links of the day

On Understanding Advanced Mathematics

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What’s it like to have an understanding of very advanced mathematics? A very detailed answer in a Quora post:

  • You can answer many seemingly difficult questions quickly. But you are not very impressed by what can look like magic, because you know the trick. The trick is that your brain can quickly decide if question is answerable by one of a small number of powerful general purpose “machines” (e.g. continuity arguments, combinatorial arguments, correspondence between geometric and algebraic objects, linear algebra, compactness arguments that reduce the infinite to the finite, dynamical systems, etc.). The number of fundamental ideas and techniques that people use to solve problems is pretty small — see http://www.tricki.org/tricki/map for a partial list, maintained by Tim Gowers.
  • You are often confident that something is true long before you have an airtight proof for it (this happens especially often in geometry). The main reason is that you have a large catalogue of connections between concepts, and you can quickly intuit that if X were to be false, that would create tensions with other things you know to be true, so you are inclined to believe X is probably true to maintain the harmony of the conceptual space. It’s not so much that you can “imagine” the situation perfectly, but you can quickly imagine many other things that are logically connected to it.
  • Your intuitive thinking about a problem is productive and usefully structured, wasting little time on being puzzled. For example, when answering a question about a high-dimensional space (e.g., whether a certain kind of rotation of a five-dimensional object has a “fixed point” which does not move during the rotation), you do not spend much time straining to visualize those things that do not have obvious analogues in two and three dimensions. (Violating this principle is a huge source of frustration for beginning maths students who don’t know that they shouldn’t be straining.) Instead…
  • When trying to understand a new thing, you automatically focus on very simple examples that are easy to think about, and then you leverage intuition about simple examples into much more impressive insights. For example, you might imagine two- and three- dimensional rotations that are analogous to the one you really care about, and think about whether they clearly do or don’t have the desired property. Then you think about what was important to those examples and try to distill those ideas into symbols. Often, you see that the key idea in those symbolic manipulations doesn’t depend on anything about two or three dimensions, and you know how to answer your hard question.
    As you get more mathematically advanced, the examples you consider easy are actually complex insights built up from many easier examples; the “simple case” you think about now took you two years to become comfortable with. But at any given stage, you do not strain to obtain a magical illumination about something intractable; you work to reduce it to the things that feel friendly.
  • You go up in abstraction, “higher and higher”. The main object of study yesterday becomes just an example or a tiny part of what you are considering today. For example, in calculus classes you think about functions or curves. In functional analysis or algebraic geometry, you think of spaces whose points are functions or curves — that is, you “zoom out” so that every function is just a point in a space, surrounded by many other “nearby” functions. Using this kind of “zooming out” technique, you can say very complex things in very short sentences — things that, if unpacked and said at the “zoomed in” level, would take up pages. Abstracting and compressing in this way allows you to consider very complicated issues while using your limited memory and processing power.
  • Understanding something abstract or proving that something is true becomes a task a lot like building something. You think: “First I will lay this foundation, then I will build this framework using these familiar pieces, but leave the walls to fill in later, then I will test the beams…” All these steps have mathematical analogues, and structuring things in a modular way allows you to spend several days thinking about something without feeling lost or frustrated. Andrew Wiles, who proved Fermat’s Last Theorem, used an “exploring” metaphor: “Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it’s completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are the culmination of—and couldn’t exist without—the many months of stumbling around in the dark that proceed them.”
  • You are humble about your knowledge because you are aware of how weak maths is, and you are comfortable with the fact that you can say nothing intelligent about most problems. There are only very few mathematical questions to which we have reasonably insightful answers. There are even fewer questions, obviously, to which any given mathematician can give a good answer. After two or three years of a standard university curriculum, a good maths undergraduate can effortlessly write down hundreds of mathematical questions to which the very best mathematicians could not venture even a tentative answer. This makes it more comfortable to be stumped by most problems; a sense that you know roughly what questions are tractable and which are currently far beyond our abilities is humbling, but also frees you from being intimidated, because you do know you are familiar with the most powerful apparatus we have for dealing with these kinds of problems.

