The Universe of Discourse


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:

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. … I will go test the tags with a magnet to see if they are ferrous.

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 2018

Utility poles

I 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:

A wooden power utility pole,
including cobra-head street light,
with three large gray cylinders mounted on it under the main
wires

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:

An old, stamped-metal identification plate nailed to a wooden
utility pole.  The plate is elliptical, and says 'PHILA ELEC. Cº 79558 B' This wooden pole has the following letters burned into it: 'BWR
CPT 51017 SPSK6 250 PECO'

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:

A close-up of an old oval tag just peeking out from behind the
corner of the metal regulation sign that was nailed to the same pole

Another thing you can see on these poles is inspection tags:

A very old pole. Three groups of tags are nailed to it.  The
bottom two groups contains an oval tag stamped with OSMOSE and an
inspection year (2001 or 2013),
and a quarter-circle tag stamped with MITC-FUME.  The top group is
missing its oval tag, and has only a rather rusty quarter-circle that
says WOODFUME

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

Weighted Reservoir Sampling

(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:

  1. Each item must be selected with equal probability
  2. We don't know ahead of time how big the list is
  3. We may only make one pass over the list
  4. We may use only constant memory

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:

# Python
from random import random
selected = inputs.next()
for item in inputs:
    if random() < 0.5:
        selected = item

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:

from random import random
n = 0
selected = None

for item in inputs:
    n += 1
    if random() < 1/n:
        selected = item

The inputs here is some sort of generator that presents the list of items, one at a time. After the loop completes, the selected item is in selected. A proof that this selects each item equiprobably is left as an easy exercise, or see this math StackExchange post. A variation for selecting !!k!! items instead of only one is quite easy.

The good part

Last 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}!!:

total_weight = 0
selected = None

for item, weight in inputs:
    if weight == 0: continue
    total += weight
    if random() < weight/total:
        selected = item

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:

Frank Olivo, the Santa Claus who got pelted with snowballs at the Eagles game that winter day in 1968, died Thursday, April 30…

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:

Was former D.A. Ed Rendell's worst mistake to (A) bet a drunken hooligan he couldn't reach the field, (B) lie about it, (C) confess, or (D) take his friend down with him?

My vote is C. Too honest. Why do you think he can't win an election?

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.

The Petersen
graph has two sets of five vertices each.  Each set is connected into
a pentagonal ring.  There are five more edges between vertices in
opposite rings, but instead of being connected 0–0 1–1 2–2 3–3 4–4,
they are connected 0–0 1–2 2–4 3–1 4–3.

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:

The Petersen
graph laid out as a tree, with a root attached to three level-1 nodes,
each attached to 2 level-2 nodes.  The six level-2 nodes are then
connected into a ring so that each level-2 node is at distance 1 or
distance 2 from each other level-2 node.

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:

  • Obviously, the root node (black) has distance at most 2 to every other node, because the tree has only depth 2.

  • Each of the three second-level nodes (red) is distance 2 from the other two, via a path through the root.

  • The six third-level nodes (blue) are linked in a 6-cycle (dotted lines), so that each third-level node is at most two steps away along the cycle from the others, except for the one furthest away, but that is its sibling in the tree, and it has a path of length 2 through their common parent.

  • And since each third-level node (say, the one with the red ring) is connected by a dotted edge (orange) to cousins in both of the other branches of the tree, it's only distance 2 from both of its red uncle nodes.

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):

narf

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:

narf

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:

narf  

But maybe it's easier to see if we look for red and blue pentagons. There are a couple of ways to do that:

narf   narf

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:

narf

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 2018

Shitpost roundup, 2018-01

As I may have mentioned, I have started another blog, called Content-type: text/shitpost.

Last month I said:

The shitposts have been suffering quality creep and I am making an effort to lower my standards. I will keep you posted about how this develops.

I think I am doing better. I will continue my efforts to emphasize quantity over quality, with a multi-pronged approach:

  • Faster production with lower production standards
  • Less filtering of possible topics for relevance, general interest, or almost anything else
  • Promote insufficiently shitty posts to The Universe of Discourse

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 2018

Sapporo has closed

Sapporo 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:

  1. If !!a!! is a clique, and !!a'\subset a!!, then !!a'!! is also a clique.
  2. Suppose !!\mathcal M!! is some family of cliques such that !!a\cup a'!! is a clique for each !!a, a'\in \mathcal M!!. Then !!\bigcup {\mathcal M}!! is also a clique.

