The Unz Review - Mobile
A Collection of Interesting, Important, and Controversial Perspectives Largely Excluded from the American Mainstream Media
 BlogviewJohn Derbyshire Archive
FEBRUARY DIARY: New Year In Chinatown; Black Math, Psychologist Math, SJW Math, and Real Math; ETC.
🔊 Listen RSS
Email This Page to Someone

 Remember My Information


Bookmark Toggle AllToCAdd to LibraryRemove from Library • BShow CommentNext New CommentNext New ReplyRead More
ReplyAgree/Disagree/Etc. More... This Commenter This Thread Hide Thread Display All Comments
These buttons register your public Agreement, Disagreement, Troll, or LOL with the selected comment. They are ONLY available to recent, frequent commenters who have saved their Name+Email using the 'Remember My Information' checkbox, and may also ONLY be used once per hour.
Ignore Commenter Follow Commenter
Search Text Case Sensitive  Exact Words  Include Comments

Nightmare on Main Street

February 5th was Lunar New Year on the Chinese system. Out with the dog, in with the pig.

It was also of course Pax 10th on the Mayan calendar. This year, however, we thought we’d forgo the human sacrifice and just have a dim sum lunch and do some pre-festival shopping in Chinatown the weekend before.

Parking-wise, Chinatown—this is the Flushing Chinatown in New York City’s Queens borough, not Manhattan Chinatown—is a nightmare even on the average weekend. The weekend before Lunar New Year things are far worse.

Tooling down Flushing’s Main Street I had the glum forboding that we would spend most of the day looking for a parking spot. I even suggested to my lady that we back off to a friendly non-Chinatown subway station and ride the rest of the way. No, she said, we’d have too many bags to carry so far.

In the event we lucked out. After just two circuits of the restaurant parking lot, a space opened up as we approached. Out of car, into restaurant. Piece of cake—bean-curd, whatever.

The dim sum was great, though I made the mistake I always make with dim sum: taking everything the first couple of trolleys have to offer, leaving no room in my digestive tract for later trolleys, which of course have stuff that looks even more appetizing. There must be an art to pacing yourself through this, but in fifty years of patronizing dim sum parlors on three continents, I’ve never mastered it.

The restaurant was clean and efficient for such a big place. There is a chaotic, sharp-elbowed, Malthusian quality to Chinese social life that in certain moods I rather like; but when they want to do brisk efficiency, they sure can. Lake Pavilion Restaurant runs like clockwork.

Then, the nightmare. Back out in the parking lot I said I’d stay by the car while Mrs walked down to the supermarket for food shopping.

Off she went. There I stood. In the street just across the low wall from our car was a food stand selling snacks. It broadcast its wares with a loud ten-second message that just kept repeating: Tianjin Taohua chao xiao chi! Chao huasheng, chao guazi—xian chao! xian mai! … (“[Business name] fried snacks! Fried peanuts, fried melon seeds—frying now! buy now! …”) And repeating … and repeating …

I was stuck there waiting by the car. I didn’t want Mrs to come back loaded with bags and me not be there. I didn’t know what store she’d gone to, so couldn’t follow her.

I got into the car and sat there with the windows closed. No good: the message was too loud. I got out, walked off across the lot, keeping the car in sight. No good: the damn thing followed me. Chao huasheng, chao guazi—xian chao! xian mai! … I wished I could smoke a cigarette to calm my nerves, but I’ve quit.

Chao huasheng, chao guazi—xian chao! xian mai! … how on earth did the guy running the food stand keep his sanity? The expression Chinese water torture was making a lot of sense, vivid sense.

Tianjin Taohua chao xiao chi! Chao huasheng, chao guazi—xian chao! xian mai!

Tianjin Taohua chao xiao chi! Chao huasheng, chao guazi—xian chao! xian mai!

Tianjin Taohua chao xiao chi! Chao huasheng, chao guazi—xian chao! xian mai!

Tianjin Taohua chao xiao chi! Chao huasheng, chao guazi—xian chao! xian mai!

Mrs Derbyshire showed up just as I was trembling on the edge of homicide. “What’s the matter? You look stressed out.”

“Nothing, honey. I’ll take the bags, get in the car.”

The rest of the New Year holiday went well.

Museum of the Dog

The Museum of the Dog, from the American Kennel Club, opened Friday a couple of blocks south of Grand Central, after relocating from St. Louis, Missouri. Its 11,000-square-foot gallery is filled with pooch-centric paintings, sculptures, photos and artifacts, some going back centuries. [Museum of the Dog is a fun, fitting tribute to man’s best friend by Eric Hegedus; New York Post, February 8 2019.]

On the dog-loving spectrum, Mrs. Derbyshire is way over at the extreme right end. When she saw that story in the New York Post she near swooned. “We must go see that!”

So off we went. Yes, the place is fun. The walls are hung with dog paintings—originals, not reproductions—some from surprisingly far back in the 19th century, the dogs looking perfectly up-to-date. The only one of the artists’ names that rang a bell with me was Landseer.

I wondered if perhaps we might donate Elizabeth Cockey’s portrait of our late beloved Toby to the museum, but the Mrs doggedly refuses to let it go.

There are dog-movie posters, too: Lassie, Old Yeller, Beethoven, … And sculptures, and bronzes, and porcelain dogs; and some cute hi-tech “installations.”

The installation we liked best was one that takes your picture, listens to you bark, then tells you what breed of dog you most resemble. I am either a French Bulldog (“Adaptable, Playful, Smart”) or a Boston Terrier (“Friendly, Bright, Amusing”), depending on whether or not I am smiling.

The Museum includes a library of dog books—I never knew there were so many. They are all properly Deweyed, so you can easily find anything you’re looking for. Literature is well-represented: Jack London of course, Richard Adams, and that curious little biography of Elizabeth Barrett Browning’s dog Flush by Virginia Woof … sorry, Woolf.

Some curiosities, too. There’s a book of dog poems titled, of course, Doggerel; a dog’s history of the world, subtitle “Canines and the Domestication of Humans”; and books of dog jokes. How many dogs does it take to put in a light bulb, by breed?

Lab: Me, me, oh let me do it! Please, pleeeeaase, let me! Here, I’m here! Let me do it!

Border collie: Just one, and I’ll replace any wiring that’s not up to code.

Etc., etc. And yes, some of the older books are regrettably dog-eared.

Book ends

“Of making many books there is no end,” said the preacher. I wouldn’t be too sure, pal.

Walking back to Penn Station from the dog museum, we took a detour to New York Public Library, which often puts on small, interesting exhibitions good for killing a half hour in midtown Manhattan.

On February 17th it only had a show celebrating homosexuality, which neither of us is interested in. We mooched around the magnificent building anyway.


The Mrs wanted a picture taken in the big main reading room, so we went there. Standing by the endless shelves of reference materials while she manipulated her smartphone camera, I got to thinking how redundant these books are. Collier’s, Britannica, Larousse, … aren’t they all online now? Does anyone actually pull one of these volumes down and open it?

Nobody did while we were there, and the reading desks were full of readers … well-nigh every one with a laptop or iPad in front of him. Nor did anyone go up to the gallery that runs round the room.

There is something melancholy, something forlorn about libraries now. I get that feeling even in my own much-less-grand provincial library, which I used to spend hours in but now rarely visit. How long will libraries still be around? Are school guidance counsellors warning youngsters not to embark on a career in librarianship?

I guess it’s geezerish and futile to think like that. Probably some Sumerian Derb around 3000 b.c. was bemoaning the decline of good solid baked-clay tablets and scoffing at the new-fangled papyrus—so perishable! so flammable!

It didn’t help that I am a fan of junky sci-fi-apocalypse movies like Beneath the Planet of the Apes and The Day After Tomorrow, in which a scene in the shattered ruins of New York Public Library seems to be compulsory.

Is our civilization on its last legs? Well …

Life cycle of civilizations

If you enjoy contemplating the collapse of civilization, envy the the scholars at Cambridge University’s Center for the Study of Existential Risk, who ponder the civilizational End Times all day long and get paid a salary for it.

Luke Kemp is one of those scholars. He got some headlines of the lesser sort this month with an essay titled Are We on the Road to Civilizational Collapse? [BBC, February 19, 2019] His answer: Could be.

The world is worsening in areas that have contributed to the collapse of previous societies. The climate is changing, the gap between the rich and poor is widening, the world is becoming increasingly complex, and our demands on the environment are outstripping planetary carrying capacity …

Good flesh-creeping stuff. Kemp includes a chart showing the lifespan of 86 civilizations. The average lifespan, he says, is 336 years.

This quantitative approach to civilizational rise and fall has a long pedigree. A big name here in the 20th century was Arnold Toynbee, who Kemp refers to in passing. Toynbee’s humongous 12-volume Study of History was known to every educated person in the generation before mine—to most of my schoolmasters, for example. (I don’t say they all read it. It was known to them, that’s all.)

