Enthusiasms

My favorite deep, elegant and beautiful explanation.

Few things truly blow my mind. It’s not that I’m so intelligent and perceptive that any surprising fact seems unsurprising to me: it’s more that I like to reserve hyperbole for special occasions, and not every mildly surprising fact is a special occasion. One thing that truly blows my mind, though, is math. Not math in general, but the orgasmic parade of mindblowing facts and theories that exist within it. I’m meta-mindblown by the sheer amount of mind-blowing that goes on in mathematics. It’s such a shame that this isn’t in mathematics education: instead, math is taught variously as a practical, but boring toolkit, or as a series of highly theoretical, but uninteresting abstractions. The real joy of math is nowhere to be found in mandatory schooling anywhere in the world, as far as I know.

I feel confident that this is not simply frustration of a burnout—I don’t have any mathematical education beyond high school and scattered outside reading—because many professional mathematicians have lamented the same thing. However, clearly the solution to this problem isn’t obvious. In the 1960s, there was an attempt to bring this abstract but interesting stuff into elementary math education. It was called New Math, and it was an abject failure.

A while ago I linked to Edge’s question for 2012: what is your favorite deep, elegant or beautiful explanation? I didn’t really consider the question myself, content to revel in the explanations of others smarter than me. But then today, I stumbled on a link, and suddenly it hit me. My favorite explanation that is both deep, elegant and beautiful is Georg Cantor’s infinite set theory in general, and his diagonal argument in particular.

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Representing a wide range of real numbers in finite space is tricky business. Computers use floating point numbers, which behave like real numbers in most ways, but sometimes lead to surprising results. Around zero, not all numbers are representable, and some are only representable in a special form that sacrifices precision and processing speed: the denormalized floating-point numbers.

It shouldn’t come as a surprise that not all real numbers in a given range can be represented in finite space. That’s because between any two reals, or indeed any two rational numbers, there’s infinitely many more. When we start talking about infinite sets, we’ve abstracted so far away from the puny origins of numbers used for counting that our intuitions break apart.

I keep returning to this self-contained little crash course in infinite set theory. It’s really quite readable and assumes no prior knowledge of set theory, but the learning curve is steep. It’s also a guaranteed mind-blower. For a little ape brain used to counting things one by one, the idea of a set of numbers that can’t be counted is pretty hard to grasp in itself. But it gets better, or perhaps worse: there are no more rational numbers than there are natural numbers, even though the natural numbers are clearly a subset of the rationals.

Georg Cantor is the founder of set theory. Cantor’s big insight was that when comparing the sizes of groups, counting isn’t the essential thing. When we want to know if we have more apples than oranges, or if Scrooge McDuck has more money than the bum on the street corner, we count them separately and arrive at neat numbers that we can compare in the usual way. But this clearly doesn’t work on infinite sets. However, there is another way to go about this. We could put the apples in one bag and the oranges in another, and then keep making pairs, one from each bag, and then see which bag empties first. This notion of one-to-one correspondence does generalize to uncountable sets, and it forms the basis of Cantor’s extension of the concept of size to infinite sets: cardinality. Two sets have the same cardinality (which reduces to “size” in finite sets) when they can be put into a one-to-one correspondence with each other.

There is some recent evidence that correspondence, or matching things up with each other to estimate magnitude, is a more basic cognitive strategy for humans than is counting. The widely studied Pirahã tribe of the Amazon appear not to have any words to express exact quantities, not even “one.” However, as experiment has shown, they perform well on one-to-one matching tasks. This also refutes, or at least strongly suggests the falsehood of previous claims that the Pirahã’s lack of words for number has left them completely without the concept of exact quantity. Of course, Cantor died long before the Pirahã came to scientific attention.

Cantor demonstrates a clever way to put the rationals in a one-to-one correspondence with the natural numbers, thus showing that in an important sense—perhaps the only sense that makes sense—they are of equal magnitude. From this result, we may be tempted to think that all infinite sets are the same cardinality. But this is a mistake. The crash course states it succinctly: “infinity is not synonymous with ‘totality’, a clarification which alone dispells many of the ancient conundrums and paradoxes surrounding the infinite.” Cantor has an ingenious method called the diagonal argument that shows that you cannot put the real numbers in a one-to-one correspondence with the natural numbers, and thus that the reals have a larger cardinality.

