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Wed, 21 Mar 2018 A couple of years ago I was reading Wikipedia's article about the the 1943 Bengal famine, and I was startled by the following claim:
It was cited, but also marked with the “not in citation” tag, which is supposed to mean that someone checked the reference and found that it did not actually support the claim. It sounded like it might be the sort of scurrilous lie that is widely repeated but not actually supportable, so I went to follow it up. It turned out that although the quotation was not quite exact, it was not misleadingly altered, and not a scurrilous lie at all. The attributed source (Tharoor, Shashi "The Ugly Briton". Time, (29 November 2010).) claimed:
I removed the “not in citation” tag, which I felt was very misleading. Still, I felt that anything this shocking should be as well-supported as possible. It cited Tharoor, but Tharoor could have been mistaken. So I put in some effort and dug up the original source. It is from the journal entry of Archibald Wavell, then Viceroy of India, of 5 July 1944:
This appears in the published version of Lord Wavell's journals. (Wavell, Archibald Percival. Wavell: The Viceroy's journal, p. 78. Moon, Penderel, ed. Oxford University Press, 1973.) This is the most reliable testimony one could hope for. The 1973 edition is available from the Internet Archive. A few months later, the entire article was massively overhauled by a group of anglophiles and Churchill-rehabilitators. Having failed to remove the quotation for being uncited, and then having failed to mendaciously discredit the cited source, they removed the quotation in a typical episode of Wikipedia chicanery. In a 5,000-word article, one sentence quoting the views of the then-current British Prime Minister was deemed “undue weight”, and a failure to “fairly represent all significant viewpoints that have been published by reliable sources”. Further reading: In Winston Churchill, Hollywood rewards a mass murderer. (Tharoor again, in last week's Washington Post.) [Other articles in category /wikipedia] permanent link Tue, 20 Mar 2018In English we can sometimes turn an adjective into a verb by suffixing “-en”. For example:
But not
(Note that I am only looking at -en verbs that are adjective-derived present tenses. This post is not concerned with the many -en verbs that are past participles, such as “smitten” (past participle of “smite”), “spoken” (“speak”), “molten” (“melt”), “sodden” (“seethe”), etc.) I asked some linguist about this once and they were sure it was purely morphological, something like: black, red, and white end in stop consonants, and blue, green, and yellow don't. Well, let's see:
There are some fine points:
but clearly the morphological explanation wins. I'm convinced. [ Addendum: Wiktionary discusses this suffix, distinguishing it from the etymologically distinct participial “-en”, and says “it is not currently very productive in forming new words, being mostly restricted to monosyllabic bases which end in an obstruent”. ] [Other articles in category /lang] permanent link Mon, 19 Mar 2018I had a fun idea this morning. As a kid I was really interested in polar coordinates and kind of disappointed that there didn't seem to be any other coordinate systems to tinker with. But this morning I realized there were a lot. Let !!F(c)!! be some parametrized family of curves that partition the plane, or almost all of the plane, say except for a finite number of exceptions. If you have two such families !!F_1(c)!! and !!F_2(c)!!, and if each curve in !!F_1!! intersects each curve in !!F_2!! in exactly one point (again with maybe a few exceptions) then you have a coordinate system: almost every point !!P!! lies on !!F_1(a)!! and !!F_2(b)!! for some unique choice of !!\langle a, b\rangle!!, and these are its coordinates in the !!F_1–F_2!! system. For example, when !!F_1(c)!! is the family of lines !!x=c!! and !!F_2(c)!! is the family of lines !!y=c!! then you get ordinary Cartesian coordinates, and when !!F_1(c)!! is the family of circles !!x^2+y^2=c!! and !!F_2(c)!! is the family !!y=cx!! (plus also !!x=0!!) you get standard polar coordinates, which don't quite work because the origin is in every member of !!F_2!!, but it's the only weird exception. But there are many other families that work. To take a particularly simple example you can pick some constant !!k!! and then take $$\begin{align} F_1(c): && x & =c \\ F_2(c): && y & =kx+c. \end{align} $$ This is like Cartesian coordinates except the axes are skewed. I did know about this when I was a kid but I considered