Category Archives: Philosophy

The Opposite of the Princess Bride

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It’s odd, I know. But this morning I was sure that my right foot was home to six toes. Like many of you, I imagine, I fall out of bed—which is actually a couch despite my having access to a bed à la Fox Mulder—onto the floor. I then slump across the room to shut off my radio alarm clock before waddling over to the bathroom for a morning shower. However, this morning, after I had found the appropriate water temperature, I slipped my right foot into the tub. Because my footing is not especially sure in wee hours of the early dawn, I have to muster my full concentration not to fall. And then I saw it: a sixth toe. It was tucked nicely between my middle toe and I guess what would be called my ring toe. It wasn’t especially offensive, just, well, weird. I looked up and back down to refresh my visual sense. Yep, still there: six.

There was a problem with the way I was counting, though. I do this a lot. People think that because I have a degree in math, I’m naturally good at arithmetic. I’d like to dispel that rumor here. In fact, I’m going to let you in on a secret. You’ll be privy to some of the inner-workings of my mind. Most of the time, I don’t actually add or count: I see and intuit. Let me explain. When I was much younger, I used to stare down at the tiles on my bathroom floor for tens of minutes. Sure, it doesn’t sound like a lot, but you try it. It’s a lot harder to do than it sounds. Each tile measured about an inch and a half across and varied in color from a dark brown with a smooth finish to a rough, speckled beige and white. My current bathroom floor reflects a similar choice in design, except the palette runs across the blues. There in the bathroom, I’d sit on the toilet and stare down at the floor, trying first to focus on one tile, then a group of two, then three, and so on. Then I’d close my eyes and try to visual groups of dots. I’d work hard to make sure that I had the right number by arranging the pattern, rather than counting. I played other tricks, too. I still do. I try to focus only on one color or to spell things with the contours of the tiles. I wanted my visual arithmetic to become automatic. Unfortunately, I was unaware of certain natural cognitive impediments, and so, I must report that like most others, I was stuck within the realm of 7+/-2. (I could distinguish the numbers zero through nine; now I’m up to twelve. After that, the dots in my head become unwieldy.)

I do the same thing with written arithmetic—up to a point. I try to intuit the response, though it is much harder to explain just how I go about doing it. Especially this morning. The point is, I didn’t count my toes, I just recognized a pattern that represented six. (What is the number six? It’s six fire trucks without the fire trucks.) At last I resorted to convential counting. To my relief I counted only five. Moving among the toes individually had done the trick. So much for that Gestalt business. It’s too complicated.

But this is interesting. I had long been aware that my audio senses can become numb to stimuli. As a kid I discovered that if I repeated the word “enter” quickly and without pause it would temporarily lose its meaning even though I knew what I was saying. So often I forget that these sorts of rules ought to apply more generally. In fact, my sixth toe convinces me more of a fact (which is really a conjecture) that I stumbled upon the other night after tutoring introductory physics: training a computer to see and understand a scene in motion is probably a lot easier than training it understand a static scene.

Last Friday I met my tutee Katy, for the first time. She’s a pre-med post-back with an MFA in the visual arts. Somehow vectors and derivatives have been taken out of the art history curriculum. So we drew lots of position-time, velocity-time, and acceleration-time diagrams. And we talked about jerk (the slope of acceleration) and higher order derivatives, though we didn’t quite use that language. But what I stessed most is this: Absolute position doesn’t make a whole lot of sense. In fact, there’s a lot of ambiguity in the way we measure. Because of the ghosts of our coordinate systems, we should be careful only to pay attention to changes in quantities we measure, and not to the quantities themselves. If I’m here now, and five feet over there later, I’ll have moved five feet no matter if you started measuring my position from the fire hydrant across the street or from that sketchy all-night barber shop in the back corner of Beijing. It’s possible that your eye knows this, too.

I’ve since learned that some video compression techniques take advantage of change to keep the space it takes to store video down. If a pixel doesn’t change its color, why record it twice? Only report the things that aren’t the same from frame to frame. It makes sense. And when I attempted to learn the piano, my teacher said to watch the jumps an interval, not the notes. I bet computer eyes (if not your human eyes) can track trajectories. Most visual landscapes are complicated. Last night’s asparagus dinner occluded most of my dish at the beginning, yet I was not surprised to find it at the end of the meal once the asparagus was gone. My eye (and my mind) were able to reconstruct the plate even though I couldn’t see most of it. It’s like it was there the entire time. And so it was! Computers have a harder time with that sort of thing. How are they to know what to fill in behind or under or around? (I love using prepositions without objects. Where is it? Oh, it’s between.)