It’s a proof by negative. Assume we have some pairing of each real number to each natural number. Express the real numbers as infinite decimal expansions. Put them in a table, so that the naturals (1, 2, 3, …) run in the left column, and the corresponding real numbers in the right column. Then start with the first digit of the first real, and change that. Walk diagonally down to the second digit of the second real, and change that. Continue diagonally down through all the real numbers. Here is an illustration from Wikipedia:

The resulting infinite string of digits is demonstrably not in the list, since it differs from the first real in the first digit, from the second real in the second, and so on. Thus the real cannot be in the list, and the one-to-one correspondence can’t be complete. Therefore, the infinity of real numbers is larger than the infinity of natural numbers. Quod erat demonstrandum.

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The most mindblowing sciences are those that are in some sense foundational. There is physics, which explains the most fundamental nature of reality. There is math (and some technical philosophy), which explains the most fundamental nature of rational thought. There’s the cross-section of neurology and psychology, which explains the most fundamental nature of subjective experience. And finally, there’s linguistics, which explains the most fundamental nature of human communication. (Which is to say, they all attempt to explain these things.) Everything else in some way builds on these, and while there’s plenty of interesting stuff in other sciences, there’s nothing quite like turning the basic elements of reality and of experiencing reality as a human being upside-down, or blowing to pieces our most cherished intuitions about the world. Or, for that matter, trying on for size entirely novel categories of thought.

I have a natural attraction to explanations and stories. All my interests beyond those that are purely emotional and physical (martial arts, sex) in some way reduce to stories and explanations. They are what I want to do with my life, in some sense or fashion. Fundamental explanations are especially attractive, since they tend to be the deepest, most elegant and beautiful. It makes me wonder, why didn’t I go into science? It’s still not too late: why don’t I? But then I remember my attraction to smaller stories that aren’t particularly deep or elegant, but are frequently beautiful: the human stories that are best told in literature and in photographs and films. And it saddens me that it seems impossible for me to seriously pursue both kinds.

Sentences about Photography

Photography is easy. Good photography is hard.

Photographs can be true.

Photographs are made, not taken as if by force.

Photography is about consistency. Consistency is talent, and talent is hard work.

Photographs say things that cannot be said in words.

Photographs and words are not enemies.

Photography is light. Photons, not electrons, are the elementary parts of photographs.

Photographs are enough.

Look At All These Links. Week 6.

This is the latest edition of an irregularly scheduled roundup.

The weirdness that is North Korea: Orion Choco-Pie is a popular South Korean snack food. Kaesong is an industrial park just on the North Korean side of the border where North Koreans work for South Korean companies. What happens when you put these two together? A black market. When it’s one of the few foreign commodities around, and workers can double their monthly salary by selling just six of their allotted daily “morale-boosting” snacks, sweet capitalism goes like, well, hot cakes on the black market. But Kim Jong-il appears to have mandated a crackdown before he keeled over. For more weirdness, see also: Tracking Pyongyang.

East Asian age reckoning, used primarily in Korea, counts newborns as one and adds another year not with the passing of a birthday, but with the passing of a lunar year. Thus a baby born on a strategic (or maybe not) date may be two years one day after birth.

You may be familiar with the old Norse system of age reckoning, often used to cheaply evoke viking times: age as counted in the number of winters one has survived. Should you have any interest in such matters, I wrote a thing about martial arts in the viking era a while ago. Features holmgang, Icelandic folk wrestling, berserk-going and flyting.

Have a nice book, video presentations of photobooks.

If you’ve never heard of it, rest assured, you will hear about the Baader-Meinhof phenomenon again. Soon. This despite its lack of a Wikipedia page.

Drowning doesn’t look like drowning. The instinctive drowning response looks remarkably calm to the untrained eye, quite unlike the dramatic thrashing about as seen on tv.

Why in-person socializing is a mandatory to-do item. Should be obvious, but the point is worth repeating. Includes a nice tidbit about a study showing that the most widely-cited scientific papers are almost always written by people who work right next to each other.

Breaking Bad as SNES RPG.

The average font. Creating an average font that isn’t just a blurry mess is actually an interesting theoretical problem. I wish the technical discussion was more thorough.

Iran worried US might be building 8500th nuclear weapon. “Iranian intelligence experts also warned of the very real, and very frightening, possibility of the U.S. providing weapons and resources to a rogue third-party state such as Israel.”

Hex Fiend, a free hex editor for Mac.

What’s the base rate of Kant? How often do we encounter a figure with the influence of a Kant, and why have there been none in a century? The question could be asked of many fields, and the question becomes whether or not it is a subjective or an objective phenomenon: are there objective factors, such as the increasing specialization of fields and subfields, that make such wide-ranging influence impossible, or are we just too far removed from the past and too close to the present to see them both clearly? Is there, perhaps, a Kant or a Darwin working right now, whose influence will not be evident until a hundred years hence?