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(Hat tip: Chris Dixon)

Social Media During the Reformation, or How Luther Went Viral

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It’s hard to put the words “social media” and “Reformation” together, yet this brilliant piece in The Economist explains how Martin Luther’s 95 Theses went viral.

Although they were written in Latin, the “95 Theses” caused an immediate stir, first within academic circles in Wittenberg and then farther afield. In December 1517 printed editions of the theses, in the form of pamphlets and broadsheets, appeared simultaneously in Leipzig, Nuremberg and Basel, paid for by Luther’s friends to whom he had sent copies. German translations, which could be read by a wider public than Latin-speaking academics and clergy, soon followed and quickly spread throughout the German-speaking lands. Luther’s friend Friedrich Myconius later wrote that “hardly 14 days had passed when these propositions were known throughout Germany and within four weeks almost all of Christendom was familiar with them.”

The unintentional but rapid spread of the “95 Theses” alerted Luther to the way in which media passed from one person to another could quickly reach a wide audience. “They are printed and circulated far beyond my expectation,” he wrote in March 1518 to a publisher in Nuremberg who had published a German translation of the theses. But writing in scholarly Latin and then translating it into German was not the best way to address the wider public. Luther wrote that he “should have spoken far differently and more distinctly had I known what was going to happen.” For the publication later that month of his “Sermon on Indulgences and Grace”, he switched to German, avoiding regional vocabulary to ensure that his words were intelligible from the Rhineland to Saxony. The pamphlet, an instant hit, is regarded by many as the true starting point of the Reformation.

You probably learned in your world history class that the 95 Theses were a precursor to the Reformation. So why did Luther’s message spread?

Unlike larger books, which took weeks or months to produce, a pamphlet could be printed in a day or two. Copies of the initial edition, which cost about the same as a chicken, would first spread throughout the town where it was printed. Luther’s sympathisers recommended it to their friends. Booksellers promoted it and itinerant colporteurs hawked it. Travelling merchants, traders and preachers would then carry copies to other towns, and if they sparked sufficient interest, local printers would quickly produce their own editions, in batches of 1,000 or so, in the hope of cashing in on the buzz. A popular pamphlet would thus spread quickly without its author’s involvement.

As with “Likes” and retweets today, the number of reprints serves as an indicator of a given item’s popularity. Luther’s pamphlets were the most sought after; a contemporary remarked that they “were not so much sold as seized”. His first pamphlet written in German, the “Sermon on Indulgences and Grace”, was reprinted 14 times in 1518 alone, in print runs of at least 1,000 copies each time. Of the 6,000 different pamphlets that were published in German-speaking lands between 1520 and 1526, some 1,700 were editions of a few dozen works by Luther. In all, some 6m-7m pamphlets were printed in the first decade of the Reformation, more than a quarter of them Luther’s.

Another interesting point is that the spread of Luther’s message wasn’t limited to printed media:

It was not just words that travelled along the social networks of the Reformation era, but music and images too. The news ballad, like the pamphlet, was a relatively new form of media. It set a poetic and often exaggerated description of contemporary events to a familiar tune so that it could be easily learned, sung and taught to others. News ballads were often “contrafacta” that deliberately mashed up a pious melody with secular or even profane lyrics. They were distributed in the form of printed lyric sheets, with a note to indicate which tune they should be sung to. Once learned they could spread even among the illiterate through the practice of communal singing.

The piece is interesting throughout, and has a very good conclusion: Today’s social-media systems do not just connect us to each other: they also link us to the past.

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Question for the reader: what other events/messages in history, do you think, spread virally in a similar fashion? I can think of a few.

