Posted by: Александър Макелов | October 19, 2013

## Less Wrong / HPMoR party at MIT

Jorge Luis Borges

1. Introduction

If you don’t know what HPMoR is, now you do, so go reread it! I really like it so far. There are no spoilers here though, except for… There’s time travel! But you already kind of will knew that.

I came in a bit late, must have been 7:45, since I had class until 7:30 (it’s that super awesome Ramsey Theory tutorial, more on that some other time), so maybe I missed something that happened in the beginning. On the other hand, the party ended at about 11:30, and you can ask your Occam’s razors to guess how I know that. To make things clear from the start,

Definition 1. A party is a gathering of people talking about cool stuff (e.g., math). According to the “Hitchhiker’s guide”, a party is widely considered to be the meaningful way to spend a Friday night.

Example 2. The event I’m talking about, more or less.

Definition 3. Eliezer Yudkowsky is a research fellow at the Machine Intelligence Research Institute. The problem he’s working on is building an artificial intelligence that is superior to us humans and yet friendly. If you think you have a meaningful goal in life, think again! (I’m not joking)

2. HPMoR stuff

Knowing nobody in the room, I proceeded straight to the small cloud of people surrounding Yudkowsky (he was the person I knew the most, after all, I’ve read a thousand pages of his writing, and that’s like him talking to me), and started listening to the conversations.

For those who are curious, he did not appear as any of: crackpot; scary; terrible, terrible, evil person; arrogant bastard; failing at math; spitting out the same old stuff from LessWrong. What he did seem to be was someone having a lot of fun talking about HPMoR and mathematical logic (going a bit over the top sometimes). And it was pretty clear he had a lot of fun writing it – well, most of it anyway; reliable sources tell me chapter 45 was his favorite. People had lots and lots and lots of questions about the book. One thing that kept coming up was pretty much “How the hell do time turners work? What if I’m surfing, glued to the surfboard, on a spaceship, with handcuffs, and the handcuffs are conscious but I’m not aware of that? Do I end up freely surfing in space?” He unfailingly came up with answers on the fly and explained he has no full theory of how magic works. Then there were some more time travel questions. Eliezer and Borges agree that one of the major takeaways is that HPMoR is not the kind of book you want to read once. Moreover, Eliezer says that everything that happens can be figured out within the world of the book, so don’t worry – no unsolvable mysteries here. Except maybe for time travel. He also talked about other fun stuff he plans to write, but I won’t give anything away, except that there are some.

3. Math and AI stuff

At some points the discussion digressed to math. Eliezer, just like Harry Potter, does really like recursion. Apparently there is an equivalence between ordinal numbers and logical systems. For example, see this. It seems to be related to the largest computable number you can name. It seems to also be related to his talk yesterday at MIT about recursively self-improving rational agents – in a very basic (well, in fact, probably the first) theoretical model for that based on mathematical logic, there is the unfortunate obstacle that any next stage of self-improvement needs to be based on a mathematical system that has strictly less expressive power, so you might “run out of math”… but! there are ways around that.

Also, I got to ask him some questions (yay). Yesterday I had this conversation over lunch about what it means to build a being more intelligent than yourself . If we define intelligence as the ability to answer questions, then we’ll be at least as smart as such an AI, by just asking it (under the assumption it will be friendly). If we define intelligence as the ability to answer questions without having access to other beings (ugh… not being very formal here), a person having the knowledge how to build the AI will, in theory, still be as smart – in the sense of a Turing machine simulating another Turing machine. However, we run into efficiency issues here, as the boost in running time might be huge. We see the question at hand is non-trivial. Clearly, time has to fit somewhere in the picture – and the definition the folks at MIRI are using is, as far as I understood, this: the ability to maximize utility while acting in diverse situations given some constant amount of time. Here, by maximizing utility, we mean bringing about events that have utility values close to the maximum, and the closer the better. By diverse situations, we mean that an AI should be ‘general’ enough and accomplish tasks at least as varied as the ones people can achieve (the more varied the better). This seems to be a good, down to earth, practical, behaviorist definition.

4. Conclusions

I don’t know all the math Eliezer was referencing these last two days, but from what I do know and the way he talks about it, my undoubtedly biased and overly constrained perspective tells me he’s doing it right. The MIRI had a somewhat nonexistent reputation in the academic community in the past, but I hope things will change (and I think they are changing already). Maybe the AI problem will be solved in 40 years, maybe in 4000, but the people working on it at the MIRI are, to my best estimates, legit. Also, I have to read more LessWrong and learn more logic. Speaking of which, this seems like a very interesting mixture of areas – I wonder who put it together and how they decided it to be that way?

Posted by: Александър Макелов | October 11, 2013

## Inevitability as a common perspective on “The idiot” and “The little prince”: essay

This is something I wrote for a class on Russian literature that I took in the spring. SPOILERS AHEAD: If you haven’t read “The idiot” by Dostoevsky, or “The little prince” by Saint-Exupery, bear in mind that this essay discusses the plots of the two works in depth.

“The inevitable blow of the knife”: Inevitability as a common perspective on “The Idiot” and “The Little Prince”

There are two well-known works of literature that undertake, each in its own way, the challenge of depicting the encounter of a perfectly good and beautiful man with the rest of the not-so-perfect world. As much as we would expect both to treat similar universal ideas of human ethics and aesthetics, they share another striking thing in common: both “The Idiot” and “The Little Prince” tell the story of a childlike prince.

