Panopticon

After I received my first paycheck from my summer job, I went out and bought a CD, my first in a few months. I picked up Panopticon, by Isis. It wasn't the first Isis album I'd heard; I bought The Red Sea a while back and liked it quite a bit. It is grandiose and abrasive and punishing. Maybe it's just the title of the record (and that of their 2nd most recent album, Oceanic), but it reminds me of a hostile ocean, inspiring fear and awe as giant waves of guitar, thundering drums, and acrid vocals crashed down in my ears. The only respite was a few samples from a David Lynch project, including a desperate recitation of some William Blake. It's a good listen if you're in the mood to be dazed and a little scared by your music.

5 years (and 5 releases) went by between The Red Sea and Panopticon, and the sound has changed. Compositions are calmer, more drawn out, and include more electronic bits. This is not necessarily a bad thing. Occasionally you may hear Isis called "post-metal", as in "post-rock" with metal touches, and Panopticon earns this title much more than The Red Sea, which had more in common with Neurosis than Godspeed You Black Emperor. Now songs begin with meandering guitar lines that become powerful chords; the "angry bear" vocals of previous albums are much rarer. There are still moments when the guitars threaten to crush you, holding you down for the drum hits to pummel you, but overall the album is a more atmospheric affair. Not indistinct or lazy, no. But where The Red Sea was an exhausting trip through hostile waters, Panopticon calms down long enough for you to get your bearings, and decide that you have no idea where the hell you are, but it's dark, and you'd rather be home. Then it shows its teeth.

But it's not just Isis's sound that can be considered artsy (if you believe that metal must be fast). Isis has certain lyrical fixations throughout their albums. As mentioned previously, the ocean comes up frequently. They also often refer to a central female figure, occasionally identified with a tower. It is this theme that gets a nod on Panopticon. It turns out that panopticon isn't just a cool word. Back in the early 19th century, Jeremy Bentham¹ tried his hand at prison reform. His design was simple. A circular prison, with cells whose doors faced the center and windows faced outwards. In the center, there was a giant watchtower².



The design hinged on two complementary principles. The prisoner can always be seen, not only from the watchtower but also by other prisoners across the circle, backlit by light coming in their windows. The guards in the central tower can never be seen, their presence hidden by blinds and the like. So as far as the prisoners know, they are always being watched. This leads to better behavior, and paradoxically allows the prison to have guards watching less often.

Bentham had high hopes for his design, but it's not best known as a practical prison layout. Instead, it is as an idea that it has proven to resonate strongest. It gained most of its notoriety when Michael Foucalt focused his attention on it, in Discipline and Punish. As I understand it, he argued that the Panopticon structure was a sort of ultimate version of hierarchy, with one (always unseen) party being able to exert control at all times over the other (always observed) party. Read more here. Later social theorists have pointed out that the modern prevalence of surveillance technology allows for a Panopticon-type social structure to be put into place (to be fair, George Orwell totally saw this coming). There's a lot to be said for this idea; surveillance cameras are pandemic (moreso in England, that den of brutes and thieves). Head over to Google Maps and you can see satellite images of just about anywhere on earth, generally of pretty high resolution. I can, in fact, see my house from here³.

Isis is full of smart guys, and they know all this stuff. I haven't mentioned these things completely randomly; the liner notes contain quotes from Bentham and Foucalt, as well as one regarding new technology. Not to mention the album art is composed entirely of satellite images. And hey, here's some lyrics from the song Backlit:

Always object
Never subject
...
Always upon you, light never ceases
Lost from yourself, light never ceases
Thousands of eyes, gaze never ceases
Light is upon you, life in you ceases

If the idea of a total surveillance society seems far-fetched, then you underestimate the incentive governments, corporations, extortionists, and Hollywood have to get you on camera. If the idea of a total surveillance society seems like a reality to you, then you should cut back on coffee and watching Enemy of the State. The big question is, quis custodiet ipsos custodes⁴? In the Panopticon, no one does. And it is this secrecy that is so threatening. I'm sure we can all think of some secrets that have come to light recently that made us feel the "unequal gaze".


Just saying. And the answer to the big question is obviously, "the people". Well, that's the idea, anyways. Here's hoping that continues to more or less work out for us.

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1. Fun fact: Jeremy Bentham had himself stuffed, and his corpse "attended" board meetings from time to time at the college he founded.
2. See? It's a tower.
3. Here being my computer, of course.
4. Who watches the watchmen? Sorry, not trying to be fancy, it just sounds cool. Also it's in one of my favorite comics, Watchmen.