The Contagiousness of Yawning

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I remember seeing a Mythbusters episode where they weren’t able to bust the myth that yawns are contagious (they ruled it was “plausible”). In a paper published this year, two Italian researchers, Ivan Norscia and Elisabetta Palagi, carried out a more thorough experiment. The goal was to determine what are some variables that play a factor in yawn contagion. Their abstract:

The ability to share others’ emotions, or empathy, is crucial for complex social interactions. Clinical, psychological, and neurobiological clues suggest a link between yawn contagion and empathy in humans (Homo sapiens). However, no behavioral evidence has been provided so far. We tested the effect of different variables (e.g., country of origin, sex, yawn characteristics) on yawn contagion by running mixed models applied to observational data collected over 1 year on adult (>16 years old) human subjects. Only social bonding predicted the occurrence, frequency, and latency of yawn contagion. As with other measures of empathy, the rate of contagion was greatest in response to kin, then friends, then acquaintances, and lastly strangers. Related individuals (r≥0.25) showed the greatest contagion, in terms of both occurrence of yawning and frequency of yawns. Strangers and acquaintances showed a longer delay in the yawn response (latency) compared to friends and kin. This outcome suggests that the neuronal activation magnitude related to yawn contagion can differ as a function of subject familiarity. In conclusion, our results demonstrate that yawn contagion is primarily driven by the emotional closeness between individuals and not by other variables, such as gender and nationality.

The gist? You’re more likely to yawn after seeing a relative (rather than a stranger) yawn.

Also: did reading this blog post make you yawn? Research has shown that even reading about yawning can trigger yawning in humans. Not boring at all.
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Facebook Timeline and Advertising

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Did you think Facebook’s latest move of unveiling Timeline was a way of improving user experience? Perhaps, but as this BetaBeat post posits, it’s also designed to woo advertisers:

In what seemed like an unrelated move, in September, Facebook announced a brand new type of profile called Timeline, where your whole personal history is laid out by month-by-month, all the way back to your birth. At the time, Facebook described it to consumers as a chance to: “Share and highlight your most memorable posts, photos and life events on your timeline. This is where you can tell your story from beginning, to middle, to now.” By the end of this year all 800 million plus Facebook profiles will have been converted to this new interface.

What most users don’t know is that the new features being introduced are all centered around increasing the value of Facebook to advertisers, to the point where Facebook representatives have been selling the idea that Timeline is actually about re-conceptualizing users around their consumer preferences, or as they put it, “brands are now an essential part of people’s identities.”

The name itself is cleverly designed to conceal the fact that your profile no longer arranges information chronologically. Yes, things are laid out by year and by month. But, when it comes to what’s displayed to your social circle at any given time, other metrics, including direct payments to Facebook itself, will now influence the ranking and placement of stories. This payola will be a crucial part of the graph rank, the new metric for placement that the social network uses to determine what appears on your profile.

If this seems a bit disheartening, remember that you aren’t a Facebook customer: its advertisers are.

The Histrionics of Grief in North Korea

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Following the death of Kim Jong-Il, North Koreans took to the streets to mark their grief of their beloved Leader. I watched, mouth aghast, not believing if these scenes were real:

Amy Davidson has a brief post in The New Yorker about the hysterics of the North Koreans:

How does a whole crowd fake tears? Barbara Demick, in “Nothing to Envy,” her book on the ravaged social landscape of North Korea, collected accounts of how ordinary North Koreans set themselves to just that task after the death of Kim’s father, Kim Il-sung, back in 1994: “It was like a staring contest. Stare. Cry. Stare. Cry,” a student told her. “Eventually, it became mechanical. The body took over where the mind left off and suddenly he was really crying. He felt himself falling to his knees, rocking back and forth, sobbing just like everyone else.”

Cue Barbara Demick’s explanation of how the North Koreans grieve:

Those waiting in line would jump up and down, pound their heads, collapse into theatrical swoons, rip their clothes and pound their fists at the air in futile rage. The men wept as copiously as the women.