Yet this seeming superficiality is only the first of many co-occurrences that transcend the mere choice of words, and extend to the fundamental imagery and tone of both books. I will try to convince you that this similarity suggests not the whim of coincidence being at play here, but a perspective for understanding both works as being driven by the same all-powerful feeling: that of inevitability. Set against the heavy, recurrent premonition of an unavoidable final event that makes the present constantly fall towards it, they can be seen as a desperate attempt to capture and reflect on these last fleeting moments of life – which encompass, in their extreme vividness and clarity of perception, the entire spectrum of human existence. Indeed, an attempt, as Dostoyevsky himself articulated it, “…to paint the face of a condemned man a minute before the guillotine falls…” (Dostoyevsky 75)

To set the stage for this comparison and provide a foundation for my analysis, I first outline the primary ways in which “The Idiot” and “The Little Prince” are alike – as well as the ways in which they differ. Central is the notion of the morally and physically beautiful prince, “falling” in the real world as if from a fairy tale – and indeed, as both stories imply, perhaps such princes only belong in fairy tales. In a sense, both the little prince and Myshkin come from a different world – whether it the asteroid B-612 or the tranquil fields of Switzerland. And similarly, in the end, both depart, in grief, to where they came from – whether literally (going back to the clinic and to idiocy) or symbolically (the death of the body that sets the invisible essence free). In their interactions with people, both strike at first glance with an expression of childlike simplicity and beauty. Even the ways in which this impression is gradually built up mimic one another. Recall the “golden curls” (Exupery 22) of the Little Prince and Myshkin’s “fair hair” (Exupery 6) – defining characteristics of their physical beauty. Or the “lovely peal of laughter” (Exupery 9) of the Little Prince, and Myshkin’s habit of honestly laughing at his own misfortunes, which both have the power to instantaneously convey to other people their innocent nature. Whereas the Little Prince is indeed little in appearance – a child, the corresponding aspect of Myshkin is constructed in a symbolic, although no less frquently recurring, pattern. He is repeatedly called a “child” by pretty much every major character in the novel, and he talks at length about his strong love for children. Recall the touching story about Marie (Dostoyevsky Part I, Chapter 6), and the special connection Myshkin established with the children from the village, an idea dear to Dostoyevsky, and one he employs again in “The Brothers Karamazov” through Alyosha. Most notably, there the prince says: “… I am indeed not fond of being with adults, with people, with grown-ups […] my companions have always been children.” (Dostoyevsky 87-88). The quality of being childlike is combined, in both our heroes, with a lack of fixed ideology, a manner of taking the conversation straight to the point, and an ability to see through people. Just as Myshkin articulates matters like the death penalty simply, eloquently and convincingly, so does the little prince speak simply and concisely about the meaning of love. And just as the Little Prince can see “what is essential”, which is “invisible to the eye” (Exupery 60), so can Myshkin understand the thoughts and feelings of others through their “physiognomy” (a most recurring word in “The Idiot”), to the point of wielding an influence over the very roots of Nastasya Filippovna’s soul: “You’re not like that, not like the person you pretended to be just now, are you?” (Dostoyevsky 138). The princes share a thoughtful, reflective soul, and a corresponding oblivity to their surroundings. Preoccupied with the eternal war between sheep and flowers (Exupery 22), the Little Prince cares little about the constant death threat posed by the desert; and Myshkin’s body often wanders around while his mind is wondering around even further (cf. Dostoyevsky, before the fit in Part II), not believing in his heart that anyone could try to hurt him or deceive him. And finally, as if all of this is not enough, both princes share above all an insatiable capacity for finding and feeling beauty. Amazingly, it is precisely beauty that they first find in their two corresponding heroines: compare “‘So that’s Nastasya Filippovna?’ he said quietly, looking at the portrait attentively and inquisitively for a moment. ‘Astonishingly good looking!’ he added at once, with ardour.” (Dostoyevsky 36) and “But the little prince could not restrain his admiration: ‘Oh! How beautiful you are!’” (Exupery 24). And, similarly, beauty of nature is something of which both princes cannot get enough. For example, recall the wonderful passage (Dostoyevsky 69-70), especially “… and I kept thinking that if I were to walk straight, walk for a very long time and go beyond that line, the line where earth meets sky, there the whole riddle around me would be solved and instantly I would see a new life” and compare to the little prince’s love of sunsets: “’I am very fond of sunsets […] One day, […] I saw the sunset fourty-four times!” (Exupery 19). It is as if both heroes are personifications of the same – and to a great depth of detail – human type.