The Imaginary, the Hyperbolic, and Mathematics

Ah. This post is going to be a long one. But then, I haven't really done a short one yet. What follows is the final paper that I wrote for my Philosophy of Mathematics class. It makes reference to all the readings we were given in that class, so it moves around a bit, but that should just serve to keep it interesting, right?

The conclusion is more or less what I believe about mathematics. The practicality of mathematics is often vastly overstated, and yet there is something that I find fulfilling about it. Over Spring Break I read A Mathematician's Apology, by G.H. Hardy, and since then, I have come to agree more and more that mathematics is a field concerned mostly with aesthetics. When dealing with messy fractions involving square roots and the like, one often calls the result ugly; certain proofs are called elegant almost without fail. I have a textbook which compares spectral theorem to a symphony. The language of beauty is tied into mathematics more deeply than most realize.

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If you asked an average person to describe mathematics, they would likely respond by saying that mathematics is the study of numbers. Some might remember to mention geometry, a topic most don’t deal with beyond one class in high school. Although this is a limited view of mathematics, it’s not wrong. And how hard can numbers be? Or plane geometry, for that matter? While some may say that they are not mathematically inclined, most people will tell you that they know what a number is, and they certainly know how shapes work. So 22/7, or the square root of 2? Definitely numbers. And two straight lines side by side will obviously never meet. Then what about the square root of -1? Is that a number? And what about hyperbolic geometry? Is it useful, or simply a warped version of what is real? Mathematical constructs like these raise questions about the very nature of mathematics.

The square root of -1, more commonly referred to as i, gives one pause. The term “square root” highlights the difficulty. The square root of 2 is the length of a side of a square with area 2. The square root of -1, then, is the length of a side of a square with area -1. Such a square makes absolutely no physical sense, which is why i is often called the imaginary number.

Is mathematics concerned with physical reality, though? This discussion goes back quite a ways. Plato, in his Republic, discussed mathematics at some length. At one point, Socrates says in reference to geometers (the mathematicians of their day), “They talk of squaring, applying, adding, and the like; whereas, in fact, the entire subject is practiced for the sake of acquiring knowledge.” (Book VII, 527a-b) So the physical impossibility of i is of no concern to mathematicians. Earlier in the Republic, Socrates takes the geometers to task for using physical objects (specifically, “the animals around us, every plant, and the whole class of manufactured things” (Book VI, 510a)) as images on which they base their studies. But no such physical object exists for i. Instead, one might say that when dealing with i, mathematicians are using other numbers as images. In the same way that geometers make arguments about squares and diagonals, drawing inspiration from the imperfect squares and lines that they see in nature, a mathematician might make an argument for the existence of i by looking at more conventional numbers. -1 is really not that different from 2. They are both integers. So if 2 has a square root, than why wouldn’t -1? This is how the square root of 2 acts as an image for i. When dealing with i, one must work completely through reason, and although still falling short of dialectic, one is still operating in the third portion of Socrates’ divided line, referred to as thought, well into the intelligible section of the line[1]. It is ironic that such numbers are called imaginary, when it is the lowly first section of the line that is referred to as imagination.

Although the concept of i has been around for centuries, it was ignored and disparaged for quite some time. Leonard Euler put in a lot of work to make i respectable. He receives the credit for introducing the notation of i for the square root of -1, and the famous equation e^iπ+1=0 bears his name. Starting in the 19^th century, mathematicians began to discover a number of surprising properties of functions involving i. This was part of a general movement in mathematics, mainly spurred by developments in geometry (which I will return to shortly), toward increased rigor and examining the foundations of mathematics. The question inevitably arose: What is mathematics? Kant classified mathematics as a synthesis of a priori knowledge with knowledge from the content of specific experience. In response, Dedekind, Frege, Russell, and Whitehead attempted to show that mathematics (in Dedekind’s case, just arithmetic) is entirely based on a priori knowledge: that is, that it is entirely analytic and concerned only with pure logic. This school of thought is known as logicism and is, at least initially, an attractive explanation of what mathematics is and with what it is concerned.

Of course, i fits into all of this. If anything, the existence and study of i in mathematics would seem to provide an excellent counterexample to Kant’s argument that mathematics is synthetic a priori, for there is no specific experience from which one might get the concept of i. As stated earlier, no squares have area -1. How could a field of study that deals with such a concept be synthetic? Then again, as also stated earlier, one might draw the concept of i from previous experience with other square roots by simply generalizing to the non-physical. I have studied no Kant (unfortunately), but I imagine something known through this type of mental experience is still considered a posteriori knowledge.

