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

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.

"...you just get used to them"

“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 A^TA = 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 A^T; 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 , which requires one more excursion into the realm of abstraction. , 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, . If you do the same thing, but with a 3-sphere (again, remember that this is the sphere living in four dimensions), you get . Of course, you can’t even really visualize or, for that matter, a 3-sphere, so really visualizing 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 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 , 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.

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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).

The anthropomorphism of religion

I might deduce one final consequence of a skepticism in regards to temporality and causality. If our only experience of the world is of an existent reality, such that something uncreated or destroyed is literally unimaginable, the superfluity of religion becomes very evident. Since it is on the basis of a parallel between finite objects, which are presumed to be necessarily created, and the universe in its totality, which in turn therefore needs its Creator, that modern religions ultimately justify themselves, if creation, rather than lack of creation, is taken to be the phenomenon unjustified by experience then the concept of God is unwarranted.

A beginner's guide to producing new results in mathematics

• First, choose a problem that nobody’s ever solved before. Be careful, though: the most common beginner’s mistake is to choose a problem that’s much too difficult. Fermat’s Last Theorem was an old favorite, but since Andrew Wiles proved that a while back, the new favorites are the Riemann Hypothesis and the Poincaré Conjecture. Rather, as a beginner, you want to pick something that’s within your grasp. I suggest picking two random 200-digit numbers (no need to make them up on your own; get your computer to do that) and adding them together. With probability 1, nobody in the history of the world has ever added these two numbers together before.

• Next, plug it into Mathematica or Maple. No red-blooded, God-fearing mathematician would ever dream of trying to solve a problem without plugging it into a computer and seeing what the answer is first. Sure, the numerical solution will probably be so vague as to be worthless, but this is how things are done.

• Now it’s time to get down to business: write the problem up on the chalkboard in your office. Of course, you could just write it down on a sheet of paper, but there are several objections to that. First, paper is much too permanent; if you end up getting stuck and you’re working on a blackboard, you can always just claim the janitor erased your chalkboard at a crucial stage. Second, it’s extremely important to always have cryptic and incomprehensible scribbles filling your chalkboard to intimidate students and colleagues that might drop by. It’s even better if you have three or four calculations overlapping eachother on the board.

• You’ve got the problem written down on your chalkboard, so now there’s nothing left to do but to solve the damn thing. Remember to take your time. Cross pieces out, even though you could just erase them. Once you’ve made some progress (and built up a goodly amount of chalk on your fingers), step back to ponder the next step. Rub your chin. Run your hands through your hair. Smooth down your shirtfront. Take a bathroom break. Do whatever it takes to transfer the chalk on your hands to various other parts of your anatomy and attire. Under no circumstances should you try to remove any of this chalk before going to teach your next class. Remember, having chalk smeared all over your face, hair, shirt and crotch is all part of the cachet.

• At some point throughout the course of the above, we’ll assume you’ve actually solved the problem. Contemplate the beauty of your solution. Write it down on paper in an indecipherable hand. If you feel like it, consider typing it up, but be sure to include a few errors. This is necessary so as to confuse the readers of your solution and throw them off your track. Whenever there’s a difficult calculation, simply gloss over it and state the end result; odds are none of your readers will actually expend the effort to do the calculation themselves, so you’ve covered your ass in case you flipped a sign somewhere.

Congratulations! You’ve just proved a new result in mathematics. Of course, if you really just added two random gigantic numbers, the probability that anybody cares is 0. Better luck next time.

Take that, logical positivists!

A propos of my brother’s last post and the comments regading it, here’s another example of a flaw in the application of logical principles to reality. Context: in my philosophy of science class I’m reading “The Philosophy of Natural Science” by Carl Hempel. Hempel was a member of the Vienna Circle, much devoted to Carnap, etc. Today he is probably most famous for formulating the so-called “paradox of confirmation” for logical statements, which as I understand goes as follows (I’m using his example): given two logically equivalent statements, such as for example (1)”all crows are black” and (2)”all non-black objects are not crows,” any evidence p which supports one of the statements supports the other as well. Hence, for example the statement p “object x is not black and is not a crow” supports statement (1) as well as statement (2), therefore any non-black non-crow, a fish, a book, a blueberry pie, all provide evidence that crows are black. The obvious response is that this is ludicrous, since p has nothing to do with crows and is therefore irrelevant to the question of whether crows are all black. But let us re-imagine the question. While it may seem that taking non-black non-crows at randomn would provide no evidence regarding the color of crows, if all of the non-black objects in the world were gathered and recorded and none of them were crows, would one not have to concede that the complementary point, “all crows are black,” would have been proved? Or take another example: say we had a box with 10 objects in it, of which an uncertain number werere crows and 4 of the objects were black, and say one decided to test the two statements “all crows in the box are black” and “all non-black objects in the box are not crows.” If an object were pulled out of the box at randomn and proved to be not black and not a crow, then we would know that more than 15% of the non-black objects were not crows, which definitely provides indicative evidence for both of the statements. And by the time 5 non-black non-crows had been pulled out, we would be all but certain that all the crows in the box were black. Therefore, I think that the seeming paradox is simply an illusion of scale. Of course, on a practical level, the paradox holds true, at least in this case: the category of non-black non-crows is so huge that finding examples of them probably won’t provide much evidence of anything. Therefore, the principle of logical equivalences does virtually nothing to advance the investigation. And in fact, I think it is likely that the problem would exist not only in empirical investigations but also in the investigation of certain mathematical properties, for example. Something to keep in mind for those who are convinced of the infinite power of logic to solve both abstract and practical problems.

From politics to mathematics and back

Last night I found myself with an unusually large chunk of time on my hands and, after doing some maintenance work around here that I’d been putting off, decided to catch up on some blog-reading. I read Colby Cosh’s excellent analysis of the ALCS from a week or so ago, enjoyed Billy Beck’s musings on book addiction and rantings on the justice system, caught up on the No Treason/Karen DeCoster/Thomas DiLorenzo shitstorm, uncovered the latest links that appear below in the “External Links” section, and enjoyed Scribbling’s pomegranate pictures. Somewhere along the way, I came across Cosmic Vortex’s “First political diatribe,” which suggests the notion of “political shock levels” as a complement to the future shock levels which extropians go on about. The author lays out a sort of political spectrum, ranging from communist to fascist, and then suggests the following:

Now, its very easy for a socialist and a progressive to discuss issues and come to an understanding, but try to get a socialist and a right wing republican together, and nothing will get accomplished except frustration and anger. Where does this leave us? Not in a good situation really - as theirs no real way to drag anyone more then 1 level away. Even if they did want to try to understand your position, they just couldnt map the concepts over if you jump too far. The cognitive differences would be un-breachable and it would require starting at the begining of their conceptual “tree”, validate every concept along the way, and maybe then something could be worked out.