Toynbee is not much read nowadays, but more recent public intellectuals have worked the theme: guns-germs-and-steel guy Jared Diamond, Tiger Mom Amy Chua, polymath David Goldman, Toynbee epigone Stephen Blaha, and others.

For big-idea intellectuals, it’s a hard theme to resist. It sure looks as though civilizations age the way individual human beings do. When young, they have a vitality, a can-do spirit, that lets them accomplish wonders. In middle age they get fat and complacent. Then comes senility—peevish, quarrelsome decline. Si jeunesse savait, si vieillesse pouvait. Analytically-minded people naturally want to try quantifying that.

Where is the U.S.A. on this timeline? It depends where you start from. If we take Luke Kemp’s average lifespan of 336 and start from the adoption of the Constitution in 1789, we’re middle-aged—fat and complacent—but we should be good until 2125.

I dunno, though. From where I’m sitting “peevish, quarrelsome decline” looks like a better fit.

Quantified history bites back

You might think that China is a problem for these civilizational theorists. Hasn’t China been a continuous civilization since deep antiquity—well back in the second millennium B.C.? How does that square with Kemp’s average of 336 years for the lifetime of a civilization? Is not China, as sinologist Lucian Pye famously remarked, “a civilization pretending to be a state”?

Toynbee, who favored civilizational cycles up to a thousand years long, got round the problem by grouping successive Chinese dynasties together.

Kemp just treats each important Chinese dynasty as a separate civilization; although since the main civilizational characteristics—language, literature, philosophy, social and political arrangements—have indeed changed only little and slowly across millennia, and with Lucian Pye in mind, I’d put quotes around the word “civilization” there.

The Chinese themselves take Kemp’s point of view, with those quotes. They have done so for a very long time, and even had a go at quantifying “civilizational”—that is, dynastic—rise and fall two thousand years ago.

A great-uncle of Kuei Hung’s contemporary Lu Wen-shu had, in fact, through astrological calculations, arrived at the conclusion that the dynasty would last for three times seventy years, and since its beginning had been fixed at 206 b.c., it was, accordingly, to end in a.d. 4.

That’s from Rudi Thomsen’s biography of Wang Mang, a brilliant but very unscrupulous man who took over the Han dynasty as it sputtered out in a sad trail of emperors who were debauched, under-aged, sickly, and/or short-lived without direct issue from 48 b.c. to a.d. 6.

That calculation of a 210-year dynastic span was one of the propaganda points Wang used to bolster his claim to rule. Time was up for the Han dynasty! he argued. Heaven said so!

What followed should be an object lesson for would-be quantifiers of history. Wang Mang’s new dynasty—it was actually called Xin, which means “new”—was one of the shortest on record, fourteen years and nine months. There were natural catastrophes and rebellions. Wang Mang was dethroned and chopped up:

The head was sent as a trophy to the Keng-shih Emperor, who displayed it in the market-place in his provisional capital, Yüan. Here the people hurled things at it, and, according to Pan Ku, someone even cut out Wang Mang’s tongue and ate it. [Ibid.]

An outlier member of the Han ruling family was put on the throne and the Han dynasty resumed as the Later or Eastern Han, a.d. 25-220.

Human affairs on the historical scale don’t yield easily to numerical analysis.

Code archeology

OK, let’s swing technical for the rest of the diary: Computer Science and Math.

First, a nostalgia trip for old code jockeys.

After Radio Derb ran a segment advising listeners not to learn to code, I got queries about my own coding history. What languages have I coded in?


That’s not easy to answer. For one thing, what counts as a language? I’ve spent a lot of time coding SQL, for example. It’s a standard set of instructions for communicating with databases. Is it a coding language? It says it is: SQL stands for “Structured Query Language”; but we code grunts never considered it to be in the same category of thing as COBOL, LISP, C, or PERL. SQL commands were usually just imbedded in real code.

Similarly with MVS-JCL. MVS was (is?) an operating sytem for the big old IBM mainframes; JCL was its Job Control Language for managing batch input and output, job scheduling, and some low-level data manipulations like file sorting. Again, JCL says it’s a language, but none of us believed it. It was just stuff you had to know to get your job run.

You can throw in HTML—HyperText Markup Language, which I’m using to write this diary. It’s mighty handy for putting web pages together, but a language? Nah.

What about scripting languages, like the one that comes with the text editor I’m using to input my HTML (the scripting language is called KEXX)? I don’t think so. After twenty years of using KEXX, my biggest macro is only 334 lines. That’s just clearing your throat in a real language.

And then, do I count languages I’ve just played with for amusement, or because everyone was talking about them? C and C++ fall into that category. I never coded them for money, even to teach them. I’m going to take a strict approach and just include languages I’ve earned a living coding or teaching.

With those limitations, here’s the tally. Chronological order.

  • While a trainee programmer at the U.K. Post Office, I learned ALGOL from a manual and coded up some tables for my boss.
  • The assembly language for the Leo 326 machines, basic second-generation mainframe workhorses for the Post Office (which in the late 1960s also ran the U.K. telephone system).
  • A COBOL-type high-level language for the Leo. We used it for telephone billing.
  • The assembly language for ICL System 4 machines. The System 4 was a clone of the IBM 360, so Usercode was just 360 BAL in a dress. I did not know this at the time; but later, when I had to code 360 BAL, I was pleasantly surprised—no learning curve! I’ll count Usercode and BAL as one language; and throw in their macro language, which as a systems programmer I worked with some.
  • Dartmouth BASIC was a big draw for colleges in the early 1970s. I worked with Jack Harwell’s firm developing a BASIC interpreter for System 4 machines, testing out the math functions.
  • I lectured in Computing at a trade school in Hong Kong. We taught from approved IBM texts. We started the students with RPG, then advanced to …
  • COBOL, then to …
  • I didn’t know RPG or COBOL before teaching them. I learned them from the teaching texts, racing to keep ahead of the students. (No mean feat: these were Chinesestudents.) From knowing CLEO I quickly got the hang of COBOL. I knew FORTRAN from having played with it at a former installation—code monkeys like to play—but had never before made money from knowing it.
  • Having learned COBOL, which everyone in the business world used, I coasted on it through the later 1970s and early 1980s. In the later eighties I switched from MVS to VM (the hipper IBM mainframe operating system, MVS being blue-collar). REXX was a very neat interpreted language for VM. KEXX is one of its descendants.
  • PC Assembly Language. When PCs came in for office work, I learned the Assembly language from Peter Norton’s books; had fun around the office taking a break from COBOL and REXX to code up little PC routines for the users.
  •,0,0,0_FMpng_AC_UL320_SR246,320_.jpgApple IIGS Assembly Language. At some point in the late 1980s I did a job on the side for my boss, who wanted it done on an Apple IIGS. I read up the durn thing’s assembly language (Merlin? Or was that just the name of the IDE? The memory is badly faded — mine, not the machine’s), coded the job in that, and got paid for it.
  • Visual Basic. Assembly language is a chore after a while, like brewing your own beer. I graduated to VB for PC work, and stayed with it until I quit coding for a living in 2001. I taught it for a semester at Baruch College in New York City.

That’s a neat dozen languages I’ve coded for money. I have a longstanding vague idea to bring my coding skills into the 21st century, if only to keep cognitive decline at bay. I’ve actually bought Zed Shaw’s book to learn Python from, but … have not yet cracked the spine …

No justice, no math!

I generally reserve my mathematical inclinations for the last segment of the diary. This month, however, math has been in the news HBD-wise, thanks to the efforts of New York Times reporter Amy Harmon.

As I noted in the February 22nd Radio Derb:

Ms Harmon recently contributed two pieces for the Times on the shortage of black mathematicians.February 18th she profiled a black mathematician who feels he don’t get no respect from his peers.February 20th she expanded to a general theme about how “bigotry,” “racial exclusion,” “unconscious bias,” and so on, keep blacks out of math.

The black mathematician in that first article was Edray Goins of Pomona College. My Radio Derb commentary passed over Prof. Goins in silence and focused on the second of Amy Harmon’s articles.

Is Prof. Goins the real thing—i.e. a decently productive mid-level math academic—or just an affirmative-action token? I didn’t pass an opinion for the excellent reason that I’m not qualified to judge.

There’s a list of Prof. Goins’ talks and refereed papers here. To answer the question in the previous paragraph you’d need some feel for:

(a) Quantity-wise, is that a good steady level of productivity for a tenured math professor? and
(b) Quality-wise, are those topics appropriate for same, and does Prof. Goins treat them in appropriate depth?

I don’t know enough to pass judgment on either point. I could take a fair stab at (b) by actually reading the papers and talk transcripts, but … life is short. I therefore stand agnostic on the main question.

