The Universe of Discourse


Fri, 22 May 2020

Catalytic converter theft

There have been several reports of the theft of catalytic converters in our neighborhood, the thieves going under the car and cutting the whole thing out. The catalytic converter contains a few grams of some precious metal, typically platinum, and this can be recycled by a sufficiently crooked junk dealer.

Why weren't these being stolen before? I have a theory. The catalytic converter contains only a few grams of platinum, worth around $30. Crawling under a car to cut one out is a lot of trouble and risk to go to for $30. I think the the stay-at-home order has put a lot of burglars and housebreakers out of work. People aren't leaving their houses and in particular they aren't going on vacations. So thieves have to steal what they can get.

[ Addendum 20200522: An initial glance at the official crime statistics suggests that my theory is wrong. I'll try to make a report over the weekend. ]


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Thu, 07 May 2020

Cleaning up 404 errors

Yesterday I went through the last few months of web server logs, used them to correct some bad links in my blog articles.

Today I checked the logs again and all the "page not found" errors are caused by people attacking my WordPress and PHP installations. So, um, yay, I guess?


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Tue, 05 May 2020

Article explodes at the last moment

I wrote a really great blog post over the last couple of days. Last year I posted about the difference between !!\frac10!! and !!\frac00!! and this was going to be a followup. I had a great example from linear regression, where the answer comes out as !!\frac10!! in situations where the slope of computed line is infinite (and you can fix it, by going into projective space and doing everything in homogeneous coordinates), and as !!\frac00!! in situations where the line is completely indeterminite, and you can't fix it, but instead you can just pick any slope you want and proceed from there.

Except no, it never does come out !!\frac10!!. It always comes out !!\frac00!!, even in the situations I said were !!\frac10!!.

I think maybe I can fix it though, I hope, maybe. If so, then I will be able to write a third article.

Maybe.

It could have been worse. I almost published the thing, and only noticed my huge mistake because I was going to tack on an extra section at the end that it didn't really need. When I ran into difficulties with the extra section, I was tempted to go ahead and publish without it.


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Fri, 01 May 2020

More about Sir Thomas Urquhart

(Previously)

I don't have much to add at this point, but when I looked into Sir Thomas Urquhart a bit more, I found this amazing article by Denton Fox in London Review of Books. It's a review of a new edition of Urquhart's 1652 book The Jewel (Ekskybalauron), published in 1984. The whole article is worth reading. It begins:

Sir Thomas Urquhart … must have been a most peculiar man.

and then oh boy, does it deliver. So much of this article is quotable that I'm not sure what to quote. But let's start with:

The little we know about Urquhart’s early life comes mostly from his own pen, and is therefore not likely to be true.

Some excerpts will follow. You may enjoy reading the whole thing.

Trissotetras

I spent much way more time on this than I expected. Fox says:

In 1645 he brought out the Trissotetras … . Urquhart’s biographer, Willcock, says that ‘no one is known to have read it or to have been able to read it,’ …

Thanks to the Wonders of the Internet, a copy is available, and I have been able to glance at it. Urquhart has invented a microlanguage along the lines of Wilkins’ philosophical language, in which the words are constructed systematically. But the language of Trissotetras has a very limited focus: it is intended only for expressing statements of trigonometry. Urquhart says:

The novelty of these words I know will seeme strange to some, and to the eares of illiterate hearers sound like termes of Conjuration: yet seeing that since the very infancie of learning, such inventions have beene made use of, and new words coyned, …

The sentence continues for another 118 words but I think the point is made: the idea is not obviously terrible.

Here is an example of Urquhart's trigonometric language in action:

The second axiom is Eproso, that is, the sides are proportionall to one another as the sines of their opposite angles…

A person skilled in the art might be able to infer the meaning of this axiom from its name:

  • E — a side
  • Pro – proportional
  • S – the sine
  • O – the opposite angle

That is, a side (of a triangle) is proportional to the sine of the opposite angle. This principle is currently known as the law of sines.

