February 26, 2005

Bentham's mummified corpse, like Lenin's, remains fresh in appearance

Posted by Curt at 08:27 AM in Geek Talk | permalink | 4 comments

It’s almost comforting that such invidious fluffy-minded sludge as this is floating around, as it seems, like religion, to keep the middle-brows hypnotized by “beautiful sentiments” which are so vague as to keep them from actually getting together and doing anything. It’s sort of weird to hear this weakly Marxist social-democratic pap which used to be shouted from the rooftops now being whispered in a low monotonous whine. The author avows his fealty to Jeremy Bentham, not Marx, and calls it utilitarianism not Marxism, but there are many illegitimate fathers along this line of thought.

The root of the idea is that, now that neuroscience has supposedly made it possible to actually identify what makes us happy, the idea of happiness has become quantifiable, and hence a program of providing the greatest happiness to the greatest number of people has become objectively possible. However, the author does not make the slightest effort to apply these wonders of modern science to actually determining what the alleged sources of human happiness are. The neuroscience tack is really just a defensive ploy to ward off the eternal charges that utilitarinism is simply a euphemism for an authoritarian imposition of values. As for espousing his positive program for what constitutes human happiness, it is simply the usual liberal middle-class canards, with not surprisingly a socialist edge: more time to spend with family, a decent wage for everyone, blah blah blah. But he seems to make two pretty criminally unsubstantiated assumptions: one is these sources are essentially the same for everyone, or at least could be under certain conditions, and the other is that they do not inherently conflict with anyone else’s.

I say under certain conditions could be, because in evaluating our current society he seems to privilege envy of other’s material well-being as the principal determinant of happiness. His theory is that above a certain level of material subsistence people are motivated primarily by status-seeking and the desire for a high rank within their social group. Therefore, the increasing wealth of the society will not increase happiness because people measure their well-being relative to the group, not by their absolute prosperity. This is always been a flaw in the concept of the “war against poverty”; I’m not sure it’s much of an argument for socialist economic redistribution. But actually if you read his section on the value of income taxes carefully, he doesn’t even seem to be arguing that they are useful insofar as they can be redirected to the less prosperous, although he does evidently believe that a certain amount of money contributes more to the happiness of a poor person than to a rich one’s. Rather, he seems to think that taking money away from the properous is valuable in and of itself, because it will supposedly make them less focused on the “rat race,” more family-oriented, etc., etc. In short he seems to be advocating a net impoverishment of society.

All of which may be consistent with the program of a good little socialist, but does not necessarily accord marvelously with his own evidence about the supposedly quantified happiness of humanity. The research that he cites non-specifically supposedly indicates that people’s feeling of happiness has not risen in the last half-century, but he does not cite anything which indicates that it has necessarily declined. He cites rising rates of depression and crime as presumably implicit indicators of greater unhappiness, but he does not seem to acknowledge the possibility that in our hyper-medicated and surveillance-based society perhaps people simply report depression and crime more. In any event, if roughly similar numbers of people today as in the ‘50’s report themselves happy (and we believe them), despite the increase in prosperity, that might perhaps indicate that happiness is not fixed to material well-being. Which may be consistent with his general point, but not with his idea of increasing happiness by manipulating income levels.

And even if it did, it seems rather difficult to countenance any social program predicated upon appealing to one of humanity’s most depraved instincts, namely envy. The author acknowledges that his ideal of taxation is mainly motivated by the desire to pander to people’s envy, but he seems to think that their envy will be sated by the loss of prosperity of those around them and that after that point there will be no more. So the envy of the less prosperous will be satisfied by the losses accrued by the more prosperous, which will somehow not be counter-balanced by the chagrin of the more prosperous at the prospect of seeing their status diminished. Very logical.