Anyway, I’d just like to report that my right foot only has five toes. That is all.

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Grading Woes

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Earlier this week my friend Emma the Grecian Sailor lamented that grading for her fluid mechanics class takes up too much of her time—there are only precious few good sailing days left this season, you know. And she’s right. How does one give a responsible, formative assessment fast enough to have time left over to bat around on the Charles? This question is as old as grading itself. Likewise, Verena has complained that Physics 15b has taken over her life. So, to my grad school buddies, I have an illustrative anecdote and suggestion: Like a magician would, confuse your students—divert their attention.

A few years ago, I helped teach what has since become a notoriously large freshman honors math course: Math 23ab tries to train those heretofore uninitiated in mathematics to think, write, and speak like a mathematician. We try to sneak in a little multivariable calculus and linear algebra along the way. (I threw in a little geometry and physics, too, when I could.) But being high-performing over-acheivers, these kids were super-aware of their grades. When one of the teaching staff gave one student a full zero out of ten points on her set, she objected and cried—in class! We had an emergency staff meeting to discuss the matter. The grading, it seemed, had to change.

Isadora and I complained that the last set had taken us each more than twenty hours to grade. (And for those of you economists and computer scientists out there, we weren’t doubling up on the work. Each course assistant graded exactly one problem from the set each week. If the problems were shorter, sometimes two.) As is usually the case in any collaborative venture, we couldn’t come to a consensus. We did have some wonderful lunches together at the faculty club, though.

To address the problem personally, I devised the following tactics:

  • Never grade out of a small, round number. Tens, twentys, and hundreds are strictly out. Kids can figure out their percentage right pretty easily if you do. And that means they’ll protest their grades more often. You don’t want them to pass their sets back in once you’ve passed them out. It’s just no fun.
  • Use outrageously large, unround numbers instead. Typically, I’d increase the worth of each question the further we were into the semester. Say, for example, a question might be work 268 points in September; 94760 in January. When the point values climb, students are less likely to care if you took 79 points off for something. Also, they almost never divide out to find their percentages anymore. Who cares what 6432/7356 is? Those are mean-looking numbers, after all!
  • Vary point values throughout a single set. If I had more than one question to grade, or one multi-part question, I’d mix it up. Part A might be worth 2305 points, whereas Part B was out of 7342. It shifts the focus off the grades, and off of you.
  • Don’t worry about strict consistancy. Setting up and maintaining a rubric is hard work, especially if you change your mind thirty-three sets into a hundred. Because the point values are so wacky, the students will assume that your grading schema is complicated and often won’t challenge or compare grades. It risks entering the rigamarole of your mind. That said, try hard to be fair.
  • Write encouraging remarks on their work. My favorites were wizard, way-to-go, good effort, and ingenious. Always follow your comments up with an exclamation mark.
  • Never grade in red. It puts them in a bad place, psychologically. And by bad, I mean nervous and contemptuous. I prefered orange Crayola markers. If I felt especially fiesty, I’d use purple.
  • Give explanatory feedback when appropriate. One former student reminded me that I had once graded her set, “This is impossible. [short explanation] See solution set. 10/10.” To be fair to me, she had it conceptually correct, except for, of course, the part that was impossible, which I circled in orange.
  • Write clear solution sets and post them in a timely manner. Another student, who had turned her set in late, told me that she used my solution set for help. I asked if she cited it. She had. My answers looked right to me. And so she got full-credit.

Hopefully, these techniques will redirect the students’ attention from their grades (product) onto their arguments (process). The idea is to retrain them, many of whom hold strongly developed outcome-orientations, to care about how they got the answer, and not merely that they got the answer. (Last night Michelle told me that a biologist told her that you can train just about anything larger than an amoeba, and that includes people. Of course people in social learning theorists have been saying that for years.) [Check out an old post for more.] And with any luck, it’ll make your life as a grader a little more comfortable.