Urquhart's idea of constructing mnemonic nonsense words for basic laws was not a new one. There was a long-established tradition of referring to forms of syllogistic reasoning with constructed mnemonics. For example a syllogism in “Darii” is a deduction of this form:

  • All mammals have hair
  • Some animals are mammals
  • Therefore some animals have hair.

The ‘A’ in “Darii” is a mnemonic for the “all” clause and the ‘I’s for the “some” clauses. By memorizing a list of 24 names, one could remember which of the 256 possible deductions were valid.

Urquhart is following this well-trodden path and borrows some of its terminology. But the way he develops it is rather daunting:

The Directory of this second Axiome is Pubkegdaxesh, which declareth that there are seven Enodandas grounded on it, to wit, foure Rectangular, Upalem, Ubeman, Ekarul, Egalem, and three Obliquangular, Danarele, Xemenoro, and Shenerolem.

I think that ‘Pubkegdaxesh’ is compounded from the initial syllables of the seven enodandas, with p from upalem, ub from ubamen, k from ekarul, eg from egalem, and so on. I haven't been able to decipher any of these, although I didn't try very hard. There are many difficulties. Sometimes the language is obscure because it's obsolete and sometimes because Urquhart makes up his own words. (What does “enodandas” mean?)

Let's just take “Upalem”. Here are Urquhart's glosses:

  • U – the Subtendent side
  • P – Opposite, whether Angle or side
  • A — an angle
  • L — the secant
  • E — a side
  • M — A tangent complement

I believe “a tangent complement” is exactly what we would now call a cotangent; that is, the tangent of the complementary angle. But how these six items relate to one another, I do not know.

Here's another difficulty: I'm not sure that ‘al’ is one component or two. It might be one:

  • U – the Subtendent side
  • P – Opposite, whether Angle or side
  • Al — half
  • E — a side
  • M — A tangent complement

Either way I'm not sure what is meant. Wait, there is a helpful diagram, and an explanation of it:

A right triangle,
labeled ‘Upalem’.  Let us call the legs X, Y, and the hypotenuse H,
although these names do not appear in the diagram. <br />
H is crossed by a single mark, Y two marks.  Inside the figure, the
hypotenuse is labeled ‘p.’ and with something that might be capital
‘H’ or lowercase ‘u’.  The angle between H and X is labeled ‘sapy 3.’.  The right angle
between X and Y is labeled “Rad”.  Near side Y are the notations “4”
and “yr”.

The first figure, Vale, hath but one mood, and therefore of as great extent as it selfe, which is Upalem; whose nature is to let us know, when a plane right angled triangle is given us to resolve, who subtendent and one of the obliques is proposed, and one of the ambients required, that we must have recourse unto its resolver, which being Rad—U—Sapy ☞ Yr sheweth, that if we joyne the artificiall sine of the angle opposite to the side demanded with the Logarithm of the subtendent, the summe searched in the canon of absolute numbers will afford us the Logarithm of the side required.


This is unclear but tantalizing. Urquhart is solving a problem of elementary plane trigonometry. Some of the sides and angles are known, and we are to find one of the unknown ones. I think if if I read the book from the beginning I think I might be able to make out better what Urquhart was getting at. Tempting as it is I am going to abandon it here.

Trissotetras is dedicated to Urquhart's mother. In the introduction, he laments that

Trigonometry … hath beene hitherto exposed to the world in a method whose intricacy deterreth many from adventuring on it…

He must have been an admirer of Alexander Rosse, because the front matter ends with a little poem attributed to Rosse.

Pantochronachanon

Fox again:

Urquhart, with many others, was taken to London as a prisoner, where, apparently, he determined to recover his freedom and his estates by using his pen. His first effort was a genealogy in which he names and describes his ancestors, going back to Adam. … A modern reader might think this Urquhart’s clever trick to prove that he was not guilty by reason of insanity …

This is Pantochronachanon, which Wikipedia says “has been the subject of ridicule since the time of its first publication, though it was likely an elaborate joke”, with no citation given.

