Dover

John Jones, over at the USDCMDP, recently posted a stinging critique of Intelligent Design, as well as a thorough recap of the events leading to the Dover, PA, creationism trial. No, Jones is not a liberal blogger. John E. Jones III is the U.S. District Judge for the Middle District of Pennsylvania, a Republican nominated by George W. Bush and confirmed unanimously by the Senate, and he presided over the trial. His ruling (pdf) is fascinating, and long (over a hundred pages). I suggest you skim it in its entirety — I did.

I'd like to highlight the conclusion of his ruling, both for its cogency and for its acerbity (I was going to use "piquancy", but The American Heritage Dictionary tells me that such use is archaic, and I'm anything if not with the times):

Conclusion
The proper application of both the endorsement and Lemon tests to the facts of this case makes it abundantly clear that the Board’s ID Policy violates the Establishment Clause. In making this determination, we have addressed the seminal question of whether ID is science. We have concluded that it is not, and moreover that ID cannot uncouple itself from its creationist, and thus religious, antecedents.

Both Defendants and many of the leading proponents of ID make a bedrock assumption which is utterly false. Their presupposition is that evolutionary theory is antithetical to a belief in the existence of a supreme being and to religion in general. Repeatedly in this trial, Plaintiffs’ scientific experts testified that the theory of evolution represents good science, is overwhelmingly accepted by the scientific community, and that it in no way conflicts with, nor does it deny, the existence of a divine creator.

To be sure, Darwin’s theory of evolution is imperfect. However, the fact that a scientific theory cannot yet render an explanation on every point should not be used as a pretext to thrust an untestable alternative hypothesis grounded in religion into the science classroom or to misrepresent well-established scientific propositions.

(Underlined citations in original; emphasis added.)

I've only read the beginning of On Pandas and People; its introduction, at least, allows that microevolution and divine creation are consistent. Indeed, microevolution has been observed and measured, and most IDers/Creationists endorse it, but they reject macroevolutionary theories of the origins of life. When Jones describes "evolutionary theory", however, he means it in the same sense as the IDers do (a la his Finding of Facts): the idea that modern species evolved over millions of years from nothing. What is elucidated in the ruling is not whether this claim is consistent with ID (and the Court is careful to make no ruling on the truth of the claims of the Theory of Intelligent Design proper), but that ID proponents misinterpret and misidentify scientific disagreements about the details of evolution's mechanisms as evidence against evolution, and moreover they misinterpret and misidentify evidence against evolution as evidence for intelligent design.

Jones goes on, getting more and more astringent as he approaches the end:

The citizens of the Dover area were poorly served by the members of the Board who voted for the ID Policy. It is ironic that several of these individuals, who so staunchly and proudly touted their religious convictions in public, would time and again lie to cover their tracks and disguise the real purpose behind the ID Policy.

With that said, we do not question that many of the leading advocates of ID have bona fide and deeply held beliefs which drive their scholarly endeavors. Nor do we controvert that ID should continue to be studied, debated, and discussed. As stated, our conclusion today is that it is unconstitutional to teach ID as an alternative to evolution in a public school science classroom.

Those who disagree with our holding will likely mark it as the product of an activist judge. If so, they will have erred as this is manifestly not an activist Court. Rather, this case came to us as the result of the activism of an ill-informed faction on a school board, aided by a national public interest law firm eager to find a constitutional test case on ID, who in combination drove the Board to adopt an imprudent and ultimately unconstitutional policy. The breathtaking inanity of the Board’s decision is evident when considered against the factual backdrop which has now been fully revealed through this trial. The students, parents, and teachers of the Dover Area School District deserved better than to be dragged into this legal maelstrom, with its resulting utter waste of monetary and personal resources.

To preserve the separation of church and state mandated by the Establishment Clause of the First Amendment to the United States Constitution, and Art. I, § 3 of the Pennsylvania Constitution, we will enter an order permanently enjoining Defendants from maintaining the ID Policy in any school within the Dover Area School District, from requiring teachers to denigrate or disparage the scientific theory of evolution, and from requiring teachers to refer to a religious, alternative theory known as ID. We will also issue a declaratory judgment that Plaintiffs’ rights under the Constitutions of the United States and the Commonwealth of Pennsylvania have been violated by Defendants’ actions. Defendants’ actions in violation of Plaintiffs’ civil rights as guaranteed to them by the Constitution of the United States and 42 U.S.C. § 1983 subject Defendants to liability with respect to injunctive and declaratory relief, but also for nominal damages and the reasonable value of Plaintiffs’ attorneys’ services and costs incurred in vindicating Plaintiffs’ constitutional rights.