The histrionics of grief took on a competitive quality. Who could weep the loudest? The mourners were egged on by the TV news, which broadcast hours and hours of people wailing, grown men with tears rolling down their cheeks, banging their heads on trees, sailors banging their heads agains the masts of their ships, pilots weeping in the cockpit, and so on. These scenes were interspersed with footage of lightning and pouring rain. It looked like Armageddon.

Seems like history repeats itself.

Your Credit Score and Social Media

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Well, this is slightly unnerving. In a BetaBeat post titled “As Banks Start Nosing Around Facebook and Twitter, the Wrong Friends Might Just Sink Your Credit,” we learn about a new wave of start-ups that is…

working on algorithms gathering data for banks from the web of associations on the internet known as “the social graph,” in which people are “nodes” connected to each other by “edges.” Banks are already using social media to befriend their customers, and increasingly, their customers’ friends. The specifics are still shaking out, but the gist is that eventually, social media will account for at least the tippy-top of the mountain of data banks keep on their customers.

“There is this concept of ‘birds of a feather flock together,’” said Ken Lin, CEO of the San Francisco-based credit scoring startup Credit Karma. “If you are a profitable customer for a bank, it suggests that a lot of your friends are going to be the same credit profile. So they’ll look through the social network and see if they can identify your friends online and then maybe they send more marketing to them. That definitely exists today.”

And in the last year or so, financial institutions have started exploring ways to use data from Facebook, Twitter and other networks to round out an individual borrower’s risk profile—although most entrepreneurs working on the problem say the technology is three to five years away from mainstream adoption.

Here’s what I am thinking. If you have a solid credit rating, then exposing your social media outlets could potentially hurt you. On the other hand, these algorithms may be devised such that you take a bigger hit if you don’t divulge your information. If you have a poor credit rating but a strong network of friends, then divulging your social media crumbs could help you in your overall credit score. One thing is for certain, however: if there is any way that a bank could find out more information about you to better predict your ability to repay a loan, the more aggressively it will try to implement the schema into its arsenal of judging your credit score.

We live in a brave new world.

How To Be a Better Blogger

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Dan Frommer’s post on how to blog better is one of the most helpful blog posts I’ve read this year. Here are some of Dan’s ten tips to be a better blogger:

  1. Above all else, factual accuracy and attention to detail. That’s the easiest and best way to build and maintain trust over the long-term. If a fact is wrong, fix it and don’t be shy about it. If an opinion or prediction is wrong, learn from it and consider explaining how you got it wrong.
  2. Write the site that you want to read. That covers story selection, length, frequency, style, vocabulary, attitude, humor, level of sensationalism, and more. Don’t publish anything you’re not proud of. Be yourself.
  3. Be more skeptical. Companies and people have no interest in telling any side of the story but their own. Often, that side is flawed, invalid, or incorrect. Let someone else be the gullible one who looks silly later: Always question everything. (But don’t let it turn you into too much of a conspiracy theorist.)
  4. Try new things, all the time. Especially those that are a little outside your comfort zone. This is the Internet — don’t act like you’re writing for Time Magazine in the 80s. Stories can be pictures, charts, lengthy essays, numbered lists, or 140 characters. Measure how your experiments do, and take the results into account for the future.

However, I think the most important lesson for me is regarding attribution. I make it very clear when I quote or paraphrase, but what I have to get better at is referring you, the reader, to click over to the original piece I link to. Writes Dan:

Aim to become as big of a traffic referrer as you possibly can — not only is that good policy, but it’s a great business asset.

Amen. With that in mind, you should read Dan’s post to see his other tips for better blogging.

Top Ten Physics Breakthroughs of 2011

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Physics World begins thusly:

The two physics stories that dominated the news in 2011 were questions rather than solid scientific results, namely “Do neutrinos travel faster than light?” and “Has the Higgs boson been found?”