Where There’s No I

We live in the age of narcissism. I don’t believe for one second that this generation is substantially worse than the one before, or the five hundred before that. The young were immoral and lazy and unwise when Socrates walked the streets of Athens, and they are no different now. As were the old. But this generation has developed the technology to perfectly express the same narcissism our parents and our ancient forebears carried.

Nevertheless, in this age of narcissism, there exists an opposite impulse. Matt Webb quotes Borges: “All men, in the vertiginous moment of coitus, are the same man. All men who repeat a line of Shakespeare are William Shakespeare.” In another story, Borges writes: “I am god, I am hero, I am philosopher, I am demon and I am world, which is a tedious way of saying that I do not exist.” There is a saying, everyone’s favorite word is their own name, and there’s some truth to that. But isn’t there something exhilarating about Borges’s statement, too?

There is no actor, there is only the act. All men are every man, and every man is all men. It is the ultimate denial of narcissism: there is no ego. Much has been said about the so-called online disinhibition effect. When online and anonymous, we act differently, and frequently malevolently. But couldn’t this be a healthy antidote to the narcissism of social networks? Many people claim that anonymity is simply an easy way for people to act like assholes with no consequences. But I think part of it, at least for some people, is also a healthy impulse that is opposed to the egoistic drive.

I don’t know if there are any other animals that have such contradictory instincts: are there any creatures as fiercely individualistic, and at the same as fiercely social as humans? And while we celebrate our own egos, while we prop up celebrity mega-egos and build technologies designed for the generation of ego-worship, shouldn’t we also be celebrating this contrary impulse? Both individualism and collectivism can be ugly, but both can also be beautiful. It would be a beautiful thing if we could have arenas for the expression of both. When I get tired of fapping to the magnificence of my own ego, can’t I have somewhere to go where I can get lost in the masses, where I am everyone and everyone me?

This is one of the reasons I’m such a big proponent of anonymity, which seems to be evolving into public enemy number one. Governments want to kill anonymity so they can keep tabs on us. Companies want to kill anonymity so they can make money off us. Individuals want to kill anonymity so they can worship and be worshipped. I want to preserve anonymity so there can be a place where there is no I.

Cyberdérive

No one surfs the web anymore. Increasingly, we get everything we want from an endless stream provided by a small selection of websites: Reddit, Google, Tumblr, Facebook, Wikipedia and Amazon would be my list. Those six sites probably constitute a frightening percentage of my web browsing, and most of the content that originates on other sites can be consumed through them—or will be consumable through them in the future.

Evgeny Morozov points this out in The death of the cyberflâneur. He might have a book out about “The Dark Side of Internet Freedom”, but he’s right. Morozov compares the early web with 19th century Paris, where flâneurs would stroll the streets aimlessly for the simple pleasure of seeing where one ended up and taking in the sights and sounds that presented themselves. This was how we experienced the early web, but, just like modernism made Paris less friendly to the flâneur, so Facebook and Google and the “app paradigm” and frictionless sharing make the internet less conducive to “cyberflânerie”:

This is the very stance that is killing cyberflânerie: the whole point of the flâneur’s wanderings is that he does not know what he cares about. As the German writer Franz Hessel, an occasional collaborator with Walter Benjamin, put it, “in order to engage in flânerie, one must not have anything too definite in mind.” Compared with Facebook’s highly deterministic universe, even Microsoft’s unimaginative slogan from the 1990s — “Where do you want to go today?” — sounds excitingly subversive. Who asks that silly question in the age of Facebook?

But the author fails to point out that the tradition of aimless strolling for its own sake didn’t die out in Paris. It was briefly resurrected in the 1950s as the dérive: “… an unplanned journey through a landscape, usually urban, where an individual travels where the subtle aesthetic contours of the surrounding architecture and geography subconsciously direct them with the ultimate goal of encountering an entirely new and authentic experience.” The dérive was invented precisely because human experience—in this case, the experience of living in a real-world city—was becoming increasingly limited to a small set of locales. Here’s Guy Debord:

In his study Paris et l’agglomération parisienne (Bibliothèque de Sociologie Contemporaine, P.U.F., 1952) Chombart de Lauwe notes that “an urban neighborhood is determined not only by geographical and economic factors, but also by the image that its inhabitants and those of other neighborhoods have of it.” In the same work, in order to illustrate “the narrowness of the real Paris in which each individual lives … within a geographical area whose radius is extremely small,” he diagrams all the movements made in the space of one year by a student living in the 16th Arrondissement. Her itinerary forms a small triangle with no significant deviations, the three apexes of which are the School of Political Sciences, her residence and that of her piano teacher.