Interesting idea and stated in a somewhat unique way. What really caught my attention in reading, though, was the sentence I’ve taken the liberty of italicizing. I have to admit, the very first thought that popped into my head upon reading that sentence was: “Sounds like a chain complex!” For those too lazy to click the link, a chain complex is basically a sequence of maps between objects such that moving two steps along always takes you to zero. They arise a lot in topology and homological algebra (for example, I first ran across them while learning about simplicial homology in an algebraic topology class). The connection with shock levels being that if you try to map more than one level down the line you can’t go anywhere but zero, just as the conversation between socialist and republican goes nowhere.

“A nice little analogy,” I though to myself, not quite realizing, for the moment, how loony it would have sounded had I tried to explain it (at this point tenses completely break down, given that I just have tried to explain it). Consider, in addition, how one of my office-mates and I had laughed earlier in the day when she described having just caught herself before asking two of her students (who are identical twins) if they were “isomorphic”. I know I’ve talked about this before (that time when a friend referred to this Strong Bad song as a “canonical techno song”), but I still find the way in which the accumulation of a new vocabulary shapes my outlook either amusing or frightening, depending on the time of day.

Of course, in a sense, the vocabulary is the least important part of what I’ve (hopefully) learned in the last year or so of grad school, but applying the vocabulary outside of its mathematical context is perhaps the most obvious outward sign. Well, one of the most obvious, anyway. Perhaps the other obvious sign of what might be called my increasing mathematical sophistication (or confusion, depending on your perspective) manifests itself in how I answer the questions of my students.

I’m currently teaching four recitation sections of a class innocuously called “Ideas in Mathematics” in the course catalogue, but of which the unofficial course title bestowed by the professor is “Mathematics and Politics”.¹ A friend rather uncharitably characterizes it as “math for morons”, in that it’s the only freshman-level non-calculus course that fulfills the college’s math requirement. Anyway, the point is that I spend most of my time answering questions about the homework or the lectures, and, in answering, I often find myself engaging in impromptu monologues about how intuitionists would object to proofs by contradiction or how mathematics only describes the world insofar as it simplifies away the hard bits. And, most importantly, I have a very difficult time answering conceptual questions definitively.

Needless to say, I imagine my students find it frustrating when, for example, they ask “Why is a conditional true when its antecedent is false?” and I have to say something along the lines of the following:

Well, the short answer is because it’s defined that way, and the long answer is still because it’s defined that way, but it’s defined that way because that’s really the most reasonable way to define it. You see, when we’re thinking about whether a logical statement is “true” or “false”, it’s probably best not so much to think actually in terms of “true” and “false”, but rather in terms of compatibility with the world. In other words, can you believe the statement while also believing in reality without contradicting yourself? We only say the statement is “false” if not; otherwise we call it “true” even though it may be counterfactual, absurd, or completely irrelevant to reality.

At this point, I’m usually lucky if the looks I’m getting are merely quizzical. So I try again:

Well, let’s think in terms of an example. Suppose, back in 2002, a friend told you “If the U.N. approves a war in Iraq, France will go to war.” Now, we know that, in reality, the U.N. didn’t approve the war and that France didn’t go to war. So this is a situation where the antecedent and consequent are both false, so, if we’re thinking in terms of logic, we would say the conditional is true. Why? Well, because you can believe what your friend said and also believe in reality. That is to say, you can believe the statement without contradiction. So the assignment of “true” or “false” is more or less like how you treat a friend: because he’s your friend and you trust him not to mislead you, you assume he was telling the truth unless you can definitely prove that he was lying. In this case, the only way you could know he was lying would be if the U.N. had approved the war and France had stayed home (i.e. the antecedent is true and the consequent is false), since that’s not what actually happened, we would say that the statement is “true”.

Having given this explanation more than a half dozen times, it’s been mildly surprising that nobody has actually called me (and, by extension, math) out on it all being a bunch of convoluted bullshit, but I have to imagine some were thinking it. Usually, at this point, seeing the pained expressions on some of the faces staring out at me, I say something along the lines of “Of course, you could just think of this as the definition,” which seems to be a great relief to some. Which is ironic, given that, without the explanation, the notion that things could be this way just because that’s how they’re defined seems entirely unsatisfactory (which, by the way, I completely agree with. Definitions suck without context).

Having spent ten minutes writing about the conditional, I’m not sure it really illustrates the point I’m trying to make. Perhaps more appropriate would be the times that I’ve had to catch myself before I start ranting about epistemology, theories of logic, reductionism and how mathematics education is, essentially, a system of useful lies. Just as a calculus teacher extolls the virtues of the definite integral, talking about how useful it is and how many amazing physical properties it explains without mentioning that, in any actual application integration is not only difficult but usually impossibly difficult, I find myself teaching material which is useful in certain cases but usually too simplistic to be applicable to the real world. I try to point this out as much as possible, but I think it’s still probably misleading.

That having been said, the underlying concepts are, in fact, incredibly deep. It’s difficult, though, to emphasize that what’s important are the concepts, the fundamental ideas which lead us to particular formulas or computations, especially when midterms are looming and homework is due on Friday. I remember one student asking, the night before the midterm, if she ought to memorize a particular counter-example listed on the review sheet. My honest answer was “No, I don’t think you should memorize it; I think you should understand it,” which I don’t think she liked very much.

That question, though, lies at the heart of the topic that I’m apparently (finally) coming around to, which is that there seems to be a fundamental dichotomy in most people’s minds between, say, the humanities and mathematics. I doubt if anybody would ask an English professor, the night before a midterm, if he ought to memorize Joyce’s “The Dead” for the test, but in a math class it seems like a perfectly legitimate question (incidentally, I’m not trying to say that memorizing is completely worthless; in learning a foreign language, for example, unless one is lucky enough to be living in the country where the language being learned is spoken there’s really no way to make progress without memorizing verb conjugations, vocabulary, etc.). The fact that, for whatever reason, mathematics seems to be equated with rote memorization and plugging values into a formula seems to me to be one of the primary reasons that so many people have such a strong aversion to math.

Which is completely understandable, in a way. Memorizing is boring and almost completely lacking in cognitive content, which most people instinctively recognize, and the fact that math is equated with this boring activity is, I think, one of the primary reasons why an aversion to mathematics is considered acceptable even among people who would strongly decry stunted development in other intellectual pursuits. As John Allen Paulos puts it in Innumeracy: “In fact, unlike other failings which are hidden, mathematical illiteracy is often flaunted: ‘I can’t even balance my checkbook.’ ‘I’m a people person, not a numbers person.’ Or ‘I always hated math.’”