Fox mentions that Urquhart claims Alcibiades as one of his ancestors. He also claims the Queen of Sheba.

According to Pantochronachanon the world was created in 3948 BC (Ussher puts it in 4004), and Sir Thomas belonged to the 153rd generation of mankind.

The Jewel

Denton Fox:

Urquhart found it necessary to try again with the Jewel, or, to to give it its full title, which in some sense describes it accurately…

EKSKUBALAURON [Εκσκυβαλαυρον]: OR, The Discovery of A most exquisite Jewel, more precious then Diamonds inchased in Gold, the like whereof was never seen in any age; found in the kennel [gutter] of Worcester-streets, the day after the Fight, and six before the Autumnal Aequinox, anno 1651. Serving in this place, To frontal a Vindication of the honour of SCOTLAND, from that Infamy, whereinto the Rigid Presbyterian party of that Nation, out of their Covetousness and ambition, most dissembledly hath involved it.

Wowzers.

Fox claims that the title Εκσκυβαλαυρον means “from dung, gold” but I do not understand why he says this. λαύρα might be a sewer or privy, and I think the word σκυβα means garden herbs. (Addendum: the explanation.)

[The book] relates how… Urquhart’s lodgings were plundered, and over 3200 sheets of his writings, in three portmanteaux, were taken.… One should remember that there is not likely to be the slightest bit of truth in this story: it speaks well for the morality of modern scholars that so many of them should have speculated why Urquhart took all his manuscripts to war with him.

Fox says that in spite of the general interest in universal languages, “parts of his prospectus must have seemed absurd even then”, quoting this item:

Three and twentiethly, every word in this language signifieth as well backward as forward; and how ever you invert the letters, still shall you fall upon significant words, whereby a wonderful facility is obtained in making of anagrams.

Urquhart boasts that where other, presumably inferior languages have only five or six cases, his language has ten “besides the nominative”. I think Finnish has fourteen but I am not sure even the Finns would be so sure that more was better. Verbs in Urquhart's language have one of ten tenses, seven moods, and four voices. In addition to singular and plural, his language has dual (like Ancient Greek) and also ‘redual’ numbers. Nouns may have one of eleven genders. It's like a language designed by the Oglaf Dwarves.

Two panels from the “Oglaf”
comic feature two dwarves enthusiastically describing the wondrous
sword they have forged: “Okay, so first up, it's the size of a plow,
which is awesome!  This is the bit that flies up and cuts of their
faces!!  And here is where you strap a live viper!  To bite while you
smite!”

A later item states:

This language affordeth so concise words for numbering, that the number for setting down, whereof would require in vulgar arithmetick more figures in a row then there might be grains of sand containable from the center of the earth to the highest heavens, is in it expressed by two letters.

and another item claims that a word of one syllable is capable of expressing an exact date and time down to the “half quarter of the hour”. Sir Thomas, I believe that Entropia, the goddess of Information Theory, would like a word with you about that.

Wrapping up

One final quote from Fox:

In 1658, when he must have been in his late forties, he sent a long and ornately abusive letter to his cousin, challenging him to a duel at a place Urquhart would later name,

quhich shall not be aboue ane hunderethe – fourtie leagues distant from Scotland.

If the cousin would neither make amends or accept the challenge, Urquhart proposed to disperse copies of his letter

over all whole the kingdome off Scotland with ane incitment to Scullions, hogge rubbers [sheep-stealers], kenell rakers [gutter-scavengers] – all others off the meanist sorte of rascallitie, to spit in yor face, kicke yow in the breach to tred on yor mushtashes ...

Fox says “Nothing much came of this, either.”.

I really wish I had made a note of what I had planned to say about Urquhart in 2008.