The real web in which each individual “lives” is becoming increasingly narrow. Perhaps what we need is a ressurection of the dérive: occasionally setting aside time to drift aimlessly along the psychogeographical contours of the landscape in order to more fully experience everything it has to offer.

Alternatively, we could all get off our asses and leave the internet. But that’s not going to happen, so how about the next best thing?

Literature as Freedom

Norwegian author Stig Sæterbakken recently took his own life. Apart from a couple essays, I’ve never read anything he’s written. Judging by the literati’s response to his death, that was probably an oversight on my part. It feels vaguely shameful to discover an author because of their death: the fact that it takes losing someone to appreciate them says that given different circumstances, you would never have taken note at all.

Anyway, here’s an excerpt from an essay Sæterbakken wrote called Everything Could Have Been Different. The thrust of the essay is that literature is an existential matter: it shows us ways the world could have been different. And with this realization comes freedom, but the kind of scary freedom that also opens up the possibility of failure. My translation:

I still ask myself, sometimes, why looking at photographs of my parents in their youth always brought with it disgust. Was this disgust brought on solely by the reminder and confirmation of a life before my own? Or was it more specifically connected with the carefree manner and the joy these pictures always bore witness to, and which in no way corresponded with the people I knew as “mother” and “father”, as if it was a strange couple standing there, a strange woman and a strange man, who only years later would undergo the complete transformation of personality that must necessarily have taken place to bring them to where they were now, here, with me, like a safe fort surrounding me, not happy and joking around, but serious, strict and compassionate, sighing vaguely under the load of time and responsibility? (…)

Maybe it’s the inarguable conclusion you need to draw at some point growing up as an oversensitive child, that most of the fun and parties that enrich an adult human life more or less happened BEFORE THEY HAD YOU, or in other and harder words: that everything they are given to see as the content and meaning of life after your arrival on Earth is just a lousy afterparty. (…)

These pictures broke something. A rift was created in my until then prostrate devotion to authority: the thought that the way things appear to me, that’s not necessarily the way they are after all. In other words: the appearance of things and people doesn’t necessarily cover their complete and full identity. And the same thought, taken a little further: things don’t necessarily need to be the way they are. And further: the way things are is random!

A connection to the enduring order was broken. And I was terrified. Why? Because—I didn’t get it then, but I get it now—I had been liberated. I had been liberated from what was enduring, what I thought was determined by nature, unchangeable, unwavering. I was liberated by being shown the possibility of a different world, a different life, a different way to be present as an adult human, a radically different edition of Helge Sæterbakken and Ingjerd Sæterbakken, née Olsen, than the one mother and father had pretended was the only one, an aspect of my parents’ needs in life hidden by their parenting and child-rearing duties, a shady, cheerful and dangerous backyard somewhere inside the main building of deprivation. I’d thought, until then, that I lived in the best of all possible worlds. No, more than that. I thought I lived in the only possible world. (…)

I was terrified. Because freedom is always scary. Freedom that isn’t scary, that precludes the possibility of failure, expulsion and loneliness, is no freedom, it’s only a freedom that has never been threatened, that we have never been denied the enjoyment of, said in another way: a freedom we take for granted, said in a third way: no freedom. (…)

Reading is like drinking. Something unravels. An anchor slips. Thus we’re reunited with childhood’s shock and anguish at being liberated from the apparent order of things, and realize, once again, that what we submit to is frail, that it’s a political and social and psychological system among many possible political and social and psychological systems.

The disturbing message of literature is that everything could be different. That’s why it doesn’t matter where the author gets his material from. Growing up persecuted under Ceausescu’s terror regime in Romania or thirty uneventful years in two rooms and a kitchen in Oslo, both are equally rich as starting points for writing: the deciding factor is the author’s ability to transform that experience into language, that is to say: making language itself an experience, touching the outer edges of the humanly possible. (…)

The cupboard [where I found the old photos of my parents] introduced me to the most radical thought of all: everything could have been done differently. What is law could have been crime. What is true could have been lies. What is society could have been chaos. What is society probably is chaos, only hidden under a lid.

Writing, to me, is making use of the terrifying freedom the cupboard gave me. To thus attempt to carefully close in on the fragility of the enduring order of things.

Controversially, Sæterbakken invited holocaust denier David Irving to a literature festival whose theme was “Truth”. What is true could have been lies. What is law could have been crime.

What is a long life could have been a tragically short one. A tragically short life could have been long.