As I look back on the above, I hope I’m not giving the wrong impression about my students. They’ve been wonderful, certainly much more perceptive and good-natured than I had any reason to expect, and I hope they’re learning as much as I am. What it comes down to, I think, is that it’s virtually impossible to interact on a daily basis with people whose level of expertise in a given field is significantly less than one’s own without having to think quite a bit both about the nature of that expertise (imperfect though mine still is) and the misconceptions about the field that will inevitably come to light.

Anyway, I’ve now strayed quite far afield of what I originally intended to write, which was a self-deprecatory post about how I’ve become almost stereotypically geeky in grad school, but I guess the above sort of illustrates that point.

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¹ Actually, a very interesting class. Aside from learning some basic logic and doing some simple proofs, we’ve talked a lot about different voting systems, leading up to the proof of a simplified version of Arrow’s Impossibility Theorem, the full version of which says that there is no voting system (other than a dictatorship, which everybody pretty much agrees isn’t much of a voting system) which satisfies both the Pareto condition (which says that if everybody prefers candidate X to candidate Y, then Y will not win the election) and independence of irrelevant alternatives (i.e. there is no “spoiler effect”). Also, we’ve learned a bit about power indices, namely the Shapley-Shubik and the Banzhaf indices, and are now starting on some basic probability.

Geek ink

So the other day I’m walking down the street behind a girl in a sundress and gigantic sunglasses (ironically, it wasn’t a particularly sunny day), when I happen to look down at her feet and notice she has an unusual tattoo across the back of her left ankle. It’s very small, but as I’m about to pass her I finally realize that it’s the quadratic formula, tattooed right across her Achilles tendon. Which was a bit of a shock. I mean, lame Celtic scrollwork, flames, barbed wire strands, Chinese characters and various other designs are pretty standard for people to permanently etch on their skin these days, but the quadratic equation?

Being the geek that I am, I admit it posed a bit of a quandary; I mean, that’s a dedication to geekiness that I can admire, but, at the same time, the quadratic equation is sort of, well, middle-schoolish, don’t you think? Let’s just say I would have been more impressed if it had been the Gauss-Bonnet Formula or the Riemann Zeta Function or something.

Why social scientists need to throw away their classical paradigms

Brian Doss has responded to my response to my initial response to his critique of Steven Strogatz’ book Sync (whew! Have I broken the record for hyperlinks in a single sentence yet?). There’s a lot to cover here, but I’ll try to do what I can with it.

First, he rightly points out that much of his original point is uncontroversial:

My small beef with the concept that there was any sort of ‘the emerging’ science of spontaneous order was in the (I thought) uncontroversial point that the fields of Biology (macro, micro, and molecular) and Economics both concern themselves with spontaneous order and have done so for centuries (more or less) prior to the publication of ‘Sync’. As that was the case, I further noted that since we have 2 sciences studying specific kinds of spontaneous order and that neither science requires mathematics to either understand the subject matter or to gain knowledge in the first place, that perhaps the author of ‘Sync’ should take some hints and possible insights.

This is all true, but at the same time misleading. While biology has generally done a good job describing the spontaneous order processes that come into play in evolution, the “pure” biological approach is not well-suited to, for example, explaining the tertiary structure of proteins. The tertiary structure of a protein is the general shape of the protein, which is determined by the sequence of amino acids of which the protein is made, but which can be surprisingly complex and three-dimensional. One of the big revelations to come since proteins were first understood as sequences of amino acids is that knowing the higher-order structure of a protein is crucial to understanding what it does and how. And scores of mathematicians are intimately involved in trying to understand exactly how these higher-order structures arise.

In fact, mathematical biology is one of the hottest fields in mathematics today, and much of the research in that area stems from attempts to understand the structures of both proteins and of nucleic acids (i.e. DNA and RNA). And, perhaps surprisingly, advances in that field have had bidirectional influences on supposedly “abstract” areas like knot and braid theory.

Also, to borrow the definitive example from Strogatz’ book, biology did a very poor job of explaining the spontaneous order evident in the simultaneous flashing of Thai fireflies. It took some very hard-core mathematics (and some extreme simplifying assumptions) to even begin to explain how millions of fireflies could all flash in unison without having some “master” firefly. Even verifying those explanations (or discrediting them, for that matter) is something that should be experimentally possible, but such an experiment would be very difficult and has not, as yet, been carried out. This fact, along with the fact that the explanation required significant assumptions, is what led me to say in my previous post that “the mathematics of spontaneous order is both several steps ahead of and well behind the real world.”

The point of this digression is simply to suggest that the classical approaches are reaching their limiting points even in biology and that the days where one didn’t need to know mathematics to do chemistry are swiftly fading. Which isn’t to say that mathematics might not benefit from incorporating the techniques of biology and economics as they relate to spontaneous order, but, based on my admittedly very limited understanding of those two subjects, I have my doubts as to how much fruit such an attempt would bear.

Why do I have doubts? Quite simply, because biology and economics have generally done a good job noting that spontaneous order does arise in the relevant areas, but have not done a particularly good job by themselves of explaining why. Which is not to say that the why is not understood, but the best explanations I’ve seen (here I would cite, for example, Dawkins’ The Selfish Geneor pretty much any microeconomics course) derive, ultimately, from game theory, which is itself a distinguished mathematical discipline, dating back to at least [Fermat].

Back to Doss’ post: he rightfully points out that I unfairly posed the following parenthetical question:

(as a side note, both Doss and Swanson, in the original Catallarchy post linked above, seem to reject mathematics because it conflicts with the principles of Austrian economics and the Austrians’ rejection of empirical economics is well-known; so my question is this: if Austrians reject empiricism as well as mathematics (i.e. deduction), how, exactly, do they advocate gaining knowledge? (Of course, I know the answer, but the Austrian-sympathizers would do well, in my opinion, to keep this question in mind)).

Of course, mathematics does not conflict with the principles of Austrian economics/praxeology (although some of the more zealous and narrow-minded Austrian sycophants seem to think it does); rather, it is the application of mathematical methods to economics in parallel to the classical application of mathematics to physics that conflicts with Austrianism. Or, as Doss puts it, the classical “methods appropriate for studying the physical sciences are inappropriate for studying thinking, acting, subjective humans.”

However, I do think that many Austrians have a fairly poor grasp of exactly what mathematics is, which is why I added the disclaimer that they would do well to keep the above question in mind. I grant that this strikes of pedantry, but I think it’s an important point, and Doss seems to be one of those Austrians who doesn’t seem to understand mathematics very well:

Mathematical methods work in the physical sciences (and to a lesser extent in life sciences) because (a) there is an objective, unchanging reality to the physical laws of the universe and therefore it is (b) possible to design experiments where aspects of reality can be held constant, and thus strict, formal, mathematical relationships can be inferred from the data.