After that disclosure, the Physics World editorial team has compiled a nice end-of-the-year list. The first place goes to Aephraim Steinberg and colleagues from the University of Toronto in Canada for their experimental work on the fundamentals of quantum mechanics:

Steinberg’s work stood out because it challenges the widely held notion that quantum mechanics forbids us any knowledge of the paths taken by individual photons as they travel through two closely spaced slits to create an interference pattern.

This interference is exactly what one would expect if we think of light as an electromagnetic wave. But quantum mechanics also allows us to think of the light as photons – although with the weird consequence that if we determine which slit individual photons travel through, then the interference pattern vanishes. By using weak measurements Steinberg and his team have been able to gain some information about the paths taken by the photons without destroying the pattern.

See the rest of the breakthroughs and links to the original articles.

Original New York Times Review of George Orwell’s 1984

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From the 1949 New York Times review of George Orwell’s 1984 (which was written in 1948 and published in 1949):

James Joyce, in the person of Stephen Dedalus, made a now famous distinction between static and kinetic art. Great art is static in its effects; it exists in itself, it demands nothing beyond itself. Kinetic art exists in order to demand; not self-contained, it requires either loathing or desire to achieve its function. The quarrel about the fourth book of ”Gulliver’s Travels” that continues to bubble among scholars — was Swift’s loathing of men so great, so hot, so far beyond the bounds of all propriety and objectivity that in this book he may make us loathe them and indubitably makes us loathe his imagination? — is really a quarrel founded on this distinction. It has always seemed to the present writer that the fourth book of ”Gulliver’s Travels” is a great work of static art; no less, it would seem to him that George Orwell’s new novel, Nineteen Eighty-Four, is a great work of kinetic art. This may mean that its greatness is only immediate, its power for us alone, now, in this generation, this decade, this year, that it is doomed to be the pawn of time. Nevertheless it is probable that no other work of this generation has made us desire freedom more earnestly or loathe tyranny with such fullness.

”Nineteen Eighty-four” appears at first glance to fall into that long-established tradition of satirical fiction, set either in future times or in imagined places or both, that contains works so diverse as ”Gulliver’s Travels” itself, Butler’s ”Erewhon,” and Huxley’s ”Brave New World.” Yet before one has finished reading the nearly bemused first page, it is evident that this is fiction of another order, and presently one makes the distinctly unpleasant discovery that it is not to be satire at all.

In the excesses of satire one may take a certain comfort. They provide a distance from the human condition as we meet it in our daily life that preserves our habitual refuge in sloth or blindness or self-righteousness. Mr. Orwell’s earlier book, Animal Farm, is such a work. Its characters are animals, and its content is therefore fabulous, and its horror, shading into comedy, remains in the generalized realm of intellect, from which our feelings need fear no onslaught. But ”Nineteen Eighty-four” is a work of pure horror, and its horror is crushingly immediate.

The motives that seem to have caused the difference between these two novels provide an instructive lesson in the operations of the literary imagination. ”Animal Farm” was, for all its ingenuity, a rather mechanical allegory; it was an expression of Mr. Orwell’s moral and intellectual indignation before the concept of totalitarianism as localized in Russia. It was also bare and somewhat cold and, without being really very funny, undid its potential gravity and the very real gravity of its subject, through its comic devices. ”Nineteen Eighty-four” is likewise an expression of Mr. Orwell’s moral and intellectual indignation before the concept of totalitarianism, but it is not only that.

It is also — and this is no doubt the hurdle over which many loyal liberals will stumble — it is also an expression of Mr. Orwell’s irritation at many facets of British socialism, and most particularly, trivial as this may seem, at the drab gray pall that life in Britain today has drawn across the civilized amenities of life before the war.