[ Addendum 20200502: Brent Yorgey has explained Εκσκυβαλαυρον for me. Σκύβαλα (‘skubala’) is dung, garbage, or refuse; it appears in the original Greek text of Philippians 3:8:

What is more, I consider everything a loss because of the surpassing worth of knowing Christ Jesus my Lord, for whose sake I have lost all things. I consider them garbage, that I may gain Christ…

A tiny screenshot of an entry
from Liddell and Scott's Greek-English Lexicon of 1897, glossing ‘αύρον’
as ‘gold’

And while the usual Greek word for gold is χρῡσός (‘chrysos’), the word αύρον (‘auron’, probably akin to Latin aurum) is also gold. The screenshot at right is from the 8th edition of Liddell and Scott. Thank you, M. Yorgey! ]



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Thu, 30 Apr 2020

Geeky boasting about dictionaries

Yesterday Katara and I were talking about words for ‘song’. Where did ‘song’ come from? Obviously from German, because sing, song, sang, sung is maybe the perfect example of ablaut in Germanic languages. (In fact, I looked it up in Wikipedia just now and that's the example they actually used in the lede.)

But the German word I'm familiar with is Lied. So what happened there? Do they still have something like Song? I checked the Oxford English Dictionary but it was typically unhelpful. “It just says it's from Old German Sang, meaning ‘song’. To find out what happened, we'd need to look in the Oxford German Dictionary.”

Katara considered. “Is that really a thing?”

“I think so, except it's written in German, and obviously not published by Oxford.”

“What's it called?”

I paused and frowned, then said “Deutsches Wörterbuch.”

“Did you just happen to know that?”

“Well, I might be totally wrong, but yeah.” But I looked. Yeah, it's called Deutsches Wörterbuch:

The Deutsches Wörterbuch … is the largest and most comprehensive dictionary of the German language in existence. … The dictionary's historical linguistics approach … makes it to German what the Oxford English Dictionary is to English.

So, yes, I just happened to know that. Yay me!

Deutsches Wörterbuch was begun by Wilhelm and Jakob Grimm (yes, those Brothers Grimm) although the project was much too big to be finished in their lifetimes. Wilhelm did the letter ‘D’. Jakob lived longer, and was able to finish ‘A’, ‘B’, ‘C’, and ‘E’. Wikipedia mentions the detail that he died “while working on the entry for ‘Frucht’ (fruit)”.

Wikipedia says “the work … proceeded very slowly”:

Hermann Wunderlich, Hildebrand's successor, only finished Gestüme to Gezwang after 20 years of work …

(This isn't as ridiculous as it seems; German has a lot of words that begin with ‘ge-’.)

The project came to an end in 2016, after 178 years of effort. The revision of the Grimms’ original work on A–F, planned since the 1950s, is complete, and there are no current plans to revise the other letters.


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Tue, 28 Apr 2020

Urquhart, Rosse, and Browne
[ Warning: I abandoned this article in 2008 and forgot that it existed. I ran across it today and decided that what I did write was worth publishing, although it breaks off suddenly. ]


A couple of years ago, not long before I started this blog, I read some of the works of Sir Thomas Browne. I forget exactly why: there was some footnote I read somewhere that said that something in one of Jorge Luis Borges' stories had been inspired by something he read in Browne's book The Urn Burial, which was a favorite of Borges'. I wish I could remember the details! I don't think I even remembered them at the time. But Thomas Browne turned out to be wonderful. He is witty, and learned, and wise, and humane, and to read his books is to feel that you are in the company of this witty, learned, wise, humane man, one of the best men that the English Renaissance has to offer, and that you are profiting thereby.

The book of Browne's that made the biggest impression on me was Pseudodoxia Epidemica (1646), which is a compendium of erroneous beliefs that people held at the time, with discussion. For example, is it true that chameleons eat nothing but air? ("Thus much is in plain terms affirmed by Solinus, Pliny, and others...") Browne thinks not. He cites various evidence against this hypothesis: contemporary reports of the consumption of various insects by chameleons; the presence of teeth, tongues, stomachs and guts in dissected chameleons; the presence of semi-digested insects in the stomachs of dissected chameleons. There's more; he attacks the whole idea that an animal could be nourished by air. Maybe all this seems obvious, but in 1672 it was still a matter for discussion. And Browne's discussion is so insightful, so pithy, so clear, that it is a delight to read.