The key misunderstanding, I think, derives from a conflation of the terms “mathematical” and “computational”. Not that this is an uncommon confusion: my non-math friends occasionally ask me what it is, exactly, that I do, occasionally jesting that I must be adding some really big numbers. In fact, mathematics is, ultimately, the discipline devoted to determining the abstract structure that logically follows from a particular axiom set. Mathematicians aren’t, generally speaking, taking a particular equation and plugging a bunch of different values into it to see what results.

In this sense, in fact, mathematics is remarkably similar to Austrian economics itself. Doss links to a Mises article which explicitly compares economists to mathematicians, a comparison I’ve made myself many times before. In fact, in my view, the Austrian school is the most mathematical of all schools of economics by a wide margin. As Doss points out, though, “[t]he difference, of course, is that Austrian scholars have followed a verbal logical formalism instead of a mathematical one.” Which is something I have never well understood. I simply cannot understand why the Austrians consistently reject symbolic logic (which I would call “formal logic”, though obviously an Austrian would contend that my definition is incomplete). Which isn’t to say I haven’t seen the arguments, it’s just that I don’t understand them. For example, here’s what Rothbard has to say on the matter in “Toward a Reconstruction of Utility and Welfare Economics” (which parallels his argument in Man, Economy and State):

The suggestion has been made that praxeology is not really scientific, because its logical procedures are verbal ( literary ) rather than mathematical and symbolic. But mathematical logic is uniquely appropriate to physics, where the various logical steps along the way are not in themselves meaningful; for the axioms and therefore the deductions of physics are in themselves meaningless, and only take on meaning operationally, insofar as they can explain and predict given facts. In praxeology, on the contrary, the axioms themselves are known as true and are therefore meaningful. As a result, each step-by-step deduction is meaningful and true. Meanings are far better expressed verbally than in meaningless formal symbols. Moreover, simply to translate economic analysis from words into symbols, and then to retranslate them so as to explain the conclusions, makes little sense, and violates the great scientific principle of Occam s Razor that there should be no unnecessary multiplication of entities.

In a sense, Rothbard is correct to point out that the Austrians’ deduction is slightly different from standard formal logic, because, in formal logic, you are free to use valid propositions in the deduction even though those propositions may not make any sense when translated into natural language, whereas the Austrians want to use propositions that are both valid and meaningful at every step along the way. However, I disagree with his suggestion that, as such, translating into a formal language is a pointless and superfluous step. If Austrian deductions are, in fact, valid, then they should be translatable into formal language and still hold as valid deductions; the fact that many other deductions might be possible in that formal language that would be invalid to the Austrian would be irrelevant. Rothbard’s objection serves, it seems to me, as a valid rationale as to why an Austrian wouldn’t want to deduce in the formal language itself, but I don’t think it at all justifies the apparent disdain the Austrians have for confirming the deductions they’ve already made in this rigorous way (such confirmation being something I’ve asked for before).

Now, Rothbard is quite right that meaning is better expressed in words than in symbols, which is why it would, presumably, be difficult to translate Austrian axioms/deductions into formal language, but difficulty, in and of itself, shouldn’t serve as justification for not attempting the task. I also find it curious that Rothbard employs Occam’s Razor as a scientific principle, as the Austrians seem to be so disdainful of scientific methodology in economics. Besides the fact that Occam’s Razor is totally inapplicable in this realm, anyway, as the notion that “the simplest explanation is usually the best” says nothing about how to go about confirming conclusions that one has reached.

Having digressed again, I want to make the point Doss (and many other Austrians, for that matter) may be closing his mind to mathematical insights that actually buttress his position because he views mathematics through a classical lens. As a matter of fact, modern mathematics, with its investigations into chaos and complexity, is actually making the case that predictive determinism is essentially impossible. As commenter buck40 points out:

One of the main insights [of modern mathematics] is that prediction and control are in most cases false hopes. Those who apply the insights of complex adaptive systems to social sciences do not seek control, do not counsel control. Quite the opposite, they help policy makers understand why efforts to control will surely fail. You might find that they are your allies in a way, that they are all Hayekians in a manner of speaking.

In that light, consider, for instance, the recent No Treason post “Butterflies and Sweatshops,” where John T. Kennedy suggests that not only is the effect of an individual purchase on third world working conditions too small to predict, but that the effect of that purchase simply cannot be predicted to any level of certainty. I think we would all agree that the world economy is more complicated than the three body problem, yet even the three-body problem cannot, in general, be solved explicitly. Hayek and Mises argued that central planners suffered from a knowledge problem, that someone trying to direct the economy could not, practically speaking, acquire enough knowledge to accurately predict how their interventions would affect the economy. Chaos theorists have extended that further, essentially demonstrating that this is not merely a practical problem, but that, in fact, such a prediction is manifestly impossible, even in the abstract. So, chaos/complexity theorists are “all Hayekians in a manner of speaking”.

Similarly, insights in network theory are helping to explain, for example, both the scale-free aspects of internet hyperlinks and the resiliency of the power grid. One can only imagine that a solid grounding in network theory coupled with an understanding of economics might well yield new insights into economic phenomena.

And, finally, it should be pointed out that the work being done by Strogatz and others is demystifying spontaneous order, demonstrating that there’s nothing supernatural about markets or evolution, but rather that the fruits of spontaneous order are all around us and that the mechanisms that underlie this order are often very simple. To return to the fireflies, the simultaneous flashing that is almost certainly a result of the interaction of coupled oscillators is more extensive than could ever be coordinated by some master firefly keeping time.

Body-snatching Lorenz equations

First off, a bookkeeping note: Curt will be traveling around Europe for the next month, so he likely won’t be posting much, if at all.

Now, I promised last week, in my review of Strogatz’ Sync, that I would devote an entire post to an extended quotation from the book that I found very interesting. Reading the comment thread associated with John Sabotta’s post denouncing evolutionary psychology at No Treason, I was reminded of that promise, so this is that post. In the pertinent passage, Strogatz is discussing a chaos-based encryption system first envisioned by Lou Pecora. Pecora was trying to figure out a way to devise an encryption system based on Lorenz equations, wherein three variables are related to each other in a particular way (specifically, by way of a system of differential equations).