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:

The aim is to interpret a type by a coherence space !!\mathcal{A}!!, and a term of this type as a point [clique] of !!\mathcal{A}!!, infinite in general…

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:

  1. Read ahead and see if there is more explanation. I tried this but I still wasn't getting it. The next section seemed clear: the cliques define a “coherence” relation on the tokens, from which the cliques can be recovered. Consider a graph, where the vertices are tokens and there is an edge !!a—b!! exactly when !!\{a, a'\}!! is a clique; we say that !!a!! and !!a'!! are coherent. Then the cliques of the coherence space are exactly the cliques of the graph; hence the name. The graph is called the web of the space, and from the web one can recover the original space.

    But after that part came stable functions, which I couldn't figure out, and I got stuck again.

  2. Read ahead and see if there is a more complicated specific example. There wasn't.

  3. Read ahead and see if any of the derived concepts are familiar, and if so then work backward. For instance, if I had been able to recognize that I already knew what stable functions were, I might have been able to leverage tha tinto an understanding of what was going on with the coherence spaces. But for me they were just another problem of the same sort: what is a stable function supposed to be modeling?

  4. Read someone else's explanation instead. I tried several without much success. They all seemed to be written for someone who already had a clue what was going on. (That is a large part of the reason I have written up this long and clueless explanation.)

  5. Try to construct some examples and see if they make sense in the context of what comes later. For example, I know what the coherence space of booleans looks like because Girard showed me. Can I figure out the structure of the coherence space for the type of “wrapped booleans”?

    -- (Haskell)
    data WrappedBoolean = W Bool
    

    Can I figure it out for the type of pairs of booleans?

    -- (Haskell)
    type BooleanPair = (Bool, Bool)
    

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:

In section 8.4 Girard defines a direct product of coherence spaces, but it doesn't look like the direct product I need to get a product type; it looks more like a disjoint union type. If the coherence space for Pairbool is the square of the coherence space for !!{{\mathcal B}ool}!!, how? It has 4 2-cliques, but if those are the total elements of !!{{\mathcal B}ool}^2!!, then what are do the 1-cliques mean?

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:

If !!{\mathcal A}!! and !!{\mathcal B}!! are two coherence spaces, we define !!{\mathcal A}\&{\mathcal B}!! by:

!!|{\mathcal A}\&{\mathcal B}| = |{\mathcal A}| + |{\mathcal B}| = \{1\}×|{\mathcal A}| \cup \{2\}×|{\mathcal B}|!!

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.
For the WrappedBoolean type, for example, the type has four elements: !! W\ \text{True}, W\ \text{False}, W\ \bot,!! and !!\bot!!. I think the coherence space for WrappedBoolean is just like the one for !!{{\mathcal B}ool}!!:

Presented with a value from WrappedBoolean, you don't initially know what it is. Then you examine it, and you know whether it is !!W\ \text{True}!! or !!W\ \text{False}!!. You are now done.

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:

Many programmers have had the experience of explaining a problem to someone else, possibly even to someone who knows nothing about programming, and then hitting upon the solution in the process of explaining the problem.

[“Rubber duck”] is a reference to a story in the book The Pragmatic Programmer in which a programmer would carry around a rubber duck and debug their code by forcing themselves to explain it, line-by-line, to the duck.

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.)


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Thu, 25 Jan 2018

Samuel Johnson and Ossian

For me, a little of Samuel Johnson goes a long way, because he was a tremendous asshole, and the draught is too strong for me to take much at once. But he is at his best when he is in opposition to someone who is an even bigger asshole, in this case James Macpherson.

Macpherson was a Scottish poet who perpetrated a major hoax for his own literary benefit. He claimed to have discovered and translated a collection of 3rd-century Gaelic manuscripts, written by a bard named Ossian, which he then published, with great commercial and critical success.

Thomas Bailey Saunders, in The Life and Letters of James Macpherson (1894), writes:

[Macpherson] was six-and-twenty, and flattered to the top of his bent. What happened with him was only what happens with most literary adventurers whose success is immediate and greater than their strict deserts; he contracted the foolish temper in which a man regards all criticism as ignorant and impertinent, and declines to take advice from any one.

Ossian and Macpherson did receive a great deal of criticism. Much of it was rooted in anti-Scottish bigotry, but many people at the time correctly suspected that Macpherson had written the "translations” himself, from scratch or nearly so. There was a great controversy, in which nobody participated more forcefully (or impolitely) than Johnson, a noted anti-Scottish bigot, who said that not only was the poems’ claimed history fraudulent, but that the poems themselves were rubbish.