There are suggestive hints in some of the comments on the Times article, and even in the article itself:

Dr. Goins’s colleagues at Purdue said his receipt of tenure and subsequent promotion to full professor signaled the university’s willingness to overlook a sparse research portfolio in light of his extraordinary work with undergraduates, as well as the summer programs he organized for minority students.

[My italics.] For sure there are affirmative-action token hires in college math departments. Several readers emailed in to tell me I was much too kind to Assistant Professor Piper Harron at the University of Hawaii in my May 2017 diary. The mathematician who blogs as PostTenureTourettes was less restrained towards Prof. Harron, and took another swing at her this month while commenting on Amy Harmon’s pieces.

Blackety-blackety black scholarship

The following may be relevant. I’ve heard it down the years from friends in many of the hard sciences. To paraphrase them:

You go to an academic conference on, oh, say, microbiology. A white guy steps up and reads a paper on the role of histone acetylation in the control of gene expression. Applause, questions, discussion.

An East Asian lady steps up and reads a paper on phospholipid biosynthesis in mammalian cells. Applause, questions, discussion.

A South Asian guy steps up and reads a paper on the regulation of sphingomyelinases. Applause, questions, discussion.

A black guy steps up and reads a paper titled: “How can we get more underprivileged youth interested in microbiology?” …

The SJW-ification of math

For sure the SJW-ification of math proceeds apace. A great host of academic administrators, and a few actual academics—with the eager assistance of fake journalists like Amy Harmon—are striving to drag this most rational, most profound, most demanding, and purest of intellectual disciplines down to the infantile subjective feelgood level of Raza Studies or Queer Legal Theory.

(I have just noticed that Ms. Harmon apparently believes that “pure math” belongs in mockery quotes while the applied variety does not. I guess she’s telling us that math is, like any other style of intellection, au fond just a manifestation of the oppressive power relations in capitalist, white supremacist, patriarchal, heteronormative society. Nothing pure about that!)

For a random data point on that SJW-ification, see the editorial by Jacqueline Jensen-Vallin in the current (Feb/Mar 2019) issue of Focus, “Newsmagazine of the Mathematical Association of America.” It’s online here—scroll down past the cover page—but you may need to be a subscriber, so here’s the content:

From the Editor [Shouldn’t that be “Editrix”? … never mind—JD]

I’m very excited about this issue of MAA FOCUS. Preparing for this issue, I realized that I had a number of submissions about issues of inclusivity in our community. This is one of MAA’s core values—advocating for and celebrating diversity by promoting mathematics for all and broadening access through initiatives to engage diverse audiences. I hope that you will see these themes throughout this issue.


While compiling this issue, I wanted to give voice to groups of people who have been historically silenced or heavily edited. To this end, I have tried to only lightly edit most of the articles appearing in this issue. My goal was for people to be heard as they intended. Therefore, the opinions in the articles belong to the authors and not necessarily to the MAA as an organization. I have not managed to hear all voices, but my goal is to continue to listen and give access to groups historically under-represented in mathematics. We hope that sharing these opinions leads to open communication and a conversation about ideas.

Thanks to all of you who have contributed to this issue. I am grateful for those who lent their voices, thoughts, opinions, and ideas.

Yecchh! I don’t suppose being the editrix of a math newsletter pays worth a damn, but plainly Ms Jensen-Vallin is positioning herself for higher things: HR Director at Google or Twitter, perhaps.

And I can’t help but admire the deft way she has absolved herself of the need to do much actual editing.

Immatheracy in the human sciences

Last year I gave a passing mention to Eric Turkheimer in my opinion piece on Charles Murray. Turkheimer is a field commander in the long rearguard action to defend the no-such-thing-as-race position against the steady advance of behavioral genetics.

I had always thought that Turkheimer was one of the sharpest knives in that particular drawer—a brilliant tactician, to flip back to the military metaphor—with a good understanding of all the main points (including, I mean, the ones he disagrees with) and formidable skill in the manipulation of data.

Well, there’s another bubble burst. It turns out Turkheimer can’t do middle-school math. On February 11th he tweeted:

Dumb (but real and research-related) question:

Y 1 = aX² + bX + c
Y 2 = dX² + eX + f

What is Y 1 in terms of Y 2?

r/t if you know a good high school math teacher

(I’ve sub- and super-scripted to make the tweet more readable.)

For Heaven’s sake, man: Solve the second equation for X in terms of Y 2, d, e, and f, using the standard formula for solution of a quadratic equation. Then substitute those values of X (there are two, of course—it’s a QUADRATIC EQUATION) in the first equation. Next!

Turkheimer is a Professor of Psychology. Can you really be a professor of anything without knowing basic algegraic manipulations? All right, anything in the sciences: but that should certainly include soft sciences like psychology and sociology. All the results out of the soft sciences that strike me as interesting involve heavy-duty statistics, i.e. math.

The responses to Turkheimer’s query are even more depressing. There is massive math-ignorance—”innumeracy” doesn’t quite catch it; this is algebra, not arithmetic; “immatheracy”?—in the comment threads of even the best human-science blogs, like Greg Cochran’s.

Intersecting parabolas—and in one case hyperbolas—seem to be particularly popular, I cannot fathom why. So is confusion between coefficients and variables. So is the notion that since the two equations are really the same, just with different names for the coefficients, Y 1 and Y 2 must be equal. Ye gods!

Is elementary math really so remote from the understanding of educated people?

Greg’s post on Turkheimer’s math problem is titled “An extra sense.” One of the commenters tells us that this comes from a Darwin quote: Darwin felt that people who were good at math had an extra sense, which he envied. Perhaps the old boy was on to something.

Silver lining: I’m feeling much better right now about my own Class III degree.

Math Corner

All right, enough of the stupid girly social stuff. Let’s have some actual math.

Here’s a nifty number:

f1 = 2.920050977316…

It’s irrational (see the last sentence below) so the decimal form never recurs. The ellipsis at the right-hand end means “go ahead and compute a few more—or a few dozen more, or a few trillion more—decimal places, if you feel so inclined and have the resources.”

So what’s nifty about it? It contains within itself all the prime numbers, that’s what.

Why is that nifty? Lotsa numbers contain all the primes: Euler’s product formula for the zeta function gives you an infinity of such numbers, one for every value of s that is greater than 1. Heck: 0.235711131719232931… contains all the primes.

Sure; but try extracting the primes from the decimal expression of such numbers. You can work up algorithms, but they are way cumbersome and knotty.

The nifty thing about f1 is that the primes drop out one by one, via some very simple arithmetic. f1 packages up all the primes, but in a way that makes it exceptionally easy to unwrap the package. Thus:

First, break f1 into its integer part and its decimal part. I’ll call them int(f1) and dec(f1).

Plainly int(f1) = 2 and dec(f1) = 0.920050977316…

Now carry out the following extremely simple arithmetic algorithm, taking k of course to be 1 in this instance. I shall call the algorithm Al (no, it’s not one of his, I just like looking at the old boy):

  • Add 1 to dec(fk). Then
  • Multiply the answer by int(fk).

Well, that’s not hard. Adding 1 to dec(f1) gets you 1.920050977316… If you now multiply that by int(f1), which is to say by 2, you get 3.840101954632…

Let’s call that number f2. Then let’s apply Al to f2.

int(f2) is of course 3; dec(f2) is 0.840101954632… Adding 1 to the latter and multiplying by 3 gets you 5.520305863896… I’m going to call that—guess what? — f3.

Applying Al to f3 means multiplying 1.520305863896… by 5. That gets you 7.601529319480…, which of course I shall call f4.

Applying Al to f4 means multiplying 1.601529319480… by 7. Answer: f5 = 11.210705236360…

See what’s happening? The integer parts of f1, f2, f3, f4, … are just the prime numbers p1, p2, p3, p4, … in order.

This goes on being the case for as long as you keep going, although of course you will need more and more decimal places of f1 to keep it on the rails. As I said, all the prime numbers are contained within that one starting number, f1, and can be extracted via simple arithmetic. Nifty, or what?

While indubitably nifty, the result is not very surprising when you know how f1 is defined. It’s defined as an infinite sum a1 + a2 + a3 + a4 + …, where each of the a‘s is a fraction. The numerator of an is pn − 1, where pn is the n-th prime number. The denominator is the primorial of pn-1.


(Which is to say, the product of all the primes up to the (n − 1)-th. The primorial of 2 is 2; the primorial of 3 is 2×3, the primorial of 5 is 2×3×5; the primorial of 7 is 2×3×5×7, and so on. For convenience, although 1 is never included among the primes, the zero-th primorial is defined to be 1.)


a1 = (2 − 1) / 1, which is 1.
a2 = (3 − 1) / 2, which is also 1.
a3 = (5 − 1) / (2×3), which is 4/6, i.e. 2/3.
a4 = (7 − 1) / (2×3×5), which is 1/5.
a5 = (11 − 1) / (2×3×5×7), which is 1/21.
a6 = (13 − 1) / (2×3×5×7×11), which is 2/385.
… …

And so on. If you add those up you get 2.919479… (I think), so the sum of the a‘s is already closing in fast on f1.