(As a side note, I would point out, apropos my earlier comments on Strogatz book, that this passage serves as an excellent example of both the strengths and weaknesses of the book. The primary strength, aside from Strogatz’ obvious depth of knowledge of the material, is his ability to describe complicated technical mathematics by way of metaphors that make it highly accessible to a lay reader. The primary weakness, at least in my view, is that he doesn’t ever give any of the technical details. Admittedly, systems of differential equations are a bit intimidating, but the fact that the entire book, basically, is written in metaphor is a bit grating)

Anyway, on to the quotation, followed by one or two of my own thoughts:

In technical terms his scheme can be described as follows: Take two copies of a chaotic system. Treat one as the driver; in applications to communications, it will function as the transmitter. The other system receives signals from the driver, but does not send any back. The communication is one-way. (Think of a military command center sending encrypted orders to its soldiers in the field or to sailors at sea.) To synchronize the systems, send the ever-changing numerical value of one of the driver variables to the receiver, and use it to replace the corresponding receiver variable, moment by moment. Under certain circumstances, Pecora found that all the other variables—the ones not replaced—would automatically snap into sync with their counterparts in the driver. Having done so, all the variables are now matched. The two systems are completely synchronized.

This description, though technically correct mathematically, does not begin to convey the marvel of synchronized chaos. To appreciate how strange this phenomenon is, picture the variables of a chaotic system as modern dancers. By analogy with the Lorenz equations, their names are x, y, and z. Every night they perform onstage, playing off one another, each responding to the slightest cues of the other two. Though their turns and gestures seem choreographed, they are not. On the other hand, they are certainly not improvising, at least not in the usual sense of the word. Given where the others are at any moment, the third reacts according to strict rules. The genius is in the artfulness of the rules themselves. They ensure that the resulting performance is always elegant but never monotonous, with motifs that remind but never repeat. The performance is different from minute to minute (because of aperiodicity) and from night to night (because of the butterfly effect), yet it is always essentially the same, because it always follows the same strange attractor.

So far, this is a metaphor for a single Lorenz system, playing the role of the receiver in Pecora’s communication scheme. Now suppose that time stands still for a moment. The laws of the universe are suspended. In that terrifying instant, x vanishes without a trace. In its place stands a new variable, called x’. It looks like x but is programmed to be oblivious to the local y and z. Instead, its behavior is determined remotely by its interplay with y’ and z’, variables in a transmitter far away in another Lorenz system, all part of an unseen driver.

It’s almost like the classic horror movie Invasion of the Body Snatchers. From the point of view of the receiver system, this new x would seem inscrutable. “We’re trying to dance with x but suddenly it’s ignoring all of our signals,” think y and z. “I’ve never seen x behave like that before,” says one of them. “Hey, x,” the other whispers, “is it really you?” But x wears a glazed expression on its face. Just as in the movie, x has been taken over by a pod. It’s no longer dancing with the y and z in front of it—its partners are y’ and z’, unseen doppelgängers of y and z, remote ones in the parallel universe of the driver. In that faraway setting, everything about x’ looks normal. But when teleported to the receiver, it seems oddly unresponsive. And that’s because the receiver’s x has been hijacked, impersonated by this strange x’ coming from out of nowhere. Sensitive souls that they are, y and z make adjustments and modify their footwork. Soon, all becomes right again. The x, y, z trio glides in an utterly natural way, flowing through state space on the Lorenz attractor, the picture of chaotic grace.

But what is so sinister here, and so eerie, is that y and z have now been turned into pods themselves. Unwittingly, they are now dancing in perfect sync with their own doppelgängers, y’ and z’, variables they have never encountered. Somehow, though the sole influence of the teleported x’, subtle information has been conveyed about the remote y’ and z’ as well, enough to lock the receiver to the driver. Now all three variables x, y, and z have been commandeered. The unseen driver is calling the tune.

— pp. 196-7

Unfortunately:

So far, chaos-based methods have proved disappointingly weak. Kevin Short, a mathematician at the University of New Hampshire, has shown how to break nearly every chaotic code proposed to date. When he unmasked the Lorenzian chaos of Cuomo and Oppenheim, his results set off a mini-arms race among nonlinear scientists, as researchers tried to develop ever more sophisticated schemes. But so far the codebreakers are winning.

— pg. 204

The obvious conclusion is that each variable in these specialized systems in fact encodes the entirety of the system. From a mathematical perspective, I admire the ingenuity required to come up with this sort of scheme. From a metaphysical perspective, though, it’s hard not to find this whole thing vaguely unsettling. After all, people usually respond in fairly predictable ways to outside stimuli. The point I guess I’m stumbling towards is this: the idea that a chaotic system in which the actors respond in predictable ways to the other actors could lead to the sort of lockstep mirroring described above seems, at least to me, to say something very pertinent to the whole determinism/free will debate. Admittedly, the eerieness is largely due to Strogatz’ metaphor, but the fact that the y and z variables, thinking themselves to be operating independently of the y’ and z’ variables, would ultimately end up mirroring those same variables simply because they followed the same rules in reacting to the x/x’ variable is pretty fascinating.

What I’m not trying to do is say that this sort of thing settles the determinism/free will debate. Rather, I’m just pointing out that these new mathematics give new insight into ways in which seemingly independent activity can yield identical results. And, although these are admittedly specialized cases, it’s somewhat surprising (at least on an intuitive level) that there are any circumstances in which variables seemingly reacting to another variable’s independent activities would, in fact, end up exactly duplicating the variables to which that wild variable was itself reacting. The result, of course, being a system in which all three seemed to be mutually reacting, while in fact one was paying no attention at all to the other two. In other words, one has to wonder, at least a little bit, in what direction causality points, exactly.

Spontaneous order revisited

Back in September, I wrote a post critiquing the responses of Tim Swanson and Brian Doss (of Catallarchy fame) to Stephen Strogatz’ book Sync: How Order Emerges from Chaos in the Universe, Nature, and Daily Life. Those responses an be found in Brian’s initial post and the ensuing comments thread and suggest that those studying spontaneous order would be best served by following Alfred Marshall’s advice to “burn the mathematics”. My post (which apparently lost the first paragraph or two in the switch to a new domain back in December), is called “Spontaneous Order” and also spawned a response from Neil.

Okay, with the citations out of the way, the issue of the day is whether I still agree with what I said back in September, now that I’ve just finished reading the book; and the answer is largely “yes”. In fact, I don’t think I went nearly far enough in my criticism of the notion that those studying spontaneous order should avoid mathematical formalism. For example, the following was my conclusive summary of the post:

My point is not to demonstrate that the study of spontaneous order is a mathematical discipline, nor that it should be. Rather, I just want to make the point that it has certain similarities to mathematics and, of course, will necessarily need to use mathematical tools in many instances. In fact, though I admit to not knowing nearly enough to be able to have any insights, it seems like mathematics, especially areas of study like graph theory and networks, might be able to shed some light on some of the applications of spontaneous order mentioned by Strogatz

Not very bold, right? Well, I hadn’t actually read the book yet. Now that I have, it’s abundantly clear that anybody who has actually read Strogatz’ book already knows that it is, at heart, a math book; the fact that it’s not publicized as such has more to do with the irrational fear most people have of mathematics. Virtually every result Strogatz cites is a result in pure or applied mathematics, with all the usual deduction and separation from empirical strategies that that entails. Many of these are proofs about idealized models of coupled oscillators, results which probably help explain how, for example, Thai fireflies flash in sync, but building an experiment to actually test this is so difficult that it apparently hasn’t been done to any degree of satisfaction yet (or, if it has, Strogatz doesn’t mention it). The same holds for, say, three-dimensional synchronicity, which has applications to cardiac arrhythmia, but which is discussed in the book purely in terms of mathematics and chemical reactions in a very special kind of fluid.