One of the more egregious presumptions of utilitarians is that non-utilitarian social systems somehow aren’t concerned with seeking the greatest good for the greatest number of people. On the contrary, that’s the defining problem of practically every social and political theory I can think of, and they all either seek or claim to have found the answer—whether such a solution exists, I have my doubts, but that’s why I’m a skeptic about politics. This is a handy trick by utilitarians: they say “I believe in the greatest good for the greatest number of people.” Which is practically begging the question: “As opposed to whom?” It’s useful because it tends to conceal the fact that their real agenda is generally somewhat more specific, and tends to consist in the autocratic notion that one or two measures of social living can be authoritatively determined to be the sources of happiness, and then divided up in a centralized fashion. Those that are the most insistent on the idea of liberty are generally those that are the most skeptical about the possibility of the notion of happiness being either quantitatively defined or generalizable. In other words, only indviduals can determine their own sources of happiness.

For the author, on the other hand, the fact that certain stimuli trigger certain areas of the brain at the times when test subjects profess pleasure has solved the problem of determining happiness. Of course, as mentioned, he never really bothers with the results that those studies have yielded. Somehow the fact that he considers envy to be a principal element of human happiness does not place very severe limits on the harmoniousness of individual happiness. Nor does it constitute a tyranny of the majority, because he claims that in an ideal utilitarian society the happiness of the most unhappy would be considered of pre-eminent importance. Of course, at the beginning of the article he cited the equal importance of each individual’s happiness as the fouding tenet of his theory, but I’m sure it all sorts out in the end.

Among social factors responsible for unhappiness, he cites divorce and unemployment as of pre-eminent importance. Of course, rates of both divorce and unemployment in the crassly materialistic and religious United States are much lower than in the much more overtly utilitarian-embracing Europe, but it would be a bit embarassing for him to admit this after avowing that all traditional value-systems outside of utilitarianism and “individualism” are dead.

Personally the question of the greatest happiness for the greatest number of people doesn’t exactly compel me constantly, although the issue of personal happiness tends to impose itself intransigently. I would have thought that evolutionary biology would have provided an adequate explanation of this, as well as the recurrence of what we call altruism. But such an idea of course suggests that happiness, whatever that is, is not really the point of our little existences, and that the more imperious competitiveness of life will ultimately subvert all of these little trifles of pleasure and pain. But in the meantime, we have these debased statistical notions of happiness to amuse us in an idle hour.

It seems to me that if one’s “objective” measure of happiness is electrical stimulation in the cerebral cortex, the most efficient utilitarian solution to the problem of human happiness would be strap everyone onto hospital gurneys and stimulate the “happiness” part of their brain all day long. If one does not wish to be this deterministic about it, perhaps one should allow more latitute to individuals to discover their own conception of happiness. Personally, I have found happiness generally to be an idea for the unhappy and something rarely spoken of by the happiness; mention of practically guarantees that it is not present in the environment where it is uttered. I don’t deny that what you might call love is the real bridge between personal happiness and moral obligations, and the only true means by which the desires of oneself and of others are united, but such a sentiment can never be mandated; it is entirely resistant to intellectual compulsion. Utilitarianism, which sometimes does a decent job of faking morality, is nevertheless ultimately predicated on the pleasure principle, and hence is wholly inadequate to uniting the moral and the pleasurable except when love truly pertains. In that case, of course, political theory is entirely superfluous, which is why this is all a waste of time.

p.s. I don’t claim that people’s behavior necessarily reflects what really would make them happy, but presumably it does at least reflect what they consciously value. Hence, if I were the author I would have been a bit skeptical of using the results of “surveys” of what people claim to value when the results don’t correlate with their behavior, i.e. they claim that spending time with family is most important, but they spend a disproportiante amount of time working (at least according to him). So either people are not really being forthright (consciously or unconsciously) in responding to surveys, or there is not actually a problem of priorities. In either case, he’s way over-valuing surveys as a guide to what will make people happy.