Since this definition of f1 involves all the primes, it’s not astonishing that you can squeeze all the primes back outof f1. What’s nifty is that you can do it by such a simple arithmetic manipulation. Thanks, Al!

(I lifted this from a paper in the January 2019 issue of The American Mathematical Monthly, which can be found in any college library. The paper is “A Prime-Representing Constant” by Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime, and Massi Tron Florentin. It includes some interesting generalizations, and a proof that f1 is irrational.)

(Republished from VDare by permission of author or representative)
• Category: Ideology • Tags: American Media, Political Correctness 
Hide 61 CommentsLeave a Comment
Commenters to FollowEndorsed Only
Trim Comments?
  1. Daniel H says:

    Derb, consider learning Clojure. It’s essentially LISP, but way cooler. It runs on the Java Virtual Machine. Seamless interface with any/everything ever written, or that ever will be written, in Java. But why would you want to program in Java when you can program in Clojure? I just fool around with the language. It’s a lot of fun. As a LISP with the Macro feature you can customize the language in a way that is not possible with procedural languages. Code, data: same thing. Cool. Important: apparently it’s very efficient. Execution is fast.

    • Replies: @Cloudbuster
  2. supertjx says:

    Your section on Chinese civilizational history reminds me of the opening lines of the Romance of the 3 Kingdoms:

    The empire, long divided, must unite; long united, must divide. Thus it has ever been.

  3. The following may be relevant. I’ve heard it down the years from friends in many of the hard sciences. To paraphrase them:

    You go to an academic conference on … oh, say, microbiology. A white guy steps up and reads a paper on the role of histone acetylation in the control of gene expression. Applause, questions, discussion.


    A black guy steps up and reads a paper titled: “How can we get more underprivileged youth interested in microbiology?” …

    It’s not just academic conferences, and for some reason the black guy is almost always a guy.

  4. macilrae says:

    I had not realized we had so much in common – the approximate birth place and year, the Chinese connection (as well as its duration) and even the interest in mathematics – except that I came to “C” in my latter sixties and did little ‘coding’ before that. We also tend to see eye to eye on the ‘other matters’ – which is no real surprise.

    It’s likely we both did pure mathematics at “A” Level – did you take a look at today’s equivalent and compare? We used to muck around with cubics and left quadratics to “O” level – as written and spoken English has gone, so also has mathematics. Talk about a dumb-down.

  5. John says: • Website

    I like Python but I couldn’t give a slam-dunk reason why. It’s just been handy to write little and not-so-little utilities. And get paid for it! My job is programming simulations of applications for computer-based training, and I found it useful in the case of one such application to obtain (openly and honestly, from the people who wrote it) the source code in Visual Basic, then translate it to the Java our simulations are mostly written in, using a translator I myself wrote in Python. It ain’t pretty but it works, is easy to maintain, and – as we move our simulations out of Java and into HTML5 – was easily modified to output that instead. But this all could have been done in other languages.

    Maybe more interesting is to make a list of programming languages to be avoided. Lisp; Haskell. Those feel like the results of somebody’s stupid compulsions. They made me think of something from Richard Brautigan’s short story “Hangover as Folk Art”: the part about undesirable tainted trinkets made of rusty beer cans and coal, if not the part about at last recovering to the point there is only an abstract chalky awareness that one is still breathing. Lisp and Haskell ain’t THAT bad!

    I have considered studying Lua, simply because Brazilians invented it. And I have considered inventing a programming language modeled on Turkish grammar. But the feelings always pass.

    • Replies: @Cloudbuster
  6. Polymath says:

    They should teach the cubic formula, not just the quadratic formula. It’s not VERY hard if you get the equation into the right form first with a linear substitution:
    x^3 = 3px + 2q
    Case 1: q^2 >= p^3: 1 real root
    x1 = cbrt(q + sqrt (q^2 – p^3)) + cbrt(q – sqrt(q^2 – p^3)
    Case 2: q^2 < p^3 : 3 real roots
    x1 = 2 sqrt(q) cos(A),
    x2 = 2 sqrt(q) cos(A + 2pi/3),
    x3 = 2 sqrt(q) cos(A + 4pi/3)
    where A = (1/3)arccos(q/(p*sqrt(p))

    If you also want the two complex roots in Case 1, let A and B be the cube roots that were summed to get the real root, and take
    x2 = -(A+B)/2 + sqrt(3)(A-B)i/2
    x3 = -(A+B)/2 – sqrt(3)(A-B)i/2

    Here's a puzzle I invented (stolen by a later author) that can be solved using the formula. The probability of the server in a tennis game winning the next point is such that he is just as likely to win a game in which he has the "advantage" after "deuce" as he was at the beginning of the game. (That is, "the advantage isn't an advantage".) What is the probability he wins a point?

    Hint: the answer also arises in the apparently unrelated context of "Tribonacci numbers", although I'm not sure it's just a coincidence. (Tribonacci:Fibonacci::3:2)

  7. I’d argue that some extensions of SQL: T-SQL and PL/SQL are real languages because theyve got procedural statements. If so I have 13 under my belt.

  8. Daniel H says:


    I have Zed Shaw’s book too. Not the right book for you. You are way beyond the point for Shaw’s pedagogical style. Shaw’s book is good for one thing: he goes through the motions of setting up the development environment for your project, creating the optimal directory structure. The rest of the book is just tedious. There are better tutorial resources on Youtube. Excellent, free, Python focused development environment is PyCharm. Download, install, fire up Youtube for a little guidance and you will feel like your are 30 years old again, coding to blazes.

    • Agree: Kratoklastes
  9. Anon[378] • Disclaimer says:

    Great piece Derb, I thought I was your biggest fan, but these commenters are speaking some moon man language that I cannot understand, are you some kind of alien?

    • Replies: @anonymous
  10. The Museum includes a library of dog books—I never knew there were so many.

    Do they have the ones about collies by Albert Payson Terhune? Those were my favorite dog books as a kid.

  11. As regards the tyranny of the loudspeaker as practiced in China, there’s only one solution. Allow me to introduce three of America’s finest intellectuals as they adroitly deal with this pernicious technology:

  12. Rich says:

    “The climate is changing…our demands on the planet are outstripping…” Can someone, somewhere explain why all these Chicken-Littles with their apocalyptic visions feel the best way to “save the planet” is to tax me into penury and force me into an unheated shack?

    • Replies: @Oleaginous Outrager
  13. anonymous[340] • Disclaimer says:

    Yes. He’s from England.

    Which seems especially appealing to a segment of his readers. It’s the same presumed superiority long traded on by Alistair Cooke, David Frost, Graham Kerr, that recent obnoxious chameleon whose name eludes me that helped stage a meltdown over gun control with Alex Jones, Jon Oliver — Heritage Americans Plus.

    • Replies: @Anonymous
  14. The dim sum was great, though I made the mistake I always make with dim sum: taking everything the first couple of trolleys have to offer, leaving no room in my digestive tract for later trolleys, which of course have stuff that looks even more appetizing. There must be an art to pacing yourself through this, but in fifty years of patronizing dim sum parlors on three continents, I’ve never mastered it.

    Breaking News: Derb confesses to high time preference! 🙂

  15. but the Mrs doggedly refuses to let it go…. And yes, some of the older books are regrettably dog-eared.

    Limiting himself to only two dog puns in that section must have been difficult.

  16. @Daniel H

    I have a special, irrational hatred in my heart for Java. I worked for a company for about 13 years that was involved in, to put it very simply, high speed packet collection and analysis.

    Performance was critical and all the core code was written in C++.

    Very late in my tenure (and near the end, for the company), new product development was put in the hands of the most incompetent, unreasonable, arrogant and evil manager I’ve ever worked for.

    He pretty much guaranteed the company’s death by insisting that 13 years of development simply be tossed away and the entire thing rewritten in Java by people who didn’t know the core product. Not only didn’t he understand the product or the code, he didn’t WANT to. He had me in his office one day to review the applications as they currently existed and even if he had allowed me to explain, the time allocated was woefully short, but instead he interrupted me after a couple minutes and spent the rest of the time bitching about the product and how his new Java engine was going to solve all the problems (as perceived by him).

    He did not like me disagreeing with him or bringing up objections at meetings. He believed in credentials over results (my actual diploma’d development credentials are virtually non-existent. I only had my 25 years of experience to fall back on). :/

    I shut up and started to teach myself Java. I was laid off within a couple months. The company sold its intellectual property and went out of business within two years, never having produced anything for sale after I left.