Believe Again

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Yes, yes. We’ve all heard that the pen is mightier than the sword. Somehow it’s easy to forget, though, just how powerful those silly little words can be. The Republicans seem to know. They’ve sent out now ubiquitous catch-phrases—who doesn’t know to Support Our Troops?—to rally Americans to their causes without actually giving any cause to do so. These slogans are short, to the point, and entirely devoid of content. And still they have proven to be incredibly powerful. Remember when Colbert talked Geoffrey Nunberg, linguist and author of Talking Right: How Conservatives Turned Liberalism into a Tax-Raising, Latte-Drinking, Sushi-Eating, Volvo-Driving, New York Times-Reading, Body-Piercing, Hollywood-Loving, Left-Wing Freak Show, into the ground with only three carefully crafted phrases? (If not, search through the archive tapes for the show originally aired August 21, 2006. Comedy Central has clips: Part 1, Part 2, Part 3.)

Last night, I pointed out to my roommate DJ that a Democrat has finally smartened up and done the same. Massachusetts gubernatorial candidate Deval Patrick, whose website browser icon is funnily DP—I wonder if his marketing team are aware of this—, has used similarly effective however empty campaign slogans. The weakest of his tag lines claims that Patrick is No Ordinary Leader. Now this is good, sure, but it’s not great. It tries to exploit the constant dissatisfaction that most of us harbor against whatever we currently have (be it our government, job, or any other part of life). More than that, it presumes that ordinary is bad and that unsual is good. Just to keep us in line, I’d like to point out—and I know that I’m using an unfair extreme–that Hitler was No Ordinary Leader. I’m not going to argue with you now, so take it at face value when I say that Hitler was bad. A good leader, sure; a bad man, certainly. But like I said, Patrick’s got better ones.

Next in order of efficacy, I think, comes his invitation to join him. Together We Can his posters say. My sister’s boyfriend Andrew finds this one particularly stirring. Last night he told me, “It evokes a partnership between me, the common man, and the candidate for the leadership embodied in the State’s chief magistrate,” or something. “Also, this guy went to some farmers out west somewhere and told them, ‘I’m not a farmer. I don’t know about this stuff. Tell me what I should do to help you.’ He’s really thinking out of the box,” he went on to tell me. My roommate DJ nearly drowned in his own tears (of laughter) upon hearing this.

Andrew proves my point. Perhaps now I should make it.

Together We Can is genius simply because it promises nothing. Patrick’s team were very careful never to use punctuation after any of their slogans on any of their signs. Of course not. They’re fragments. You can’t put a period after a fragment, after all. Doing so might point out raise the attention of a lazy reader. Then he’d realize that you haven’t said anything at all. To Andrew I asked, “Together we can what?” Patrick doesn’t tell us. Instead, he lets our imaginations run wild. That’s right, I am going to help run this State. I am important. Wrong. This slogan is so compelling because it calls on the reader to finish the sentence according to his personal whims and then pretend that it’ll happen, that he’s effected the change, and it spares him the hassle of doing any, real work. People love to feel like they’ve contributed something useful; on the other hand, they hate to exert themselves. This slogan let’s you think you can have your cake and eat it, too. (I’ve never understood that saying.)

But undoubtedly the best slogan I’ve heard so far, Patrick saved for until after he won the primary. Now it’s showing up on bumper stickers. Patrick asks us to Believe Again. I can’t begin to explain how impressed I was when I read this slogan. I wanted to run up and shake him and cry and clap my hands uncontrollably. It’s really quite amazing. This slogan reaches the largest audience possible. Being the most devoid of content, it has the greatest reach. Believe Again entices the voter to conjure up the most romantic, idealized form of government possible. But it doesn’t stop there, the implications are unstoppable. It’s an easy jump from government to general quality of life. Improving one naturally improves the other, right? No matter what you believe in, Patrick does, too—at least according to this slogan. And shouldn’t you support someone who holds such a coincident and intimate commitment to those things you hold so dear? It’s hard to argue against him, because you’d have to argue against yourself. Imagine a leader who would allow you to Believe Again.

To test my claims that these are, indeed, worthy of the Republicans, DJ asked quite blankly, “Are you suggesting we Cut and Run?”

To which I answered, “It’ll take No Ordinary Leader.”

To which he countered, “But don’t you Support Our Troops?”

But then I hit him full-force with, “Together We Can. I want to Believe Again.”

It was over. The conversation left both of us stunned.

