by Carl Pierer

***

M.C. Escher: Three Spheres II.

*

If we take an infinite collection of sets, all of which contain at least one element, is there a way to choose exactly one element from each of them? This seems to be just obviously true. After all, there is something in every set, so we can just take any of those things. Certainly, it holds for finite collection of sets. Even if this number is very large, we could just go to each set and pick an element. However, if the collection becomes infinite, this becomes more worrisome. We would need a principled way of choosing one element of each set. Consider, for instance, all non-empty subsets of the natural numbers (the natural numbers being those we use to count: 1, 2, 3, …). By the nature of the natural numbers, such a set will always have a least element. Hence, we could simply pick the least element of each set in our collection. Compare this with an infinite collection of sets of pairs of gloves. As a pair of gloves always consists of a right and a left glove, we can always pick the left glove, say. But this is not so straightforward with other sets. For example, considering all non-empty subsets of the real numbers (containing the natural numbers, but also ¼, e, √π,…) it is far from obvious how we could ensure that we can pick precisely one element from each of those sets. Contrast this with an infinite collection of sets of peas. You know that each set will contain at least 1 pea, but which one are you going to pick if there are more than 1?

The axiom of choice, then, claims that this is always possible. It is, what some may call, somewhat theological, as it asserts the existence of something (a choice function) – even though we might have no idea at all how to construct it, as for instance in the real number example. Yet, it just seems very obvious that this principle should hold. After all, there is something in every set to choose from. The axiom is probably one of the most contentious of the standard axioms of set-theory[i]. On the one hand, it is very powerful and many important proofs in mathematics – explicitly or implicitly – use it. On the other, it leads to some highly counter-intuitive consequences.

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by Hari Balasubramanian

Is probabilistic analysis of any use in analyzing text – sequences of letters or sequences of words? Can a computer generate meaningful sentences by learning statistical properties such as how often certain strings of words or sentences occur in succession? What other uses could there be of such analysis? These were some questions I had this year as I collected material to teach a course on a special class of probability models called Markov chains. The models owe their name to the Russian mathematician Andrey Markov, who first proposed them in a 1906 paper titled “Extension of the law of large numbers to dependent quantities”.

The key phrase, as we shall see, is ‘dependent quantities'. Broadly speaking, Markov models are applications of that basic rule of conditional probability, P(A|B): the probability of Event A happening, given that B occurs. The uses of Markov chains are many and varied – from the transmission of genes through generations, to the analysis of queues in telecommunication networks, to the movements of particles in physics. In 2006 – the 100th anniversary of Markov's paper – Philipp Von Hilgers and Amy Langville summarized the five greatest applications of Markov chains. This includes the one that is unknowingly used by most of us on a daily basis: every time we search on the internet, the ranking of webpages is based on the solution to massive Markov chain.

The focus of this piece, however, is the analysis of letter and word sequences as they appear in text. In what follows, I'll look at four examples where Markov models play a role.

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by Tasneem Zehra Husain

It just so happened that when the discovery of gravitational waves sent ripples through the world, Valentine's Day was right around the corner. News stories about this victorious verification of ‘Einstein's last remaining prediction' were as ubiquitous as the pink and red cards, making grandiose claims – or extravagant promises – about an undying love. It was an interesting juxtaposition.

With gravitational waves very much on my mind, I was wandering the aisles of a local store, when suddenly I came across a display full of cards dripping saccharine sweetness, and a stray thought wandering into my head almost made me laugh out loud. These cards would be completely appropriate, I thought, if sent from one black hole to another! After all, it would not be out of place for a couple of black holes, spiraling into each other, to exchange ardent promises that they will merge and become one; they would be entirely justified in claiming that the story of their love will reverberate across galaxies and through millennia, and that people far away will delight in discovering their tale. As LIGO proved last month, all these claims are true!

As you have doubtless read many times by now, gravitational waves are ripples in spacetime, caused by accelerating masses. What does that mean? Using the cliched, but still useful, metaphor of spacetime as a rubber sheet, a heuristic description of general relativity is as follows: mass deforms spacetime, much as a heavy ball would a rubber sheet – the heavier the mass, the deeper the ‘dip'. Other, smaller, objects in the vicinity (assuming they are far less massive and the dips they cause are not appreciable in comparison) are then obligated to modify their paths as they approach this mass. So as to avoid falling into the valley created by the larger object, they steer around it, and in effect, end up tracing an arc. This mechanism explains the planetary orbits, as well as all other phenomena encompassed by Newton's theory of gravitation – and it goes much further.

