Nonfiction of the month: 1984 June 8th marked the 70th anniversary of the publication of Nineteen Eighty-Four, one of the half-dozen greatest pieces of imaginative fiction the last century produced, and the most politically potent.

My own feelings about the book are more personal than the average. For one thing I am the same age as the protagonist Winston Smith, as precisely as we can deduce from the text. ("He believed that he had been born in 1944 or 1945.") For another, I read the book in my teens—with pleasure and understanding, not as a chore. Reading experiences of that kind leave a permanent deep impression.

And for another I believe very strongly in the reality of the external world. So did Winston Smith; so did Orwell himself; so did Kipling—one of Orwell's heroes, and one of mine too.

This is an eccentric, unpopular point of view. We stone-kickers are a small, despised and mostly ignored minority of party poopers who sit apart—check your etymology, please—jeering at the fashionable cant of the times when we should be making our proper prostrations to the Gods of the Market Place.

DIVERSITY IS OUR STRENGTH

says the preacher from the pulpit, and

THERE IS NO SUCH THING AS RACE.

Yeah, yeah, we snicker from the rear pews, and summon up the Party slogans from Nineteen Eighty-Four:

WAR IS PEACE

FREEDOM IS SLAVERY

IGNORANCE IS STRENGTH.

There are deeper, darker personal affinities, too. The degree to which Orwell's physical condition—he was dying of TB when he wrote Nineteen Eighty-Four—colored the book's narrative has been much discussed. My own view is that the attitude to death that the author brought to the novel was one that he had carried with him from his childhood, not much heightened, if at all, by thoughts of his own approaching demise.

Each one of us is, in life, separated by a screen or membrane from the Other Place. The membrane is thicker for some of us than for others. For most of us, most of the time, it is blessedly opaque. For Orwell, it was the mere skin of a soap bubble.

Now, approaching the tail-end of life, and with an incurable condition of my own, I am more than ever sure I know the identity of Big Brother.

(A dear Russian friend, when I discussed this with him several years ago, observed that in some pagan cultures, including that of the Slavs, "Big Sister" would be more appropriate. My friend could, if communication were only possible, now give me a more definitive ruling.)

Well, to commemorate this year's anniversary Picador has brought out a new book about the book: The Ministry of Truth: A Biography of George Orwell's 1984. After reading some reviews and Googling the author, I think I shall pass on it.

The author, Dorian Lynskey, is a young British democratic socialist, a regular at CultMarx outlets like the Guardian and Observer. Orwell himself was a democratic socialist, of course, so there is a broad ground of sympathy between Lynskey and Orwell. A lot of water has flowed under many, many bridges since 1949, though, and all of it has flowed right past Lynskey without his noticing.

This comes out in the June 4th interview Lynskey [Tweet him] gave to Pacific Standard. First you get a slab of sound Orwellian common sense:

I do find it dismaying that parts of the left continue to make the same old mistakes: a failure to understand the range of fears and desires that motivate ordinary people, a refusal to confront their own flaws, a weakness for authoritarians abroad, and so on. The British Labor Party under Jeremy Corbyn is guilty of this.

Then, in the next sentence, a lapse into CultMarx cluelessness:

I'm more hopeful about the United States, where people like Bernie Sanders, Elizabeth Warren, and Alexandra Ocasio-Cortez have so far avoided most of these traps and focused on bold domestic policy ideas.

Bold policy ideas like … what? Open borders? Sure, says Lynskey, Orwell would have been right on board with that!

He disliked xenophobia and welcomed immigrants, especially refugees.

Yo, Mr Lynskey: George Orwell is an old friend of mine from long, long acquaintance. I can tell you with high confidence what his shade, if it still exists somewhere (a thing he did not believe possible) would say about the polyglot boarding-house that his beloved England has become:

"Thank God I died in 1950!"

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Fiction of the month: When They See Us In the general zone of fictional productions, the most outrageous this month was the Netflix TV docudrama When They See Us. (Actually released May 31st but all over the news in early June.) It was an imaginative rewrite, by a Trump-hating black nationalist, of the 1989 Central Park Five case, portraying the young muggers and rapists prosecuted in that case as angelic innocent victims of a racist law-enforcement establishment.