The argument raged for some time. Johnson took up the matter in Journey to the Western Islands of Scotland (1775), in which he said:

I believe [the poems] never existed in any other form than that which we have seen. The editor, or author, never could shew the original; nor can it be shewn by any other; to revenge reasonable incredulity, by refusing evidence, is a degree of insolence, with which the world is not yet acquainted; and stubborn audacity is the last refuge of guilt. It would be easy to shew it if he had it; but whence could it be had?

Macpherson was furious to learn, before it was published, what Journey to the Western Islands would say, and attempted to prevent the passage from appearing in the book at all. When he discovered he was too late for this, he suggested that a slip of paper be inserted into the printed copies, apologizing and withdrawing the paragraph. This suggestion was ignored, and Johnson's book was published with no alteration.

Macpherson then challenged Johnson to a duel, and then, Johnson having declined, sent him a final letter, now lost, which a contemporary described as telling Johnson:

as he had declined to withdraw from his book the injurious expressions reflecting on Mr. Macpherson's private character, his age and infirmities alone protected him from the treatment due to an infamous liar and traducer.

(John Clark, An Answer to Mr. [William] Shaw's Inquiry, p. 51. Reprinted in Works of Ossian, vol. 1, 1783.)

Johnson's famous reply to this letter, quoted by Boswell, was:

MR. JAMES MACPHERSON.

I RECEIVED your foolish and impudent letter. Any violence offered me I shall do my best to repel; and what I cannot do for myself, the law shall do for me. I hope I shall never be deterred from detecting what I think a cheat, by the menaces of a ruffian.

“What would you have me retract? I thought your book an imposture; I think it an imposture still. For this opinion I have given my reasons to the publick, which I here dare you to refute. Your rage I defy. Your abilities, since your Homer, are not so formidable; and what I hear of your morals inclines me to pay regard not to what you shall say, but to what you shall prove. You may print this if you will.

“SAM. Johnson.”

Both Macpherson and Johnson are buried in Westminster Abbey. Some say Macpherson bought his way in.


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Mon, 22 Jan 2018

The gods of Stackexchange karma are fickle

A few years ago an se.math user asked why their undistinguished answer to some humdrum question had gotten so many upvotes. I replied:

It's the gods of Stackexchange Karma evening your score for that really clever answer you posted two weeks ago that still has 0 upvotes.

If you're going to do Stack Exchange, I think it's important not to stress out about the whys of the votes, and particularly important not to take them personally. The karma gods do not always show the most refined taste. As Brandon Tartikoff once said, “All hits are flukes”.

Today I got an unusually flukey hit. But first, here are some nice examples in the opposite direction, posts that I put a lot of trouble and effort into, which were clear and useful, helpful to the querent, and which received no upvotes at all.

Here the querent asked, suppose I have several nonstandard dice with various labeling on the faces. Player A rolls some of the dice, and Player B rolls a different set of dice. How do I calculate things like “Player A has an X% chance of rolling a higher total”?

Mathematicians are not really the right people to ask this to, because many of them will reply obtusely, informing you that it depends on how many dice are rolled and on how their faces are actually labeled, and that the question did not specify these, but that if it had, the problem would be trivial. (I thought there were comments to that effect on this question, but if there were they have been deleted. In any case nobody else answered.) But this person was writing a computer game and wanted to understand how to implement a computer algorithm for doing the calculation. There is a lot one can do to help this person. I posted an answer that I thought was very nice. The querent liked it, but it got zero upvotes. This happens quite often. Which is fine, because the fun of doing it and the satisfaction of helping someone are reward enough.

Like that one, many of these questions are ignored because they aren't mathematically interesting. Or sometimes the mathematics is simple but the computer implementation is not.

Sometimes I do them just because I want to know the particular answer. (How much of an advantage does Player A get from being allowed to add 1 to their total?)

Sometimes there's an interesting pedagogical problem, such as: how do I give a hint that will point in the right direction without giving away the whole secret? Or: how do I wade through this person's confused explanation and understand what they are really asking, or what they are really confused about? That last person was given a proof that glossed over six or seven steps in the reasoning, focusing instead on the technically interesting induction proof in the middle. The six or seven steps are straightforward, if you already have enough practice with logical reasoning about quantified statements. But this querent didn't follow the reasoning that led up to the induction, so they didn't understand why the induction was useful or what it was for.