    He didn’t even have the guts to lay me off personally. He had some poor newbie female HR drone do all the talking, over the phone (I worked remotely). I gave them both an earful. Apparently stories of my rant spread widely around the company. Thanks poor newbie HR drone. At least you did that.

    Anyway, every now and then I have to get that off my chest. I really hate the guy. He made his fortune suing big companies for theft of intellectual property and then settling.

    Oh, BTW, he was Chinese.

    • Replies: @Achmed E. Newman
  17. @John

    Lua’s useful to know because it’s the default utility language for most computer game add-ons and a lot of home automation software APIs.

  18. Mr. Anon says:

    The American Institute of Physics (AIP) and The American Physical Society (APS) are getting in on the diversity racket in a big way. Just about very issue of Physics Today and APS News have an article or two on this or that “marginalized” group, how physics is not sufficiently inclusive, and how the profession can futher bend over backwards to accomodate them.

  19. Anonymous[403] • Disclaimer says:

    Derbyshire fills his rice bowl by being a professional Englishman overseas. That’s why he cultivates his accent even though he prances around calling himself an American when he is clearly not, and clearly does not want to be.
    He doesn’t even really believe there is such a thing as an American nationality; if he did, he would know not to call himself one of us. He gives it away when he says things like “our civil war,” yet refers to the English civil war as “the civil war.” Of course, for any nationality, the only civil war is the one their country had, thus the definite article when referring to it. The use of “our” implies that there are other civil wars with equal claim on our attention. But who cares about foreigners fighting among themselves?
    Derbyshire has no connection to and no understanding of this country at all; even his wife is a foreigner. Thus he misunderstands common terms like “home town,” and marvels over routine phrases he discovers such as “flop sweat.”
    Read him for amusement only — if you bother to read him at all.

    • Troll: Cloudbuster
  20. Dear Mr. Derbyshire, sir:

    All parabolas are similar, just as all circles are similar.

    A circle is the locus of all points a distance r from the center C. You can write a transform from one circle to another with a translation of the center. With C1, C2 and T written as vectors (ordered pairs), (C1 + T = C2) and a scaling factor r2/r1.

    A parabola is the locus of all point equidistant from the focus and the directrix (line). Thus F1 -> F2 and scaling factor = p2/p1 where p is the distance from the vertex to the focus. Perhaps he wanted to know how to transform one parabola to another? Naw.

    This is what the guy was asking for: Express Y2 in terms of Y1.

    If you have two linear equations, you can express one in terms of the other by writing Y2 = Y1 + (Y2 – Y1), but that’s trivial, right? But, in general, you would need two orthogonal lines.

    I don’t think you can express one quadratic in terms of another quadratic. I think you can express a quadratic in terms of a series of quadratics, just as you can express one line in terms of two orthogonal lines.

    James “It’s called Linear Algebra” Speaks

    • Replies: @blake121666
  21. @Mr. Anon

    Every profession that involves a modicum of math has become obsessed with getting more blacks and Hispanics.

    It’s a Kabuki dance. A black guy gets paid extremely well to give speeches at a conference or as a consultant. Business get their “Get out of jail free” card. The rank-and-file whites get to not feel guilty about not actually liking blacks and wanting to live around whites.

    Strange times.

  22. @James Speaks

    A “series of quadratics”?

    One can obviously manipulate the equations to find that:

    A * Y1^2 + B * Y2^2 + C * Y1 * Y2 + D * Y1 + E * Y2 + F = 0

    where (A,B,C,D,E,F) are derived from the (a,b,c,d,e,f) of the original equations.

    So the Y1 and Y2 have a general quadratic relation with each other and therefore can be viewed as points on a conic section if you like.

    While this is not technically a “function” relationship, it’s not a “series of quadratics” relation. It’s simply a quadratic relation – plain and simple.

    Y1 in terms of Y2 would in general give 2 points – as Derbyshire said.

    • Replies: @James Speaks
  23. As I posted elsewhere (on Steve Sailer’s blog where this came up), that Turkheimer quadratic problem I thought to be an interesting way to view trajectories of projectiles under a constant force (such as gravity).

    Of course a projectile on the surface of the earth would be a parabola (ax^2 + bx +c) due to the constant gravity. So 2 projectiles in the same plane (say one starts at a different angle and with a different initial velocity than the other) would give the two quadratic equations such as Turkheimer states. Then Turkheimer’s question becomes: What is the other projectile’s trajectory from the point of view of the first projectile? And the answer is: some type of conic section – since those equations can be manipulated to show that Y1 and Y2 have a general quadratic relationship to each other.

    So if I were shot from a cannon at a certain initial velocity and angle and viewed someone else shot from the same cannon at a different velocity and angle, I’d see that person’s trajectory as a conic section!

    • Replies: @blake121666
    , @James Speaks
  24. @Rich

    Because they’re too pussy to come straight out with what they really want: you rotting away in a shallow grave.

  25. Daniel H says:

    OT: This HAD to be DESTROYED.

    • Replies: @Truth
  26. @blake121666

    A “series of quadratics”?A “series of quadratics”?

    Yes. Take a course in linear algebra and then get back to me.

    • Replies: @blake121666
  27. @James Speaks

    I know linear algebra quite well. What does your “series of quadratics” have to do with it?

    The Y1 and Y2 share a quadratic relationship to each other and can therefore be seen as points on a conic section. Why anyone would wish to view that as a “series of quadratics” is beyond me. Would you view the points on a circle as a “series of quadratics” for each ordinate? Of course not. Your “series of quadratics” concept is more complicated than the original problem – which is simply to state the relationship between ordinates parameterized by quadratic equations. For the circle, you’d parameterize like:

    Y1 = cos(theta)
    Y2 = sin(theta)

    In this instance, the Y’s are parameterized by quadratics. Saying they therefore can generally be seen as points on a conic section is a hell of alot easier to think about than a quadratic for each point – simply because each point would map to 2 points. The (a,b,c,d,e,f) simply determines the particular conic section. Easy peasy. No need for your “series of quadratics” – which of course has nothing whatsoever to do with linear algebra!

    Do you even know what LINEAR algebra is? This problem is quadratic!

    • Replies: @James Speaks
  28. Is the letter “S” in short supply?

  29. @blake121666

    It just occurred to me that I should have used “t” insead of “x” for clarity here. Or “delta_t” actually since the trajectories don’t need a common starting time of course.

    h = h0 + v0 * delta_t – 1/2 g (delta_t)^2

    The height of the trajectory is quadratic in time.

  30. @Anonymous

    It’s really hard to lose an accent after 15 – 18 years old, #403. Just from memory, after heading to China, where I really doubt he’d change his English accent to a Chinese one, he came to NY well into his 30’s (correct me, Mr. Derb., if I’m wrong). I know a friend who’s been here 40-something years, but still says certain words that betray his Canadian background.

    Therefore, notwithstanding the rest of your comment, your accent remark is bogus. I do know a guy, though, who speaks with an English accent on purpose just to help get chicks. He tells us he’s English, but I think we was there for 2 months as a baby!

    • Replies: @Anonymous
  31. @Cloudbuster

    Good story, Cloudbuster. I had a couple of jobs in which I regret not raising ENOUGH hell when I left. I had this “burning your bridges” thing in my mind. It’s not that I don’t like speaking out, but sometimes it’s the friends in your own group and the low-level boss that you hate to have trouble come down on afterwards.

    BTW, I thought I was getting laid off at one computer job due to my login suddenly not working (and a guy I knew confirmed my name was off the pw-file(?)). I took all my and my company’s stuff and went home, already making plans for the future. I was ready to hold onto the company phone/computer etc. as ransom, just as a matter of pride and fun. Then I found out someone had made a mistake and I was not terminated. In a way, I was a little disappointed, I gotta say.

    “Why’d you leave so early in a hurry on Friday?”, my low-level boss said on Monday. “Oh, I thought I was part of the lay-off, so I took all ‘my’ stuff home.” I got a kind of worried look back.

  32. This is embarrassing.

    First there’s the typical middle-class failure to account for actual costs. Derbyshire is able to focus on the extreme irritation of parking, but there’s no mention of the cost of driving there and back (also the cost of parking–not free in NYC). Remember that the cost of driving is not just the cost of fuel, but also depreciation.

    Here’s what the IRS has to say about that:

    A low of 14 cents (service or charity) and a high of 58 cents (business). Derb being proficient in both math and programming could likely calculate a fairly accurate cost.

    Everything Mrs. Derbyshire bought in Chinatown (or wanted to buy) is available online with cheap or even free delivery. Any savings she believed she was getting in Chinatown are fictitious owing to the cost of transportation.