DJ then noted that we should write for the Colbert Report, or, maybe I should write for the Colbert Report, or, possibly, just to them, to let them know that someone else figured out how to play the word game.

What’s worth mentioning is that Patrick’s slogans are even more sinister than the Republican’s because they aren’t immediately negative. (No Ordinary Leader comes closest to being overtly aggressive, but is pretty sissy when flanked by Cut and Run and Support Our Troops. Notice, however, that Support Our Troops also makes the people who say it feel like they’ve really accomplished something even though they’ve taken no physical action.) Patrick’s tag lines get stuck in your ear, and, while there, make you feel better about him and about yourself. How empowering! I really can’t get over just how brilliant they are.

Moral: If don’t want people to disagree with you, don’t say anything that they can disagree with.

The Writing on the Wall

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I have some spare time in between my summer job and my fall classes. So I’ve spent the past five days learning to program in PHP and MySQL. My focus has been on the development of so-called large scale web applications. Luckily my dad has agreed to accommodate my unemployment, taken me back in, and even found me a room in the apartment so that I no longer have to sleep on the couch in the living room. In process of learning good organization and coding practices, I came across the idea of templates. And then I realized why graffiti never became an art form, despite its introduction into high-class New York galleries in the early 90s. Stephanie, this post is dedicated to you.

Templates are pretty natural and, nowadays, pretty common. Anyone who has used Microsoft Word or Excel has probably seen a template, some have maybe even used one. They’re empty containers which you can fill in with your own information to produce a finished product without too much effort. What they do, though, is subtle. I hadn’t realized just how subtle they are until last night. Templates allow you to separate content from presentation. This is important. The same thing works in programming, except in web development it’s a little bit more complicated.

A web application has three parts: the content, the presentation—both of which you, the user, see—and the business logic—the code which does the actual heavy-lifting in the silently backroom in the dark. It’s a good idea to keep these guys as far away from each other as possible. You enter the content, more or less, in HTML. Fortunately and unfortunately, the paragraph tag < p > is blind to your content. You wouldn’t format the address in a letter the same way you format a recipe, for example. HTML, however, can’t distinguish between the two. It treats everything similarly. Luckily, that’s where another standard, Cascading Style Sheets (CSS), comes in.

CSS allows you to tell the browser exactly how you’d like a certain type of text to look. You can control placement, font face, font weight, behavior in response to events (like when the mouse cursor hovers over a link), and more! This is the presentation part of it all.

A clever little package, aptly named Smarty, lets you keep your PHP scripts from mingling with your HTML and your CSS. That way you can redesign the look of your pages without having to update the guts which control the functionality, too. Your copy editors and content managers stay happy, too, because they aren’t effected, either, and can continue doing what they like to do best: write content.

And all this got me thinking about my friend Stephanie and her undergraduate honors thesis. It straddled the divide between literature and art history; she wrote about the rise and subsequently fall of graffiti in the art world. She argues that the art world rejected graffiti, actually, a particular type called Writing—writers would never call their Writing graffiti, so why should we?—because it was written and people got caught up in trying to read it. And that makes sense. Try to take in the artistic value of the following:

Do not read this.

You can’t. If you know how to read in English, then you read and processed the above statement, even though I intended it as a purely visual object. Writing is a little bit more subversive. A single author didn’t always tag with one name, and often the script was so stylized that it was impossible to read in the popular sense of the word. Yet other Writers had no problem identifying authorship. The trick is, they were able to distinguish between the presentation and content of a Writer’s tag. Critiques strove to find meaning in the words the Writers presented—meaning that was never there. And the style discovered its author, not the name. Writers had deconstructed the written word, extracting only a visual idiom while leaving the word’s referent alone to fend for itself fully detatched from its referrer.

Few people in the academia of literature, it seems, study the effects typography and layout have on a written work for better or for worse. Perhaps it is more important to distinguish the two when investigating Islamic writing (and its calligraphy) or medieval, illuminated texts. Too bad, though, that presentation has been relegated to the design world. Everyone interacts with layout. It affects so much of what we do everyday.

ImagineifIhadalternatedcolorratherthanusedspacestobreakbetweenwords?

What are the implications of my scheme?

I don’t even want to mention what a meta-language like XML might mean to literature academes. At least not now.