Browne's list of topics is fascinating in itself. Some of the many issues he deals with are:

  • That Crystall is nothing else but Ice strongly congealed;
  • That a Diamond is made soft, or broke by the blood of a Goate;
  • That Misseltoe is bred upon trees, from seeds which birds let fall thereon;
  • That an Elephant hath no joints;
  • That Snayles have two eyes, and at the end of their hornes;
  • That men weigh heavier dead then alive;
  • Of the pictures of Adam and Eve With Navels [a classic question, that; Saint Augustine took it up in City of God];
  • Of the pictures of our Saviour with long haire;
  • Of the falling of salt;
  • That Children would naturally speak Hebrew [another question of perennial interest. I have heard that Frederick the Great actually made the experiment and raised children in isolation to see if they would learn Hebrew];
  • Of the blacknesse of Negroes;
  • That a man hath one Rib lesse then a woman;
  • Of Crassus that never laughed but once;
  • Of the wandring Jew;
  • Of Milo, who by daylie lifting a Calfe, attained an ability to carry it being a Bull.
There are many reasons why I enjoy reading books from this period, and that list makes me realize one of them. It was a time when science was new, and there were huge tracts of unexplored territory. Every question was open for investigation, including whether storks will live only in republics. A quick perusal of the table of contents from Richard Waller's Posthumous Works of Robert Hooke will give a similar impression, although with a somewhat different flavor to it. And nobody knew what was going to work and what wasn't. Can you make gold by subliming away all the impurities from iron? (No.) Hey, can you learn anything about vision by sticking a metal spike in your eye socket? (Yes.) I have written before about how the Baroque philosophers often chased ideas that seem crackpot to us now—but we can see these ideas as crackpot only because they were tried by the Baroque guys three hundred years ago, and didn't work.

Well, I digress. To return to that list of topics I quoted, you might see "of the blacknesse of Negroes", and feel your heart sink a little. What racist jackass thing is the 1646 Englishman going to say about the blackness of negroes?

Actually, though, Browne comes out of it extremely well, not only much better than one would fear, but quite well even by modern standards. It is one of the more extensive discourses in Pseudodoxia Epidemica, occupying several chapters. He starts by rebutting two popular explanations: that they are burnt black by the heat of the sun, and that they are marked black because of the curse of Ham as described in Genesis 9:20–26.

Regarding the latter, Browne begins by addressing the Biblical issue directly, and on its own terms, and finds against it. But then he takes up the larger question of whether black skin can be considered to be a curse at all. Browne thinks not. He spends some time rejecting this notion: "to inferr this as a curse, or to reason it as a deformity, is no way reasonable". He points out that the people who have it don't seem to mind, and that "Beauty is determined by opinion, and seems to have no essence that holds one notion with all; that seeming beauteous unto one, which hath no favour with another; and that unto every one, according as custome hath made it natural, or sympathy and conformity of minds shall make it seem agreeable."

Finally, he ends by complaining that "It is a very injurious method unto Philosophy, and a perpetual promotion of Ignorance, in points of obscurity, ... to fall upon a present refuge unto Miracles; or recurr unto immediate contrivance, from the insearchable hands of God." I wish more of my contemporaries agreed.

Another reason I love this book is that Browne is nearly always right. If you were having doubts that one could arrive at correct notions by thoughtful examination of theory and evidence, Pseudodoxia Epidemica might help dispel them, because Browne's record of coming to the correct conclusions is marvelous.

Some time afterward, I learned that there was a rebuttal to Pseudodoxia Epidemica, written by a Dr. Alexander Rosse. (Arcana Microcosmi, ... with A Refutation of Doctor Brown's VULGAR ERRORS... (1652).) And holy cow, Rosse is an incredible knucklehead. Watching him try to argue with Browne reminded me of watching an argument on Usenet (a sort of pre-Internet distributed BBS) where one person is right about everything, and is being flamed point by point by some jackass who is wrong about everything, and everyone but the jackass knows it. I have seen this many, many times on Usenet, but never as far back as 1652.


