# The Universe of Discourse

Tue, 30 Jan 2018

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.

Mon, 29 Jan 2018

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.)

Thu, 25 Jan 2018

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.

Mon, 22 Jan 2018

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!) 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! ] Sat, 20 Jan 2018 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. Fri, 19 Jan 2018 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. ] Thu, 18 Jan 2018 In autumn 2014 I paid for something and got$15.33 in change. I thought I'd take the hint from the Universe and read Wikipedia's article on the year 1533. This turned out unexpectedly exciting. 1533 was a big year in English history. Here are the highlights:

• January 25: King Henry VIII marries Anne Boleyn
• March 30: Thomas Cranmer becomes Archbishop of Canterbury
• May 23: Cranmer annuls Henry's marriage to Catherine of Aragon
• June 1: Boleyn crowned queen consort by Cranmer
• July 11: Henry excommunicated by Pope Clement VII
• September 7: Princess Elizabeth, later Queen Elizabeth I, is born to Boleyn

A story clearly emerges here, the story of Henry's frantic response to Anne Boleyn's surprise pregnancy.

The first thing to notice is that Elizabeth was born only seven months after Henry married Boleyn. The next thing to notice is that Henry was still married to Catherine when he married Boleyn. He had to get Cranmer to annul the marriage, issuing a retroactive decree that not only was Henry not married to Catherine, but he had never been married to her.

In 2014 I imagined that Henry appointed Cranmer to be Archbishop on condition that he get the annulment, and eventually decided that was not the case. Looking at it now, I'm not sure why I decided that.

While Cranmer was following Charles through Italy, he received a royal letter dated 1 October 1532 informing him that he had been appointed the new Archbishop of Canterbury, following the death of archbishop William Warham [on 22 August]. Cranmer was ordered to return to England. The appointment had been secured by the family of Anne Boleyn, who was being courted by Henry.

Cranmer had been working on that annulment since at least 1527. In 1532 he was ambassador to Charles V, the Holy Roman Emperor, who was the nephew of Henry's current wife Catherine. I suppose a large part of Cranmer's job was trying to persuade Charles to support the annulment. (He was unsuccessful.) When Charles conveniently went to Rome (what for? Wikipedia doesn't say) Cranmer followed him and tried to drum up support there for the annulment. (He was unsuccessful in that too.)

Fortunately there was a convenient vacancy, and Henry called him back to fill it, and got his annulment that way. In 2014 I thought Warham's death was just a little too convenient, but I decided that he died too early for it to have been arranged by Henry. Now I'm less sure — Henry was certainly capable of such cold-blooded planning — but I can't find any mention of foul play, and The Divorce of Henry VIII: The Untold Story from Inside the Vatican describes the death as “convenient though entirely natural”.

[ Addendum: This article used to say that Elizabeth was born “less than seven months” after Henry and Boleyn's marriage. Daniel Holtz has pointed out that this was wrong. The exact amount is 225 days, or 32 weeks plus one day. The management regrets the error. ]

Tue, 16 Jan 2018

If I were in charge of keeping plutonium out of the wrong hands, I would still worry about [people extracting it from plutonium-fueled pacemakers].

This turns out to be no worry at all. The isotope in the pacemaker batteries is Pu-238, which is entirely unsuitable for making weapons. Pu-238 is very dangerous, being both radioactive and highly poisonous, but it is not fissile. In a fission chain reaction, an already-unstable atomic nucleus is hit by a high-energy neutron, which causes it to fragment into two lighter nuclei. This releases a large amount of nuclear binding energy, and more neutrons which continue the reaction. The only nuclei that are unstable enough for this to work have an odd number of neutrons (for reasons I do not understand), and Pu-238 does not fit the bill (Z=94, N=144). Plutonium fission weapons are made from Pu-241 (N=147), and this must be carefully separated from the Pu-238, which tends to impede the chain reaction. Similarly, uranium weapons are made from U-235, and this must be painstakingly extracted from the vastly more common U-238 with high-powered centrifuges.