Yet, there is a crucial difference between the ways the stories of the princes are constructed. Whereas “The Little Prince” has the flavor of a passive, allegorical recollection on love and human nature in what could be the last moment before death, “The Idiot” draws us into a literal, painful, dynamic mixture of deception, passion, sickness, and murder. To see the distinction on the allegorical-literal axis, recall that in Saint-Exupéry’s story all the physicality of death is reduced to the tenderest, most aesthetic three lines: “There was nothing but a flash of yellow close to his ankle. […] There was not even any sound, because of the sand.” (Exupery 75) The rest is a dialogue, a recollection on the little prince’s love with the rose and the encounters during his journey. Because of the tale’s omnipresent allegorical nature, it is not clear if the little prince died or magically made it back to his planet, or if the drawing of the sheep ever ate the flower. And indeed, this uncertainty is in part even needed for Saint-Exupéry’s message. To the contrary, as much symbolism as there is in “The Idiot”, in his typical fashion Dostoyevsky nonetheless portraits the reality of death and sickness, in all their conclusiveness. Prime examples of this are the striking, somatic descriptions – of Myshkin’s epileptic fits, Ippolit’s final stage of consumption, and Nastasya Filippovna’s dead body, among others. The contrast on the passive-dynamic axis is produced not only by the difference in balance between reflection and action, as we discussed, but by the way the characters themselves are dispersed spatially and temporarily in the narrative. In “The Little Prince”, it is the prince who is travelling and entering into dialogue. Notice that he has only one interlocutor at any given moment – the king, the conceited man, the drunkard, etc. are all separated from one another, each in a different world (sometimes quite literally!); and by the time the narrative takes place, all but two of his encounters lie in the past (apart from the pilot and the snake). He is even the one who approaches the snake, and chooses to be bitten by him. Conversely, after his arrival in Russia, Myshkin is continuously subjected to the rapid fire of all the other characters that repeatedly come to him, share, challenge, deceive, mock, insult – and step back to give way for yet others to do the same. Recall (Dostoyevsky Part I, Chapter 8), where Kolya, Varya, Ganya, Ferdyshchenko, General Ivolgin and Nina Alexandrova – indeed, the entire Ardalionovni household, visit the prince’s room in the span of six pages and what must be no more than half an hour; or the visit of Antip Burdovsky, the false “Pavlishchev’s son” in (Dostoyevsky Part II, Chapters 7-8); or Ippolit’s recurring sharp intrusions, accompanied by his ideological antagonism and “unexpectedly shrill voice” (Dostoyevsky 303). And finally, Rogozhin’s attempt at Myshkin’s life (Dostoyevsky 274), the ultimate attack. Whereas the little prince is undertaking a journey and finds himself in the numb, inanimate personification of death that the desert is, Myshkin has to constantly defend himself from the ceaseless assaults of other characters. Thus, in “The Idiot”, all the antithetic social elements are mixed together in a vortex of actions and interactions that ultimately execute the speculations of death and sadness raised in “The Little Prince” for real.

Where is inevitability to be found in the above speculations and actualities, and how does it manifest itself in the two literary worlds we described? I shall recognize two “physiognomies” of it, and analyze the ways in which they interact. The first is rather obvious – one might say the human synonym for inevitability – death. The second is a more subtle one – yet, perhaps, the cause for the first – the constancy of human character. Let me first outline how they are depicted in “The Little Prince”, and then go through “The Idiot” in more depth.

The place of the novella’s present, the desert, is an unchanging, passive reminder of the proximity of death, and this is alluded to at the very beginning: “It was a question of life or death for me: I had scarcely enough water to last a week” (Exupery 5) Another personification of mortality is the snake, the first and last companion of the prince on Earth (Exupery, Chapters 17&26). The concrete premonition of the death of the prince gradually builds up from Chapter 24 (“And I felt him to be more fragile still. I felt the need of protecting him, as if he were a flame that might be extinguished by a little puff of wind…”) and permeates the narrative with anxiety. Among these clear manifestations of the inevitable are the more subtle ones. The baobab is one vivid metaphor for an unalterable force of annihilation: “A baobab is something you will never, never be able to get rid of if you attend to it too late. It spreads over the entire planet. It bores clear through it with its roots. And if the planet is too small, and the baobabs are too many, they split it in pieces” (Exupery 16). The constancy of the “warfare between the sheep and the flowers” is another: it has lasted “for millions of years” (Exupery 22). And then come all the disturbing, grotesque descriptions of the people the little prince meets along his journey. The king, the conceited man, the tippler, the businessman, the lamplighter, the geographer,… – all stuck in an infinite loop (this idea is most strikingly executed in the story of the tippler (Exupery Chapter 12)), inhabiting their lonely, narrow worlds – just as narrow as their personalities have become. Preoccupied with a single ambition, a single destination, it is as if they are not people, but parts of people. In this context, inevitability means the impossibility of them ever escaping the gravitational pull of these worlds, of them ever changing. This combines with the reocurring lack of understanding and true dialogue between the little prince and these other people, the “grown-ups”, to impress on the reader a feeling of utter hopelessness. Thus we see how these two faces of inevitability reemerge throughout the novella and account for its persistent tone of grief. Yet: the end is open. Whether the sheep ever ate the rose is left as a “great mystery” (Exupery 77).

“The Idiot” is, then, its apocalypse. Because apocalypse – the object of the book of Revelations, so frequently alluded to in the novel – literally means the unveiling of a great mystery. We will see how the imagery and tone of each part gradually brings us closer to the final lifting of this veil.

The concept of inevitability enters the narrative in an unequivocal way through the discussion of death penalty. As tangential and unmotivated as it may seem at a first reading, it turns out to concisely represent the novel’s essence, and I shall identify allusions to its imagery in the parts to follow. The use of such revelations (pun intended) seems to be a favorite of Dostoyevsky, as I shall keep demonstrating. Most notably, recall the early appearance of “Well, he might marry her tomorrow; might marry her, and a week later, perhaps, cut her throat” (Dostoyevsky 43). Through the vivid, eloquent speech of Myshkin, capital punishment comes to life. The passages about it share a clear message: “…and you’ll no longer be a human being, and that this is certain; the main thing is that it’s certain.”, “… he lived in the unquestionable conviction that in a few minutes’ time he would face sudden death”, “… when your head lies on the block, waits, and knows, and suddenly hears above it the sliding of the iron!” (Dostoyevsky 27, 71, 77) After these quotes, there is no further need to prove that unavoidability is the driving force behind this imagery. More relevant to the novel’s immediate plot, in Part I we also have all the characters constantly reiterating the conclusiveness of the birthday soiree: “… she will deliver her final word”, “Today my fate will be decided…”, “and this evening it’s all to be decided between them”, “The matter is settled”, … (Dostoyevsky 35, 99, 115, 117) All eyes are turned towards that certain future. Another recurring construct of anxiety is the warning, communicated from the other characters to Myshkin: “Then beware of him, I warn you…”, “I’ve come to warn you:…”, “I warn you in advance:…” (Dostoyevsky 100, 110, 113) The apparent reason for uttering these words is always different, yet the synchrony and proximity in the text is so remarkable that we are inclined to feel they additionally serve the same higher purpose. Is the prince about to stage his own story about the condemned man?