February 22, 2005

February 13, 2005

"...you just get used to them"

Posted by shonk at 04:37 AM in Geek Talk | permalink | 10 comments

“Young man, in mathematics you don’t understand things, you just get used to them.” —John von Neumann1

This, in a sense, is at the heart of why mathematics is so hard. Math is all about abstraction, about generalizing the stuff you can get a sense of to apply to crazy situations about which you otherwise have no insight whatsoever. Take, for example, one way of understanding the manifold structure on SO(3), the special orthogonal group on 3-space. In order to explain what I’m talking about, I’ll have to give several definitions and explanations and each, to a greater or lesser extent, illustrates both my point about abstraction and von Neumann’s point about getting used to things.

First off, SO(3) has a purely algebraic definition as the set of all real (that is to say, the entries are real numbers) 3 × 3 matrices A with the property ATA = I and the determinant of A is 1. That is, if you take A and flip rows and columns, you get the transpose of A, denoted AT; if you then multiply this transpose by A, you get the identity matrix I. The determinant has its own complicated algebraic definition (the unique alternating, multilinear functional…), but it’s easy to compute for small matrices and can be intuitively understood as a measure of how much the matrix “stretches” vectors. Now, as with all algebraic definitions, this is a bit abstruse; also, as is unfortunately all too common in mathematics, I’ve presented all the material slightly backwards.

This is natural, because it seems obvious that the first thing to do in any explication is to define what you’re talking about, but, in reality, the best thing to do in almost every case is to first explain what the things you’re talking about (in this case, special orthogonal matrices) really are and why we should care about them, and only then give the technical definition. In this case, special orthogonal matrices are “really” the set of all rotations of plain ol’ 3 dimensional space that leave the origin fixed (another way to think of this is as the set of linear transformations that preserve length and orientation; if I apply a special orthogonal transformation to you, you’ll still be the same height and width and you won’t have been flipped into a “mirror image”). Obviously, this is a handy thing to have a grasp on and this is why we care about special orthogonal matrices. In order to deal with such things rigorously it’s important to have the algebraic definition, but as far as understanding goes, you need to have the picture of rotations of 3 space in your head.

Okay, so I’ve explained part of the sentence in the first paragraph where I started throwing around arcane terminology, but there’s a bit more to clear up; specifically, what the hell is a “manifold”, anyway? Well, in this case I’m talking about differentiable (as opposed to topological) manifolds, but I don’t imagine that explanation helps. In order to understand what a manifold is, it’s very important to have the right picture in your head, because the technical definition is about ten times worse than the special orthogonal definition, but the basic idea is probably even simpler. The intuitive picture is that of a smooth surface. For example, the surface of a sphere is a nice 2-dimensional manifold. So is the surface of a donut, or a saddle, or an idealized version of the rolling hills of your favorite pastoral scene. Slightly more abstractly, think of a rubber sheet stretched and twisted into any configuration you like so long as there are no holes, tears, creases, black holes or sharp corners.

In order to rigorize this idea, the important thing to notice about all these surfaces is that, if you’re a small enough ant living on one of these surfaces, it looks indistinguishable from a flat plane. This is something we can all immediately understand, given that we live on an oblate spheroid that, because it’s so much bigger than we are, looks flat to us. In fact, this is very nearly the precise definition of a manifold, which basically says that a manifold is a topological space (read: set of points with some important, but largely technical, properties) where, at any point in the space, there is some neighborhood that looks identical to “flat” euclidean space; a 2-dimensional manifold is one that looks locally like a plane, a 3-dimensional manifold is one that looks locally like normal 3-dimensional space, a 4-dimensional manifold is one that looks locally like normal 4-dimensional space, and so on.

In fact, these spaces look so much like normal space that we can do calculus on them, which is why the subject concerned with manifolds is called “differential geometry”. Again, the reason why we would want to do calculus on spaces that look a lot like normal space but aren’t is obvious: if we live on a sphere (as we basically do), we’d like to be able to figure out how to, e.g., minimize our distance travelled (and, thereby, fuel consumed and time spent in transit) when flying from Denver to London, which is the sort of thing for which calculus is an excellent tool that gives good answers; unfortunately, since the Earth isn’t flat, we can’t use regular old freshman calculus.2 As it turns out, there are all kinds of applications of this stuff, from relatively simple engineering to theoretical physics.