Descartes, Urban Planning, and Chaotic Systems

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Motivated by shame, I’ve taken up Dan Aaron’s challenge to read more original work by giants who laid the foundations of my field. Being a hack mathematician with physicist tendencies, I’ve turned to René Descartes—the man who married geometry and algebra to give us Cartesian coordinates, among other very important things—and his Discourse on Method. In the first part of the Discourse, he situates the reader a bit, explains his personal history, and how it led him to his approach to problem-solving. He tries very hard to be humble and sometimes he almost succeeds, but we know Descartes was a genius. He knew it, too. And even he can’t hide his high opinion of himself.

Descartes claims that the work of several authors is usually inferior in quality to that wrought by a single hand: it’s the old “too many cooks spoil the soup” theory. That’s how he justifies denouncing centuries of philosophers and their work: he had to do it start on his own from scratch; it’s impossible to sort through the ideas of others with any systematic clearity. But he stays out of the kitchen in his metaphor, instead traveling to the city:

Thus we notice that buildings conceived and completed by a single architect are usually more beautiful and better planned than those remodeled by several persons using ancient walls that had originally been built for quite other purposes. Similarly, those ancient towns which were originally nothing but hamlets, and in the course of time have become great cities [think London], are ordinarily very badly arranged compared to one of the symmetrical metropolitan districts which a city planner has laid out on an open plain according to his own designs [Brasilia, say: it looks like a bird or airplane]. It is true that when we consider their buildings one by one, there is often as much beauty in the first city as in the second, or even more; nevertheless, when we observe how they are arranged, here a large unit, there a small; and how the streets are crooked and uneven, one would rather supppose that chance and not the decisions of rational men had arranged them.

Now, Descartes’ philosophy aside, there’s a pretty interesting observation in there. Even though a very complicated system, such as a city, maybe be very orderly on small-scales, the over-all effect is painfully complex. I’ve declared more than a few times, proudly to visitors, that the streets of Boston were once the random paths of wandering cattle. So this Fathers’ Day, when my dad and sister came to visit my new digs, I had to remind them to look in all—not both—directions when crossing the street. “The streets around here are so wacky, that the cars going one way can be stopped at a red and still you can be hit in three other directions.” We all made it across, a little bit hurried but safe. Meanwhile, Manhattans live on a grid and seem to like it. Some have even claimed to prefer it.

A similar phenomenon exists in economics. A while back someone was awarded a Nobel Prize for showing that even though each person may act rationally, the market as a whole still might act irrationally. I don’t know much about how this works, but I can tell you about an analogue in math.

There are several competing definitions of chaos in mathematics, depending on just which subfield you subscribe to. One of the most tractable definitions comes from one-dimensional, real, discrete dynamics; that is, the study of iterated functions that live on the real number line. Simply, take a function, which is only a fancy word for a rule, which takes in a number (like 3) and spits out another (say 1.5) and repeat it over and over again. We say that a function is chaotic if it exhibits the following three behaviors:

  • The function has a dense set of periodic points. Periodic points are orderly; their trajectories are predictable. They start at one location. The function moves a periodic point to another point; another application of the function moves it again to another, and another, and another. Eventually, though, the point needs to wind up where it started. Think of periodic points as travelers on a multi-city, round-trip itinerary. A salesman might start in Boston, go to Denver, follow-up in Phoenix, stop short to family visit in D.C., but at the end of it all, he’s got to go home to Boston. In the language of discrete dynamics, our salesman is a periodic point. Dense is just a mathematician’s way of saying most. So, if you blindly pick out a number, it’ll most likely be a periodic point. And if it’s not, there are plenty of periodic points nearby. On a plane, not every passenger is needs to be a travelling salesman like the one in our example. But near each passenger there should be a few of them close by.

    • Notice that the definition of a chaotic map (or system or function—these terms all refer to the same thing) demands lots of order. Periodic points are simple. We know exactly where they go and exactly where they’ll end up: they travel in loops forever. However, chaos requires a little bit more.

  • The function displays sensitivity with respect to initial conditions. This requirement ensures that points which start out close don’t stay that way forever. You can think of functions which are sensitive to initial conditions as those maps which mix points up. Sensitive functions are not very tolerant of approximation. They hate playing horse-shoes, for example. And they’re very hard to plot on computers due to rounding errors. Even though we might be very accurate, a chaotic function will churn the points about so wildly that we cannot guarantee that anything we learn about one point will shed any insight on the whereabouts its neighbor.