In Part II, the stakes go up, with the premonitions of Myshkin’s fit and Rogozhin’s attempt at his live intertwining with painful clarity, ultimately converging to the same moment in time. In the span of fifty pages, the eyes, the headache, the “physiognomy” of the house, the storm, the “same eyes” weave a thrilling gradation of apprehension (Dostoyevsky 223, 227, 236, 239, 266, 270, 273). But most notably, it is the knife, constantly approaching the narrative and asserting itself, that is the principal manifestation of inevitability here. First from the past: “She was like a madwoman all that day, now weeping, now preparing to kill me with a knife,…”, then as a possibility: “…because to put it bluntly you may cut her throat”, as a physical presence, materializing on the mention of jealousy, playing back and forth between Myshkin and Rogozhin; and, finally, as the carrier of certainty: “the inevitable blow of the knife” (Dostoyevsky 246, 249, 253, 274). The language that was used in Part I for seemingly different matters is now employed again, almost verbatim: “… a total and overwhelming impression that led involuntarily to the most complete conviction?”(p.272); “’It will all be decided in a moment!’ he said to himself with strange conviction.” (Dostoyevsky 272, 273) The only difference is that the prince is stubbornly oblivious to this certainty: “Parfyon, I don’t believe it!” (Dostoyevsky 274)

Shortly after this temporary denouement, inevitability puts on the mask of consumption in Part II and Part III, leading up to Ippolit’s necessary explanation – even the word “necessary” itself is a clear allusion to the certainty of his death. In this explanation we find the most grotesque personification of that idea in “The Idiot” – Ippolit’s creature. Because who created it but him? With mathematical precision, he describes the unearthly monster’s body, its size, number of limbs, feelers, its motion around the room; a precision that closely mirrors his own inclination of measuring the minutes and seconds of life (Dostoyevsky Part III, Chapter 5). Ippolit is painfully aware of his position, to the point that he describes himself re-using the exact imagery of Part I: “take me for […] most probably of all, a man condemned to death” (Dostoyevsky 460) Parallel to that, there is a continuous allusion to the book of Revelations: “it had appeared in my room on purpose, and in this there was some kind of secret”, “…what was the secret behind it?”, “…mystical terror…” (Dostoyevsky 454, 455) And the apocalypse here, as well as in the novel itself – the apocalypse that Myshkin acknowledges but refuses to believe in, and the one that ultimately drives him to insanity – lies in the actuality: yes, it will certainly happen. Despite all the courage of Norma, there is no getting away from the sting: “With a yelp and a howl she opened her mouth in pain, and I saw that the chewed-up reptile was still moving across it, emitting from its half-crushed body a large quantity of white fluid…” (Dostoyevsky 456)

Dispersed among the above faces of death and certainty, in the first three parts we also discover the subtle presence of constancy in the human soul. Many characters are driven in a high degree by a collection of single ideas, bearing striking similarity to the ones encountered in Exupery’s novella (chapters 10 to 15): money and vanity (Ganya), alcohol and forgetfulness (Ivolgin), passion and violence (Rogozhin), measuring time and mortality (Ippolit). Myshkin remains generally misunderstood, just as the little prince, and instead all his companions attempt to assign him to some ideology or other. The impression that Myshkin is conclusively unable to change the world for the better is created; the most notable proof for this is his assessment of Rogozhin: “… if there was a certain awkwardness in his [Rogozhin’s] gestures and conversation, it was merely external; in his soul this man could never change” (Dostoyevsky 424).

The fears and premonitions raised in the first three parts are gradually executed by Dostoyevsky in Part IV, ending with Nastasya’s murder. A notable symbolic interlude to this is the breaking of the vase. It is, along with the portrait of the condemned man, the most vivid and concise allegory for the plot of the novel. Firstly, it signifies the onset of Myshkin’s own disintegration. The seed of the idea is innocently planted by Aglaya, but greatly distresses the prince: “’…You must at least break the Chinese vase in the drawing room!…’ […] ‘On the contrary, I shall try to sit as far from it as possible…[…]”; “I am sure I’ll start talking out of fear, and break the vase out of fear […] I shall have dreams about it all night; why did you have to mention it?’” (Dostoyevsky 612, 613) And, in accordance to the laws of the novel, this is exactly how it happens. In a strikingly similar way, the former harmony of Myshkin’s eloquence is perturbed, broken to pieces by all the ellipses, abrupt exclamations and questions that mark his speech on pages 629 through 637. “‘You saw me when I was a child?’, asked the prince with some surprise”, “Oh, but I didn’t say it because I … doubted it … and, anyway, how could one doubt it (heh-heh!) … in the slightest? … I mean, even in the slightest!”; “‘Pavlishchev … Pavlishchev went over to Catholicism? That cannot be!’, he exclaimed in horror”, … No collection of a few quotations is able to fully express the tension imprinted on these eight pages! Secondly, the breaking of the vase is the final confirmation that the apocalypse in “The Idiot” is about the meandering but certain to occur actuality: “The vase swayed slightly, as if at first uncertain whether to fall on the head of one of the elderly gentlemen, but suddenly inclined in the opposite direction, towards the little German, […] and crashed to the floor” (Dostoyevsky 638); “But we cannot fail to mention another strange sensation that struck him [Myshkin] at precisely that moment and suddenly manifested itself to him out of the throng of all the other strange and troubled sensations: […] the realized prophecy!”(Dostoyevsky 639). From this point on, the conclusion is no surprise. The scene with Myshkin and Rogozhin gradually approaching Nastasya’s body completes the portrait of the inevitable. The veil – for Dostoyevsky granted us a literal veil with his last stroke of the brush, the curtain in Chapter 11 – is lifted, and the secret – not much of a secret anymore – revealed. The description does not contain an actual statement along the lines of “Nastasya Filippovna was dead” – and indeed there is no need for Dostoyevsky to say what had already been said many times.