So, anyway, the point is that manifolds look, at least locally, like plain vanilla euclidean space. Of course, even the notion of “plain vanilla euclidean space” is an abstraction beyond what we can really visualize for dimensions higher than three, but this is exactly the sort of thing von Neumann was talking about: you can’t really visualize 10 dimensional space, but you “know” that it looks pretty much like regular 3 dimensional space with 7 more axes thrown in at, to quote Douglas Adams, “right angles to reality”.

Okay, so the claim is that SO(3), our set of special orthogonal matrices, is a 3-dimensional manifold. On the face of it, it might be surprising that the set of rotations of three space should itself look anything like three space. On the other hand, this sort of makes sense: consider a single vector (say of unit length, though it doesn’t really matter) based at the origin and then apply every possible rotation to it. This will give us a set of vectors based at the origin, all of length 1 and pointing any which way you please. In fact, if you look just at the heads of all the vectors, you’re just talking about a sphere of radius 1 centered at the origin. So, in a sense, the special orthognal matrices look like a sphere. This is both right and wrong; the special orthogonal matrices do look a lot like a sphere, but like a 3-sphere (that is, a sphere living in four dimensions), not a 2-sphere (i.e., what we usually call a “sphere”).

In fact, locally SO(3) looks almost exactly like a 3-sphere; globally, however, it’s a different story. In fact, SO(3) looks globally like RP3, which requires one more excursion into the realm of abstraction. RP3, or real projective 3-space, is an abstract space where we’ve taken regular 3-space and added a “plane at infinity”. This sounds slightly wacky, but it’s a generalization of what’s called the projective plane, which is basically the same thing but in a lower dimension. To get the projective plane, we add a “line at infinity” rather than a plane, and the space has this funny property that if you walk through the line at infinity, you get flipped into your mirror image; if you were right-handed, you come out the other side left-handed (and on the “other end” of the plane). But not to worry, if you walk across the infinity line again, you get flipped back to normal.

Okay, sounds interesting, but how do we visualize such a thing? Well, the “line at infinity” thing is good, but infinity is pretty hard to visualize, too. Instead we think about twisting the sphere in a funny way:

You can construct the projective plane as follows: take a sphere. Imagine taking a point on the sphere, and its antipodal point, and pulling them together to meet somewhere inside the sphere. Now do it with another pair of points, but make sure they meet somewhere else. Do this with every single point on the sphere, each point and its antipodal point meeting each other but meeting no other points. It’s a weird, collapsed sphere that can’t properly live in three dimensions, but I imagine it as looking a bit like a seashell, all curled up on itself. And pink.

This gives you the real projective plane, RP2. If you do the same thing, but with a 3-sphere (again, remember that this is the sphere living in four dimensions), you get RP3. Of course, you can’t even really visualize RP2 or, for that matter, a 3-sphere, so really visualizing RP3 is going to be out of the question, but we have a pretty good idea, at least by analogy, of what it is. This is, as von Neumann indicates, one of those things you “just get used to”.

Now, as it turns out, if you do the math, SO(3) and RP3 look the same in a very precise sense (specifically, they’re diffeomorphic). On the face of it, of course, this is patently absurd, but if you have the right picture in mind, this is the sort of thing you might have guessed. The basic idea behind the proof linked above is that we can visualize 3-space as living inside 4-space (where it makes sense to talk about multiplication); here, a rotation (remember, that’s all the special orthogonal matrices/transformations really are) is just like conjugating by a point on the sphere. And certainly conjugating by a point is the same as conjugating by its antipodal point, since the minus signs will cancel eachother in the latter case. But this is exactly how we visualized RP3, as the points on the sphere with antipodal points identified!

I’m guessing that most of the above doesn’t make a whole lot of sense, but I would urge you to heed von Neumann’s advice: don’t necessarily try to “understand” it so much as just to “get used to it”; the understanding can only come after you’ve gotten used to the concepts and, most importantly, the pictures. Which was really, I suspect, von Neumann’s point, anyway: of course we can understand things in mathematics, but we can only understand them after we suspend our disbelief and allow ourselves to get used to them. And, of course, make good pictures.