We have chewed over When They See Us pretty well here at VDARE.com and I don't have anything new to say about the show itself. I was, though, shocked by the follow-up mistreatment of Linda Fairstein, the prosecutor who won convictions of the vicious young thugs. Ms Fairstein has, as Lady Ann reported here, "been forced off a half-dozen corporate boards, dropped by her publisher and dumped by her talent agency."

(Like many others, I've been wondering why Ms Fairstein doesn't sue the pants off Netflix for defamation. Apparently she contemplated doing so but was discouraged by legal advice.

John C.P. Goldberg, a Harvard law professor and expert on defamation law, said that Ms Fairstein's position as a public figure would make it difficult for her to win a defamation suit. [Linda Fairstein, Once Cheered, Faces Storm After 'When They See Us'" by Elizabeth A. Harris and Julia Jacobs, New York Times, June 6th 2019]

At any rate, I can't find any news at month's end of a defamation suit having been filed.)

"Dropped by her publisher" refers to the fact that Ms Fairstein is a prolific and successful writer of crime fiction. Once I had learned that, and still indignant at the way her gutless publisher and spineless agent had dumped her, I resolved to read one of Ms Fairstein's novels. Crime fiction isn't at all my thing; but as a gesture of support to a victim of the Thought Police, I took one of her books at random out of my local library and settled down with it and a glass of bourbon.

This didn't go well. My native fondness for futile gestures proved to be seriously out of balance with my tolerance for unfamiliar genres of middlebrow fiction. I am ashamed to say I bailed out halfway through the book.

Silent Mercy isn't at all a bad book. If you like crime fiction and haven't yet sampled Ms Fairstein, I hope you'll do better with it than I did. Middlebrow is middlebrow; I wasn't anticipating literary fireworks or deep explorations of the human psyche. There are middlebrow genres I like a lot: historical, sci-fi, social. Ms Fairstein draws convincingly on her own wealth of experience at investigation and prosecution.

The author really knows her New York City, too. I am of course familiar with St Patrick's Cathedral on Fifth Avenue, but I've learned from Silent Mercy that there is an older one—same name, also a cathedral—in Little Italy downtown, corner of Mott and Prince. A cathedral should be hard to miss; but I lived a half-dozen blocks away on Broome for several weeks when I first landed here, and pounded those streets in unemployed desperation, without ever noticing the place. (It's behind a high wall.)

So no, not a bad book in any seriously objectionable way. It's just that the fictionalized interactions of cops, prosecutors, lawyers, and judges set me to fidgeting unless seasoned with first-rate social observation, which Silent Mercy isn't.

The book also has minor structural and stylistic shortcomings that helped weaken my resolve. It's hard to keep track of the characters; there's too much dialogue; the on-the-job guy-gal buddy banter between prosecutor and cop is painfully artificial; and Ms Fairstein has a compulsion to show a flash of feminist ankle every few pages, interrupting the main narrative with a sentence or two of Grievance Studies flapdoodle:

But I knew that the women who had come before me in the law, as in many careers, had faced insurmountable obstacles simply getting through the door of the courtroom.

Get over it, lady.

Still, I have to record this as a moral failure on my part, not a literary failure on Ms Fairstein's. Even futile gestures should be seen through to completion. I am sorry, Ma'am.

And I urge you to take action against Netflix. They told flagrant lies about you in public, causing you professional and financial loss: how can that not be defamation? Netflix's market cap is around $200 billion; they could pay out a half billion or so without noticing it, and you could buy yourself a nice spread in Maui next to Oprah's. Go for it!

I turned back with relief to the middlebrow fiction I'd been engaged with before indignation about Ms Fairstein overwhelmed me: Ken Follett's Fall of Giants. Follett really has the narrative gift, backed up with masses of research, a good general understanding of human nature, and the right balance of dialogue and description. This is still middlebrow territory, with not much depth of characterization and no dazzling literary tricks (and Follett seems to be trying for Literary Review's Bad Sex in Fiction Award) but just the thing for a rainy afternoon and an armchair.

Follett's politics are visible but not obtrusive (well, so far). He's old-stock democratic socialist, not far from Orwell; but while Orwell's social observations are romantically English, Follett's are more pedantically Welsh.

