both for the detailed discussions and to try to understand what is to me a rather foreign mindset. I encourage my more liberal-secular readers to do the same.

07 November 2005

Lots of primes, but not so many proofs

There's a classic "topological" proof of the infinitude of primes given by Fürstenberg:
Consider the an unusual topology on \Z in which our basic open sets are both-ways-infinite arithmetic progressions (i.e. sets of the form \{ ... , c-d, c, c+d, c+2d , ... \}). It's clear the the intersection of two (and hence finitely many) basic opens is a basic open (or empty), and in general any open set is either empty or the union of possibly infinitely many of these basic opens, and is thus infinite. It's also the case that arithmetic progressions are closed. Since every integer that's not 1 or -1 is a multiple of some prime, the union over all primes p of the sequences \{ ... , -p , 0 , p , ... \} is exactly \Z \setminus \{ -1 , 1 \} . But if there were only finitely many primes, then this set would be closed, so \{ -1 , 1 \} would be open but finite, a contradiction.

I'd like to unpack this proof a bit. First of all, we don't need the topological language. It's enough to say that every number is divisible by some prime, so the only numbers that are in the intersection over all primes p of \{ np+k : 1\leq k \leq p-1 \} are 1 and -1, but that this set is infinite if there are only finitely many primes. In fact, we can do even better: \{ np+k : 1\leq k \leq p-1 \} contains the set of numbers that are 1 mod p, and the intersection of these sets is infinite if there are only finitely many primes (and none of them, clearly, are divisible by any primes). But why? One answer: The Chinese Remainder Theorem. Which is a heavy piece of machinery, and clearly more than we need. A better answer: because if some number c is in two arithmetic progressions, one with step-size d and the other with step-size e, then certainly c+de is also in both arithmetic progressions, as is c+2de, etc. And since 1 \in \bigcap_{p prime} \{ ... , 1-p, 1, 1+p , ... \}, if there are only finitely many primes, then 1+n\prod_{p prime} p is also in the set. But, of course, the proof doesn't even actually need all those infinitely many things that aren't divisible by any prime; 1 + \prod_{p prime} p is not divisible by any prime but is not one or minus one. Just as Euclid said.

Fürstenberg's proof obfuscates Euclid's original proof, but ultimately reduces to it. To unpack all the little claims in Fürstenberg's proof (e.g. that the intersection of finitely many basic opens is either empty or a basic open) requires Euclid-style arguments. As a way of thinking about this integers, Fürstenberg's proof is nice — describing all open sets, for instance, is nontrivial (right?) — and it shows some of the power of using topological language. But as a proof of in the infinitude of primes, it fails to say anything more than Euclid's (and, if claims that it's less explanatory than the earlier proof, perhaps it says yes; it is extremely obfuscated). Fürstenberg's proof of the infinitude of primes is simply Euclid's proof with garnish.