    Then there’s the fact that Derb allowed his wife to dictate to him. By his own admission he’s a “Herb”. I bet he’s uttered the embarrassing line, “Happy wife, happy life,” before.

    Did it not occur to him to refuse to drive and announce that they were both taking the train and that was final? If she protested that arrangement, he could’ve charitably offered her the use of the car and announced he was taking the train (or not going at all).

  33. @blake121666

    Do you even know what LINEAR algebra is? This problem is quadratic!

    Yes, I know what linear algebra is. The LINEAR in linear algebra refers to linear COMBINATIONS of basis vectors. The unit vectors i j k are one set.

    This problem involves quadratic vector spaces, which is (only slightly) advanced linear algebra.

    You know basic linear algebra and you think you know it all. Common problem.

    I suggest you study this text:

    Learn this (Chapter Five)

    As suggested at the end of chapter 4, the vector spaces Rn are not the only vector spaces. We now give a general definition that includes Rn for all values of n, and RS for all sets S, and more. This mathematical structure is applicable to a wide range of real-world problems and allows for tremendous economy of thought; the idea of a basis for a vector space will drive home the main idea of vector spaces; they are sets with very simple structure. The two key properties of vectors are that they can be added together and multiplied by scalars. Thus, before giving a rigorous definition of vector spaces, we restate the main idea.
    A vector space is a set that is closed under addition and scalar multiplication.

  34. @blake121666

    All you’re doing is to say Y2 = Y1 + Y3 where Y3 = Y2 – Y1. Not interesting.

    • Replies: @blake121666
  35. MBlanc46 says:

    I (wisely) gave up my pursuit if mathematics after my undegraduate degree. But I still retain a soft spot (awe, in fact, for the discipline. It is heartbreaking, although not surprising, that the Neo-Stalinists now have it clearly in their sights.

  36. Truth says:
    @Mr. Anon

    LOL, you clowns spend a awful lot of your time hand-wringing over whom the-powers-that-be want to hire, in a career field in which you have no personal attachment.

    • Replies: @Mr. Anon
    , @Mr. Anon
  37. Truth says:
    @Daniel H

    Big Ol’ massive hands, neck and shoulders on that Bird there, eh, Old Bean?

    • Replies: @Mr. Anon
  38. @James Speaks

    The question was: what is Y1 in terms of Y2 when each of those is parameterized by a quadratic X? In other words, get the “X” out of the equations. It was really that simple – as Derbyshire said.

    You are the one who doesn’t understand.

    The “basis vectors” as you said in the previous reply are not linear at all. The Ys are PARAMETERIZED by quadratics (parabolas). Complicating the issue by talking of “basis vectors” only gives you a complicated mess of an affine space where your “basis vectors” change at each point – which means they aren’t proper mathematical vectors at all (they aren’t closed under addition and scalar multiplication). These “vectors” you imagine are not even affine actually (a physics vector space – as opposed to a mathematics vector space). A parabola has the weird symmetry of your original post – which is not linear in the space itself (doesn’t scale linearly).

    You are quite confused.


    I have not introduced any “Y3”. It is you who is introducing “basis vectors” into this non-vector space.

    And it is you who doesn’t know your “(only slightly) advanced linear algebra” you are musing about. In general, the Ys have a non-linear relationship, period.

    • Replies: @James Speaks
  39. Roy says:

    I would include Spengler as the #1 civilizational eschatology expert. His solution for the civilizational cycle of China (and India, and all non-Western civilizations of the day) was that in fact China was dead. Meaning, he saw Chinese civilization as a mindless sclerotic zombie running on the fumes of its past. That was in general the view from the West when gazing upon the once great civilizations of the East – chaos, fossilization and complete lack of innovation. Had Spengler lived longer he would have had to explain the rise of both Japan and China, but death had spared him of the need to explain anything further.

    One might say that perhaps Spengler was right. The recent rises in the fortunes of China and India is perhaps not the result of these civilizations’ own vitality, but of the transfusion of Western vitality, ideas and science into them, awakening the zombie with a Western electric shock.

  40. @blake121666

    The question was: what is Y1 in terms of Y2 when each of those is parameterized by a quadratic X? In other words, get the “X” out of the equations. It was really that simple – as Derbyshire said.

    No. The question was:

    Dumb (but real and research-related) question:
    Y 1 = aX² + bX + c
    Y 2 = dX² + eX + f
    What is Y 1 in terms of Y 2?
    r/t if you know a good high school math teacher

    Y1 and Y2 are functions in X. To write Y1 in terms of other 2nd order polynomials, you need a set of them, i.e. the vector space is quadratic. From the text I referenced but you refuse to acknowledge:

    pp 12-13

    1.2 What are Vectors? 13
    (C) Polynomials: If p(x) = 1 + x 􀀀 2×2 + 3×3 and q(x) = x + 3×2 􀀀 3×3 + x4 then
    their sum p(x) + q(x) is the new polynomial 1 + 2x + x2 + x4.
    (D) Power series: If f(x) = 1+x+ 1
    2!x2+ 1
    3!x3+ and g(x) = 1􀀀x+ 1
    2!x2􀀀 1
    then f(x) + g(x) = 1 + 1
    2!x2 + 1
    4!x4 is also a power series.
    (E) Functions: If f(x) = ex and g(x) = e􀀀x then their sum f(x) + g(x) is the new
    function 2 cosh x.
    There are clearly dierent kinds of vectors. Stacks of numbers are not the
    only things that are vectors, as examples C, D, and E show.

    Your rigid little mind learned that vectors are stacks of numbers and you refuse to learn anything beyond that.

    • Replies: @blake121666
    , @blake121666
  41. Anonymous[403] • Disclaimer says:
    @Achmed E. Newman

    Hello, Achmed,
    Yes, of course, a native accent rarely entirely goes away. Among writers on this website, Linh Dinh, for example, still has a noticeable accent. But he doesn’t promote it, push it forward as part of his professional persona. But Derbyshire does. A colleague of his at National Review that I know says that if you disagreed with him he would come on like an Oxford don chastising an undergrad, really pushing the accent to emphasize he was English and therefore in all ways superior to a lowly American.
    I guess that works with Anglophiles. For the rest of us, an English accent only says gay.

  42. @James Speaks

    I have a quite advanced level of knowledge in the subjects you think I have a “rigid little mind” about. That is why I showed you where you are misunderstanding these things.

    You misunderstand the quite introductory text you cite and are mis-applying vectors with what you are doing here. And that is why you think your “series of quadratics” nonsense is something other than nonsense. Your “series of quadratics” crapola is a result of you applying a linear analysis to a non-linear problem, has nothing whatsoever to do with the original problem, and has everything to do with you not understanding the methods you are using. You are chasing your own tail.

    The original problem merely asked to express Y1 in terms of Y2. And that is what Derbyshire showed how to do: solve for X in terms of Y2 and plug that into the Y1 equation. But that doesn’t really give one any insight into how Y1 and Y2 are related because you’d of course have the messy square root in Y1 from the quadratic formula. It is more informative to remove the X by simple manipulation of the equations to get that:

    A * Y1^2 + B * Y2^2 + C * Y1 * Y2 + D * Y1 + E * Y2 + F = 0

    where the (A,B,C,D,E,F) are some constants based on the initial (a,b,c,d,e,f) constants. And so this obviously shows that the initial parametric equations constrain the Y1 and Y2 to some conic section. This is all one can say given the generality of the initial problem. The Y1 and Y2 have this nonlinear relationship to each other.

    A general parabola (ax^2 + bx +c) cannot be referenced with bases vectors in a 2-dimensional linear vector space because parabolas are not linearly similar to each other – like a circle where any circle can be expressed as a translation (addition) and scaling of any other circle. So you cannot come up with some “parabolic coordinate system” akin to the “polar coordinate system” for circles. If you do, you’d find that your bases vectors will need to expand or contract depending on where they are at that particular point in the space. IOW the space is NOT a linear one. And so your thinking about it as if it IS linear is bogus and a misunderstanding of the tools you are using.

    You should have been clued into that by the very fact that your “series of quadratics” crap is actually ridiculously more involved than the original problem itself! It’s as if I asked you to tell me what “x” is if “x+5 = 10” and you respond with “well if you square the circle … blah blah blah”. You didn’t address the problem at all. You merely showed that you are quite confused about the tools you chose to misuse for the problem. Do you see that?

    • Replies: @James Speaks
  43. @blake121666

    You’re just plain stupid, and wrong.

    A general parabola (ax^2 + bx +c) cannot be referenced with bases vectors in a 2-dimensional linear vector space because parabolas are not linearly similar to each other – like a circle where any circle can be expressed as a translation (addition) and scaling of any other circle.

    Actually, they are. All parabolas are similar. Any parabola can be scaled up or down and translated to become congruent to any other parabola, and you don’t know that, stupid.