This is the point at which I stopped writing the article in 2008. I had mentioned the blockheaded Mr. Rosse in an earlier article. But I have no idea what else I had planned to say about him here.

Additional notes (April 2020)

  1. I mentioned “the table of contents from Richard Waller's Posthumous Works of Robert Hooke”. I had mentioned Waller previously, in connection with Hooke's measurement of the rate at which a fly beats its wings. The Waller book is available on the Internet Archive, but it does not have a table of contents! I realize I had actually been thinking of Hooke’s Philosophical Experiments and Observations, edited by William Derham, which in some editions, does have a table of contents. The Derham book made a couple of other appearances in this blog in the early days.

  2. I no longer have any idea who Urquhart was, or what I had planned to say about him. Searching for him in conjunction with Browne I find he was Sir Thomas Urquhart, a contemporary of Browne's. There is a Wikipedia article about Urquhart. Like his better-known contemporary John Wilkins, he tried to design a universal language in which the meaning of a word could be inferred from its spelling.

  3. I learned that Rosse was the target of sarcastic mockery in Samuel Butler's Hudibras:
    There was an ancient sage philosopher
    Who had read Alexander Ross over.
    It seems that Rosse was a noted dumbass even in his own lifetime.

  4. John Willcock's biography of Urquhart says “Ross himself is now only known to most of us from the mention made of him in Hudibras”, and that was in 1899.

  5. Thomas Browne is the source of the often-quoted suggestion that:

    What song the Sirens sang, or what name Achilles assumed when he hid himself among women, though puzzling questions, are not beyond all conjecture.
    (It appears in his book Hydrotaphia, or Urn Buriall.) This hopeful talisman has inspired many people over the centuries to continue their pursuit of such puzzling questions, sometimes when faced with what seems like a featureless wall of lost history.

  6. A few years later I revisited Milo, who by daylie lifting a Calfe, attained an ability to carry it being a Bull.

    [ Addendum 20200501: Uquhart was a very peculiar man. ]


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    Sun, 26 Apr 2020

    Jools Holland

    I ran into this album by Jools Holland:

    Cover
of jools Holland's 2017 Album “Piano” features the shape of a grand
piano outlined in blue neon, with a silhouette of Holland's face cut
out the of  the upper register, a blue keyboard, and the word PIANO in
red neon above the keyboard.

    What do you see when you look at this? If you're me, you spend a few minutes wondering why there is a map of Delaware.

    I colored
this map of Delaware in the same style as the album cover, with the
state border in neon blue, the keyboard and PIANO label placed down at
the southern edge.  The twelve-mile circle at the north end becomes the
rounded back edge of the piano.  The Maryland border is the straight
left edge. The silhouette cutout at the upper
right is Delaware Bay.  Delaware is more elongated than the piano, but
the shapres are quite similar.

    (Previously)


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    Fri, 24 Apr 2020

    Tiers of answers to half-baked questions

    [ This article is itself somewhat half-baked. ]

    There's this thing that happens on Stack Exchange sometimes. A somewhat-clueless person will show up and ask a half-baked question about something they are thinking about. Their question is groping toward something sensible but won't be all the way there, and then several people will reply saying no, that is not sensible, your idea is silly, without ever admitting that there is anything to the idea at all.

    I have three examples of this handy, and I'm sure I could find many more.

    1. One recent one concerns chirality (handedness) in topology. OP showed up to ask why a donut seems to be achiral while a coffee cup is chiral (because the handle is on one side and not the other). Some people told them that the coffee cup is actually achiral and some others people told them that topology doesn't distinguish between left- and right-handed objects, because reflection is a continuous transformation. (“From a topological point of view, no object is distinguishable from its mirror image”.) I've seen many similar discussions play out the same way in the past.