Jones' ruling is, of course, a win for the Left and a major rebuff for the Right. But let's be careful before partying too hard. Liberals ought to support teaching Intelligent Design in schools. Students should read (excerpts from) this ruling, and learn about the history of the Evolution/Creation controversy in the U.S. They should have assignments that involve critically examining claims and evidence, comparing various versions of so-called evolutionism with various versions of creationism, and understanding how much of the dichotomy is false. Pedagogical studies have shown how best to convince kids of evolution's power as an empirical fact: respect them enough to teach them everything. When you hold something back, the kids know it, like refusing to discuss birth control or homosexuality in a sex-ed class.

The Dover ID Policy was a bad one, supported by religion and poorly instigated, and "Defendants are permanently enjoined from maintaining the ID Policy in any school within the Dover Area School District." But I hope that high-school biology teachers don't interpret this and other rulings as totally barring mention of ID in science classrooms (indeed, in Edwards (landmark 1987 case against Creationism), the Supreme Court found that "teaching a variety of scientific theories about the origins of humankind to school children might be validly done with the clear secular intent of enhancing the effectiveness of science instruction"). Of course, an already overloaded science curriculum does not need such extra material, so teachers who decide not to broach the subject in favor of other more worthwhile material are not shirking their duty. Still, a nice extra-credit assignment or option for a term paper would be such a compare-contrast exposition.

On this and other major national issues, Liberals are becoming stodgy and entrenched, giving up the role of social-change activists to the Radical Religious Right. We should be out there, empowering students to research and criticize lots of ideas (because such empowerment is a value in its own right, and incidentally those students will be more likely to decide that Conservatives' religious beliefs are wrong). As soon as IDers can claim that their policies are "for the secular purposes of improving science education and encouraging students to exercise critical thinking skills" (even though the Court found in this case that "Defendants’ previously referenced flagrant and insulting falsehoods to the Court provide sufficient and compelling evidence for us to deduce that any allegedly secular purposes that have been offered in support of the ID Policy are equally insincere.") — as soon as IDers can claim to support critical thinking and academic freedom, we have lost the moral high ground.

But it doesn't stop there. We're reduced to defending Choice by pointing to the power of precedent, when there are many precedents (e.g. Plessy) requiring reversal. There are valid and strong arguments for why precedent should be respected, but that's not what we need — instead, we must effect a radical change in the understanding of life and rights of creatures like fetuses (I have no doubt that a neonatal fetus is a feeling mammal, and a person as much as my cat is; the point isn't life per se, because I would hold a private funeral for a fetus who undergoes late-term abortion, but rather who should make decisions over whether a woman remains pregnant). We should present reasoned, impassioned arguments for the Equal Rights Amendment, for a more progressive tax system, for Medicaid and Social Security overhauls (resulting in more tax-funded healthcare for everyone, especially members of the lower classes, and in a standard retirement age closer to 80) and for a new understanding of environmental rights. And don't get me started on foreign policy, where the "conservatives" and "liberals" switch sides every few administrations.

Instead, the disquieted left is putting much of its formidable creativity into ultimately unhelpful (if not down-right hindering) negative personal attacks.

At least we can still advocate for gender-neutral marriage.

One Lion to rule them all

It's been, oh, probably ten years since I last read C.S. Lewis' The Chronicles of Narnia — I did read all of them, some more than once, and enjoyed about half — more recently, I've read many reviews and discussions of the new film and the old books. I'd like to respond to a few published thoughts, and record a few of my own (with the caveat that my memory for the original texts is patchy at best, so take my claims with a pinch of salt thrown over your shoulder). Which is to say, to the extent that it's possible to post spoilers to a film like this, I will try to the utmost to do so.

Andrew Adamson's new film is not the first time Narnia has been on screen. The low-budget made-for-TV 1988 production is really quite wonderful, and clearly motivated the current rendition. I've never seen the 1979 cartoon — is it any good? Hopefully, Adamson and WETA will also produce remakes of Prince Caspian and the Voyage of the Dawn Treader (presumably as two separate movies this time) and The Silver Chair, as well as my favorite book of the series, The Horse and His Boy. Adamson has masterfully updated the 1988 movie, replacing its motley mix of live action, animatronics, and animation with a variegated blend of live action, animatronics, and (now computer-) animation.