Still, the logicists have a trick or two up their sleeves to talk about i in terms of pure logic. It requires only a rather simple argument to show that all numbers involving i (called complex numbers) can be represented by 2 real numbers, and any operations you can perform with complex numbers are the same as certain operations on those 2 real numbers. So if the real numbers can be said to be purely logical construction, it follows trivially that i, and indeed all complex numbers, are also purely logical constructions. But there’s the rub; it has turned out to be subtly and maddeningly difficult to give a logical formulation of the real numbers. By the time one has arrived at such a construction, its definition has enough conditions on what a number is that it seems far from a priori knowledge. And Gödel’s Incompleteness Theorem brings into question whether looking at mathematics only in terms of logic is even useful. After all, one could derive a statement whose truth is undecidable from its axioms. This is a grave limitation that naturally leads to the assertion that math is more than just logic, for good or ill. So although i has risen to be as respected as the other numbers, it turns out that this is only so far up. It would seem that even as unreal a concept as i must be taken to be known only through experience, albeit experience of a very particular sort.

All this focus on number has led us away from the other branch of mathematics, dealing with shape. Geometry has been intensely studied at least since the time of the Greeks. It is geometers that Plato alternately criticized for basing their studies on unshakeable, physically-based hypotheses, and lauded for working by means of pure thought when making their arguments. Later, probably at least in part as a response to Plato, Euclid would attempt to lay down a set of axioms and postulates from which the important geometric theorems of his day could be logically derived. Many of these were uncontroversial, perhaps because they fell into the trap of using the perceived behavior of physical objects as their basis. It seems clear from experience that “a straight line may be drawn from any one point to any other point” (Euclid’s Elements, Book 1, Postulate 1), or “The whole is greater than its part” (Book 1, Axiom 9). But one axiom stood out as being different from the others, and caused a great deal more concern among mathematicians. Euclid’s parallel line postulate can be stated in many different ways, most commonly as, “Given a point and a line, there is exactly one line through that point that will never intersect the given line,” although this is not the form found in the Elements. Though this postulate was fiercely debated, and to some seemed that it should be somehow derivable from the other axioms, it was shown much later to be indispensable when dealing with Euclidean geometry. In fact, if one removed the parallel postulate, he or she would end up with a completely alien system of geometry, one in which nearly everything diverges from everything else, given enough time. This system is called hyperbolic geometry, and given a point and a line, there are multiple lines through that point that will never meet the given line. To many, it sounds ridiculous. However, it has been shown that all of Euclid’s other axioms work just fine in hyperbolic geometry. There is even a simple model of hyperbolic geometry, a circular plane with its border at infinity, and straight lines drawn as circular arcs that cross the border at a right angle. But as simple as this model is, it seems absurd, and a little lonely. Everything ends up infinitely far away from everything else.

It was inquiry into alternate geometries such as these that helped lead to serious discussion of the nature of mathematics. In dealing with such geometries, it was shown that any theorem that could be deduced from its axioms should hold in any accurate model of those axioms. This lent itself to a logicist view of mathematics, because so long as the axioms of geometry could be considered pure logic, all of its theorems were deduced through pure logic, and so geometry itself could be considered purely logical. This ran into the same problem as the logicist view of numbers, namely, the Incompleteness Theorem. Other schools of thought arose in the late 19^th and early 20^th centuries as well. One, formalism, is in many ways similar to logicism, and so will not be discussed in this paper. But the third major school of thought, intuitionism, has serious differences from the other two. It seems to be a complete restructuring of mathematics that leaves behind some of the most important tools in mathematics to arrive at new and novel ideas.

Intuitionists believe that mathematics deals solely with mental constructions based on intuition. Based on this belief, they reject the Law of the Excluded Middle, which holds that if something is not not true, then it is true. An intuitionist would respond that by showing that something is not not true, one has not shown any construction in which it is true, so how can one draw such a conclusion? This thinking led to a radical reconstruction of analysis and logic, albeit one that appears in many ways limited and, ironically, counter-intuitive. What would an intuitionist make of the hyperbolic plane? A better question might be, what would an intuitionist make of the parallel postulate? It is doubtful that they would hold it in high regard. Although it is simple to construct two lines with a common perpendicular, an intuitionist would likely scoff at the claim that all the perpendiculars between those two lines are now common. Such a thing is not able to be shown, because it deals with an infinite number of perpendiculars. Or when dealing with Euclid’s original statement of the parallel postulate, one can certainly show where two lines that meet on a straight line at less than two right angles meet using trigonometric functions. But an intuitionist would challenge the thought that you can tell whether or not lines meet at a right angle. If one measures the angle between the two to be 90.0000^o, it still cannot be said with certainty that it’s a right angle, because it could really be 90.00001⁰. So an intuitionist would not hesitate to throw out the parallel postulate, opening the door for hyperbolic geometry. It seems likely, however, that an intuitionist would also not hesitate to throw out quite a bit more. Any theorem in hyperbolic geometry proven by contradiction would be thrown out. Any proof that uses such a theorem would be thrown out. Precious little remains of geometry after an intuitionist has his or her way.