The point is this: right now, the mathematics of spontaneous order is both several steps ahead of and well behind the real world. It’s several steps ahead in the sense that mathematical explanations of synchronous processes seems, in large measure, to be ahead of the capability of experimental science to confirm (or deny, of course). On the other hand, mathematics is obviously very far behind the real world, as we can’t yet accurately model the spontaneous order that occurs between nerve cells to make our hearts beat, let alone the presumably much more complicated processes occurring within our brains. Whatever the case, reading Strogatz’ book confirmed my suspicion that, in fact, mathematics is essential to the emerging field of spontaneous order (as a side note, both Doss and Swanson, in the original Catallarchy post linked above, seem to reject mathematics because it conflicts with the principles of Austrian economics and the Austrians’ rejection of empirical economics is well-known; so my question is this: if Austrians reject empiricism as well as mathematics (i.e. deduction), how, exactly, do they advocate gaining knowledge? (Of course, I know the answer, but the Austrian-sympathizers would do well, in my opinion, to keep this question in mind)).

This all having been said, Sync was a bit too devoid of mathematical content for my taste, in the sense that, although almost everything in the book boiled down to mathematics, Strogatz explained most of the actual mathematical machinery in terms of analogies to runners on a track or audiences clapping or whatever, whereas I would have liked to see greater mathematical rigor (not necessarily the equations themselves, which are almost certainly too complicated to mean very much to the uninitiated, but rather a more rigorous argument, with references to actual mathematical principles, theorems, etc.). For example, when Strogatz says “[u]sing a theorem from topology, Winfree proved that a twisted scroll ring was impossible, at least as a solitary entity”, it would have been nice if he had explained what theorem, exactly, even if only in the endnotes. Or, as Neil says,

In fact, as [Strogatz] chronicled the mathematical history of sync as an abstract study, I found myself wanting various symbols and equations

As an ego-boost, I’ll point out that in the above quotation from my September post, my suggestion that graph theory and networks “might be able to shed some light on some of the applications of spontaneous order mentioned by Strogatz” was right on, as Chapter Nine of Sync is titled and devoted to “Small-World Networks”.

Now, a couple of quotations to think about (I’ve omitted one interesting and very extended passage, because I want to dedicate and entire, separate post to it, hopefully some time this weekend):

In other words, a dumb rule (majority rule) running on a smart architecture (a small world) achieved performance that broke the world record.

— pg. 251 (Here, Strogatz is talking about the density classification problem for one-dimensional binary automata, where he and one of his students decided to re-wire the binary automata as a small-world network — where most of the connections between automata (think lightbulbs) are locally clustered, but a few are long-distance — and almost immediately, using the simplest algorithm imaginable, were able to solve the problem more consistently than the best algorithm using “dumb architecture”)

Barabási and his team pointed out that scale-free networks [like the Internet or protein interactions in yeast] also embody a compromise bearing the stamp of natural selection: They are inherently resistant to random failures, yet vulnerable to deliberate attack against their hubs. Given that mutations occur at random, natural selection favors designs that can tolerate haphazard insults. By their very geometry, scale-free networks are robust with respect to random failures, because the vast majority of nodes have few links and are therefore expendable. Unfortunately, this evolutionary design has a downside. When hubs are selectively targeted (something that random mutation could never do), the integrity of the network degrades rapidly—the size of the giant component collapses and the average path length swells, as nodes become isolated, cast adrift on their own little islands.

— pg. 257

Helbing and Huberman computed the long-term traffic patterns under a variety of different conditions. When there were only a few vehicles on the road, all the cars sailed past the slower-moving trucks without ever decelerating, while the trucks lumbered along at their maximum safe speed of 55 miles an hour. At higher but still moderate densities of traffic, some unlucky cars found themselves trapped behind trucks for a long time, with no room to pass or switch lanes.

At a critical density of traffic—about 35 vehicles in each lane per mile of road—all the cars and trucks spontaneously synchronized, traveling down the highway like a solid block. Remarkably, out of pure competition, with no coordinator or central authority, a large group of selfish individuals ended up in a cooperative state that was optimal for all of them. (Adam Smith would approve.) This state was optimal in the sense that the flux of traffic was as high as it could be: The number of cars and trucks passing through a given stretch of highway per hour was maximized. It was also the safest way for traffic to flow, because the drivers had no opportunities to change lanes or pass (the maneuvers associated with most accidents). Helbing and Huberman tested their model against data taken from a two-lane Dutch highway and found evidence of the predicted state. At the critical density, the car speeds were at their most stable, as measured by their velocity fluctuations, and lane changing and passing were minimized. Unfortunately—and as the model also predicted—the crystalline state proved to be delicate. At densities just above critical, it melted into a disorganized liquid state, which created opportunities for passing again, leading to unsteady, stop-and-go traffic.

— pgs. 269-70

“In individuals, insanity is rare, but in groups, parties, nations, and epochs it is the rule.”

— Nietzsche, cited on pg. 273

A blast from the past

Today, I stumbled across this Wired article on gopher, the internet protocol developed at the University of Minnesota way back in 1992. The article brought back memories, because I remember gopher from my middle school days when we spent all our time using Lynx to access gopher and read blonde jokes instead of improving our typing, learning HyperCard, or whatever other useless pursuits the teachers had in mind. Given that the paradigm of the day was the BBS, gopher was quite a revelation. Now, of course, everyone is used to everything on the internet being a mere mouse-click away, but that was all-new 12 years ago.

I was somewhat surprised to learn that not only is gopher still kicking, but a few people are actually trying to bring it into the 21st century. For example, John Goerzen, in addition to maintaining supposedly the largest active gopher server in the world at Quux.org, thinks gopher could be used as dynamic data exchange protocol like XML-RPC and SOAP. He also sees it as a good alternative to current PDA and phone browsers:

“Consider this example: Port-a-Goph, a gopher client in development for Palm OS. Cameron Kaiser wrote this in his spare time and got it working quickly on his own Palm,” he said. “Contrast that with the state of Web browsing on handheld devices: Despite many years to improve them, I still regularly run across websites that simply do not render at all, or render so poorly that they are unusable.”