But I did not know this when I spent part of the weekend thinking about the difficulties of collecting plutonium from pacemakers, and discussing it with a correspondent. It was an interesting exercise, so I will publish it anyway.

While mulling it over I tried to identify the biggest real risks, and what would be the most effective defenses against them. An exercise one does when considering security problems is to switch hats: if I were the bad guy, what would I try? What problems would I have to overcome, and what measures would most effectively frustrate me? So I put on my Black Hat and tried to think about it from the viewpoint of someone, let's call him George, who wants to build a nuclear weapon from pacemaker batteries.

I calculated (I hope correctly) that a pacemaker had around 0.165 mg of plutonium, and learned online that one needs 4–6 kg to make a plutonium bomb. With skill and experience one can supposedly get this down to 2 kg, but let's take 25,000 pacemakers as the number George would need. How could he get this much plutonium?

(Please bear in mind that the following discussion is entirely theoretical, and takes place in an imaginary world in which plutonium-powered pacemakers are common. In the real world, they were never common, and the last ones were manufactured in 1974. And this imaginary world exists in an imaginary universe in which plutonium-238 can sustain a chain reaction.)

Obviously, George's top target would be the factory where the pacemakers are made. Best of all is to steal the plutonium before it is encapsulated, say just after it has been delivered to the factory. But equally obviously, this is where the security will be the most concentrated. The factory is not as juicy a target as it might seem at first. Plutonium is radioactive and toxic, so they do not want to have to store a lot of it on-site. They will have it delivered as late as possible, in amounts as small as possible, and use it up as quickly as possible. The chances of George getting a big haul of plutonium by hitting the factory seem poor.

Second-best is for George to steal the capsules in bulk before they are turned into pacemakers. Third-best is for him to steal cartons of pacemakers from the factory or from the hospitals they are delivered to. But bulk theft is not something George can pull off over and over. The authorities will quickly realize that someone is going after pacemakers. And after George's first heist, everyone will be looking for him.

If the project gets to the point of retrieving pacemakers after they are implanted, George's problems multiply enormously. It is impractical to remove a pacemaker from of a living subject. George would need to steal them from funeral homes or crematoria. These places are required to collect the capsules for return to Oak Ridge, and conceivably might sometimes have more than one on hand at a time, but probably not more than a few. It's going to be a long slog, and it beggars belief that George would be able to get enough pacemakers this way without someone noticing that something was up.

The last resort is for George to locate people with pacemakers, murder, and dissect them. Even if George somehow knows whom to kill, he'd have to be Merlin to arrange the murder of 25,000 people without getting caught. Merlin doesn't need plutonium; he can create nuclear fireballs just by waving his magic wand.

If George does manage to collect the 25,000 capsules, his problems get even worse. He has to open the titanium capsules, already difficult because they are carefully made to be hard to open — you wouldn't want the plutonium getting out, would you? He has to open them without spilling the plutonium, or inhaling it, or making any sort of mess while extracting it. He has to do this 25,000 times without messing up, and without ingesting the tiniest speck of plutonium, or he is dead.

He has to find a way to safely store the plutonium while he is accumulating it. He has to keep it hidden not only from people actively looking for him — and they will be, with great yearning — but also from every Joe Blow who happens to be checking background radiation levels in the vicinity.

And George cannot afford to take his time and be cautious. He is racing against the clock, because every 464 days, 1% of his accumulated stock, however much that is, will turn into U-234 and be useless. The more he accumulates, the harder it is to keep up. If George has 25,000 pacemakers in a warehouse, ready for processing, one pacemaker-worth of Pu-238 will be going bad every two days.

In connection with this, my correspondent brought up the famous case of the Radioactive Boy Scout, which I had had in mind. (The RBS gathered a recklessly large amount of americium-241 from common household smoke detectors.) Ignoring again the unsuitability of americium for fission weapons (an even number of neutrons again), the project is obviously much easier. At the very least, you can try calling up a manufacturer of smoke alarms, telling them you are building an apartment complex in Seoul, and that you need to bulk-order 2,000 units or whatever. You can rob the warehouse at Home Depot. You can even buy them online.