This is my kind of middlebrow fiction, anyway. It's good to know that Fall of Giants, all 920 pages of it, is only the first of a trilogy. Weeks of effortless reading matter!

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More Waugh stories. In last month's diary I mentioned Naim Attallah's new anthology of writings by Auberon Waugh. That drew a couple of emails from readers.

Reader A wanted me to know that although Auberon Waugh died in 2001, he has apparently been tweeting from the Afterlife. That's good to know; and I am glad to be reminded of Waugh's handy usage (coinage?) "zealotocracy," which I spotted in the earlier anthology Kiss Me, Chudleigh.

The problem with democracy is that it is not democracy at all but a zealotocracy, or rule by enthusiasts.

There is half of our political problems right there.

Reader B remembered a story Waugh told in his memoir Will This Do?

The story concerns Auberon's father, the novelist and curmudgeon Evelyn Waugh. During WW2 food was strictly rationed in Britain and tropical fruit was well-nigh unobtainable. Then, after the war ended, there came a day—it seems to have been January 3rd, 1946—when the first postwar consignment of bananas arrived at the docks in Bristol, thirty miles from where the Waugh family lived.

Evelyn's wife brought three bananas home, to the great delight of the five Waugh children. Evelyn had all three bananas peeled and put on his plate. Then:

Before the anguished eyes of his children, he poured on cream, which was almost unprocurable, and sugar, which was heavily rationed, and ate all three.

My Reader B thought this "a distressing tale," and thought Will This Do? "very sad … a story of a fairly dreadful father."

It's hard to disagree. You can't fault Evelyn for lack of self-knowledge, though. There is another story about him that exists in a couple of variant forms, of which the one most-quoted goes as follows:

He was once very rude and his hostess remonstrated: "How can you behave so badly—and you a Catholic!" Waugh replied: "You have no idea how much nastier I would be if I was not a Catholic. Without supernatural aid I would hardly be a human being."

That title Will This Do?, by the way, refers to the words scrawled on every covering note—nowadays of course e-typed on every covering email—attached to every book review or opinion column ever submitted to any newspaper or magazine—or nowadays, website—anywhere at any time by any dishevelled, hungover freelance writer since pen first touched paper: "Sorry it's late, will this do?"

And I can't leave the Waughs without a mention of the memorable evening in the mid-1960s when the Tannoy—that is, public-address system—of the student union lounge at University College, London informed us scholars lounging therein that: "On Thursday this week Catholic Society will hold a discussion group on the life and works of English Catholic writer Evelyn Woff."

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The red and the blue.

The Quebec government has passed a controversial immigration reform bill which will scrap thousands of ongoing applications to live and work in the province … The legislation gives the province more authority over who receives permanent residency, which will involve a "values test" for would-be immigrants. [Quebec passes controversial immigration reform bill in rare overnight vote, by Cillian O'Brien; CTV News, June 16th 2019.]

Good for them. My tendency until recently has been to fall asleep, as every good American does, when anyone starts talking about Canada. After our trip up there in May, though (recorded in my VDARE.com Diary for that month) I'm starting to find the place interesting.

In that Diary I sketched some contrasts between Toronto and Montreal. A reader from the former city offered a very interesting perspective.

While growing up there, and until I left at the end of the 1960s, Montreal was always the biggest city in Canada. But already in the '60s there was a rising separatist sentiment in Quebec, including some terrorism (very mild terrorism: we're talking about Canada!) The separatist Parti Quebecois was elected in provincial elections, and began holding referendums about leaving the Canadian Confederation.



This put the wind up the business community, who began quietly moving their assets and headquarters out of Montreal and over to Toronto. Even the Bank of Montreal moved its headquarters to Toronto. With the headquarters went the jobs. The government tried to enforce the dominance of the French language, recruiting what Mordecai Richler memorably called "Tongue Troopers." Immigrants began leaving Montreal, because they did not want to be forced to raise their children in French. The Jews also began leaving, because they did not want to be stuck in an isolated backwater.



In the end, separatism did not win, but people were not taking chances. Now Toronto is Canada's biggest city. What happened to Montreal is exactly what the anti-Brexiters are predicting will happen to England.