it not sufficiently interesting. For a more interesting example, try $$\begin{align} F_1(c): && x^2-y^2 & =c \\ F_2(c): && xy & =c \end{align} $$ which looks like this: I've seen that illustration before but I don't think I thought of using it as a coordinate system. Well, okay, every pair of hyperbolas intersects in two points, not one. So it's a parametrization of the boundary of real projective space or something, fine. Still fun! In the very nice cases (such as the hyperbolas) each pair of curves is orthogonal at their point of intersection, but that's not a requirement, as with the skew Cartesian system. I'm pretty sure that if you have one family !!F!! you can construct a dual family !!F'!! that is orthogonal to it everywhere by letting !!F'!! be the paths of gradient descent or something. I'm not sure what the orthogonality is going to be important for but I bet it's sometimes useful. You can also mix and match families, so for example take: $$\begin{align} F_1(c): && x & =c \\ F_2(c): && xy & =c \end{align} $$ Some examples work better than others. The !!xy=c!! hyperbolas are kind of a mess when !!c=0!!, and they don't go together with the !!x^2+y^2=c!! circles in the right way at all: each circle intersects each hyperbola in four points. But it occurs to me that as with the projective plane thingy, we don't have to let that be a problem. Take !!S!! to be the quotient space of the plane where two points are identified if their !!F_1–F_2!!-coordinates are the same and then investigate !!S!!. Or maybe go more directly and take !!S = F_1 \times F_2!! (literally the Cartesian product), and then topologize !!S!! in some reasonably natural way. Maybe just give it the product topology. I dunno, I have to think about it. (I was a bit worried about how to draw the hyperbola picture, but I tried Google Image search for “families of orthogonal hyperbolas”, and got just what I needed. Truly, we live in an age of marvels!) [Other articles in category /math] permanent link Mon, 12 Mar 2018
Quick and dirty prime counting
I've been thinking for a while that I probably ought to get around to memorizing all the prime numbers under 1,000, so that I don't have to wonder about things like 893 all the time, and last night in the car I started thinking about it again, and wondered how hard it would be. There are 25 primes under 100, so presumably fewer than 250 under 1,000, which is not excessive. But I wondered if I could get a better estimate. The prime number theorem tells us that the number of primes less than !!n!! is !!O(\frac n{\log n})!! and I think the logarithm is a natural one, but maybe there is some constant factor in there or something, I forget and I did not want to think about it too hard because I was driving. Anyway I cannot do natural logarithms in my head. Be we don't need to do any actual logarithms. Let's estimate the fraction of primes up to !!n!! as !!\frac 1{c\log n}!! where !!c!! is unknown and the base of the logarithm is then unimportant. The denominator scales linearly with the power of !!n!!, so the difference between the denominators for !!n=10!! and !!n=100!! is the same as the difference between the denominators for !!n=100!! and !!n=1000!!. There are 4 primes less than 10, or !!\frac25!!, so the denominator is 2.5. And there are 25 primes less than 100, so the denominator here is 4. The difference is 1.5, so the denominator for !!n=1000!! ought to be around 5.5, and that means that about !!\frac2{11}!! of the numbers up to 1000 are prime. This yields an estimate of 182. I found out later that the correct number is 186, so I felt pretty good about that. [ Addendum: The correct number is 168, not 186, so I wasn't as close as I thought. ] [Other articles in category /math] permanent link Tue, 20 Feb 2018
Composition of utility pole ID tags
In a recent article discussing utility poles, and the metal ID plates they carry, I wondered what the plates were made of:
They are not ferrous. Probably they are aluminum. My idea that aluminum is too expensive to use for the plates was ridiculous. The pole itself costs a lot of money. The sophisticated electrical equipment on the pole costs thousands of dollars. The insulated wire strung from the pole is made of copper. Compared with all this, a ten-centimeter oval of stamped aluminum is not a big deal. 1.8mm aluminum sheet costs $100 per square meter even if you don't buy it in great quantity. Those aluminum tags probably cost no more than fifty cents each. [Other articles in category /oops] permanent link Wed, 14 Feb 2018I am almost always interested in utility infrastructure. I see it every day, and often don't think about it. The electric power distribution grid is a gigantic machine, one of the biggest devices ever built, and people spend their whole lives becoming experts on just one part of it. What is it all for, how does it work? What goes wrong, and how do you fix it? Who makes the parts, and how much do they cost? Every day I go outside and see things like these big cylinders: and I wonder what they are. In this case from clues in the environment I was able to guess they were electrical power transformers. Power is distributed on these poles at about seven thousand volts, which is called “medium voltage”. But you do not want 7000-volt power in your house because it would come squirting out of the electric outlets in awesome lightnings and burn everything up. Also most household uses do not want three-phase power, they want single-phase power. So between the pole and the house there is a transformer to change the shape of the electricity to 120V, and that's what these things are. They turn out to be called “distribution transformers” and they are manufactured by — guess who? — General Electric, and they cost a few thousand bucks each. And because of the Wonders of the Internet, I can find out quite a lot about them. The cans are full of mineral oil, or sometimes vegetable oil! (Why are they full of oil? I don't know; I guess for insulation. But I could probably find out.) There are three because that is one way to change the three-phase power to single-phase, something I wish I understood better. Truly, we live in an age of marvels. Anyway, I was having dinner with a friend recently and for some reason we got to talking about the ID plates on utility poles. The poles around here all carry ID numbers, and I imagine that back at the electric company there are giant books listing, for each pole ID number, where the pole is. Probably they computerized this back in the seventies, and the books are moldering in a closet somewhere. As I discussed recently, some of those poles are a hundred years old, and the style of the ID tags has changed over that time: It looks to me like the original style was those oval plates that you see on the left, and that at some point some of the plates started to wear out and were replaced by the yellow digit tags in the middle picture. The most recent poles don't have tags: the identifier is burnt into the pole. Poles in my neighborhood tend to have consecutive numbers. I don't think this was carefully planned. I guess how this happened is: when they sent the poles out on the truck to be installed, they also sent out a bunch of ID plates, perhaps already attached to the poles, or perhaps to be attached onsite. The plates would already have the numbers on them, and when you grab a bunch of them out of the stack they will naturally tend to have consecutive numbers, as in the pictures above, because that's how they were manufactured. So the poles in a vicinity will tend to have numbers that are close together, until they don't, because at that point the truck had to go back for more poles. So although you might find poles 79518–79604 in my neighborhood, poles 79605–79923 might be in a completely different part of the city. Later on someone was inspecting pole 79557 (middle picture) and noticed that the number plate was wearing out. So they pried it off and replaced it with the yellow digit tag, which is much newer than the pole itself. The inspector will have a bunch of empty frames and a box full of digits, so they put up a new tag with the old ID number. But sometime more recently they switched to these new-style poles with numbers burnt into them at the factory, in a different format than before. I have tried to imagine what the number-burning device looks like, but I'm not at all sure. Is it like a heated printing press, or perhaps a sort of configurable branding iron? Or is it more like a big soldering iron that is on a computer-controlled axis and writes the numbers on like a pen? I wonder what the old plates are made of. They have to last a long time. For a while I was puzzled. Steel would rust; and I thought even stainless steel wouldn't last as long as these tags need to. Aluminum is expensive. Tin degrades at low temperatures. But thanks to the Wonders of the Internet, I have learned that, properly made, stainless steel tags can indeed last long enough; the web site of the British Stainless Steel Association advises me that even in rough conditions, stainless steel with the right composition can last 85 years outdoors. I will do what I should have done in the first