Was that the only mystery? Was the apocalypse all about the inevitable ruin of a good, beautiful man in an unchanging world? I think not, for I have two more amazing similarities between Dostoyevsky’s novel and Exupery’s novella which I’ve been keeping a secret. Compare Myshkin’s words “Beauty is a riddle” (Dostoyevsky 91) with “‘What makes the desert beautiful’, said the little prince, ‘is that somewhere it hides a well…’ […] When I was a little boy I lived in an old house, and legend told us that a treasure was buried there. […] But it cast an enchantment over that house. My home was hiding a secret in the depths of its heart” (Exupery 66). Death is certainly a ‘mystery’ in both books – but so is beauty. And compare the portrait of the condemned man (Dostoyevsky 75), the portrait of the “poor knight” (Dostoyevsky 289) with the narrator of “The Little Prince” continuously painting pictures of the little prince (and indeed Exupery’s illustrations are a major part of the narrative). The combination of these two ideas leads to the emergence of an aesthetic dimension of ‘mystery’: a portrait of the beauty and vividity of life in that single moment “exactly a minute before death” (Dostoyevsky 77), when only “the last stair [of the scaffold] can be seen clearly and closely” (Dostoyevsky 77). In its brevity and beauty, it is a desperate antithesis to the constancy and ugliness of the inevitability of death.

I feel there is much more to be said about these two books than what I tried to convey in these several pages, and many more impressions struck me while I was writing this essay. By comparing the high-level structure of Exupery’s novella and Dostoyevsky’s novel, I managed to abstract away some of the complexity in the latter, and through the simplicity of the former grasp and analyze more clearly the manifestations of inevitability and the way they are constructed in the two books. The two narratives form a synergistic bond in which each empowers the understanding of the other. As we saw, while Exupery chooses to discuss the issue of inevitability from a distance, and on the allegorical level, Dostoyevsky brings it to an unequivocal conclusion in reality. At the end of both works, the dominating impression is that of sadness, of something forever lost, irreparably broken. Yet, we can ask ourselves, just like Myshkin does before his fit in Part II, is this not justified by the momentary glimpse at the perfect beauty, at the “final cause” (Dostoyevsky 264)? Indeed it is. The vase sways for a second before it falls, but it sways beautifully.

References

Dostoyevsky, F. “The Idiot”, Penguin Classics

Exupery, A. “The Little Prince”

Posted by: Александър Макелов | October 10, 2013

## Arbitrarily biasing a coin in 2 expected tosses

Here’s a neat probability trick that I learned from Konstantin Matveev and which, I think, everybody mildly interested in math should know about:

Problem. Given a fair coin, how do you (efficiently) generate an event $E$ with probability 1/5?

Solution. We can, of course, toss the coin three times, giving us a total of 8 possibilities, then discard our least favorite 3 of them, and weight the remaining  5 possibilities equally. This algorithm requires an expected number of tosses equal to $3\times 8/5=24/5$. But, what if instead of $1/5$, we have $1/1000000$? You can easily see that the expected number of tosses to emulate a probability of $1/n$ grows logarithmically with $n$. But even worse, what if we had $1/\pi$? Well, here’s a trick that gets rid of both of these problems: let

$\frac{1}{5} = \displaystyle\sum_{i=1}^\infty \frac{a_i}{2^i}$

for $a_i\in \{0,1\}$ be the binary expansion of $1/5$. Then, start tossing the coin until it lands heads, at some time $I$. If $a_I=1$, declare that $E$ has occurred; otherwise, $E$ has not occurred. Then clearly

$\mathbb{P}[E]=\displaystyle\sum_{i=1}^\infty \mathbb{P}[E\ \big| \ I=i]\mathbb{P}[I=i]=\displaystyle\sum_{i=1}^\infty \frac{a_i}{2^i}=\frac{1}{5}$

Furthermore, notice that $\mathbb{E}[I]=2$, regardless of the probability we want to emulate! Well, that seems pretty efficient. When you think about it some more, it really appears to be mind-boggling – you can emulate extremely small, or irrational, probabilities with just two expected tosses. Moreover, you don’t need to have the binary expansion of the probability in advance – you can pass the next digit depending on the status of your experiment.

Combining this with a standard unbiasing technique, say von Neumann unbiasing,, this gives you a very simple procedure that given a biased coin that lands heads with probability $0, allows you to simulate a biased coin that lands heads with probability $0 for any other $q$. Any binary source of randomness is convertible to any other such source.

But we haven’t said anything about the efficiency of unbiasing. There, we can’t do as well as in biasing: there is a fundamental obstacle, the information-theoretic limit. Roughly speaking, the amount of information a biased coin tells us is always strictly less than the amount of information we get from an unbiased coin – this is why biasing is easier than unbiasing. Fortunately, there is a procedure that lets us extract an unbiased stream of bits that on average achieves the best performance theoretically possible: see this paper by Mitzenmacher to learn more.