The hyperbolic plane and the number i both turn our focus to the unbelievable in mathematics. Their existence seems nothing more than useless mathematic abstraction, separated from reality. An intuitionist would say that mathematics is wholly based on our intuitions about reality, and although intuitionism is largely dissatisfactory to many mathematicians, if the intuitionists are wrong, then what is mathematics if it is not concerned with reality? Some would say it is concerned only with logic, but this view does not hold up to close scrutiny. Others reduce mathematics further, to simple operations on strings of symbols, but this seems to ignore the power of mathematics. Plato described mathematics as reasoning from assumptions, and few would argue with that. It is still a good description, and it is by working logically from simple assumptions that surprising results like the properties of i and non-Euclidean geometries have been discovered. But what good is such reasoning? Although Plato found the assumptions mathematicians made appalling, he still recognized that mathematicians worked through pure reason, and in doing so, turn their minds toward what he termed the Good, by which all truth and beauty is illuminated. Ultimately, mathematics is the pursuit of the Good through logic, with help from a few extra-logical assumptions. From as simple a set as possible of first principles, mathematicians work toward beauty and truth. By working with concepts like i and the hyperbolic plane which are far removed from the physical world, mathematicians come to deal more completely in pure thought, moving ever closer to that goal.

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[1] Does this mean Plato would have favored complex analysis over real analysis?

On Persistence

Persistence of Time

It's generally agreed that thrash metal as a genre is dominated by four bands: Metallica, Megadeth, Slayer, and Anthrax. These bands are, at the very least, the most successful thrash groups. Artistically, they brought different things to the thrash metal party. Metallica wasn't afraid to show off some prog rock along with speed metal in their influences, and created some fairly complex metal. And Dave Mustaine, who formed Megadeth after being kicked out of Metallica, decided that anything Metallica could do, he could do harder, faster, and more technically. Because of this (somewhat one-sided) rivalry, Metallica and Megadeth charted similar territory with their music. Slayer was darker. Dealing with Satan, murder, the Holocaust, or anything else horrific, they used simpler riffs played faster and louder to create a visceral sound. Really, Slayer is as close to death metal as you can come without the growling vocals necessary for genre membership.

Which brings us to Anthrax. How do they fit in? After working my way through the classic albums of three of the Big Four (Metallica's Master of Puppets¹, Megadeth's Rust in Peace, and Slayer's Reign in Blood), I decided that it was time to complete the tetrafecta² and find out. After consulting a few sources, I picked up Persistence of Time, which is considered by many to be Anthrax's best album. I listened to it and was dismayed.

The music in Persistence of Time isn't particularly fast. For a thrash album, that's a rather damning statement. In fact, the album drags on horribly. Persistence of Time turns out to be an appropriate name for the album; as you listen, each second of every song seems to last longer than it should, a strange dilation of time that makes you wonder how only one hour has passed when the album finishes. It seems like three times that, hours spent feeling your energy wane to nothing. But it's not just the speed. The riffs are all limp, having no power and incapable of holding your attention. The album bores me. Listening to it was a chore, and I hope to never do so again.

1. I prefer ...And Justice for All, myself, but that's a minority opinion.
2. What, were you thinking "quadfecta", or "quadrafecta", or something like that? Nuh-uh, dude.

Persistence of Memory

Persistence of Memory is Salvador Dalí's best known work, and the painting most people think of when they hear the term "surrealism". It features a barren landscape with few landmarks, and four clocks that appear to be melting. It's a striking image to be sure, one that has worked its way into our culture, and with it, the idea of surrealism.

Many people think that surrealism as a movement solely focused on presenting the strange and unreal. However, its goals were better defined than that. Surrealism sought to create an art of the subconscious, to reproduce the logic of dreams in art. And at this, I tend to think that it fails. I am, of course, no expert. My knowledge is limited to the collections of a handful of museums that I have been to. But allow me to use Persistence of Memory as an example. Never have I had a dream that presented to me an image similar to the "soft clocks" of that picture. Dalí­ paintings contain images that are completely outside the realm of my experience. Their logic is that of a mad painter, not of a sleeping individual.