place, and go test the tags with a magnet to see if they are ferrous. Here's where some knucklehead in the Streets Department decided to nail a No Parking sign right over the ID tag: Another thing you can see on these poles is inspection tags: Without the Internet I would just have to wonder what these were and what OSMOSE meant. It is the name of the company that PECO has hired to inspect and maintain the poles. They specialize in this kind of work. This old pole was inspected in 2001 and again in 2013. The dated inspection tag from the previous inspection is lost but we can see a pie-shaped tag that says WOODFUME. You may recall from my previous article that the main killer of wood poles is fungal infection. Woodfume is an inexpensive fumigant that retards pole decay. It propagates into the pole and decomposes into MITC (methyl isothiocyanate). By 2001 PECO had switched to using MITC-FUME, which impregnates the pole directly with MITC. Osmose will be glad to tell you all about it. (Warning: Probably at least 30% of the surmise in this article is wrong.) [Other articles in category /tech] permanent link Tue, 13 Feb 2018(If you already know about reservoir sampling, just skip to the good part.) The basic reservoir sampling algorithm asks us to select a random item from a list, easy peasy, except:
Maybe the items are being read from a pipe or some other lazy data structure. There might be zillions of them, so we can't simply load them into an array. Obviously something like this doesn't work:
because it doesn't select the items with equal probability. Far from it! The last item is selected as often as all the preceding items put together. The requirements may seem at first impossible to satisfy, but it can be done and it's not even difficult:
The The good partLast week I thought of a different simple variation. Suppose each item !!s_i!! is presented along with an arbitrary non-negative weight !!w_i!!, measuring the relative likelihood of its being selected for the output. For example, an item with weight 6 should be selected twice as often as an item with weight 3, and three times as often as an item with weight 2. The total weight is !!W = \sum w_i!! and at the end, whenever that is, we want to have selected each item !!s_i!! with probability !!\frac{w_i}{W}!!:
The correctness proof is almost the same. Clearly this reduces to the standard algorithm when all the weights are equal. This isn't a major change, but it seems useful and I hadn't seen it before. [Other articles in category /prog] permanent link Mon, 12 Feb 2018
Philadelphia sports fans behaving badly
Philadelphia sports fans have a bad reputation. For example, we are famous for booing Santa Claus and hitting him with snowballs. I wasn't around for that; it happened in 1968. When the Santa died in 2015, he got an obituary in the Phildelphia Inquirer:
The most famous story of this type is about Ed Rendell (after he was Philadelphia District Attorney, but before he was Mayor) betting a Eagles fan that they could not throw snowballs all the way from their upper-deck seat onto the field. This was originally reported in 1989 by Steve Lopez in the Inquirer. (Lopez's story is a blast. He called up Rendell, who denied the claim, and referred Lopez to a friend who had been there with him. Lopez left a message for the friend. Then Rendell called back to confess. Later Rendell's friend called back to deny the story. Lopez wrote:
A few years later Rendell was elected Mayor of Philadelphia, and later, Governor of Pennsylvania. Anyway, I digress.) I don't attend football games, and baseball games are not held in snowy weather, so we have to find other things to throw on the field. I am too young to remember Bat Day, where each attending ticket-holder was presented with a miniature souvenir baseball bat; that was eliminated long ago because too many bats were thrown at the visiting players. (I do remember when those bats stopped being sold at the concession stands, for the same reason.) Over the years, all the larger and harder premiums were eliminated, one by one, but we are an adaptable people and once, to protest a bad call by the umpire, we delayed the game by wadding up our free promotional sport socks and throwing them onto the field. That was the end of Sock Day. On one memorable occasion, two very fat gentlemen down by the third-base line ran out of patience during an excessively long rain delay and climbed over the fence, ran out and belly-flopped onto the infield, sliding on the wet tarpaulin all the way to the first-base side. Confronted there by security, they evaded capture by turning