    A parabola is the locus of all points equidistant from the focus (point) and directrix (line).

    You’re like a parrot, know a bunch of words but not their meaning. Oh wait, parrots understand some meaning.

  44. @James Speaks

    I had to rethink that last reply from me.

    If one wished to reference something in two dimensions using polar coordinates instead of cartesian coordinates, one would use the basis vectors r_hat and theta_hat where:

    x * x_hat = r * cos(theta) * r_hat
    y * y_hat = r * sin(theta) * r_hat

    x * x_hat in this notation is simply saying the vector “x” – has magnitude x in the direction of x_hat (right-left on a cartesian coordinate). Similarly for y in the up-down coordinate. And of course r_hat is a unit vector in the radial direction and theta_hat would be a unit vector in the angular direction.

    In the problem at hand, we have that x and y are parametric in t (let’s use x,y, and t rather than the original’s Y1,Y2, and X) for a more standard way of writing it.

    x = a * t^2 + b * t + c
    y = d * t^2 + e * t + f

    Now, if you wish to look at this as transforming the bases of x,y into a basis for t (and something else – let’s keep it 2 dimensions), what are those bases? Are you saying that you are using the focus and directrix of a parabola as the bases?

    How might looking at these parametric equations in this way help in understanding those parametric equations better?

    Is this what you are saying? And how does this help the initial person in his desire to know how x and y relate to each other when they are parameterized by a quadratic t? I can’t see why you would impose a vector space here.

    • Replies: @James Speaks
  45. @Anonymous

    Yes you deserve someone to call you a troll. The English civil war of the 1640s is part of every English speaking country’s core history.

  46. I’m sure it was Flushing Main Street that I waited in for friends who were hiring a car after arriving at JFK to pick me up en route to an old-fashioned sporting club to the NW. My impression was (?false memory is) that the businesses were all Vietnamese apart from three Korean one’s. Of about 3000 people I estimated passing in the street I guess no more than three were black and very few spoke English when I sought directions. Does that sound like your Flushing Main Street (It would have been about 6 years ago)?

    You have been too kind to Turkheimer and unkind to yourself. You omitted to notice that Turkheimer followed up immediately by getting the Y1 and Y2 equations mixed up. As for yourself, you let in those who might want to to put you down as frivolous and trivial – which you are not – by that silly bit about an “Editrix”. (That would have its place only when you were smiling at her).

    Let me testify that you are an overvachiever, not only in wit and evidence based argument, but in mathematics. Fancy a chap with a third getting me through to page 135 (from memory) of Prime Obsession actually sure that I truly understood the Riemann Hypothesis and would remember forever “all non-trivial zeros of the theta function have real part one half”. Do you BTW think there will be an Andrew Wiles to invent branches of mathematics and seclude himself for years to prove, or disprove it?

  47. anonymous[191] • Disclaimer says:

    I’ve been writing code for 37 years. I’ve written production code (well, OK, if you count the crap I wrote as a kid as “production”) in over 20 languages and 7 flavors of assembler. By far, the most difficult language for me to learn was (and continues to be): JavaScript.

    First, let’s ignore all the stupid Turing-machine joke languages like Brainfuck and so on. Those are languages developed to prove a point about Turing machines and quines, not to write code in.

    A lot of people say “the most difficult language is your first,” but that certainly wasn’t true for me and I don’t think it’s generally true. My first language was BASIC and my second language was a Z80A assembler (that I typed in from a magazine ). Getting things done in either was easy to learn with diligence and experimentation, because the ecosystem of early 80s computing was very easy to control and understand.

    Other than JavaScript, the most difficult language I’ve ever used was PL/S as an intern with IBM in the early 90s. Not because it’s a terribly difficult language to understand, but because the compiler was a buggy piece of crap and any useful documentation was generally protected by IBM’s highest intellectual property level (Registered Confidential), which made it seriously difficult to RTFM.

    As the story unfolds, you might see where this is going—ecosystem. Most professional-grade languages aren’t very difficult for a trained and intelligent programmer to grasp. What sucks is trying to navigate the conventions of other programmers and the ecosystem the code generally resides in. The ecosystem of my TS2068 or Commodore 64 was stable and easy to understand. C’s ecosystem is slightly less easy to understand, but it’s still pretty stable and manageable. Java—well, less, still, but it’s manageable. And so on down through the Rs, Pythons, SQLs, and Rubys of the world.

    When we get down to the bottom of the list, we find JavaScript. The JS ecosystem is freaking nasty. Whether you’re in Node.js or a browser, solving trivial tasks is a serious pain in the ass. There’s LOTS of documentation, and that’s actually the problem because the waning and waxing enthusiasm for JS means that most of it is out of date, and that can be a serious problem with an ecosystem that is almost entirely composed of arcane conventions.

    Finally, reading and understanding the code of other JS developers is unpleasant at best and impossible at worst. This is the real killer for JavaScript, because the best way to learn a language is to experiment with an experienced programmer’s existing code. And that’s not really a lot of fun with JavaScript, either.
    118.4k Views · View 480 Upvoters

  48. To the extent that money is important, this subject of innumeracy or mathematical illiteracy is among the most important. The following is a brief introduction to the so-called nominal method of interest calculation that is prohibited as criminal fraud in the U.K. while being required by law under U.S. federal Truth-in-Lending legislation. It’s completely absurd and insane.


    Ever continuing / compounding nominal-rate-fraud

    Concurrently, by total amount, all debt outstanding in Canada today is independently a function of the continued and ever-continuing use of the egregiously and insidiously fraudulent nominal method of interest calculation and declaration / disclosure that has been illegal in Canada since 1880 on mortgage-secured debt (under the same law (s. 6) as the No penalties provision) and since 1897 on virtually all other debt (under s. 4), and, again, notwithstanding that it has also been recognised and banned as criminal fraud throughout the U.K. since 1974 on the grounds that it is “false and seriously misleading” (and which is itself the understatement of the century).

    More toward the near inconceivably fraudulent higher-end, an average or typical payday-loan in Canada (and the U.S., and most everywhere else) defines a real interest rate of about 30,000% per annum, but which is passed off, using the nominal method, to the nominal borrowers and to the public generally, as about 300%.

    Its broader purpose and policy is to help ensure that the working-poor and the rank-and-file military remain in a state of perpetual poverty and political powerlessness by targeting their de facto working-capital.

    If the borrower receives for example, $335 today, in exchange for an obligation to pay $400 in ten days’ time, then the principal amount is $335, the interest charge is $65, and the interest rate is 64,622% per annum. Here is the spreadsheet formula (for ms excel and apple numbers):


    = 646.22 or 64,622% per annum.

    The payday-lender makes a gross rate of return of 64,622% per annum on their $335 principal investment over and for that ten-day period. If you had an effective / real-interest-rate daily accrual savings account, then it would have to pay interest at an annual rate of 64,000% for $65 to accrue on a deposit of $335 over ten days.

    If the borrower instead receives $350, then the interest charge also declines to only $50, and the rate of interest is:


    = 129.82 or 12,982% per annum.

    If instead we go the same $15 the other way, then the borrower will receive only $320 and the interest charge also increases to $80, and the rate of interest is:


    = 3444.20 or 344,420% per annum.

    Note especially the exponential sensitivity. Our base case is $65 and 64,000%. Based on the same $400 paycheque and due in the same ten days, if you give the borrower an extra $15, then the rate of interest declines to just under 13,000% per annum. But if instead you take away another $15, then the rate of interest balloons to 344,000%.

    Every competent engineer, math teacher, insurance professional, and banker on the planet knows how to do the calculation. But former bank lawyers and bank solicitors after they have been appointed judges – not so much.

    That’s policy.

    And even ignoring the staggering amount of money involved, it is an incredibly dangerous policy. Imagine the extinction of humanity beginning with the following (as yet) hypothetical news story out of equatorial Africa:

    100 million people now lie dead from the worst infectious disease outbreak in human history. An international panel of experts has identified the primary cause of the crisis getting out of control as being that the original samples of the lethal virus, with an observed actual growth rate of 5% per day, were mislabelled by the technician responsible, as having an AGM or Annual Growth Rate of 1,845% instead of the real and actual 5.4 billion per cent. The technician, a recent graduate of Bankers’ Math University, responded when asked for comment: “I’m really sorry. I honestly didn’t know.”

    A species that celebrates and rewards professional incompetence in pursuit of financial gain is a species living on borrowed time.

  49. Mr. Anon says:

    LOL, you clowns spend a awful lot of your time hand-wringing over whom the-powers-that-be want to hire, in a career field in which you have no personal attachment.