I guess the moral of all this is the following: if you’re stuck on a deserted island with $\pi -1$ other people, you need to decide who the first to be eaten is, and all you have in your random arsenal is a suspicious-looking coin handed to you by one of your shipmates, do not despair – you can still make sure you have a fair chance of surviving the day.

Posted by: Александър Макелов | September 29, 2013

## Reflections on “The library of Babel” and computational complexity

“…mirrors and copulation are abominable, because they increase the number or men.”

“Tlön, Uqbar, Orbis Tertius”, Jorge Luis Borges

You can tell that Borges was very fond of reflections, and now I intend to try to make him happy.

In short, the Cosmic Coincidence Control Center (and it seems that I’m included in that number?) was extremely busy last week. After finishing my first-ever short story, that feeble imitation of Borges, bearing the following arrogant dedication “While this story was being written, I thought I had stolen Borges’ style; but now I know – he stole my idea”, I was ruthlessly hunted down – so after all it was me who stole something, but hey, who is to say.

First, I decided to write my first paper for the science fiction class I’m taking (which is absolutely fun, thanks to this guy) on “The library of Babel”. OK, I can take that – after all, you might argue that I have free will and whatnot, so in fact it was not a coincidence.

Next, I randomly decided to watch a video by vihart called “Twelve tones”, cause, you know, it seemed to be her most popular one. And – bam! – there was “The library” again.

After that, I was even more randomly reading the chapter on randomized algorithms from the book on computational complexity by Oded Goldreich, and guess what, the quote at the beginning was:

I owe this almost atrocious variety to an institution which other republics
do not know or which operates in them in an imperfect and secret manner:
the lottery

Jorge Luis Borges, “The Lottery in Babylon”

I know, it’s not a library, it’s a lottery, but a lottery is just the closest equivalent of a library to people doing randomized algorithmis – after all, a bunch of monkeys randomly typing on a bunch of typewriters will produce the works of Shakespeare at some point. And a Babylon is like a baby Babel anyway.

Finally, it turned out that the book I blogged about last week, “Orphans of the sky”, is way too similar to “The library of Babel” – something I realized only after re-reading the library (or rather, “The library”. haha). It’s not just that both things came out in 1941 (yeah, I don’t know, it’s crazy), but they both construct extremely similar settings, visually and conceptually. Read them and you’ll know – don’t want to spoil anything!

All in all, it was pretty obvious that Borges was after me, and that he wouldn’t leave me alone unless I wrote something about the library and about computational complexity. So here we are now.

What is this library anyway? The premise of the story is simple enough: a library which contains all possible books 410 pages long, conveniently stacked in a seemingly infinite array of identical hexagonal galleries, which comprise all the world. It has the complete works of Shakespeare, the biographies of all people that have ever lived on Earth, the proofs of a bunch of conjectures in mathematics, these same proofs with the last line wrong, “The library of Babel”, etc. Sure, it’s a big place. It also has people randomly walking up and down and thinking they have it all figured, arguing that, you see, a pentagonal gallery would be fundamentally impossible, so that’s why galleries are hexagonal.

But I don’t really want to talk about the social metaphors of the library (a decent subject in its own right); rather, I like to think of it as a representative of a somewhat underrepresented part of SF, something you might reasonably call “math fiction”.  Borges wrote several other stories with a strong flavor of mathematics – “The Aleph”, “The garden of forking paths”, “Blue tigers”, “The book of sand”, to name a few amazing ones.

Is MF SF? I would argue that it is, for:

1) math is as good a science as any of your usual ‘favorites’ in SF – physics, chemistry, biology – and in fact, it is the language underlying all of them, a language of even greater expressive power

2) yes, all the ‘falsifiable hypothesis blabla’ stuff does apply to mathematics, and in fact, modern mathematics seems to rely more and more on simulations and experiments

3) MF has already sneaked in SF: there are works that arguably classify as MF which have won a bunch of awards. I know for I’ve read one such – “Permutation city” by Greg Egan, which I strongly recommend to people interested in the computational aspects of consciousness.

YAY MATH FICTION! So, “The library of Babel” uses a very simple mathematical idea – “the set of all sequences of a given length, in a given set of symbols” – to achieve very interesting and complicated effects, and that makes it great math fiction. Suppose you wanted to write a book, and you had some reasonably good idea of what you want it to be about, and you knew it wouldn’t be longer than 410 pages. It then seems very plausible that, if someone hands you a book and you read it, it will be qualitatively easier for you to tell if that’s the book (or a book) you want to write. Then, if you just go to the library and read all books (for there is a very big, but finite number of such books), you will finally find one that suits you!  So you’ll have achieved a qualitative improvement by increasing your efforts only quantitatively. Essentially, it might seem that you’ve written a book without writing it!

This has two consequences: one philosophical, one computational. First, is an author just a treasure-hunter? Does an author create a work, or has the work been there all the time, and the author is `merely’ the one who found it? What the hell?

But hey, that’s not a big deal. What if we try to write books in the way described above? What if we try to do math the way described above – if we want to prove a theorem, we just go through all possible proofs of a given length, for all lengths, until we find one that works? Then mathematical discovery will be more or less fully automated! Ideas of the sort motivated the computational revolution that was just starting at the time Borges wrote his story, and they shape much of modern computational complexity theory.