around and sliding back. These heroes were eventually run down, but only after livening up what had been a very trying evening. The main point of this note is to shore up a less well-known story of this type. I have seen it reported that Phillies fans once booed Miss Pennsylvania, and I have also seen people suggest that this never really happened. On my honor, it did happen. We not only booed Miss Pennsylvania, we booed her for singing the national anthem. I was at that game, in 1993. The Star-Spangled Banner has a lot of problems that the singer must solve one way or another, and there are a lot of ways to interpret it. But it has a melody, and the singer's interpretation is not permitted to stray so far from the standard that they are singing a different song that happens to have the same words. I booed too, and I'm not ashamed to admit it. [Other articles in category /games] permanent link Wed, 07 Feb 2018
The many faces of the Petersen graph
(Actually the Petersen graph cannot really be said to have faces, as it is nonplanar. HA! HA! I MAKE JOKE!!1!) This article was going to be about how GraphViz renders the Petersen graph, but instead it turned out to be about how GraphViz doesn't render the Petersen graph. The GraphViz stuff will be along later. Here we have the Petersen graph, which, according to Donald Knuth, “serves as a counterexample to many optimistic predictions about what might be true for graphs in general.” It is not that the Petersen graph is stubborn! But it marches to the beat of a different drummer. If you have not met it before, prepare to be delighted. This is the basic structure: a blue 5-cycle, and a red 5-cycle. Corresponding vertices in the two cycles are connected by five purple edges. But there is a twist! Notice that the vertices in the red cycle are connected in the order 1–3–5–2–4. There are different ways to lay out the Petersen graph that showcase its many interesting properties. For example, the standard presentation, above, demonstrates that the Petersen graph is nonplanar, since it obviously contracts to !!K_5!!. The presentation below obscures this, but it is good for seeing that the graph has diameter only 2: Wait, what? Where did the pentagons go? Try this instead:
Again the red vertices are connected in the order 1–3–5–2–4. Okay, that is indeed the Petersen graph, but how does it help us see that the graph has diameter 2? Color the nodes by how far down they are from the root:
Looking at the pentagonal version, you would not suspect the Petersen graph of also having a sixfold symmetry, but it does. We'll get there in two steps. Again, here's a version where it's not so easy to see that it's actually the Petersen graph, but whatever it is, it is at least clear that it has an automorphism of order six (give it a one-sixth turn): The represents three vertices, one in each color. In the picture they are superimposed, but in the actual graph, no pair of the three is connected by an edge. Instead, each of the three is connected not to the others but to a tenth vertex that I omitted from the diagram entirely. Let's pull apart the three vertices and reveal the hidden tenth vertex and its three edges:
Here is the same drawing, recolored to match the tree diagram from before; the outer hexagon is just the 6-cycle formed by the six blue leaf nodes: But maybe it's easier to see if we look for red and blue pentagons. There are a couple of ways to do that: As always, the red vertices are connected in the order 1–3–5–2–4. Finally, here's a presentation you don't often see. It demonstrates that the Petersen graph also has fourfold symmetry:
Again, and represent single vertices stretched out into dumbbell shapes. The diagram only shows 14 of the 15 edges; the fifteenth connects the two dumbbells. The pentagons are deeply hidden here. Can you find them? (Spoiler) Even though this article was supposed to be about GraphViz, I found it impossible to get it to render the diagrams I wanted it to, and I had to fall back on Inkscape. Fortunately Inkscape is a ton of fun. [Other articles in category /math] permanent link Thu, 01 Feb 2018As I may have mentioned, I have started another
blog, called Last month I said:
I think I am doing better. I will continue my efforts to emphasize quantity over quality, with a multi-pronged approach:
It will be a struggle, but I resolve to do my best! Here is a list of January's shitposts. Boldface indicates the articles that may (may) be of more general interest (ha). There are fewer of these than last month because I promoted several of the better ones, so you have seen them already.