1 This, by the way, is my second-favorite math quote of the year, behind my complex analysis professor’s imprecation, right before discussing poles vs. essential singularities, to “distinguish problems that are real but not serious from those that are really serious.”

2 As a side note, calculus itself is a prime example of mathematical abstraction. The problem with the world is that most of the stuff in it isn’t straight. If it were, we could have basically stopped after the Greeks figured out a fair amount of geometry. And, even worse, not only is non-straight stuff (like, for example, a graph of the position of a falling rock plotted against time) all over the place, but it’s hard to get a handle on. So, instead of just giving up and going home, we approximate the curvy stuff in the world with straight lines, which we have a good grasp of. As long as we’re dealing with stuff that’s curvy (rather than, say, broken into pieces) this actually works out pretty well and, once you get used to it all, it’s easy to forget what the whole point was, anyway (this, I suspect, is the main reason calculus instruction is so uniformly bad; approximating curvy stuff with straight lines works so well that those who who are supposed to teach the process lose sight of what’s really going on).

February 12, 2005

But what about the antiquarians?

Posted by Curt at 07:44 AM in Words of Wisdom | permalink | 1 comment

"Those who do not learn from the mistakes of the past are doomed to repeat them." This is good advice, especially for democracies, which tend (as even Plato and Thucydides noted) to have short-term attention spans. But we shouldn't forget an equally important lesson, articulated most forcefully by Nietzsche: The health of a person and a people also depends vitally on the capacity to forget. Forgetting is necessary to free ourselves from imperfectly understood "lessons of history," so that we can see the challenges ahead clearly, without preconceptions or prejudice. Forgetting is also the better part of forgiving, and there are whole domains of political controversy -- indeed, whole regions of the world -- where a little less history could be of service in this respect.
--Eliot Noyes, Getting Past the Past

February 10, 2005

Superstring cultists--tough luck

Posted by Curt at 08:37 AM in Science | permalink | 1 comment

At the conjunction of this critique of reductionism in physics and this interview with Benoit Mandelbrot I think one sees the same basic dynamic at work: a devaluation of simplicity and generalization in math and science, what I suppose Mandelbrot might call “smoothness,” and a preference for the complex and the multifarious. To some extent this seems to cut against the basic scientific impulse to simplify, to generalize, which is what a law or an equation generally does. In Laughlin I think there is even a certain disillusionment with realism perhaps not totally dissimilar from that in the analysis of language by dear friend Wittgenstein. Although, by encouraging investigation of the specifics and intricacies of phenomena which seem to be superficially covered by the most general and basic laws and to give up idle speculation about the far nether regions of the universe in space and time which cannot in any way be corroborated, he seems to be trying to bring physics back into the solid world of relative certainties and reasonable evidence, it seems to me that this is a tacit admission that the theories which seem to cover and explain adequately all phenomena except for those extreme edges are in actuality insufficient to represent the richness of even the most mundane levels of reality.

Just as in the case of the over-heated discoveries of Wittgeinstein and Cambridge group, this sudden realization that the broad and universal physical laws established and the abstract shapes used to represent them don’t really reflect the full multiplicity of reality seems a little phony to me. I mean, isn’t that the entire point? Isn’t that abstractness and simplicity supposed to yoke all of that complexity within a reasonable level of comprehensibility sufficient to possibly predict other phenomena, or at least relate them to what we have already seen? Now maybe we see a revision in the valuation of these ideals, and in both Laughlin and Mandelbrot a movement away from final solutions, formulations and summations. Seemingly nothing out-of-the-ordinary about that, but if one ceases to regard oneself as capturing the essence of a phenomenon in an equation or image describing it, then that necessarily leads to a re-evaluation of the type of work one is doing and the standard by which it is judged. Let’s put it this way: although there are many rules which often govern both the form and content of a form (some, granted, quite idiosyncratic and individual), it would be quite ludicrous to suggest that a listing of those rules would be an adequate reflection or description of the poem, let alone itself be a poem, or equivalent to the poem.