Sun, 14 Jan 2018

I woke up in the middle of the night wondering: Some people have implanted medical devices, such as pacemakers, that are plutonium-powered. How the hell does that work? The plutonium gets hot, but what then? You need electricity. Surely there is not a tiny turbine generator in there!

There is not, and the answer turns out to be really interesting, and to involve a bunch of physics I didn't know.

If one end of a wire is hotter than the other, electrons tend to diffuse from the hot end to the cold end; the amount of diffusion depends on the material and the temperature. Take two wires of different metals and join them into a loop. (This is called a thermocouple.) Make one of the joints hotter than the other. Electrons will diffuse from the hot joint to the cold joint. If there were only one metal, this would not be very interesting. But the electrons diffuse better through one wire (say wire A) than through the other (B), and this means that there will be net electron flow from the hot side down through wire A and then back up through B, creating an electric current. This is called the Seebeck effect. The potential difference between the joints is proportional to the temperature difference, on the order of a few hundred microvolts per kelvin. Because of this simple proportionality, the main use of the thermocouple is to measure the temperature difference by measuring the voltage or current induced in the wire. But if you don't need a lot of power, the thermocouple can be used as a current source.

In practice they don't use a single loop, but rather a long loop of alternating metals, with many junctions:

This is called a thermopile; when the heat source is radioactive material, as here, the device is called a radioisotope thermoelectric generator (RTG). The illustration shows the thermocouples strung out in a long line, but in an actual RTG you put the plutonium in a capsule and put the thermocouples in the wall of the capsule, with the outside joints attached to heat sinks. The plutonium heats up the inside joints to generate the current.

RTGs are more commonly used to power spacecraft, but there are a few dozen people still in the U.S. with plutonium-powered thermopile batteries in their pacemakers.

In pacemakers, the plutonium was sealed inside a titanium capsule, which was strong enough to survive an accident (such as a bullet impact or auto collision) or cremation. But Wikipedia says the technique was abandoned because of worries that the capsule wouldn't be absolutely certain to survive a cremation. (Cremation temperatures go up to around 1000°C; titanium melts at 1668°C.) Loose plutonium in the environment would be Very Bad.

(I wondered if there wasn't also worry about plutonium being recovered for weapons use, but the risk seems much smaller: you need several kilograms of plutonium to make a bomb, and a pacemaker has only around 135 mg, if I did the conversion from curies correctly. Even so, if I were in charge of keeping plutonium out of the wrong hands, I would still worry about this. It does not seem totally out of the realm of possibility that someone could collect 25,000 pacemakers. Opening 25,000 titanium capsules does sound rather tedious.)

Earlier a completely different nuclear-powered pacemaker was tried, based on promethium-powered betavoltaics. This is not a heat-conversion process. Instead, a semiconductor does some quantum physics magic with the electrons produced by radioactive beta decay. This was first demonstrated by Henry Moseley in 1913. Moseley is better-known for discovering that atoms have an atomic number, thus explaining the periodic table. The periodic table had previously been formulated in terms of atomic mass, which put some of the elements in the wrong order. Scientists guessed they were in the wrong order, because the periodicity didn't work, but they weren't sure why. Moseley was able to compute the electric charge of the atomic nucleus from spectrographic observations. I have often wondered what else Moseley would have done if he had not been killed in the European war at the age of 27.

It took a while to gather the information about this. Several of Wikipedia's articles on the topic are not up to their usual standards. The one about the radioisotope thermoelectric generator is excellent, though.

Thermopile illustration is by FluxTeq (Own work) CC BY-SA 4.0, via Wikimedia Commons.