      But nobody (other than me) told them that there is a whole branch of topology, knot theory, where the difference between left- and right-handed objects is a major concern. Everyone else was just acting like this was a nonissue.

    2. This category theory example is somewhat more obscure.

      In category theory one can always turn any construction backward to make a “dual” construction, and the “dual” construction is different but usually no less interesting than the original. For example, there is a category-theoretic construction of “product objects”, which generalizes cartesian products of sets, topological product spaces, the direct product of groups, and so on. The dual construction is “coproduct objects” which corresponds to the disjoint union of sets and topological spaces, and to the free product of groups.

      There is a standard notion of an “exponential object” and OP wanted to know about the dual notion of a “co-exponential object”. They gave a proposed definition of such an object, but got their proposal a little bit wrong, so that what they had defined was not the actual co-exponential object but instead was trivial. Two other users pointed out in detail why their proposed construction was uninteresting. Neither one pointed out that there is a co-exponential object, and that it is interesting, if you perform the dualization correctly.

      (The exponential object concerns a certain property of a mapping !!f :A×B\to C!!. OP asked insead about !!f : C\to A× B!!. Such a mapping can always be factored into a product !!(f_1: C\to A)×(f_2: C\to B)!! and then the two factors can be treated independently. The correct dual construction concerns a property of a mapping !!f : C\to A\sqcup B!!, where !!\sqcup!! is the coproduct. This admits no corresponding simplification.)

    3. A frequently-asked question is (some half-baked variation on) whether there is a smallest positive real number. Often this is motivated by the surprising fact that !!0.9999\ldots = 1!!, and in an effort to capture their intuitive notion of the difference, sometimes OP will suggest that there should be a number !!0.000\ldots 1!!, with “an infinite number of zeroes before the 1”.

      There is no such real number, but the question is a reasonable one to ask and to investigate. Often people will dismiss the question claiming that it does not make any sense at all, using some formula like “you can't have a 1 after an infinite sequence of zeroes, because an infinite sequence of zeroes goes on forever.”. Mathematically, this response is complete bullshit because mathematicians are perfectly confortable with the idea of an infinite sequence that has one item (or more) appended after the others. (Such an object is said to “have order type !!\omega + 1!!”, and is completely legitimate.) The problem isn't with the proposed object itself, but with the results of the attempt to incorporate it into the arithmetic of real numbers: what would you get, for example, if you tried to multiply it by !!10!!?

      Or sometimes one sees answers that go no further than “no, because the definition of a real number is…”. But a better engagement with the question would recognize that OP is probably interested in alternative definitions of real numbers.

    In a recent blog article I proposed a classification of answers to certain half-baked software questions (“Is it possible to do X?”):

    1. It surely could, but nobody has done it yet
    2. It perhaps could, but nobody is quite sure how
    3. It maybe could, but what you want is not as clear as you think
    4. It can't, because that is impossible
    5. I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question

    and I said:

    Often, engineers will go straight to #5, when actually the answer is in a higher tier. Or they go to #4 without asking if maybe, once the desiderata are clarified a bit, it will move from “impossible” to merely “difficult”. These are bad habits.

    These mathematically half-baked questions also deserve better answers. A similar classification of answers to “can we do this” might look like this:

    1. Yes, that is exactly what we do, only more formally. You can find out more about the details in this source…
    2. Yes, we do something very much like that, but there are some significant differences to address points you have not considered…
    3. Yes, we might like to do something along those lines, but to make it work we need to make some major changes…
    4. That seems at first like a reasonable thing to try, but if you look more deeply you find that it can't be made to work, because…
    5. I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question

    The category theory answer was from tier 4, but should have been from tier 2. People asking about !!0.0000…1!! often receive answers from tier 5, but ought to get answers from tier 4, or even tier 3, if you wanted to get into nonstandard analysis à la Robinson.