One key difference between Einstein's theory and Newton's, is that of philosophy. Where Newton viewed gravity as a mysterious force of attraction, wielded at a distance, Einstein revealed it to be a natural consequence of the shape of spacetime. This has ramifications. For instance, with our post-Einstein understanding of gravity, we would conclude that if a massive object were to move, the spacetime deformation it causes would propagate as well. In particular, if the mass accelerates, it causes a rhythmic disturbance that ripples outward; which is another way of saying that gravitational waves are predicted by general relativity.

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Scientists are claiming a stunning discovery in their quest to fully understand gravity.
They have observed (gravitational waves) the warping of space-time generated by the
collision of two black holes more than a billion light-years from Earth. —BBC, 2/11/16

Gravitational Waves

when I entered your orbit
and you swept into mine
I gave a gravitational wave
which you returned and I received
without the help of lasers

now a billion years have passed
black holes have come and gone
and those waves are rippling
through the universe still
clear and sharp as razors
.

by Jim Culleny
2/12/16

by Leanne Ogasawara

Last week, I finally got to meet one of my heroes, Kip Thorne. I was a real cosmology geek when I was young have been a fan of his since early junior high school. So when my astronomer offered to introduce me during a Caltech event to celebrate the Ligo discovery, I jumped at the chance! And not only did I get to meet Kip Thorne, but really enjoyed the event as well.

Listening to the Ligo scientists playing talk about their role in this forty year idea, I couldn't decide what was more amazing, that they had had detected these waves that so perfectly corresponded to Einstein's magnificent 100 year equations or that the NSF had stayed with the program for such a long time to see it through!

Really, what a project–not just in terms of long-term investment but to have over a thousand scientists working for decades on something that had such a nominal chance of success –or to put a different way, they simply had no idea it was possible until they actually had done it! That is something, isn't it?

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by Matt McKenna

There's a week in a thirteen-year-old boy’s life in which Gods of Egypt might fit somewhere within their top twenty favorite films of all time. At least, it would have been in my top twenty films when I was thirteen–there are certainly enough outrageous battles and sardonic quips to keep teenage me entertained. Strangely, the theatre where I watched Gods of Egypt was absent of teenagers, and instead contained a dozen or so adults who sat quietly through the film’s two hour running time, walked out just as quietly after it was over, and presumably went quietly home to never think about the film again. I wonder what their expectations were for Gods of Egypt, and I wonder if the film met them.

Gods of Egypt is not about the historical Egypt, but rather a fantasy version of Egypt in which the Gods live amongst the mortals. Gods can be killed just like mortals, but they are distinguished from mortals in that they are fifty percent bigger, bleed gold blood, and can occasionally morph into robot animal things that look and sound a lot like Transformers. The silliness of this premise is the best the thing film has going for it.

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by Bill Benzon

There are people who read rags like Star, Globe, The National Enquirer, and so forth to find out what’s really going on in the world of celebrities and people in the news. What’s the scoop with Madonna? Is Kanye really like that? Did Brad and Angela get another kid?

Why do we do this? It’s one thing to gossip about people in your circle of acquaintances, but why trade in gossip about people who you don’t know, likely will never meet, and who play no direct role in our daily lives?

But they DO play a role, do they not, in our imaginative lives?

The one’s I’ve mentioned – I just grabbed their names out of the air, though I suppose I could have gone to the supermarket and checked the current headlines – are entertainers. They act in movies, or on TV, or they sing, or entertain us in some other way. They are important to us for what they pretend to be. They also live lives we imagine to be impossibly and unapproachably glamorous. And so we’re curious about what they’re really like. Just what that means in a world where Coke’s the real thing, that’s another matter.

Do you think they’re curious about one another? It’s not like they all know one another; there are too many celebrities and others “in the news” for that. Do they read the tabloids? I have no idea.

But let me ask a more specific question: Do you think that Barack Obama, for example, is at all curious about Jerry Seinfeld, for example? Seinfeld, of course, is a very well known entertainer with lots of fans, many of whom must be curious about what his life is really like. Barack Obama is not an entertainer, but, as the current President of the United States, he is certainly very well known. And, as Presidents go, he’s more glamorous than most.

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by Eric Byrd

Maybe I’m inclined to what Nietzsche called “impure thought,” that is to say, a kind of thought where abstractions are so mixed with the facts of life that you can’t disentangle them. I feel thought in general, and in particular what is unfortunately called “philosophy,” should lead a sort of clandestine life for a while, just to renew itself. By clandestine I mean concealed in stories, in anecdotes, in numerous forms that are not the form of the treatise. Then thought can biologically renew itself, as it were.