Part of me had hoped for some new and radical reading of Lewis' original fable. Such hopes, of course, could only be realized by a small independent film company — Disney is much too conservative, and American audiences cling to their literal interpretations of original texts. Besides, Aslan's Passion is already a radical rethinking of an original fable. C.S. Lewis used The Lion, the Witch, and the Wardrobe to come to terms with his own burgeoning Christianity (J.R.R. Tolkien, a devout Catholic since his mother converted when he was eight, takes the credit for helping his friend and colleague see the light). Part of me would also like to read a book report comparing Lewis' Narnia with the His Dark Materials trilogy by atheist and church-critic Philip Pullman: Lion is infinitely more magical than Compass, and differences in writing quality can be attributed to the age ranges of the intended audiences; but the extremely rushed Amber Spyglass is infinitely better than the sorry excuse that was The Last Battle.

After Peter Jackson's epic experiment in condensing and editing more than a thousand pages into ten hours, the generally positive reviews have applauded Adamson's faithful translation of two hundred thin pages. Eric at Speaking Natalie correctly writes that "they didn't change anything ..., but they did add things": Adamson's decision to start with the 1940 London air raids, which Lewis mercifully omitted, has been much-discussed (I found it emotionally moving, and at the same time of inferior animation quality). The Rings-inspired battle sequences are also new — I enjoyed some of it, thought that most was boring, felt the air-raid reprise lacked panache, and found Aragorn extremely out-of-place.

Eric raises the issue of Peter's character, and to his comments I'll agree: Adamson improved upon the book by allowing Peter to be flawed as an older brother. It's now much more meaningful when he admits that "[Edmund's betrayal] was partly my fault, Aslan. I was angry with him and I think that helped him to go wrong." What Adamson leaves out is Lewis' dislike, and Peter's revulsion, of battle. Only a split-second of screen-time allows Peter to look disgusted after killing Maugrim, and he's too steely (and adroit) in battle — I'd much rather this description of one-on-one fighting:

Then came a horrible, confused moment like something in a nightmare. He was tugging and pulling and the Wolf seemed neither alive nor dead, and its bared teeth knocked against his forehead, and everything was blood and heat and hair. A moment later he found that the monster lay dead and he had drawn his sword out of it and was straightening his back and rubbing the sweat off his face and out of his eyes. He felt tired all over.

Michael Nelson, writing in The Chronicle of Higher Education in the review I linked to, defends Lewis against Pullman's claim of the former's "sadomasochistic relish for violence" by arguing that Lewis, who fought in the trenches in World War I, understood violence as "nothing romantic," and that children who read Narnia encounter "their first unglamorous act of violence, the first to make them doubt that killing, even when it has to be done, is something to celebrate." I'd ask only that Adamson live up to that tradition.

To the other Pullman critiques that Nelson counters, I say only that Rings is infinitely worse. Lewis may have allowed the good, Christian Narnia to be light-skinned and the generally evil, heathen Calormen to strongly remind readers of northern Africa (Adamson thankfully left out all other humans); Tolkien describes all those dark-skinned southerners as evil and characterless. Just as insidiously, Tolkien's races fight primarily as races — orks and urukai and hobbits each pick sides, and few have character development — whereas if memory serves me correctly Lewis allows his motley crew of animals and mythical creatures from varius mythologies to fight on both sides (as in all wars, dwarves fight dwarves and cheetahs, cheetahs). The number of women characters, even minor ones, that appear in the entire Lord of the Rings you can count on one hand; in Narnia, girls are brave instigators of geopolitical upheaval, even if they are bound by early twentieth-century norms. In short, Nelson has picked himself a straw punching bag.

The new Chronicles of Narnia provides nothing new, except a chance to go to the movies and discuss Narnian lore (for a detailed introduction, see the Wikipedia entry), and ponder questions like "In a land where animals can talk, why would the good Kings and Queens go stag-hunting?". I think I will reread The Horse and His Boy, and wait for a big-budget epic film adaptation of The Golden Compass. I'd also like to see Dealing with Dragons done big-budget, but that may have to wait, for all that it would adapt well.

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.

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.
   

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.

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.