He’s probably right, but, for whatever reason, people seem to like to re-invent the wheel instead of just re-using proven wheels, so gopher probably will never be more than a tiny geek niche. That all having been said, there’s a lot of good stuff available on gopher servers like gopher://quux.org/, which you can access directly through nice browsers like Firefox. If you’re on IE, your best bet is probably Floodgap’s public gopher proxy, which translates gopher pages to HTML. And, if you’re anything like me, you’ll probably go immediately to the jokes pages, which are filled with a vast assortment of predominately nerdy material.

On the subject of internet protocols and the like, I should mention that last night I finished Neal Stephenson’s latest, The Confusion, which just came out in bookstores this week (for those that don’t get the connection between internet protocols and a historical novel set in the 17th century, I’d suggest a thorough perusal of Stephenson’s other work, including Snow Crash or Cryptonomicon or, if you’re too cheap to spend money on books, the essay “In the Beginning Was the Command Line”). The Confusion is quite good, interleaving or “con-fusing” the two main stories much more seamlessly than did its predecessor, Quicksilver, which I’ve already reviewed. Though I’m not particularly in the book review mood right now, I will say that this trilogy is really growing on me and I’m definitely looking forward to September, when The System of the World comes out. One thing that really stands out about The Confusion is that among the diverse topics with which it deals, one of the primary issues is that of money and markets, especially how they arise and how they work. For more on that, check out the Wired interview with Stephenson (via Catallarchy).

Synchronicity

Some weird coincidences today: I finished reading The Illuminatus! Trilogy, by Robert Shea and Robert Anton Wilson, and kept stumbling across various web pages related to the book in some way during the course of my daily web browsing. Which itself ties into one of the book’s main themes, that of synchronicity.

Anyway, over at Wikipedia, the featured article of the day is that on Emperor Norton I, a thoroughly interesting, though most likely quite insane, San Franciscan from the 19th century who declared himself Emperor of the United States and even issued his own currency. Norton gets a bit of play in Illuminatus! as a sort of Discordian hero and he seems to keep popping up in my reading. Of course, his most famous connection to literature is that he was supposedly the model for the King in Twain’s Huck Finn.

Speaking of Twain, while fooling around with MathWorld, I came across an interesting entry on the beast number, 666, which couldn’t help but remind me of The Number of the Beast, an excellent book by Heinlein, an inveterate admirer of Twain’s. In The Number of the Beast, Heinlein posits a “multiverse” with 6^{6^6} different universes contained within it, many of them (perhaps all of them) created by novelists and storytellers. Which is a conceit mentioned briefly in Illuminatus! and central to another Wilson trilogy, the Schrödinger’s Cat Trilogy.

In fact, I’m a bit surprised, given the numerological bent of much of Illuminatus!, that Shea and Wilson don’t devote any attention to some of the interesting properties of the beast number. For example, 666 is equal to the sum of the squares of the first seven primes, the sum of the numbers from 1 to 6 * 6 (i.e. the sum of the numbers from 1 to 36) and

where denotes the Euler phi, or totient, function

Of course, I think my favorite beast number property is that, writing the parameters of Coxeter’s notation side-by-side, the bimonster can be denoted by 666. Which is interesting because the bimonster is the wreathed product of the monster group by Z₂. For those that have no idea what I’m talking about, just take it on faith that the monster group, as one might guess from its name, has a sort of mythical cachet among (certain types of) mathematicians.

Anyway, back to something resembling the English language. Another big theme in Illuminatus! is that of immanentizing the Eschaton, a concept somewhat badly explained in the book as “to cause the end of days”. Now, those hip to the blogosphere scene may recognize “Eschaton” as the name of the name of the blog run by Democratic cheerleader and fellow Philadelphia-dweller atrios. However, I was somewhat surprised to note that the name of the blog is a David Foster Wallace reference; though I can’t stand the blog, I have to give atrios serious props for naming it after the tennis-academy bombardment game from Wallace’s brilliant Infinite Jest (to tie this in further with the math-speak above, I should also mention that Wallace has a pretty good math book called Everything and More: A Compact History of Infinity, which would be, I think, challenging but comprehensible for an interested layman).

And now, in a desperate attempt to salvage some semblance of thematic integrity from this post, here are some interesting quotes from Illuminatus!

“We’re anarchists and outlaws, goddam it. Didn’t you understand that much? We’ve got nothing to do with right-wing, left-wing or any other half-assed political category. If you work within the system, you come to one of the either/or choices that were implicit in the system fro the beginning. You’re talking like a medieval serf, asking the first agnostic whether he worships God or the Devil. We’re outside the system’s categories. You’ll never get the hang of our game if you keep thinking in flat-earth imagery of right and left, good and evil, up and down. If you need a group label for us, we’re political non-Euclideans. But even that’s not true. Sink me, nobody of this tub agrees with anybody else about anything, except maybe what the fellow with the horns told the old man in the clouds: Non serviam.”

— Hagbard Celine, pg. 86

“Just remember: it’s not true unless it makes you laugh. This is the one and sole and infallible test of all ideas that will ever be presented to you.”

— Hagbard Celine, pg. 250

(And Semper Cuni Linctus, the very night that he reamed his subaltern for taking native superstitions seriously, passed an olive garden and saw the Seventeen…and with them was the Eighteenth, the one they had crucified the Friday before. Magna Mater, he swore, creeping closer, am I losing my mind? The Eighteenth, whatshisname, the preacher, had set up a wheel and was distributing cards to them. Now, he turned the wheel and called out the number at which it stopped. The centurion watched, in growing amazement, as the process was repeated several times, and the cards were marked each time the wheel stopped. Finally, the big one, Simon, shouted “Bingo!” The scion of the noble Linctus family turned and fled…Behind him, the luminous figure said, “Do this in commemoration of me.”

“I thought we were supposed to do the bread and wine bit in commemoration of you?” Simon objected.

“Do both,” the ghostly one said. “The bread and wine is too symbolic and arcane for some folks. This one is what will bring in the mob. You see, fellows, if you want to bring the Movement to the people, you have to start from where the people are at. You, Luke, don’t write that down. This is part of the secret teachings.”)

— pg. 324

The most thoroughly and relentlessly Damned, banned, excluded, condemned, forbidden, ostracized, ignored, suppressed, repressed, robbed, brutalized and defamed of all Damned Things is the individual human being. The social engineers, statisticians, psychologists, sociologists, market researchers, landlords, bureaucrats, captains of industry, bankers, governors, commissars, kings and presidents are perpetually forcing this Damned Thing into carefully prepared blueprints and perpetually irritated that the Damned Thing will not fit into the slot assigned to it. The theologians call it a sinner and try to reform it. The governor calls it a criminal and tries to punish it. The psychotherapist calls it neurotic and tries to cure it. Still, the Damned thing will not fit into their slots.