—Roberto Calasso's Paris Review interview

The structure of La Folie Baudelaire, the sixth book of Calasso's project, resembles that of the “brothel-museum” of which Baudelaire dreamt in the early hours of March 13, 1856, a Thursday – a dream interrupted at 5am when his mistress, Jeanne Duval, suddenly shifted a piece of furniture, in another room. In his dream, Baudelaire encountered another poor man of letters and they split a cab; they followed nocturnal versions of their daily routines, calling at editors to submit or solicit reviews, and presenting their books to possible patrons. Baudelaire stopped the cab at a brothel and went in to present his manuscript to the madam (on the surface, the first edition of Les Fleurs du Mal was then in preparation).

In a letter to Asselineau, Baudelaire said that the brothel-museum was made of “immense galleries, adjourning, poorly lit” – “like an erotic Piranesi,” Calasso comments – in which girls and clients mingled. Baudelaire was bashful of his bare feet and his penis hanging from his fly, and avoided the crowds to study the pictures on the walls: sketches of Egyptian ruins, ornithological prints with moving, “lively” eyes, and clinical photographs of the deformed children born to prostitutes. He encountered one such “monster born in the house,” “pink and green,” who squatted painfully upon a pedestal, with an elastic, serpentine appendage, starting from his nape and wound around his body. They talked – “he informs about his troubles and pains,” the greatest of which is the humiliation of dining at the same table with the girls, the ropelike coil of his neck-tail at his side. “I awake,” Baudelaire reported, “tired, enfeebled, with aching bones, my back, legs and sides painful. I presume I had been sleeping in the monster's contorted position.”

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by Jalees Rehman

A Belgian Iron 'scolds bridle' or 'branks' mask, with bell, used to publicly humiliate and punish, mainly women, for speaking out against authority, nagging, brawling with neighbors, blaspheming or lying.

Can shame be good for you? We often think of shame as a shackling emotion which thwarts our individuality and creativity. A sense of shame could prevent us from choosing a partner we truly love, speaking out against societal traditions which propagate injustice or pursuing a profession that is deemed unworthy by our peers. But if shame is so detrimental, why did we evolve with this emotion? A team of researchers led by Daniel Sznycer from the Center for Evolutionary Psychology at the University of California, Santa Barbara recently published a study in the Proceedings of the National Academy of Sciences which suggests that shame is an important evolutionary adaptation. According to their research which was conducted in the United States, Israel and India, the sense of shame helps humans avoid engaging in acts that could lead to them being devalued and ostracized by their community.

For their first experiment, the researchers enrolled participants in the USA (118 participants completed the study; mean age of 36; 53% were female) and India (155 participants completed the study, mean age of 31, 38% were female) using the online Amazon Mechanical Turk crowdsourcing platform as well as 165 participants from a university in Israel (mean age of 23; 81% female). The participants were randomly assigned to two groups and presented with 29 scenarios: The “shame group” participants were asked to rate how much shame they would experience if they lived through any given scenario and whereas the “audience group” participants were asked how negatively they would rate a third-party person of the same age and gender as the participants in an analogous scenario.

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by Jonathan Kujawa

On a Friday night two weeks ago the US Supreme Court quietly announced they wouldn't hear a challenge to a lower court's order that North Carolina should redraw it's congressional districts. There wasn't much point in hearing the case. With Scalia gone the Court was widely expected to vote 4-4 on the case and ties go to whomever won the previous round. The case revolved around North Carolina's Districts 1 and 12. The lower court ruled that they were gerrymandered to pack more black voters into these districts. While gerrymandering is now a worldwide sensation, the US invented it and are true masters of the art.

Even a glance at NC District 12 makes it obvious that some fishy business is at work. A bow-legged cowboy could walk the length of NC 12 without touching either side:

From Wikipedia.

North Carolina is hardly the only offender. Illinois District 4 is sometimes called The Earmuffs thanks to its bizarre shape.

It's one thing to eyeball a district and guess that the ghost of Governor Elbridge Gerry was at work, but I'd rather quantify somehow that a district is overly gnarled. As I pondered this last week I noticed that, in contrast to a nice, compact geometric shape like a circle or square, the long, twisted shape of NC 12 and IL 4 has a lot of perimeter for the given area. A reasonable measure of the disfigurement of a shape is the ratio of perimeter to area. In fact, for reasons we'll discuss in a moment, we want the ratio of the square of the perimeter to the area. Let's call this the distortion of the shape [1]. In formula form distortion is given by:

The reason we use the square of perimeter is that we want distortion to be what a geometer would call “unitless”. We want the distortion of a shape to be the same regardless of which units we use when measuring. Meters, miles, furlongs, and light years should all be equally good when calculating distortion.

To put it another way, we want distortion to measure an intrinsic quality of our shape and not depend on the scale we use. It shouldn't matter if we draw a square big or small when we calculate its distortion. Indeed, a square with side length x has perimeter 4x and area x². The ratio of these two would be 4/x and would get ever smaller as the square grows in size. It would also vary if were to switch from meters to centimeters to millimeters. On the other hand, the distortion of every single square in the world is 16.