    You have no idea what my personal attachments are, where I work or what I do. Because I don’t talk about them, at least not directly. You, of course, just assume all the white people here are gas station attendents (We must all live in Oregon, the only state that still doesn’t have self-serve). Why gas-station attendents would be familiar with Physics Today is just one of life’s mysteries. There are so many mysteries for stupid people like you.

    I do however know some things about your personal attachments – or lack thereof – because you are a braggart and a loud-mouth. I know, for example, that your own son (illegitimate presumably – you’re black, after all) doesn’t like you, because you’ve said as much here. Well good for him. There is – to judge from your posts here – nothing to like about you. You’re a loser and an idiot, who excuses black criminal behavior, who thinks that all women in the public eye are actually men, who thinks that the Earth is flat, and that water is a fuel.

    You’re a vile, creepy, stupid shithead

    • Replies: @Truth
    , @Truth
  50. Mr. Anon says:

    And I suppose you think she is really a man too, don’t you?

    Why don’t you take your creepy fascination with trannies elsewhere, weirdo. There are undoubtedly websites that cater to your bizarre fetishes. You should hang out there.

  51. Mr. Anon says:

    People like you can’t fathom why anybody would be interested in the state of society or feel they have any stake in it. Your only concern in life is where you can find your next 40, blunt, and shemale hooker.

    • Agree: By-tor
  52. Truth says:
    @Mr. Anon

    You’re a vile, creepy, stupid shithead

    • Replies: @Mr. Anon
  53. Truth says:
    @Mr. Anon

    You have no idea what my personal attachments are, where I work or what I do. Because I don’t talk about them, at least not directly.

    Oh. Well chin up Bro, you’ll find a job one day.

    • Replies: @Mr. Anon
  54. @blake121666

    I can’t see why you would impose a vector space here.

    Because he asked a question that doesn’t have a good answer and I answered a similar question that does. Two, actually.

    Q: If Y1 = AX + B and Y2 = CX + D how do you express Y1 in terms of Y2?

    A: You can’t, unless Y1 and Y2 are colinear, in which case the problem is trivial.

    If you want to know how Y1 relates to Y2 you can find the difference and call it Y3 and then

    Y1 = Y2 + Y3

    But note, Y3 is just Y1 – Y2 and all you have done is express Y1 = Y2 + (Y1 – Y2)

    If you solve for X and find that (Y1 – B)/A = (Y2 – D)/C

    then you may find the point where Y1 = Y2 as

    CY-CB = AY – AD … (C-A)Y = CB – AD … Y = (CB – AD)/(C-A)

    Except this is not expressing Y1 in terms of Y2, this is merely finding a point of intersection. You’ll get either no solution (parallel lines), one solution (intersecting lines) or an infinite number of solutions (colinear) depending on whether A = C and if so, if B = D.

    If you do the same with parabolas, you’ll get two points of intersection, usually, but only one if the vertices coincide, and none if they don’t and the axes of symmetry conincide. (Usually, depending on the distances between focii and directixes.)

    But again, this is not expressing one function in terms of another; this is merely find points of intersection.

    To take the original question, expressing one function in terms of another, and to make it meaningful, one must assume the ‘mathematician’ who asked it is sub-par (which I think we agree that he is) and bring in vector spaces.

    To express one line in terms of another is meaningless; to express one line in terms of two lines that are orthogonal is a question that has meaning.

    To express one parabola in terms of another is meaningless.

    To do the same thing with a set of parabolas, that is, to use a vector space of carefully chosen parabolas, gives the question meaning.

    This is why I imposed vector spaces.

    • Replies: @blake121666
  55. If you do the same with parabolas, you’ll get two points of intersection,

    Need to make it clear that zero, one or two is for two parabolas where the directices are parallel.

    If not, then you can have zero, one, two, three or four points of intersection.

    • Replies: @James Speaks
  56. @James Speaks

    On the other, other hand (Mote in God’s Eye ?) since parabolas are similar, you can find the translation of the vertex and scaling factor, but if that is what Turkheimer wanted, that is what he should have asked for. BTW, this was what I had said in my original response.

    Dear Mr. Derbyshire, sir:
    All parabolas are similar, just as all circles are similar.
    A circle is the locus of all points a distance r from the center C. You can write a transform from one circle to another with a translation of the center. With C1, C2 and T written as vectors (ordered pairs), (C1 + T = C2) and a scaling factor r2/r1.
    A parabola is the locus of all point equidistant from the focus and the directrix (line). Thus F1 -> F2 and scaling factor = p2/p1 where p is the distance from the vertex to the focus. Perhaps he wanted to know how to transform one parabola to another? Naw.
    This is what the guy was asking for: Express Y2 in terms of Y1.

    So maybe the guy was asking for the translation vector and scaling factor, but I doubt he knew that parabolas are similar, which is what makes it possible. If anything, I think his blissful ignorance made his question ambiguous.

  57. Mr. Anon says:

    And, apparently, a Peewee Herman fan.

    Not surprised.

  58. Mr. Anon says:

    Huh? That isn’t even witty.

    I’m sure you have a very rewarding job as a used car salesman or some such.


    • Replies: @Truth
  59. Truth says:
    @Mr. Anon

    Aaah Grasshopper…
    I have said what I do…
    on multiple occasions, Grasshopper.
    YOU are the one run…
    who rack CONFIDENCE…

    The Gods, Grasshopper…
    Are unkind…

  60. @James Speaks

    No, not at all.

    Taking your linear example. If you say that:

    Y1 = AX + B
    Y2 = CX + D

    is saying that both Y1 and Y2 are linear in X. And therefore they are linear to each other. You can solve for X in either equation and get a linear equation in the Ys – as you did in your example. Talking about “Y1 – Y2” or the point where “Y1 = Y2” is just your own confusion about what is being asked. If Y1 and Y2 are both linear in a parametric X then they are linear to each other:

    CY1 – AY2 = CB – AD

    This is a linear relation between Y1 and Y2. There’s no particular reason to care about “Y1 – Y2” nor the point where “Y1 = Y2”. You are confused.

    You are also confused about what a “line” is by your statement:

    “To express one line in terms of another is meaningless; to express one line in terms of two lines that are orthogonal is a question that has meaning.”

    Your two orthogonal lines are simply your imposition of a coordinate system on how you are thinking about a line. A line is one-dimensional to itself – one need only know “where” one is on a line in that one dimension. Its relation to another line would be another dimension – if it is linearly independent (that is the definition of linear independence). Orthogonality is of course not required – it just makes the referencing tidier in the math. Linear independence is all that is needed for bases in 2 dimensions – not orthogonality.

    So a line is 1-dimensional and 2 linearly independent lines is 2-dimensional.

    A parabola is merely a curved line. The relation of 2 parabolas to each other is the same as the relation of 2 lines to each other. One can one-dimensionally reference where one is on a parabola as the distance from its apex for instance. And one can 2-dimensioanlly reference another parabola. So instead of 2 “straight lines” in your imposed 2-dimensional coordinate system, you’d have a reference of the one curved line to the other curved line – 2 dimensions.

    For example.

    Say that one fires a trajectory out of a cannon on Earth’s surface and that the only force is constant gravity (neglect all other forces but gravity).

    F_x = 0
    F_y = -mg (g is constant)

    Therefore one gets that:

    x = x0 + v0_x * t
    y = y0 + v0_y * t – 1/2 g * t^2

    Eliminating t in these equations gives that

    y = a * x^2 + b * x + c

    The trajectory is a parabola for a projectile under a constant force in one dimension and no force in the other.

    Now if you were ON this trajectory, this trajectory is your basis for anything you view outside it (your reference coordinates are that you are ON this trajectory – x = y = 0 for you). In the initial coordinate system of the problem, one has your trajectory being seen as:

    y1 = a * x^2 + b * x + c

    But to you, who is ON this trajectory, x=y=0 t all times.

    Some other trajectory, which is not the one you are on, is fired from the cannon:

    y2 = d * x^2 + e * x + f.

    The question to you is: What is this other trajectory relative to YOUR coordinate system (which itself is another parabola in the original coordinate system)?

    This is what is being asked here.

    And the answer is to eliminate the common “x” in the equation to find that:

    A * y1 ^2 + B * y2^2 + C * y1 * y2 + D * y1 + E * y 2+ F = 0

    And this is a general quadratic – which is some type of conic section depending on what the (A,B,C,D,E,F) are (ellipse, hyperbola, parabola, point, or line).

    So YOU would see that other trajectory as some particular conic section.

    • LOL: James Speaks
  61. MEH 0910 says:

Current Commenter

Leave a Reply - Comments on articles more than two weeks old will be judged much more strictly on quality and tone

 Remember My InformationWhy?
 Email Replies to my Comment
Submitted comments become the property of The Unz Review and may be republished elsewhere at the sole discretion of the latter
Subscribe to This Comment Thread via RSS Subscribe to All John Derbyshire Comments via RSS