Here is the rest of that review. I first read 1984 when I was in high school. It wasn’t required reading at my school, but I think it should have been.
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(via Kottke)

Types of Commenters on the Internet

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NPR has a fun take on the hodgepodge of commenters you can expect to chime in for end-of-the-year lists (I will have one of my own soon). Which one do you think you are?

1. The Poisoned. “The fact that you included Adele on this list of 100 things you like makes it a total joke.”

2. The Really Pretty Sure Person, Who Is Really Pretty Sure. “I’ve never seen Game Of Thrones, but I’m really pretty sure it’s not as good as Boardwalk Empire.”

3. The Person Who Is Exactly Right. “It really seems like this list of things you thought were good is just your opinion.”

4. The Surprisingly Lucid Narcoleptic. “ZZZZZZZZZ” is the classic. “SNORE” and “YAWN” are acceptable variants.

5. The Mother Of Tim “Freckles” Matterley. “There is a musician in Ann Arbor named Tim Matterley who is better than all these songs! You would like his music. He has a web site at FrecklesMatterley.com, and you can get his songs free on your computer! Please check out Tim Matterley, who does not have a big record contract YET but is very very good!!!!” Two comments later, she will often come back. “Also, Tim Matterley is in this YouTube video where he plays ‘Imagine’ at a children hospital. I am just one fan but I think he is great and he will go far!!!”

6. The Read A Book Guy. “Not one of these movies is as good as reading a book.” On a list of books, by the way, he will say none of the books is as good as books used to be. He also hates Kindles, which he may or may not mention.

7. The Self-Punisher. “I always hate your tastes, so I knew this would be a miserable and useless list before I decided to click on it and read the whole thing, and now I know I was right.”

8. The Unwitting Outlier. “Has anyone really cared about George Clooney since ER?”

9. The Person With The Imperfect Grasp Of Obscurity.“These are all completely obscure picks nobody has ever heard of. The Girl With A Dragon Tattoo sounds like a Dr. Seuss book.”

10. Harry The Hipster-Hater, Who Really, Really Hates Hipsters. “This is all hipster music. I guess it’s okay for hipsters, but I’m not enough of a hipster to like hipster picks like this. Too bad I’m not hipster enough. Maybe I’d like it better if I were more of a hipster.” [His username: “notahipstersorry.”]

11. The Person Who Thinks You Were So Close. “I like all these picks, but you ranked The Descendants as your #4 and Martha Marcy May Marlene as your #5, and they should be the other way around. FAIL.”

12. The Person Who Never Says Die. “Why isn’t Arrested Development on this list?”

13. The Subject-Changer. “If we’re talking about great comedy performances, I don’t know how you can leave out [Barack Obama/the Republican debates/Occupy Wall Street/the Tea Party]. That was the funniest thing I saw all year.”

14. The Minimalist. “These blow.”

15. The Person Who Is Probably Too Hardcore For You.“The best film of the year was shot in Tokyo and the title translates to I [Bleep]ed Your Grandmother With A [Blank] And [Bleep] You Too. I guess that’s probably a little too hardcore for you. You should get outside the multiplex once in a while.”

16. The Person Who Is Laughing But Is Actually Not At All Amused. “You put Community on your list instead of It’s Always Sunny In Philadelphia? Ha ha ha yeah right.”

17. The Concerned. “What’s wrong with you? No, seriously, what’s wrong with you?”

18. The Person Who Wanted To Be Surprised. “Uh, way to go out on a limb. These are the same things you’ve been talking about all year and saying were the best things when they came out in the first place.”

19. The Disbelieving. “Really? Are you serious? Did you mean to leave off my favorite thing? Or did something happen? Did you forget it? Did you think it didn’t come out this year? Did you write it down and then accidentally delete it in a technology mishap? Do you not know how to spell it? Really? Are you kidding?”

20. The Humble Alternative. “If you really want to know what the best choices of the year were, I put my list up at my site, KnittingWithDagmarAndLaura.com — it’s not just about knitting!”

I like #6, #12, and #19 especially. You?