There in a nutshell you have the great and ancient division between the provincial and the metropolitan. Montreal may be less busy, less wealthy, less sophisticated, less connected to the rest of the world than Toronto; but as we noticed last month, it is nicer.

If you want to live a quiet life, raise a family, practice some small hobbies, and live among people very much like yourself, being "stuck in an isolated backwater" like Montreal is a good choice.

If, on the other hand, you want the noise and bustle of a World City, with a clamor of different tongues and cuisines, hijabs and dashikis jostling you in the subway, a job at a big international bank, and some street gunplay, then Toronto is a better bet.

Some of the difference is of course temperamental. There are people who thrive in the metropolis and are bored stiff in the provinces, and contrariwise. Some other of it is age-related: the metropolis has more appeal when you're 25 than when you're 75.

As usual in human affairs, there is a point of balance. A healthy and harmonious society should have outlets for both temperaments, the metropolitan and the provincial. The crisis of multiculturalism, of which Brexit is one aspect, has arisen from a determination to make one's entire country metropolitan, to effect a mass transformation of all citizens everywhere into World City types celebrating cosmopolitan diversity. We have lost the point of balance.

Attempts at the mass transformation of human nature rarely go well.

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Hot and noisy. I have an anecdote to go with that. This happened four years ago.

An old college classmate of my wife's is now a rich man in China. He sent his teenage daughter to high school in the U.S.A., fees paid. Her school was in a small town in Ohio.

Finding out somehow that her Dad has an old classmate with a New York address, the young lady got in touch with us. She expressed dissatisfaction with her Ohio school and wondered if we could help her find a school in New York she could transfer to. Sure, we said; come on over and stay for a few days, we can take you round to check out some schools.

So that's what she did. We got on the internet and looked up some schools that take fee-paying students out here in bosky Suffolk County. She arrived. We chauffered her round to look at the schools.

We quickly perceived that something was wrong. Her interest had all evaporated. After some careful inquiries we figured out what had happened.

Our young guest had assumed that with a New York address we must be in, or at worst adjacent to, the great metropolis on the Hudson. Skyscrapers! Fifth Avenue! Bustle! Noise! Designer outlets! Trendy nightspots! The precise term of approbation in Chinese is 热闹, rènào, "hot and noisy."

(That second character has an interesting etymology. Of its two components, the outer one means "fight," the inner "market." So "to quarrel as in the market place," says Father Wieger; "the noisy wrangling and confusion of a market, so dear to the Chinese.")

In point of fact the Derbs live in the outer-outer suburbs. Our environment is hardly any more metropolitan than upcountry Ohio. It suits us just fine; but for a lively, ambitious, bilingual young girl with money to burn, it's Snoozeville. She cut her visit short, and we haven't heard from her since. Her Dad tells us she is now at a major big-city university (not in New York).

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Scourby's Bible (cont.). My one-man Bible Study group, listening to Alexander Scourby's reading of the entire King James Bible, has now gotten me all the way through the wisdom books and into the prophets. Ecclesiastes was just as grimly eloquent, or eloquently grim, as I remembered (I used Ecclesiastes 7:iv as the epigraph to Chapter 1 of We Are Doomed); although it cheers up a bit in the later chapters, a thing I hadn't noticed before.

Now I'm being scolded and warned by Isaiah, also very eloquently. "By far the greatest writer in the Old Testament," says Paul Johnson, a judgment I wouldn't gainsay; and Scourby's strong masculine voice is exactly right for Isaiah's rolling sonorities.

Never having known my Bible really well, I had never before heard the weirdly automotive passage in Isaiah 3. Warning of the calamities about to befall Judah because of the misbehavior of her citizens, especially the women, the prophet thunders that:

Therefore the Lord will smite with a scab the crown of the head of the daughters of Zion, and the LORD will discover their secret parts.

In that day the Lord will take away the bravery of their tinkling ornaments about their feet, and their cauls, and their round tires like the moon,

The chains, and the bracelets, and the mufflers,

The bonnets, and the ornaments of the legs, and the headbands, and the tablets, and the earrings …

Tires? (With chains?—in case of snow, I guess.) Mufflers? And to a British reader "bonnets" is automotive, too: "bonnet" is Britspeak for the hood of a car. (Although a couple of verses later we hear that the primped-up daughters of Zion are going to lose "the glasses, and the fine linen, and the hoods, and the vails" also.)