[ Addendum 20180115: Commenters on Hacker News have pointed out that my concern about the use of plutonium in fission weapons is easily satisfied: the fuel in the batteries is Pu-238, which is not fissile. The plutonium to use in bombs is Pu-241, and indeed, when building a plutonium bomb you need to remove as much Pu-238 as possible, to prevent its non-fissile nuclei from interfering with the chain reaction. Interestingly, you can tell this from looking at the numbers: atomic nuclei with an odd number of neutrons are much more fissile than those with an even number. Plutonium is atomic number 94, so Pu-238 has an even number of neutrons and is not usable. The other isotope commonly used in fission is U-235, with 143 neutrons. I had planned to publish a long article today detailing the difficulties of gathering enough plutonium from pacemakers to make a bomb, but now I think I might have to rewrite it as a comedy. ]

[ Addendum 20170116: I published it anyway, with some editing. ]

Sun, 07 Jan 2018

Well, yesterday I wrote an article about the drinking contest in the Gylfaginning and specifically about what was in the horn. I was very pleased with it. In the article, I said several times:

Obviously, the horn was full of mead.

A couple of my Gentle Readers have gently pointed out that I was wrong, wrong, wrong. I am deeply embarrassed.

The punch line of the story is that the end of the horn is attached to the ocean, and Thor cannot empty it, because he is trying to drink the ocean. The horn is therefore not filled with mead; it is filled with seawater.

How could I make such a dumb mistake? As I mentioned, the version I read first as a child stated that the horn was full of milk. And as a child I wondered: how could the horn be full of milk if it was attached to the sea? I decided that whatever enchantment connected the horn to the sea also changed the water to milk as it came into the horn. Later, when I realized that the milk was a falsehood, I retained my idea that there was an enchantment turning the seawater into something else.

But there is nothing in the text to support this. The jötunns don't tell Thor that the horn is full of mead. Adam Sjøgren pointed out that if they had, Thor would have known immediately that something was wrong. But as the story is, they bring the horn, they say that even wimps can empty it in three draughts, and they leave it at that. Wouldn't Thor notice that he is not drinking mead (or milk)? I think certainly, and perhaps he is initially surprised. But he is in a drinking contest and this is what they have brought him to drink, so he drinks it. The alternative is to put down the horn and complain, which would be completely out of character.

And the narrator doesn't say, and mustn't, that the horn was full of mead, because it wasn't; that would be in impermissible deceit of the audience. (“Hey, wait, you told us before that the horn was full of mead!”)

I wrote:

The story does not say what was in the horn. Because why would they bother to say what was in the horn? It was obviously mead.

No, it's not. It's because the narrator wants us to assume it is obviously mead, and then to spring the surprise on us as it was sprung on Thor: it was actually the ocean. The way it is told is a clever piece of misdirection. The two translators I quoted picked up on this, and I completely misunderstood it.

I have mixed feelings about Neil Gaiman, but Veit Heller pointed out that Gaiman understood this perfectly. In his Norse Mythology he tells the story this way:

He raised the brimming horn to his lips and began to drink. The mead of the giants was cold and salty…

In yesterday's article I presented a fantasy of Marion French, the author of my childhood “milk” version, hearing Snorri tell the story:

“Utgarða-Loki called his skutilsveinn, and requested him to bring the penalty-horn that his hirðmen were wont to drink from…”

“Excuse me! Excuse me, Mr. Sturluson! Just what were they wont to drink from it?”

“Eh, what's that?”

”What beverage was in the horn?”

“Why, mead, of course. What did you think it was, milk?”

But this couldn't have been how it went down. I now imagine it was more like this:

“Excuse me! Excuse me, Mr. Sturluson! Just what were they wont to drink from it?”

“Shhh! I'm getting to that! Stop interrupting!”

Thanks again to Adam Sjøgren and Veit Heller for pointing out my error, and especially for not wounding my pride any more than they had to.

When I was a kid I had a book of “Myths and Legends of the Ages”, by Marion N. French. One of the myths was the story of Thor's ill-fated visit to Utgard. The jötunns of Utgard challenge Thor and Loki to various contests and defeat them all through a combination of talent and guile. In one of these contests, Thor is given a drinking horn and told that even the wimpiest of the jötunns is able to empty it of its contents in three drinks. (The jötunns are lying. The pointy end of the horn has been invisibly connected to the ocean.)