As for the point of the above example in this context – we might need some new, more practical definitions of quantitative and qualitative differences after all. Especially, when you’re searching for something, going through all possibilities should count as qualitatively more expensive than looking at a single one – and that’s some intuition for where the distinction between polynomial and exponential time in computer science came from. Here’s a nice paper on that topic that I don’t really understand (yeah, I don’t really understand either): Why Philosophers Should Care About Computational Complexity

Posted by: Александър Макелов | September 22, 2013

## Reflections on “Orphans of the sky”

While the book is still fresh in my mind (it’s about 1 hour, or 60 minutes, that is, 3600 seconds, behind me). You know a science fiction book is good (to you) when it constructs curious ideas and situations you haven’t ever imagined before (which are of course made possible by some kind of, well, technology; otherwise it wouldn’t be much of a SF work; or would it?). Another way you know a science fiction book was good to you is when you read it, and then (of course) go to wikipedia, see when the thing was written, and be like “What? I thought it was written in the 70s or something…”.

“Orphans of the sky” by Heinlein was good to me in both respects. If I had to summarize the insight I gained from it in a sentence, it would roughly say this. The concepts of ‘humanity’, ‘human nature’ and ‘common sense’ are highly dependent on, and extremely, short-time-scale flexible with respect to, the knowledge passed from parents to children.

And here’s a quote that is both representative of this idea and a sort of motivation for the study of general topological spaces as opposed to metric spaces (What? What did I just say?…):

Metrical time caused him as much mental confusion as astronomical distances, but no emotional upset The trouble was again the lack of the concept in the Ship. The Crew had the notion of topological time; they understood “now,” “before,” “after,” “has been,” “will be,” even such notions as long time and short time, but the notion of measured time had dropped out of the culture. The lowest of earthbound cultures has some idea of measured time, even if limited to days and seasons, but every earthly concept of measured time originates in astronomical phenomena; the Crew had been insulated from all astronomical phenomena for uncounted generations.

Posted by: Александър Макелов | September 20, 2013

## Thought I’d write a short story as well

While this story was being written, I thought I had stolen Borges’ style;

But now I know – he stole my idea

So I sat for a couple of weeks, and here it isn’t. Bwahaha! I mean, I actually wrote it, but I don’t really want to show it here right now (and anyways, it’s not yet in English (and anyways, nobody’s reading this blog). But I’d like to use the fact that I undeniably captured your attention with this provocative title, and share some other coolness.

And this week (’cause I blog every week), it’s Vi Hart! It’s something I learned about from professor Curt McMullen at Harvard, and more precisely, from some of the cool links he put on the website of his course Math 131 that I’m CAing this fall. First of all, this video is absolutely amazing.

Second of all, this video is absolutely amazing as well. It mentions John Cage, and Jorge Luis Borges (omg! Borges! He’s after me. But that’s another story for another day).

Third of all, these are all the videos by her that I’ve seen so far. As Mad Eye Moody might say, COINCIDENCE?) (you thought there was a missing parenthesis up there, didn’t ya? I’m pretty sure you did, you

Posted by: Александър Макелов | April 13, 2013

## Thought I’d try to write a song

So I sat for 2 hours and here it is, what a spontaneous thing. I recently dreamed that I’d written a song. If it’s the same one, it’d be pretty amazing. Maybe people from the future will have some way of telling that?

Anyway, note that this is my first attempt at poetry/songwriting in English, so it’s pretty lame. Also, it doesn’t feel like something that’s really mine. I’ve no idea whom I’m singing to, I kind of have an idea what I’m singing about, but in the end it didn’t quite came out the way I intended it to; it’s just… something. Chords don’t appear over the corresponding syllables, so use your intuition until I find a way to put whitespace at the beginning of a line in wordpress.

C
Scribbles in the sand, don’t
E
really help me understand,
F
what to make of all these questions,
Fm
so maybe you’d have some suggestions?

C
Me and you, an hour or two,
E
nevermind the ticking, kicking
F
through the door, you know,
Fm
forever’s where we’ll go.

C             E
And I,
F                                Fm
wanna breathe the air again,
C             E
Tonight,
F                             Fm
maybe we’ll finally comprehend

C
That they mean nothing – words and letters,
E                                                           F
feathers of a bird, and birds fly together,
Fm
but alone, mute and voiceless souls

C
And I am you and I see me,
E
Gilmour wrote in seventy,
F
But you know it’ll never be,
Fm
but we know it’ll never be…

Posted by: Александър Макелов | December 21, 2012

## We’re back…

…in the game. Aw yeah.

It’s a bit unfortunate this happens at the end of the world, but even it can be a very fun place.

I’ve always been telling you I was not really blogging, and this past term at college has been taking up all my time, so there were not that many (read: none) cool posts around since August. The fall term itself was very fun fun fun, especially towards the end, but that’s another story for another blog post. Let’s just say that if the world ends tomorrow I’ll feel very much like this guy here doing the dangerous stunt if he could feel anything at all afterwards. Among the many things I learned over these past couple of months, a major one was that Klay world rulz. Most people (including me in a considerable fraction of the time) don’t really get the humor, but that’s not a reason not to watch it. Or to call it unfunny. Um… whatever. You, whoever you are (I’m pretty sure my readers are $\emptyset$, a.k.a. the empty set, but I don’t care, I read myself, damnit) can expect more cool stuff coming out of this blog in the next couple of weeks (well, as long as winter break lasts).

Posted by: Александър Макелов | December 21, 2012

## Yossarian lives!

“They’re trying to kill me,” Yossarian told him calmly.
“No one’s trying to kill you,” Clevinger cried.
“Then why are they shooting at me?” Yossarian asked.
“They’re shooting at everyone,” Clevinger answered. “They’re trying to kill everyone.”
“And what difference does that make?”