I went to my Scofield Reference Bible seeking concordance entries for "carburettor," "alternator," and "transmission," but no hits. So I guess the prophet was only vouchsafed that one fragmentary glimpse of the far future.

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The war against laughter. That the Great Awokening of the past few years has been lethal to our sense of humor is now a well-worn theme out here on the Dissident Right. We are four years on from the public declarations by Chris Rock and Jerry Seinfeld that they no longer do comedy gigs on college campuses because what were jokes a decade ago are now "hurtful," "insensitive," "transphobic," and so on.

The war against laughter is most intense over in Britain, where the terror of giving offense to protected classes—most especially blacks and Muslims—now hangs over the landscape like a North Sea fog, muffling sound and masking vision.

On June 19th a supermarket greeter in northern England was fired for posting a Billy Connolly joke on Facebook. The joke mocked religion in general and Islam in particular:

Suicide f***ing bombing—now there's a bright idea. Every time there's a bang the world is a w***er short. F***ing idiots.

This guy getting fired was not, please note, a former big-city prosecutor who's written a shelf-full of best-sellers; this guy is a minimum-wage shlub with some kind of disability condition. That's how things are now in Airstrip One, the United Kingdom of Anarcho-Tyranny.

Billy Connolly's comedy is at least crude and obscene. The much gentler humor of the 1970s BBC sitcom Dad's Army, about a group of WW2 volunteers training to defend the British homeland against an anticipated German invasion, has been blamed by "author and TV producer Daisy Goodwin" for inciting pro-Brexit sentiments. She wants a ban on reruns of the show, which is still hugely popular over there.

I had never heard of Ms Goodwin and can't follow the thread of her argument, but I assume the all-male, all-white cast of Dad's Army, and the memories the show stirs of ethnonational unity, solidarity, and defiance of foreign tyrants, is obnoxious to her sensibilities on many levels.

Perhaps there are even deeper grounds for the lady's antipathy. One of the strongest themes in Dad's Army is national self-mockery: the English (and a token Scot, so perhaps "British") making fun of themselves. Says the show: We English are amateurish and hard to organize, not very smart and addled with petty class snobberies; but we get things right in the end!

That easygoing self-referential approach to ethnic identity—self-mockery garnished with self-congratulation—is only possible for an ethny that is sure of its permanence in a homeland where it is long and comfortably settled as a numerical supermajority. What could be more infuriating to the multiculti world-savers of the BBC?

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The end of national character. Finding ourselves funny is bad enough; finding foreigners funny—oh, dear God!

As long ago as 2007 I was lamenting the fact that we're not supposed to laugh at foreigners any more, nor even to hold common stereotypes about other nations and their silly behaviors and folkways. The cultural commissars don't want us to have any fun.

Just five years a U.S. citizen at that point, I wasn't as secure in my Americanness as I have since become, so I left out any references to foreign stereotypes about Americans. That came to mind recently when I found myself watching the 1963 slapstick comedy It's a Mad Mad Mad Mad World.

At one point in the movie British comedian Terry-Thomas is driving a van through California with Milton Berle in the passenger seat. Berle says rude things about the Brits.

MB: "You want me to tell you something? As far as I'm concerned, the whole British race [sic] is practically finished. If it hadn't been for Lend-Lease—if we hadn't kept your whole country afloat by giving you billions that you never even said 'Thank you' for—the whole phony outfit would be sunk right under the Atlantic years ago."

Terry-Thomas takes exception. He stops the car and orders Berle out.

Berle apologizes and they get moving again. The conversation resumes at a slightly lower temperature:

TT: "I must say, if I had the grievous misfortune to be a citizen of this benighted country, I should be most hesitant at offering any criticism whatever of any other."

MB: "Wait a minute. Are you knocking this country? Are you saying something against America?"