The book specified that the horn was full of milk, and as a sweet and innocent kiddie I did not question this. Decades later it hit me suddenly: no way was the horn filled with milk. When the mighty jötunns of Utgard are sitting around in their hall, they do not hold contests to see who can drink the most milk. Obviously, the horn was full of mead.

The next sentence I wrote in the draft version of this article was:

   In the canonical source material (poetic edda maybe?) the horn is full
of *mead*. Of course it is.


In my drafts, I often write this sort of bald statement of fact, intending to go back later and check it, and perhaps produce a citation. As the quotation above betrays, I was absolutely certain that when I hunted down the original source it would contradict Ms. French and say mead. But I have now hunted down the canonical source material (in the Prose Edda, it turns out, not the Poetic one) and as far as I can tell it does not say mead!

Here is an extract of an 1880 translation by Rasmus Björn Anderson, provided by WikiSource:

He went into the hall, called his cup-bearer, and requested him to take the sconce-horn that his thanes were wont to drink from. The cup-bearer immediately brought forward the horn and handed it to Thor. Said Utgard-Loke: From this horn it is thought to be well drunk if it is emptied in one draught, some men empty it in two draughts, but there is no drinker so wretched that he cannot exhaust it in three.

For comparison, here is the 1916 translation of Arthur Gilchrist Brodeur, provided by sacred-texts.com:

He went into the hall and called his serving-boy, and bade him bring the sconce-horn which the henchmen were wont to drink off. Straightway the serving-lad came forward with the horn and put it into Thor's hand. Then said Útgarda-Loki: 'It is held that this horn is well drained if it is drunk off in one drink, but some drink it off in two; but no one is so poor a man at drinking that it fails to drain off in three.'

In both cases the following text details Thor's unsuccessful attempts to drain the horn, and Utgard-Loki's patronizing mockery of him after. But neither one mentions at any point what was in the horn.

I thought it would be fun to take a look at the original Old Norse to see if the translators had elided this detail, and if it would look interesting. It was fun and it did look interesting. Here it is, courtesy of Heimskringla.NO:

Útgarða-Loki segir, at þat má vel vera, ok gengr inn í höllina ok kallar skutilsvein sinn, biðr, at hann taki vítishorn þat, er hirðmenn eru vanir at drekka af. Því næst kemr fram skutilsveinn með horninu ok fær Þór í hönd. Þá mælti Útgarða-Loki: "Af horni þessu þykkir þá vel drukkit, ef í einum drykk gengr af, en sumir menn drekka af í tveim drykkjum, en engi er svá lítill drykkjumaðr, at eigi gangi af í þrimr."

This was written in Old Norse around 1220, and I was astounded at how much of it is recognizable, at least when you already know what it is going to say. However, the following examples are all ill-informed speculation, and at least one of my confident claims is likely to be wrong. I hope that some of my Gentle Readers are Icelanders and can correct my more ridiculous errors.

“Höllina” is the hall. “Kallar” is to call in. The horn appears three times, as ‘horninu’, ‘horni’, and in ‘vítishorn’, which is a compound that specifies what kind of horn it is. “Þór í hönd” is “in Thor's hand”. (The ‘Þ’ is pronounced like the /th/ of “Thor”.) “Drekka”, “drukkit”, “drykk”, “drykkjum”, and “drykkjumaðr” are about drinking or draughts; “vel drukkit” is “well-drunk”. You can see the one-two-three in there as “einum-tveim-þrimr”. (Remember that the “þ” is a /th/.) One can almost see English in:

sumir menn drekka af í tveim drykkjum

which says “some men drink it in two drinks”. And “lítill drykkjumaðr” is a little-drinking-person, which I translated above as “wimp”.

It might be tempting to guess that “með horninu” is a mead-horn, but I'm pretty sure it is not; mead is “mjað” or “mjöð”. I'm not sure, but I think “með” here is just “with”, akin to modern German “mit”, so that:

næst kemr fram skutilsveinn með horninu

is something like “next, the skutilsveinn came with the horn”. (The skutilsveinn is something we don't have in English; compare trying to translate “designated hitter” into Old Norse.)