[Other articles in category /meta/shitpost] permanent link Tue, 30 Jan 2018Sapporo restaurant, on West 49th Street in New York, closed yesterday. There are 24,000 restaurants in New York, and for many, many years, Sapporo was my favorite. Sapporo was a ramen restaurant, probably the first in New York. I remember first hearing the word “ramen” in the early 1980s, when the Larmen Dosanko appeared near Lincoln Center. But Sapporo opened in 1975. I started going there around 1984. I didn't discover it on my own; I think my dad and I happened in one day when we were in the neighborhood. But it made a big impression on me, and I would regularly stop in whenever I was nearby, and sometimes I would walk downtown (about 45 minutes) just to eat there. When I was fifteen years old, I did somethinng fifteen-year-old boys often do: I grew six inches and added thirty pounds in one year. I ate all the time. I spent so much time eating that it wasn't enjoyable any more, and I complained that I was tired of it and didn't have enough time to do anything else. I would come home from school and eat a double-decker sandwich (sliced muenster with mayonnaise was my favorite), half a pound of feta cheese, three yogurts, and whatever leftovers I could find in the fridge, and then two hours later when my parents got home I would ask “What's for dinner? I'm starving.” I fell in love with Sapporo because it was the only restaurant where I could afford to order as much food as I could eat. I would come out of Sapporo full. Sometimes when I left Sapporo there was still some rice or noodles in my bowl, and I would stop thinking about food for an hour or two. On the table were three intriguing bottles, one brown, one pale yellow, and one bright red. The brown one was obviously soy sauce, but what were the others? I learned that the pale yellow one was vinegar and the bright red one was hot oil and had great fun trying them out in different combinations. I had never thought of using vinegar as a condiment, and I loved it. Sapporo was where I first had agedashi dofu, which is delectable cubes of soft tofu, dusted in flour or starch, fried brown and crisp, and served in savory broth. It is ephemeral: you have to eat it right away, before it gets cold and soggy. My favorite dish was the pork cutlet donburi. (They didn't call it “katsudon”; I didn't learn that until later.) The cutlet was embedded, with onions, in a fried egg that coverered and adhered to the top of the rice, and I can still remember how it tasted with the soy sauce and vinegar, and the texture of it. It was tricky to pick up the cutlet, with its attached fried egg, with chopsticks. I now use chopsticks as well as I use a fork, automatically and unconsciously, and I think Sapporo was probably where I learned to do it. Because of Sapporo, I became so enamored of vinegar that I started to put it on everything. I didn't like mustard, I thought, but one day I learned that the principal ingrediant in mustard, other than the mustard itself, was vinegar. This put a new light on things and I immediately decided I had better give mustard another try. I discovered that I did indeed like mustard (like vinegar, but with extra flavor!) and I have liked it ever since. (Around the same time of my life, I was learning about beer, and I eagerly tried Sapporo beer, which they feature, hoping that it would be as wonderful as Sapporo restaurant. I was disappointed.) I deeply regret that I missed my last chance to eat there. I was staying in Midtown with Toph last summer and suggested that we eat dinner there, but she chose to go somewhere else. It didn't occur to me that Sapporo might not always be there, or I would have insisted. Goodbye, Sapporo. I love you and I will miss you. [Other articles in category /food] permanent link Mon, 29 Jan 2018
The rubber duck explains coherence spaces
I've spent a chunk of the past week, at least, trying to understand the idea of a coherence space (or coherent space). This appears in Jean-Yves Girard's Proofs and Types, and it's a model of a data type. For example, the type of integers and the type of booleans can be modeled as coherence spaces. The definition is one of those simple but bafflingly abstract ones that you often meet in mathematics: There is a set !!\lvert \mathcal{A}\rvert!! of tokens, and the points of the coherence space !!\mathcal{A}!! (“cliques”) are sets of tokens. The cliques are required to satisfy two properties:
To beginning math students it often seems like the sorts of definitions are generated at random. Okay, today we're going to study Eulerian preorders with no maximum element that are closed under finite unions; tomorrow we're going to study semispatulated coalgebras with countably infinite signatures and the weak Cosell property. Whatever, man. I have a long article about this in progress, but I'll summarize: they are never generated at random. The properties are carefully chosen because we have something in mind that we are trying to model, and until you understand what that thing is, and why someone thinks those properties are the important ones, you are not going to get anywhere. So when I see something like this I must stop immediately and ask ‘wat’. I can try to come up with the explanation myself, or I can read on hoping for an explanation, or I can do some of each, but I am not going to progress very far until I understand what it is about. And I'm not sure anyone short of Alexander Grothendieck would have any more success trying to move on with the definition itself and nothing else. Girard explains shortly after:
Okay, fine. I understand the point of the project, although not why the definition is what it is. I know a fair amount about types. And Girard has given two examples: booleans and integers. But these examples are unusually simple, because none of the cliques has more than one element, and so the examples are not as illuminating as they might be. Some of the ways I tried to press onward were:
None of this was working. I had several different ideas about what the coherence spaces might look like for other types, but none of them seemed to fit with what Girard was doing. I couldn't come up with any consistent story. So I prepared to ask on StackExchange, and I spent about an hour writing up my question, explaining all the things I had tried and what the problems were with each one. And as I drew near to the end of this, the clouds parted! I never had to post the question. I was in the middle of composing this paragraph:
I decided I hadn't made enough of an effort to understand the direct product. So even though I couldn't see how it could possibly give me anything like what I wanted, I followed its definition for !!{{\mathcal B}ool}^2!! — and the light came on. Here's the puzzling coproduct-like definition of the product of two coherence spaces, from page 61:
That is, the tokens in the product space are literally the disjoint union of the tokens in the component spaces. And the edges in the product's web are whatever they were in !!{\mathcal A}!!'s web (except lifted from !!|{\mathcal A}|!! to !!\{1\}×|{\mathcal A}|!!), whatever they were in !!{\mathcal B}!!'s web (similarly), and also there is an edge between every !!\langle1, {\mathcal A}\rangle!! and each !!\langle2, {\mathcal B}\rangle!!. For !!{{\mathcal B}ool}^2!! the web looks like this: There is no edge between !!\langle 1, \text{True}\rangle!! and !!\langle 1, \text{False}\rangle!! because in !!{{\mathcal B}ool}!! there is no edge between !!\text{True}!! and !!\text{False}!!. This graph has nine cliques. Here they are ordered by set inclusion: (In this second diagram I have abbreviated the pair !!\langle1, \text{True}\rangle!! to just !!1T!!. The top nodes in the diagram are each labeled with a set of two ordered pairs.) What does this mean? The ordered pairs of booleans are being represented by functions. The boolean pair !!\langle x, y\rangle!! is represented by the function that takes as its argument a number, either 1 or 2, and then returns the corresponding component of the pair: the first component !!x!! if the argument was 1, and the second component !!y!! if the argument was 2. The nodes in the bottom diagram represent functions. The top row are fully-defined functions. For example, !!\{1F, 2T\}!! is the function with !!f(1) = \text{False}!! and !!f(2) = \text{True}!!, representing the boolean pair !!\langle\text{False}, \text{True}\rangle!!. Similarly if we were looking at a space of infinite lists, we could consider it a function from !!\Bbb N!! to whatever the type of the lists elements was. Then the top row of nodes in the coherence space would be infinite sets of pairs of the form !!\langle n, \text{(list element)}\rangle!!. The lower nodes are still functions, but they are functions about which we have only incomplete information. The node !!\{2T\}!! is a function for which !!f(2) = \text{True}!!. But we don't yet know what !!f(1)!! is because we haven't yet tried to compute it. And the bottommost node !!\varnothing!! is a function where we don't know anything at all — yet. As we test the function on various arguments, we move up the graph, always following the edges. The lower nodes are approximations to the upper ones, made on the basis of incomplete information about what is higher up. Now the importance of finite approximants on page 56 becomes clearer. !!{{\mathcal B}ool}^2!! is already finite. But in general the space is infinite because the type is functions on an infinite domain, or infinite lists, or something of that sort. In such a space we can't get all the way to the top row of nodes because to do that we would have to call the function on all its possible arguments, or examine every element of the list, which is impossible. Girard says “Above all, there are enough finite approximants to a.” I didn't understand what he meant by “enough”. But what he means is that each clique !!a!! is the union of its finite approximants: each bit of information in the function !!a!! is obtainable from some finite approximation of !!a!!. The “stable functions” of section 8.3 start to become less nebulous also. I had been thinking that the !!\varnothing!! node was somehow like the
!!\bot!! element in a Scott domain, and then I struggled to
identify anything like !!\langle \text{False}, \bot\rangle!!.
It looks at first
like you can do it somehow, because there are the right number of
nodes at the middle level.
Trouble arises in other coherence
spaces. Presented with a value from I think there isn't anything like !!\bot!! or !!W\ \bot!! in the coherence space. Or maybe they they are there but sharing the !!\varnothing!! node. But I think more likely partial objects will appear in some other way. Whew! Now I can move along. (If you don't understand why “rubber duck”, Wikipedia explains:
I spent a week on this but didn't figure it out until I tried formulating my question for StackExchange. The draft question, never completed, is here if for some reason you want to see what it looked like.) [Other articles in category /math/logic] permanent link |