TT: "Against it? I should be positively astounded to hear anything that can be said for it. Why, the whole bloody place is the most unspeakable matriarchy in the whole history of civilization. Look at yourself, and the way your wife and her strumpet of a mother push you through the hoop. As far as I can see American men have been totally emasculated. They're like slaves. They die like flies from coronary thrombosis while their women sit under hair-dryers eating chocolates and arranging for every second Tuesday to be some sort of Mother's Day …"

For those of us who were around and sentient in 1963, it's all perfectly familiar. Americans thought Britain was washed up, a decadent has-been Greece to America's Rome. Brits thought American men were slaves to their women, working themselves to death trying to satisfy the needs of grim-faced harpies and their vinegary mothers. (I can remember my Dad, watching The Dick Van Dyke Show around the same time, muttering from the depths of his armchair: "Why does he let her talk to him like that?")

Those were the common stereotypes: the Brit, classy but washed-up and ineffectual; the Yank, rich and mighty but whipped.

It all seems like a long time ago. Well, I guess it was a long time ago. Fifty-six years on, neither Britain nor America presents a distinctive character like that, to be admired or mocked.

Some part of the change is in our understanding. We are all more cosmopolitan now, wider-traveled and better-informed. Some other part of it is demographic change, the slow transformation of both America and Britain from distinctive nations into bickering multicultural slums.

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Death of Bawdy-ville. One more on this theme, the outlawing of humor. It's not just ethnic humor that's being killed off, it's bawdy humor.

When I complained of having dined at a splendid table without hearing one sentence of conversation worthy of being remembered, he said, "Sir, there seldom is any such conversation." Boswell: "Why then meet at table?" Johnson: "Why, to eat and drink together, and to promote kindness; and, Sir, this is better done when there is no solid conversation; for when there is, people differ in opinion, and get into bad humour, or some of the company who are not capable of such conversation, are left out, and feel themselves uneasy. It was for this reason, Sir Robert Walpole said, he always talked bawdy at his table, because in that all could join."

—Boswell: Life of Johnson

A reader of my Radio Derb transcripts scolded me for having, in my March 29th podcast, declared myself a fan of the Oughties sitcom Two and a Half Men. That show was, said my reader, a low, juvenile, and amoral entertainment, not worth the attention of a thoughtful, mature, wise, high-principled, literate and uxorious fellow like myself. (All right, he didn't say exactly that … Diarist's license.)

Well yes, it was. Sex jokes, fart jokes, dumb bimbos, drunken promiscuity, double-entendres, … Two and a Half Men was, to use a word not much favored nowadays, bawdy.

That puts it squarely in the grand Anglo-Saxon tradition. Shakespeare was bawdy; Chaucer was bawdy; I can't truthfully say I've ever gotten properly to grips with Beowulf, but I bet Alfred the Great enjoyed a good fart joke. (Although Henry VIII sometimes didn't.)

As grievance culture tightens its grip on Western society today, we are losing not just the word "bawdy" but the thing itself. We are living through the death of bawdy-ville.

We can hope that this is just a phase, like the closing of the theaters in England during the Civil War. Perhaps after twenty years will come a Restoration, and we shall be bawdy again. Societies endure these pendulum swings. While the joyless night of Puritanism lasts, though, I can quite believe that Two and a Half Men is beyond the pale.

I don't care. I think it's a superior sitcom. The principal characters are vivid and varied: louche Charlie, beta-provider Alan, the dragon Mom, the slacker kid, the white-trash housekeeper—they slot together like lego bricks, with endless possibilities for folly, deception, humiliation, and petty triumph. There are some memorable secondary characters, too: my favorite is Russell the pharmacist (Martin Mull).

I'm only speaking of the first eight seasons, of course. I stopped watching (and purchasing the downloads) when they dropped Charlie Sheen. To borrow a French culinary simile: Two and a Half Men without Charlie is like jugged hare and redcurrant jelly without the jugged hare.

Two and a Half Men is a bawdy TV sitcom, not high art. I'm not making any exaggerated claims for it. There are, I'll allow, a few moments of real drama, or something close to it. Charlie's on-off affair with Chelsea through seasons 6 and 7 is almost classically tragic, a man struggling against his own nature. Charlie doesn't struggle very hard, though, and even that is exceptional. The show's general level of psychological insight goes no deeper than mid-20th-century pop-Freudianism.

Still, it's fun. I like it. I bet Sir Robert Walpole and his dinner guests would have liked it, too.