For a laugh, I tried putting this into Google Translate, and I was impressed with the results. It makes a heroic effort, and produces something that does capture some of the sense of the passage. It identifies the language as Icelandic, which while not correct, isn't entirely incorrect either. (The author, Snorri Sturluson, was in fact Icelandic.) Google somehow mistakes the horn for a corner, and it completely fails to get the obsolete term “hirðmenn” (roughly, “henchmen”), mistaking it for herdsmen. The skutilsveinn is one of the hirðmenn.

Anyway there is no mead here, and none in the rest of the story, which details Thor's unsuccessful attempts to drink the ocean. Nor is there any milk, which would be “mjólk”.

So where does that leave us? The jötunns challenge Thor to a drinking contest, and bring him a horn, and even though it was obviously mead, the story does not say what was in the horn.

Because why would they bother to say what was in the horn? It was obviously mead. When the boys crack open a cold one, you do not have to specify what it was that was cold, and nobody should suppose that it was a cold bottle of milk.

I imagine Marion N. French sitting by the fire, listening while Snorri tells the story of Thor and the enchanted drinking horn of Utgard:

“Utgarða-Loki called his skutilsveinn, and requested him to bring the penalty-horn that his hirðmen were wont to drink from…”

“Excuse me! Excuse me, Mr. Sturluson! Just what were they wont to drink from it?”

“Eh, what's that?”

”What beverage was in the horn?”

“Why, mead, of course. What did you think it was, milk?”

(Merriment ensues, liberally seasoned with patronizing mockery.)

(In preparing this article, I found it helpful to consult Zoëga's Concise Dictionary of Old Icelandic of 1910.)

[ Addendum 2018-01-17: Holy cow, I was so wrong. It was so obviously not mead. I was so, so wrong. Amazingly, unbelievably wrong. ]

[ Addendum 2018-03-22: A followup in which I investigate what organs Skaði looked at when choosing her husband, and what two things Loki tied together to make her laugh. ]

Fri, 05 Jan 2018

Last month I wrote about the Turkish analog of “Joe Blow”. I got email from Gaal Yahas, who said

I bet you'll get plenty of replies on your last post about translating "John Doe" to different languages.

Sadly no. But M. Yahas did tell me in detail about the Hebrew version, and I did a little additional research.

The Hebrew version of “Joe Blow” / “John Doe” is unequivocally “Ploni Almoni”. This usage goes back at least to the Book of Ruth, approximately 2500 years ago. Ruth's husband has died without leaving an heir, and custom demands that a close relative of her father-in-law should marry her, to keep the property in the family. Boaz takes on this duty, but first meets with another man, who is a closer relative than he:

Then went Boaz up to the gate, and sat him down there: and, behold, the kinsman of whom Boaz spake came by; unto whom he said, Ho, such a one! turn aside, sit down here. And he turned aside, and sat down.

(Ruth 4:1, KJV)

This other relative declines to marry Ruth. He is not named, and is referred to in the Hebrew version as Ploni Almoni, translated here as “such a one”. This article in The Jewish Chronicle discusses the possible etymology of these words, glossing “ploni” as akin to “covered” or “hidden” and “almoni” as akin to “silenced” or “muted”.

Ploni Almoni also appears in the book of Samuel, probably even older than Ruth:

David answered Ahimelek the priest, “The king sent me on a mission and said to me, 'No one is to know anything about the mission I am sending you on.' As for my men, I have told them to meet me at a certain place.”

(1 Samuel 21:2, NIV)

The mission is secret, so David does not reveal the meeting place to Ahimelek. Instead, he refers to it as Ploni Almoni. There is a similar usage at 2 Kings 6:8.

Apparently the use of “Ploni” in Hebrew to mean “some guy” continues through the Talmud and up to the present day. M. Yahas also alerted me to two small but storied streets in Tel Aviv. According to this article from Haaretz:

A wealthy American businessman was buying up chunks of real estate in Tel Aviv. He purchased the two alleyways with the intention of naming them after himself and his wife, even going so far as to put up temporary shingles with the streets’ new names. But he had christened the streets without official permission from the city council.