    There is a similar hierarchy for questions of the type “can we model this concept mathematically”, ranging from “yes, all the time” through “nobody has figured that out yet” and “it seems unlikely, because”, to “what would that even mean?”. The topological chirality question was of this type and the answers given were from the “no we can't and we don't” tiers, when they could have been from a much higher tier: “yes, it's more complicated than that but there is an entire subfield devoted to dealing with it.”

    This is a sort of refinement of the opposition of “yes, and…” versus “no, but…”, with the tiers something like:

    1. Yes, and…
    2. Yes, but…
    3. Perhaps, if…
    4. No, but…
    5. No, because…
    6. I am embarrassed for you

    When formulating the answer to a question, aiming for the upper tiers usually produces more helpful, more useful, and more interesting results.

    [ Addendum 20200525: Here's a typical dismissal of the !!0.\bar01!! suggestion: “This is confusing because !!0.\bar01!! seems to indicate a decimal with ‘infinite zeros and then a one at the end.’ Which, of course, is absurd.” ]


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    Wed, 22 Apr 2020

    Mystery spam language

    This morning I got spam with this subject:

    Subject: yaxşı xəbər

    Now what language is that? The ‘şı’ looks Turkish, but I don't think Turkish has a letter ‘ə’. It took me a little while to find out the answer.

    It's Azerbaijani. Azerbaijani has an Arabic script and a Latin script; this is the Latin script. Azerbaijani is very similar to Turkish and I suppose they use the ‘ş’ and ‘ı’ for the same things. I speculated that the ‘x’ was analogous to Turkish ‘ğ’, but it appears not; Azerbaijani also has ‘ğ’ and in former times they used ‘ƣ’ for this.

    Bonus trivia: The official Unicode name of ‘ƣ’ is LATIN SMALL LETTER OI. Unicode Technical Note #27 says:

    These should have been called letter GHA. They are neither pronounced 'oi' nor based on the letters 'o' and 'i'.


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    Hidden emeralds

    Dave Turner pointed me to the 1939 Russian-language retelling of The Wizard of Oz, titled The Wizard of the Emerald City. In Russian the original title was Волшебник Изумрудного Города. It's fun to try to figure these things out. Often Russian words are borrowed from English or are at least related to things I know but this one was tricky. I didn't recognize any of the words. But from the word order I'd expect that Волшебник was the wizard. -ого is a possessive ending so maybe Изумрудного is “of emeralds”? But Изумрудного didn't look anything like emeralds… until it did.

    Изумрудного is pronounced (approximately) “izumrudnogo”. But “emerald” used to have an ‘s’ in it, “esmerald”. (That's where we get the name “Esmeralda”.) So the “izumrud” is not that far off from “esmerad” and there they are!


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    Fri, 17 Apr 2020

    Earlier dumpster fires

    In my previous article I claimed

    the oldest known metaphorical use of “dumpster fire” is in reference to the movie Shrek the Third.

    However, this is mistaken. Eric Harley has brought to my attention that the phrase was used as early as 2003 to describe The Texas Chainsaw Massacre. According to this Salt Lake Tribune article:

    One early use found by Oxford Dictionaries' Jeff Sherwood was a 2003 movie review by the Arizona Republic's Bill Muller that referred to that year's remake of "The Texas Chainsaw Massacre" as "the cinematic equivalent of a dumpster fire — stinky but insignificant."

    If Sherwood is affiliated with Oxford Dictionaries, I wonder why this citation hasn't gotten into the Big Dictionary. The Tribune also pointed me to Claire Fallon's 2016 discussion of the phrase.

    Thank you, M. Harley.


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    Thu, 16 Apr 2020

    Dumpster fires

    Today I learned that the oldest known metaphorical use of “dumpster fire” (to mean “a chaotic or disastrously mishandled situation”) is in reference to the movie Shrek the Third.

    The OED's earliest citation is from a 2008 Usenet post, oddly in rec.sport.pro-wrestling. I looked in Google Book search for an earlier one, but everything I found was about literal dumpster fires.

    I missed the movie, and now that I know it was the original Dumpster Fire, I feel lucky.

    [ Addendum 20200417: More about this. ]


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