— Hagbard Celine, from Never Whistle While You’re Pissing, pg. 385

It was the chains of communication, not the means of production, that determined a social process; Marx had been wrong, lacking cybernetics to enlighten him.

— pg. 388

“Everybody was lying to the FBI and CIA, sir. They were all afraid of punishment for various activities forbidden by our laws. No variation or permutation on their stories will hang together reasonably. Each witness lied about something, and usually about several things. The truth is other than it appeared. In short, the government, being an agency of punishment, acted as a distorting factor from the beginning, and I had to use information-theory equations to determine the degree of distortion present. I would say that what I finally discovered may have universal application: no governing body can ever obtain an accurate account of reality from those over whom it holds power. From the perspective of communication analysis, government is not an instrument of law and order, but of law and disorder. I’m sorry to have to say this so bluntly, but it needs to be kept in mind when similar situations arise in the future.”

— Fred Filiarisus, pgs. 423-4

DEFINITIONS AND DISTINCTIONS

FREE MARKET: That condition of society in which all economic transactions result from voluntary choice without coercion.

THE STATE: That institution which interferes with the Free Market through the direct exercise of coercion or the granting of privileges (backed by coercion).

TAX: That form of coercion or interference with the Free Market in which the State collects tribute (the tax), allowing it to hire armed forces to practice coercion in defense of privilege, and also to engage in such wars, adventures, experiments, “reforms,” etc., as it pleases, not at its own cost, but at the cost of “its” subjects.

PRIVILEGE: From the latin privi, private, and lege, law. An advantage granted by the State and protected by its powers of coercion. A law for private benefit.

USURY: That form of privilege or interference with the Free Market in which one State-supported group monopolizes the coinage and thereby takes tribute (interest), direct or indirect, on all or most economic transactions.

LANDLORDISM: That form of privilege or interference in the Free Market in which one State-supported group “owns” the land and thereby takes tribute (rent) from those who live, work, or produce on the land.

TARIFF: That form of privilege or interference in the Free Market in which commodities produced outside the State are not allowed to compete equally with those produced inside the State.

CAPITALISM: That organization of society, incorporating elements of tax, usury, landlordism, and tariff, which thus denies the Free Market while pretending to exemplify it.

CONSERVATISM: That school of capitalist philosophy which claims allegiance to the Free Market while actually supporting usury, landlordism, tariff, and sometimes taxation.

LIBERALISM: That school of capitalist philosophy which attempts to correct the injustices of capitalism by adding new laws to the existing laws. EAch time conservatives pass a law creating privilege, liberals pass another law modifying privilege, leading conservatives to pass a more subtle law recreating privilege, etc., until “everything not forbidden is compulsory” and “everything not compulsory is forbidden.”

SOCIALISM: The attempted abolition of all privilege by restoring power entirely to the coercive agent behind privilege, the State, thereby converting capitalist oligarchy into Statist monopoly. Whitewashing a wall by painting it black.

ANARCHISM: That organization of society in which the Free Market operates freely, without taxes, usury, landlordism, tariffs, or other forms of coercion or privilege. RIGHT ANARCHISTS predict that in the Free Market people would voluntarily choose to compete more often than to cooperate. LEFT ANARCHISTS predict that in the Free Market people would voluntarily choose to cooperate more often than to compete.

— Hagbard Celine, from Never Whistle While You’re Pissing, pgs. 622-4

A tale of a theism scorned--continued!

This continuation of the earlier discussion of the merits of atheism will no doubt attract the interest of only the most devoted pedants, but the epistemological issues run so deep here that they have already exhausted many minds and millions of words through the course of history, so another couple may not be overly gregarious here. First, I should like to thank The Serpent for pointing out an error in my definition of solipsism in the comment box to my previous entry on this subject. I defined the solipsist as a sort of empirical agnostic, unsure of the existence of the outside world and believing that such knowledge is probably unattainable.

In fact, as was pointed out to me, this is not the commonly accepted definition of solipsism, which is, according to the OED, the “view that self is the only object of real knowledge or the only thing really existent.” So one can see that the first definition, that “self is the only object of real knowledge,” is essentially comparable to my definition, but the second definition, that oneself is “the only thing really existent,” goes rather beyond it.

However, I have always felt that the movement from acceptance of the first proposition, that “self is the only object of real knowledge,” or in other words that the existence of the outside world cannot be verified, to belief in the second, that self is “the only thing really existent,” is just as logically unjustified as the atheistic movement from acceptance of the idea that the existence of God cannot be ascertained to the belief that God does not exist. Consequently, I have tended to take the first definition as my personal definition of solipsism, which fairly well describes my own personal feelings, and reject the second. However, I am aware that this is somewhat idiosyncratic and confusing for those who associate solipisism with the second definition; my apologies for any confusion created.

However, even this “agnostic” form of solipsism is subject to the criticism that I insinuated before, that it would seem to be debilitating and anti-useful if on the basis of it, qua the most rigorous standards of reason, one affected to refuse to accept the existence of anything in the external world. Now of course one might ask why even after reaching this point one would be prevented from accepting the existence of the external world.

The reason is that the the vey essence of reason, in my opinion, is the principle of refusing to accept that which cannot be established absolutely as true. Now, I accept Descartes’ argument that only the existence of our own thoughts can be absolutely estabished to be true (well, actually only the existence of my thoughts). The existence of the external world, on the other hand, cannot be. As I have said before, I do not necessarily subscribe to the view that one should only accept that which has been absolutely established as true, but if one one wishes to do so, then one, to the best of my knowledge, cannot accept the existence of the external world either on the merits of our perceptions or based on the existence of our thoughts.

One could perhaps accept the existence of the world provisionally if some other self-evident certain belief justified it; Descartes, for example, offers the existence of God, who would not deceive us as to the existence of the outside world, as this justifying belief, but his argument for the existence of God is much less convincing than his argument for the existence of mind. In any case, the immediate point, in any case, is that the existence of the external world cannot be justified on the basis of our perceptions, and the ultimate point is that if atheists wish to apply this rigourous standard of verification to justify their denial of the existence of God, for the sake of consistency they should also apply that standard to the external world and deny its existence.

Since I do not know anyone who has managed to exist for 10 minutes without accepting at some level the existence of the outside world, the verification principle, reason, does not seem to be a sufficient criterion for accepting or rejecting the most fundamental concepts of our existence. Now it could be opposed to this that while it is obviously necessary to accept the existence of the world in order to continue living, belief in the existence of God is not critical to our lives and so we can uphold our intellectual integrity in denying its existence in a way that we cannot with the outside world.

But this type of argument actually just proves my point, because here this distinction is not a logical one, but simply a matter of priorities. We find it necessary to accept the existence of food, for example, so we do; God’s existen