    • So far, we require chaotic maps to be, on the one hand, very orderly—by way of a multitude of periodic points—, and simulatenously jumbled, on the other hand—in an indirect way, through its sensitivity. In a modern treatment, we could stop here. But to really drive things home, let’s add in a third requirement.

  • The function is topologically transitive. Topological transitivity is a mathematician’s way of saying that the function meanders. Pick any two points A and B. If the function is topologically transitive, then I can find another point as close to A as you want that eventually makes its way as close to B as you want.

    • One of the simplest examples of a chaotic maps is the so-called logistic map. It’s a quadratic (there are squares in there):

    xn+1= xn (1-xn)/2.

    Its continuous conterpart crops up in population dynamics as one of the simplest, foundational models. Some examples within the logistic equation’s domain include colonies of bacteria, blades of grass on a lawn, or frogs in a pond with no predators to eat them.

    So, as long ago as 1637, Descartes noticed, very carelessly, that order can breed chaos. Let this be a warning to those of you neat-freaks who work tirelessly to assure everything is in its place. Also, Descartes might argue that cities provide evidence against the precepts of Intelligent Design. I know that’s how I read it.

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Avon is Rural.

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Thursday was one of those grey days, when the ceiling hangs very low and the clouds rather than the grass seems to come up from the ground. I relish these days. On my way to work, to Leverett, I watched the Charles misting as I drove. The winter wouldn’t be so bad if we had more days like these, or, if we had snow.

A moment ago a few flakes teased my winter sensibilities. They have since stopped. The heat was shut off for the break, and I have bundled up in a coat, scarf, and floppy hat. Rather than read math, which is what I had intended to do today, I picked up Gordon S. Wood’s short history of the The American Revolution again. I’ll put it down at the end of the chapter.

The history got me thinking. It’s something I would read to my children, should I someday be allowed to raise children, before tucking them into bed. My father, after all, would cycle between a library of books written to supplement Sesame Street and the Bible. At least the language Wood uses is more tractable than that in King James. But then, and here’s the thinking part, I thought about my father’s reading me the Bible and whether I would do the same. You see, there are two competing forces: on the one hand, I think that religion is a very grown-up affair. Its practitioners should demonstrate an informed faith [something I don’t really believe I have, actually], and such an education requires a mature mind. Really, Piaget would back me up. So most children simply aren’t even biologically equipped to process the implications of their religion, especially not my hypothetical ones. But then on the other hand, if I really believe, let’s say — and now we’re getting theoretical, not personal. My beliefs are more nuanced, but this works for the present — that Christianity is the key to salvation, then it only makes sense to introduce it to my children as soon as possible. And here St Augustine would back me up. And how can I, neither a celebrity developmental psychologist nor a Christian saint, hope to reconcile education with religion?

My friend Michelle offers an anecdote while I continue to ponder. Her childhood friend was born to Quaker parents. Faced with the same problem, they refused to bring their daughter to church until she was eighteen. She was welcomed to go to church with other families if she wished but not her own. Rather than denying my children church, and I confess I haven’t been regularly since I started college, I suppose I would supplement verses not only with discussion, but also with readings from historical exegeses and treatises from the medieval church. I better be careful before I say that we should present children with every possible vantage and then let them choose for themselves. Don’t worry, I don’t think that. But I do want them to be better read than me. And maybe I can use them as an excuse to read more.

If ever I do have kids, it’ll be terrifying. I’ll have to think about things I haven’t even thought to think about then. Good thing grad school is so long.

Head of the Charles

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Patrick Lenehan, T. Sean McKean, and others from Pfoho came back for Cambridge’s most time-honored and celebrated event, Head of the Charles. Just its mention sends me tripping down Memory Lane. Oh, those time we looked at the boats. And those other times we didn’t. I remember those other times mostly. Whatever their reasons, my almost blocking group is here this weekend. Diana Rosenthal, who was never almost in my blocking group, however, is here, staying with me. At least her suitcase is here. She seldom is. Yesterday she napped in my bed while I read math a few hours. In my mind that counts as quality bonding time. Read that as an indication of what you like.