Joseph Heller, “Catch 22″

A METAPHORICAL SEARCH ENGINE.

A METAPHORICAL SEARCH ENGINE.

A METAPHORICAL SEARCH ENGINE.

It’s not a metaphor for a search engine. Get it? It’s a thing that finds metaphors for you. And it’s called “Yossarian Lives!”. Now how cool is that. I saw this sometime this summer, and recently the alpha’s image search has become operational. I tried it briefly and I have to admit the images it returned were pretty crazy and I could see a meaningful connection to my query in only a couple of them – but still, it was an interesting analogy. I was yossarianlives!ing for “moon” and I got devils. I then realized the shape of their horns resembled that of the moon crescent and was “wow”. Of course, there might have been many other connections I missed, due to the Stephen Fry problem (oh man I love these guys, they like everything I like).

Is the Stephen Fry problem just a convenient excuse? Is the final version going to be much better than the alpha? Are the images returned plain random? The future will show. However, the idea itself is amazing. “Outsourcing our minds”, bla bla bla. Shut up. Nothing (or at least, nothing yet) can outsource your mind, it can only inspire you to think deeper. Yossarian Lives!, if it lives up to its promise, will be a free, automated, non-stop service for blowing people’s minds. Remember the good old hanging around and not thinking about anything, just staring at the emptiness, when suddenly an amazing idea flashes through you mind, and you’re like “OMG THIS IS SO EPIC HOW COULD I NOT SEE THE CONNECTION BEFORE”? No more waiting for it to randomly happen – you just go to Yossarian Lives!, strike a couple of keys and be like “Wow. Wow. Wow. Wow….” for hours. An intellectual bomb. A mind-bender. In terms of a simple metaphor (haha), if the thing works as promised, we’ll have

$\frac{\text{Yossarian Lives!}}{\text{hanging around waiting for a flash}}=\frac{\text{taking the roller coaster}}{\text{walking to grandma's house}} \ \ \ (1)$

Is (1) good? Is it bad? Is it unnatural? Well, it’s tempting, and it could bring great change to the way people think about the world. And I think this is always good. If you don’t like it you just don’t use it.

Some other arguments the creators bring up (as if (1) is not enough) can be found in this very interesting essay. A great one is the following: search engines nowadays are, by nature, predicting their users and pointing them either to knowledge the majority of other people found useful (like, autocompletion), or to knowledge that is similar to what they searched for before. This may have many obvious advantages, but they come at a price: your knowledge horizon becomes conformist and hard to change. Search engines are trying to kill everyone. So they’re trying to kill you! It’s a pretty unfortunate Catch-22, isn’t it? On the other hand, the more subjective nature of the experience of understanding a metaphor has the potential to turn all this around (and confuse entire nations, I’m guessing, but there’s no other way).

Another reason why I find all this amazing is that this project seems to lie at the intersection of… everything. You notice this post is in almost all categories on my blog, as well as in Meta. Yossarian Lives!’s very heart is an objective process that seeks highly subjective results. The idea relates mathematics with art, determinism and rigorous theoretical ideas with the mind’s inner, sometimes seemingly arbitrary, associations and feelings. All fields of knowledge are basically clustered around these two poles, which often leads to people entirely dedicating themselves to one and forgetting the other, and consequently to lack of communication, understanding, and, I’m pretty sure, many interesting ideas. Well, I believe the poles are much closer than they appear to be to most people, and this project has a great potential to make me more right (:

So, what are you waiting for??? Go ahead and try it!

Posted by: Александър Макелов | November 3, 2012

## За математическите гимназии

Преместено от http://blog.parlamentaren-kontrol.com/ на 03.11.2012

Както казахме, новият закон за училищното образование е това, което ни подтикна да започнем да се занимаваме с www.parlamentaren-kontrol.com.

Проектозаконът привлече вниманието ни още през март и може да бъде намерен ето тук. Частта, която има значение, е в Чл. 36 (2). Забележете, че възможността “V-XII” клас отсъства. Според закона, единствените училища, които могат да обучават точно този интервал от класове, са спортните (виж Чл.37 (3)).

Проблемът с това е, че математическите гимназии в България традиционно предлагат прием след четвърти клас за най-способните деца. Преимуществата на тази система (спрямо прием след седми) са, според нас, големи и лесно видими; за това може да се говори още много в някоя от следващите ни публикации. За нас, като възпитаници на ПМГ-Бургас, е ясно че годините между четвърти и осми клас са незаменими в развитието на основите на математичеко мислене. След много проведени разговори с учители установихме, че ако този закон влезе в сила, математическите гимназии ще трябва или да се откажат от приема след 4-ти клас (вредно), или да въведат прием след първи (трудно и ненужно, а в някои случаи и физически невъзможно), или да сменят името и статута си (пак вредно). Според нас, никой от тези варианти не е приемлив – системата работи добре от десетки години и тези промени със сигурност няма да я подобрят.

Още през февруари/март бяха предприети действия от страна на родители и учители (виж например тук); събра се подписка (ето тази например; на нея има доста полезна информация за това как се развиваха нещата до юни) и ситуацията се успокои. През юли отново стана дума (например тук) и се оказа, че нищо не се е променило. Събрахме много съмишленици от различни общности и организирано изпращахме писма до народните представители, за да изразим несъгласието си; уви, в отговор не получихме почти нищо. Това, което знаем сега, е, че законът е бил приет на първо четене и през септември трябва да се гласува окончателно.

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