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Math corner. The May issue of the MAA Monthly includes a nifty discussion by Dmitri Fomin of what he has christened "Moser's multiset recovery problem." I shall leave Moser aside for a moment and unpack "multiset" and "recovery."

"Multiset" just means a bunch of numbers. For simplicity I'll stick to non-negative whole numbers. An example of a multiset of two numbers would be {3, 8}.

Here's the idea of "recovery." Add the two numbers in that multiset together: 3+8 = 11. Now let's try to work back from that addition. I pose the following question to you.

I added two (non-negative whole) numbers and got answer 11. Can you tell me what my two numbers were? Can you recover them from the fact that their sum is 11?

The answer of course is no. For all you can tell I might have added 4 and 7, or 1 and 10. Unique recovery of the things sought, the quaesita, is not possible from the things given, the data.

Let's advance. Here's a multiset of three numbers: {3, 8, 12}. If I add them in pairs I get a new multiset: 3+8 = 11, 3+12 = 15, 8+12 = 20. So from the multiset {3, 8, 12} I got a new multiset, the multiset of pair-sums: {11, 15, 20}. Again I ask:

Can you recover the original multiset from the multiset {11, 15, 20} of its pair-sums?

Somewhat surprisingly the answer this time is yes, you can.

Writing the original multiset—the quaesita—as unknowns {a, b, c}, the multiset of pair-sums is {a+b, a+c, b+c}. If you add up the three numbers in that new multiset you get 2(a+b+c). If you add 11, 15, and 20 you get 46. So now we know that a+b+c = 23.

Since the data 11, 15, and 20 are a+b, a+c, b+c in some order, subtracting them in turn from 23 gives the original multiset in some order: {12, 8, 3}. Voilà!

Conclusion: In problems of this general type, recovery is sometimes possible, sometimes not.

Things are actually a bit knottier than that. Between the perfect unrecoverability of my {3, 8} example and the clean uniqueness of recovering {3, 8, 12} from the pair-sums {11, 15, 20} there is a zone of ambiguity.

For example: A multiset of k numbers gives k × (k−1) / 2 pair-sums, so a multiset of eight numbers will give you 28 pair-sums. OK, here's a multiset of 28 pair-sums: {5, 6, 7, 9, 10, 11, 11, 12, 13, 14, 15, 15, 16, 16, 16, 16, 17, 17, 18, 19, 20, 21, 21, 22, 23, 25, 26, 27}. Can I recover the original 8-multiset?

Yes, I can, but there turn out to be three of them:

{0, 5, 6, 7, 9, 10, 11, 16}



{1, 4, 5, 6, 10, 11, 12, 15}



{2, 3, 4, 7, 9, 12, 13, 14}

Any of those three will give you the same multiset of 28 pair-sums.

This is the kind of thing that gets mathematicians excited. When is recovery possible? When not? What are the conditions? What rules govern the ambiguities?

Suppose for example I have a multiset of four numbers, {a, b, c, d}. There are six ways to add them in pairs: {a+b, a+c, a+d, b+c, b+d, c+d}. Can I recover the first multiset from the second?

How about five numbers {a, b, c, d, e}? Here there are ten pair-sums: {a+b, a+c, a+d, a+e, b+c, b+d, b+e, c+d, c+e, d+e}. Recovery?

Those were the topics of a two-part problem posed back in 1957 by Leo Moser, a mathematician at the University of Alberta. The solutions to Moser's problems as stated are not difficult. You can read them for free at doi.org/10.2307/2308470; but there's a registration fandango to go through, so to spare you the trouble—and hoping the custodians of JSTOR won't mind—I've posted a screenshot here.

Moser's rather simple 1957 problem has since generated a fair-sized literature. The topic has been generalized off in several different directions. We might, for example, drop my restriction to just non-negative integers and allow any kind of numbers—negative, fractional, real, complex, …

Or we might expand our inquiries beyond just pair-sums to triplet-sums, quadruplet-sums, …, s-sums.

Generally: Given a multiset of n numbers and the multiset of their s-sums, for what values of n and s is recovery possible? When is it unique? When it's not unique, how many recovered multisets can there be?

Dmitri Fomin gives a good comprehensive survey.

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