Philosophically, I’m not troubled by this as many scientists seem to be. Despite the many declarations that Newton had discovered the very mechanism by which God controlled the universe, he himself complained famously that he felt like a dilettante on the shoreline picking up stones and shells that amused him while neglecting the vast ocean before him. This seems unnecessary if one regards theorems as essentially creations, not mirrors of nature, and hence judge the cathedral of scientific knowledge by its height above the ground rather than as an incomplete ladder to the heavens.

February 05, 2005

Abuse? I'll show you abuse!

Posted by shonk at 02:19 AM in Economics | permalink | 17 comments

Note to Curt:

Just because the state claims the authority to apprehend and punish rapists doesn’t mean that apprehending and punishing rapists is a form of state coercion. Nor is the notion that rape is bad an example of state coercion. Depending on your perspective, this is either a moral truth derived from God/reason/whatever or a widely-accepted social convention. Similarly, the notion that one can own property is (again, depending on your perspective) either morally necessary or a widely-accepted social convention that seems to work pretty well (here I’m dispensing with Communists and other fools who have nothing intelligent to say on the matter). Either way, the fact that the state claims ultimate authority to adjudicate property disputes does not make private property a form of state coercion. (Further reading)

February 01, 2005

The abuse of a college education

Posted by Curt at 07:25 PM in Economics | permalink | 5 comments

“Perhaps you’re familiar with “the tragedy of the commons,” a social dilemma outlined by the late biologist Garrett Hardin in a famous 1968 essay of the same name. The dilemma is that when individuals pursue personal gain, the net result for society as a whole may be impoverishment. (Pollution is the most familiar example.) Such thinking has fallen out of fashion amid President Bush’s talk of an “ownership society,” but its logic is unassailable.”

That response seems like a pretty damn obtuse interpretation of the essay, simply because the essay is nothing if not a plea for the creation of property rights. Furthermore, while it is true that Hardin claims that pursuing individual gain leads to group catastrophe, the word “when” in the paragraph above implies that there are times when the individual doesn’t, whereas Hardin claims that individuals basically always pursue their own interest, which is the problem in high-density situations where some amout of coordination is necessary. However, upon re-reading it, I realize that for Hardin property rights only forms a part of a wished-for imposition of coercive measures which will prevent individuals from pursuing personal gain at the expense of their environment. Which makes sense, because property rights, for all this may get lost in the ceaseless ideological wrangling today, are themselves forms of state-imposed coercion. Dismiss the semi-metaphysical nonsense in Locke and Kant about gaining “just propriety” over an object by making a visible mark on it. Think about it: animals control exactly as much “property” as they can defend; cheetahs peeing on trees only works because they will fight to defend what they have claimed. By contrast, think about who adjudicates the (in theory) incontestable property rights: the authorities, i.e. in our society, the State. The corollary of this, of course, is that nationalized or federal property is not “public property,” in the sense of property owned by the public—quite the contrary. The dichotomy between it and “private property” is spurious. “Public property” is simply property owned by the government. This no doubt seems obvious and intuitive, but based on the foolishness I cited above, it bears repeating that property rights, whether granted to others by the government or to itself, are not opposed to coercive state power but are in fact the very essence of it. That fact is perhaps more apparent in regards to so-called “intellectual property.”

As a marginal note, Hardin’s essay, despite the pithiness of its central analogy, is rather dispiriting insofar as it takes Hegel’s statement that “Freedom lies in the recognition of necessity” as its motto and guiding spirit. That formulation is, as I believe I have said before, perfectly monstruous. Freedom means nothing if it is not the absence of restriction, and it is perhaps a sign of the evasive confusion of priorities in Western culture that one would pretend to celebrate this value in such a way while in fact describing its opposite. Freedom is not an act or a thought, but rather a set of conditions under which action and thought occur. This is the same idealistic debasement of the language that has turned love into a deed: making love.