Certainly, though, I know what is meant by "dream logic". My dreams are in no way a perfect mirror of reality. Strange things are often accepted at face value, and people and places are connected in unlikely ways. I remember one dream that I had in which I was allergic to color. Nothing about this struck me as odd; I simply spent as much time in the dark as possible. And I'm not the only one who has had a dream in which the front door to your house leads to a building across town. At which point you simply make note that you won't have to mow the lawn this week, seeing as how it no longer exists, and continue on your way. Yet these ideas are absurd while awake. If Dalí, Magritte, and the rest fail to capture this distinctive view of reality, then is there anyone who does?

Yes. Of course yes. There are too many artists out there for there not to exist such a thing. In fact, I'm sure that there are more good examples of dream logic in art than I could ever possibly know about. So I'll just use this space to talk about two of my favorite examples.

The first example would be the episode of Buffy the Vampire Slayer titled "Restless". In it, we see into the dreams of Willow, Xander, Giles, and Buffy. Xander's segment in particular seems very dreamlike: his insecurities have a central role, and all paths seem to eventually lead him to his parents' basement, even if he was just on a playground or driving. Add in a touch of sex (two women make out, off screen, and Buffy's mom is strangely...forward), and you have a bona fide dream.

Second, we have the excellent webcomic A Lesson is Learned, but the Damage is Irreversible. Each strip is its own self-contained vignette, and each seems to take place in a world where things work in ways that they probably (definitely) shouldn't. Certain strips hit the perfect note: bullets that only pierce the flesh of your one true love, philosophical Yeti, and Satan marrying your mom. The art works exactly as it needs to: Panels flow into one another as required by the events in the comic, pulling off some excellent visual effects that create meaning beyond the text. Such excellence truly is something of a dream, because it's too great to exist in the real world.

Persistence of Vision

Persistence of vision is the name given to a physiological phenomenon of the retina. Well, sort of. Some people use the term to refer to the process by which one appears to see motion when images are flashed in quick succession (e.g. in movies). This is incorrect, and kind of complicated³. And honestly, the phenomenon I'm going to talk about is actually called something else nowadays (to avoid confusion); something along the lines of "afterimage", depending on who you ask.

You see, when you look at something, your retina keeps a part of the image for some time after the stimulus is removed. The most obvious example of this is the red and green dots that float around in your field of vision after a flash goes off. This happens essentially because your retina is tired. The chemicals in your retina that move around to create the sensation of color, they take a little time to build back up sometimes. This is a simplification, and probably wrong in some way. If you're really interested, you should do some research. Joseph-Antoine Plateau did some research of his own on the subject back in the mid 19th century. At one point, he stared at the sun for twenty-five seconds; this eventually led to blindness.

It's kind of poetic, actually. He stared at the sun so long that it was scarred on his retina, giving his sight for his science, and carrying the mark with him for the rest of his life.

3. If you want details, check out this link.

Persistence

persistence: Continuance of an effect after the cause is removed.
The American Heritage Dictionary of the English Language, Fourth Edition

Nothing persists; not really. Not in the above sense, anyway. Ask Ozymandias.

Actually, that's not fair. Oz stuck around for longer than we credit him. We still look upon his (or at the very least, Shelley's) works and despair. Not for the same reasons as we might have originally, I suppose. It's a paradox: We despair that we no longer despair. Because if even mighty Ozymandias can be laid low be time, then we certainly will be as well. But eventually we'll forget even that we've forgotten Ozymandias.

This is all very basic stuff: mortality, temporality, entropy. It's the way the universe runs. At the very least, it's the way this universe runs; some might say there are others where such things don't exist⁴. Which brings us to infinity. What a trip, eh? It goes on forever. Which is cool.

Well, I'm not going to tackle that stuff here. I'm underqualified, and it doesn't make for good blog reading⁵. I tried writing some fiction on the theme of persistence for this section of the post, but it was all horror, and either lost the theme or sucked or would make me sound way too creepy. So instead you get this. But it's up for the world to see now, so it can stop kicking about the back of my brain. Thus, a persistent idea sees the light of day, in which it will eventually wither. And if that sounds really harsh, then a) you must have really liked the post, and b) you totally were not paying attention to the last part.

4. Plato, I'm looking at you. And the Christians too, I guess.
5. Not like an excerpt from a news article and a snarky comment, no way!