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A Revolution of Tenderness

The pope suggests love in his
“Revolution of Tenderness and Love,”
and though popes have been on the wrong side of love,
this one, this Francis, is right

But a revolution of tenderness
which goes against the zeitgeist grain
to be rapacious is the Marianas Trench
of dashed hopes and, therefore,
the drowning place of dreams.
This is what the dark side says

The light side’s take is that tenderness
can be a fierce ally in our duel
with death
.

Jim Culleny
12/2/15
.
.

by Scott F. Aikin and Robert B. Talisse

William James's classic essay, “The Will to Believe” purports to be a “justification of faith.” James's argument is driven by an analogy between, on the one hand, the activities of making friends and pursuing love interests and, on the other hand, that of sustaining religious commitment. The acceptability of James's analyses of the former is supposed to show that religious commitment is also acceptable. James's essay is notoriously convoluted, and thus widely criticized. The most frequent challenges to James's line of reasoning have fixed on the appropriateness of his leading analogy. The critic often accepts James's analysis of how to win friends and lovers, but then asks: How are friendships and romances at all like faith? Perhaps to the religiously committed the analogy is obviously too thin? We wouldn't know. So we propose a criticism that instead challenges James's account of how to form friendships and woo potential lovers. In a nutshell, James argues that religious commitment is justified even in the absence of decisive proof of the existence of a deity for the very same reason that, when trying to form a friendship or attract a lover, it is to one's benefit to assume that one's efforts toward achieving those ends will be successful. Your believing that the person you'd like as your friend or lover already likes you (or finds you attractive) can help to bring it about that you, indeed, are liked or found attractive. So James thinks it is with religious commitment: “faith in the fact can help create the fact,” he says. It is our contention that this line of reasoning yields a particularly noxious sense of entitlement in those who accept James's account; for this reason, James's views about friendships and romances are unacceptable. If these views are indeed analogues with religious commitment, it, too, is unacceptable.

James's argument draws on the observation that there are cases of what we may call doxastic efficacy, cases where it does seem that belief in advance of the requisite justification is instrumentally efficacious in making the belief true. When one adopts a belief for the sake of making that belief true, one thereby commits an act of assumption: one self-reflectively endorses holding a belief with the plan being that in holding the belief, one has reason to expect that one will behave in a way that will contribute to making the belief true.

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by Misha Lepetic

“Facts all come with points of view/
Facts don't do what I want them to.”
~ Talking Heads

What is it with Silicon Valley and the “disruption” of education? Is it just another sector of public life that is moribund and therefore in need of a serious intervention, as if it were ‘that friend’ who used to be fun and successful but is now just depressed and drinking too much? Or do Silicon Valley types have a chip on their shoulder – perhaps they were forced to sit through one too many pointless lectures on Kant or Amazonian tribes or feminist critiques of Florentine art, and now that they’re calling the shots they’re going to fix this giant mess that’s called higher education once and for all? (Trigger warning: the only people mentioned in this post are venture capitalists).

In any case, into the ever-narrowing sweepstakes of who can make the absolutely dumbest assertions about the value of education steps Vinod Khosla, elder statesman and patron saint of tech bros in Silicon Valley and beyond. Khosla, a fabulously successful venture capitalist, has waded into the education wars with a broadside so breathtaking in its myopia that you would be forgiven for thinking that it was lifted from the satirical pages of The Onion. But before getting into Khosla’s piece, let’s set the stage with a look at a fellow-disruptor’s contribution to the debate.

Libertarian investor Peter Thiel, also fabulously successful, has put forward $100,000 scholarships fellowships for “young people who want to build new things instead of sitting in a classroom”. Thiel’s mission is to pluck potential John Galts out of the stream of college-bound lemmings and give them the latitude to realize their entrepreneurial potential. He believes that college, as it is currently constituted, leads to stagnant thinking and a narrowing of one’s horizons and potential. Which is odd, considering that most people go to college to have exactly the opposite experience. Be that as it may, anyone under the age of 22 is welcome to apply, which is a fairly dramatic, late-capitalist re-write of the countercultural edict to “not trust anyone over 30.”

I actually don’t have much of a problem with this, because Thiel is not trying to rewire the university system. He is providing more options for a vanishingly small group of people (104 so far since the fellowship’s 2010 inception), and I’ve always been convinced that college – or more specifically, a liberal arts education – is not for everyone. It never has been, and it never will be. That’s not to say that it shouldn’t be available for anyone who wants it. But it is a prime example of overreach when the system screws into people’s heads that “everyone needs a college degree” and that subsequently people waste their money getting a BA in communications, whatever that is. There are certainly people who don’t need to go to college, and I like the fact that Thiel is providing more options, not less.

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