Some people are just confused about what the question is. That person has a complicated-seeming homework exercise about the behavior of the logistic map, and need helps interpreting it from someone who has a better idea what is going on. The answer didn't require any research effort, and it's mathematically uninteresting, but it did require attention from someone who has a better idea of what is going on. The querent was happy, I was happy, and nobody else noticed.

I never take the voting personally, because the gods of Stackexchange karma are so fickle, and today I got a reminder of that. Today's runaway hit was for a complete triviality that I knocked off in two minutes. It currently has 94 upvotes and seems likely to get a gold medal. (That gold medal, plus $4.25, will get me a free latte at Starbuck's!)

The question asks how to find a prime factor of 7,999,973 without using a calculator.

It's one of those easy-if-you-happen-to-know-the-trick things, and I just happened to get there before one of the (many) other people who happens to know the trick.

As we used to say in the system administration biz, some days you're an idiot if you can't explain how to do real-time robot arm control in Unix, other days you're a genius if you fix their terminal by plugging it in.

[ Addendum 20180123: I got the gold medal. Woo-hoo, free latte! ]


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Sat, 20 Jan 2018

I forgot there are party conventions

Yesterday I made a huge mistake in my article about California's bill requiring presidential candidates to disclose their tax returns. I asked:

Did I miss anything?

Yes, yes I did. I forgot that party nominees are picked not by per-state primary elections, but by national conventions. Even had Ronnie won the California Republican primary election, rather than Trump, that would not be enough the get him on the ballot in the general election.

In the corrected version of my scenario, California sends many Ronnie supporters to the Republican National Convention, where the other delegates overwhelmingly nominate Trump anyway. Trump then becomes the Republican party candidate and appears on the California general election ballot in November. Whoops.

I had concluded that conceding California couldn't hurt Trump, and it could actually a huge benefit to him. After correcting my mistake, most of the benefit evaporates.

I wonder if Trump might blow off California in 2020 anyway. The upside would be that he could spend the resources in places that might give him some electoral votes. (One reader pointed out that Trump didn't blow off California in the 2016 election, but of course the situation was completely different. In 2016 he was not the incumbent, he was in a crowded field, and he did not yet know that he was going to lose California by 30%.)

Traditionally, candidates campaign even in states they expect to lose. One reason is to show support for candidates in local elections. I can imagine many California Republican candidates would prefer that Trump didn't show up. Another reason is to preserve at least a pretense that they are representatives of the whole people. Newly-elected Presidents have often upheld the dignity of the office by stating the need for unity after a national election and promising (however implausibly) to work for all Americans. I don't care to write the rest of this paragraph.

The major downside that I can think of (for Trump) is that Republican voters in California would be infuriated, and while they can't directly affect the outcome of the presidential election, they can make it politically impossible for their congressional representatives to work with Trump once he is elected. A California-led anti-Trump bloc in Congress would probably cause huge problems for him.

Wild times we live in.


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Fri, 19 Jan 2018

Presidential tax return disclosure

The California state legislature passed a bill that would require presidential candidates to disclose their past five tax returns in order to qualify for California primary elections. The bill was vetoed by Governor Brown, but what if it had become law?

Suppose Donald Trump ran for re-election in 2020, as seems likely, barring his death or expulsion. And suppose he declined once again to disclose his tax returns, and was excluded from the California Republican primary election. I don't see how this could possibly hurt Trump, and it could benefit him.

It doesn't matter to Trump whether he enters the primary or wins the primary. Trump lost California by 30% in 2016. Either way he would be just as certain to get the same number of electors: zero. So he would have no incentive to comply with the law by releasing his tax returns.

Most candidates would do it anyway, because they try to maintain a pretense of representing the entire country they are campaigning to lead, but Trump is really different in this way. I can easily imagine he might simply refuse to campaign in California, instead dismissing the entire state with some vulgar comment. If there is a downside for Trump, I don't see what it could be.

Someone else (call them “Ronnie”) would then win the California Republican primary. Certainly Ronnie is better-qualified and more competent than Trump, and most likely Ronnie is much more attractive to the California electorate.

Ronnie might even be more attractive than the Democratic candidate, and might defeat them in the general election, depriving Trump's challenger of 55 electoral votes and swinging the election heavily in Trump's favor.

Did I miss anything?

[ Addendum 20180120: Yeah, I forgot that after the primary there is a convention that nominates a national party candidate. Whooops. Further discussion. ]


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