The point is this. Patrick tricked me into meeting them at Shay’s last night. Worse still, they left after two beers and a shot of drambuie. In another effort to drown mi hispanicidad with Scottish liquors, I’ve taken to whiskey derivatives. At this time, they all left to go home, or did they? Felipe’s as you well know is open so long as there are hungry, drunk Harvard students milling about the streets. And so there we were, along with Barusch. She left, and was replaced with name-brand tie designer Baruch. We introduced ourselves to one another. Then I saw Natalie, to whom I apologized for my public display of social ineptitude last Sunday. She forgave me, and explained that I had really done nothing wrong, as it was unlikely that I knew she and Mark had broken up. Then, there’s always a then at Felipe’s, Summer and Janie showed up. They invited me to sit down with them, but I was on my way home. But, and there’s always a but after each then at Felipe’s, Diana Rosenthal, the very same who naps in my bed and says I looks like Abe Lincoln, appears. Before we can say three words, Ian and Berch attack us in what looked like some sort of sexual assault. The cops took care of them, asking if they were sure they had not jumped on some cars earlier that night. I told the cops that the boys were here and so it was impossible. I was waved away. By this time Summer and Janie had snuck out and ran when I crossed the street, only to stop in front of the Signet, where, as luck should have it, JC and Clint were, smoking a cigarette. They invited me inside for a cast party. There upon they called tie designer Baruch. Come four o’clock I was home. Being almost the start of a new day, I decided to groom. There’s a small nick on my upper lip. Let this be a lesson: no matter what, don’t shave while drunk.

Benoit Mandelbrot.

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The Limit Set of a Schottky Group

Last night I dragged Venera and her, and now our, friend Anahita to the Museum of Science to the first in their Lectureology series. To kick things off, they brought in mathematician Benoit Mandelbrot to talk about roughness, cauliflower, and fractals. I’m not sure he did any of these things. He did manage to tell us just how great he knows himself to be more than a few times, however.

I was fairly shocked just how quickly he was to dismiss, mostly be flagrant omission, the other “fathers of fractals.” True, Mandelbrot did coin the phrase. He told us so three times in less than one hour. He was not, as he claimed last night to be, the first one to consider roughness. Julia, famed for his aptly named Julia sets, was Mandelbrot’s own teacher. In fact, the Mandelbrot set is a catalogue of Julia sets. Of course we could throw in Douaday, Hubbard, and Fatou while we’re at it. Then, there was another branch of complex dynamics going on which resulted in Bers slices, the Maskit slice, Fucshian groups and all that. I happen to think that the other school made cooler pictures. (Compare the limit set of a Schottky group [top] with the Julia set [bottom].) The point is Mandelbrot didn’t go it entirely alone.

Perhaps the only thing I took away from my tenth grade English teacher Mr. Tony Baxter is this: when writing a story, you can’t just tell the reader what’s going on. It is incumbent upon the author to show the reader what’s going on. Mandelbrot should’ve taken Tony’s class. He’s still a teller, not a shower. And then, not even a good one. He didn’t once tell us what the Mandelbrot set is, let alone a fractal. I do remember his throwing around the term Hausdorff dimension somewhere in the middle of his talk. But he didn’t spend nearly as long on that as he did some honorary doctorate he received from somewhere in Germany.

Verena and Anahita and I left for the food court at the Galleria in the middle of the question and answer session, just after he was explaining how much metereology he knows and named-dropped infinite dependence.

At least we got into the Museum for free.

An example of a Julia set

Even String Theorists Can Be Nice.

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I just emailed Clifford Johnson, a professor at USC who specializes in string theory and gravity. More importantly he is one of the permanent bloggers at Cosmic Variance. Recently he recounted a story from his advisor at the University of Southampton. So, I started thinking, hey! that probably means Clifford, who teaches in the States, probably got his degree in England.

Previously I had had some reservations about postgraduate degrees from abroad. But I had also had the feeling that most of the big general relativity things happen in England. After all, it’s tradition.

Sir Arthur Eddington, a pacifist, quaker, humanitarian, and Chief Astronomer at the Observatory of Cambridge University, single-handedly brought Einstein’s relativity to the Allied World — proving that science transcends political boundaries even in a time of war. Anyway, it was Eddington who made Einstein famous. He not only understood and explained the theory — which, at the time, was a remarkable feat in and of itself — he also gathered the money and manpower to execute two expeditions to put GR to its first experimental test. Eddington himself led the team in South America, while another headed to Africa, both to observe the bending of light by the sun during a solar eclipse.