The mayor was so incensed by the businessman’s chutzpah that he decided to temporarily name the alleyways Simta Almonit and Simta Plonit.

And so they remain, 95 years later.

(M. Yahas explains that “Simta” means “alley” and is feminine, so that Ploni and Almoni take the feminine ‘-it’ ending to agree with it.)

Wikipedia has not one but many articles on this topic and related ones:

My own tiny contribution in this area: my in-laws live in a rather distant and undeveloped neighborhood on the periphery of Seoul, and I once referred to it as 아무데도동 (/amudedo-dong/), approximately “nowhereville”. This is not standard in Korean, but I believe the meaning is clear.

Tue, 02 Jan 2018

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

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 don't think the quality creep was the cause of lower volume this month; rather, I was on vacation for a week.)

Here is a list of last month's shitposts. I have added a short blurb to those that may be of more general interest.

I plan to continue to post monthly summaries here.

I was on vacation last week and I didn't bring my computer, which has been a good choice in the past. But I did bring my phone, and I spent some quiet time writing various parts of around 20 blog posts on the phone. I composed these in my phone's Google Docs app, which seemed at the time like a reasonable choice.

But when I got back I found that it wasn't as easy as I had expected to get the documents back out. What I really wanted was Markdown. HTML would have been acceptable, since Blosxom accepts that also. I could download a single document in one of several formats, including HTML and ODF, but I had twenty and didn't want to do them one at a time. Google has a bulk download feature, to download a zip file of an entire folder, but upon unzipping I found that all twenty documents had been converted to Microsoft's docx format and I didn't know a good way to handle these. I could not find an option for a bulk download in any other format.

Several tools will compose in Markdown and then export to Google docs, but the only option I found for translating from Google docs to Markdown was Renato Mangini's Google Apps script. I would have had to add the script to each of the 20 files, then run it, and the output appears in email, so for this task, it was even less like what I wanted.

The right answer turned out to be: Accept Google's bulk download of docx files and then use Pandoc to convert the docx to Markdown:

for i in *.docx; do
echo -n "$i ? "; read j; mv -i "$i" $j.docx; pandoc --extract-media . -t markdown -o "$(suf "$j" mkdn)" "$j.docx";
done


The read is because I had given the files Unix-unfriendly names like Polyominoes as orthogonal polygons.docx and I wanted to give them shorter names like orthogonal-polyominoes.docx.

The suf command is a little utility that performs the very common task of removing or changing the suffix of a filename. The suf "$j" mkdn command means that if $j is something like foo.docx it should turn into foo.mkdn. Here's the tiny source code:

    #!/usr/bin/perl
#
# Usage: suf FILENAME [suffix]
#
# If filename ends with a suffix, the suffix is replaced with the given suffix
# otheriswe, the given suffix is appended
#
# For example:
#   suf foo.bar baz    => foo.baz
#   suf foo     baz    => foo.baz
#   suf foo.bar        => foo
#   suf foo            => foo

@ARGV == 2 or @ARGV == 1 or usage();
my ($file,$suf) = @ARGV;
$file =~ s/\.[^.]*$//;
if (defined $suf) { print "$file.$suf\n"; } else { print "$file\n";
}

sub usage {
print STDERR "Usage: suf filename [newsuffix]\n";
exit 1;
}


Often, I feel that I have written too much code, but not this time. Some people might be tempted to add bells and whistles to this: what if the suffix is not delimited by a dot character? What if I only want to change certain suffixes? What if my foot swells up? What if the moon falls out of the sky? Blah blah blah. No, for that we can break out sed.

Next time I go on vacation I will know better and I will not use Google Docs. I don't know yet what instead. StackEdit maybe.

[ Addendum 20180108: Eric Roode pointed out that the program above has a genuine bug: if given a filename like a.b/c.d it truncates the entire b/c.d instead of just the d. The current version fixes this. ]