The rest, as they say, is history. England has continued to produce excellent relativists, and not just in the philosophical sense. Hawking and Penrose are, perhaps, the most famous. But there are also Gibbons, d’Inverno, Tod, Geroch, and so many more!

After reading Clifford’s post, I decided to email him about his Southampton experience. Minutes later I received a response.

Add Warwick, Durham, and Southampton to my English school list.

But Is It Logical?

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Monday night is free apps night at Sunset Grill & Tap in Allston. So the Yale Beer Society (formerly the Harvard Beer society) gathered who it could for a meeting. Among us were a student astrophysicist, physicist, mathematician, and recently graduated intellectual philospher. [I’m told that the intellectual part isn’t simply arrogance, but actually designates a particular type of philosopher. Or, I could be making this entire story up. Given what follows, it’s not clear even to me.]

Mistake number one: the appetizers are free on Monday’s, but not until after midnight. We got there at 8pm. With about one hundred twenty beers on tap and just over three hundred in total, we had no problem waiting around. We are YBS, after all.

Mistake number two: someone, maybe a few someones, ordered two flights of mead. For those who haven’t been, a flight is a standard unit of measurement. It is equivalent to four beers. In fact, it is four beers. This is not to be confused with a yard of beer, which is also four beers tall. The yard comes in a long, silly cylinder. The flight is served on a mat and in four glasses.

But the point is not the quantity, but rather, the contents. Mead is about the most vile stuff known to man. [Not true.] It definitely gives me some insight into viking culture. In no time flat, we were screaming on the top of our lungs at one another.

But what were we screaming? Well, the astrophysicist was Ian. And the student mathematician; ah, well that’s me. It’s not hard to guess just what were were screaming about. What is the nature of the universe; where does math exist; are virtual possibilities physically manifest; is there a God; can nothing exist; does its non-existence necessitate the full existence of physically viable possibilities? It went on. And on. And on.

By the end of the night, the bar had cleared out. Except for us, of course. Having missed the rain and the last bus home, we walked. Continuing our philosophy.

Dan, the philosopher, and I maintained our distance from the physicists. We had paired off into loud sides in the restaurant and had no intention of letting things go. He and I had accidentally reconstructed William James’ arguments about the noumenal and phenomenal, communities of inquiry, and problems with bivalent vs multivalent logic. When we ordered the two flights of mead I thought that this was one of those non-standard college moments. Now I was sure that wasn’t especially normal.

By the time we reached Kirkland, Dan’s destination, we hadn’t finished. We had only really just begun. But Dan didn’t have a light for his cigarette, so we had to stop, at least for that. There were some kids hanging in front of the entrace to the House. They were nearly scared away as they heard discussing, no longer arguing, first and higher order logic. “No school talk,” one of them demanded. I’m not sure why I didn’t tell them that this was just our fun drunk talk.

We thanked them for the light and across the diagonal of the MAC quad toward Leverett. Then, at about 3:30am, we promised to exchange reading lists. A few hours later we did. Right now I have lecture notes from when James’ taught Philosophy 9: Metaphysics and 20c: Metaphysical Seminary. Dan has a review article on axioms and belief systems in math from the Journal of Symbolic Logic.

Check out Penelope Maddy: “Believing in Axioms, I,” Journal of Symbolic Logic 53 2 (1988) pp. 481-511, if you have the chance.

[There’s a follow up on determinancy, which an axiom that is not consistent with the axiom of choice. “Believing in Axioms, II” 53 3 (1988) pp. 736-764.]

In other news, Bush is single handedly trying to ruin science in this nation for at least a generation. Earlier this week, Bush said in a press conference that:

“Both sides [evolution and intelligent design] ought to be properly taught…so people can understand what the debate is about,” he said, according to an official transcript of the session. Bush added, ”Part of education is to expose people to different schools of thought. . . . You’re asking me whether or not people ought to be exposed to different ideas, and the answer is yes.”

These comments drew sharp criticism yesterday from liberals, who said there is no scientific evidence to support the intelligent design theory and no educational basis for teaching it.

I’d like to point out that not only liberals but scientists, too, cannot find any evidence for intelligent design. According to what my fifth grade science text said about the scientific method, no untestable theory qualifies as a scientific theory. That’s the whole point about scientific inquiry, experiments. We don’t go around voting on what we think the truth is. Every once in a while, we do poll for opinion, though.