My dream Wednesday night was really very interesting, and so was Raph Levien's.
"If you talk that way and don't make that distinction, they won't believe you. They aren't going to believe you, they'll think you're lying."
It's good that I returned to this subject after quite some time away from it. Back in 1995, I got interested in why the well-known digital root divisibility test works, and also in generalizations to other bases. I managed to prove a pretty good generalization which shows clearly why it works, and later I also showed when it won't work.
Yesterday I also examined the last digit divisibility test, and once again showed why it works and when. Interestingly, these results can be used to establish the relatively obvious conclusion that the two tests never both work for divisibility by a particular digit.
Another simple corollary: the last digit test works to tell whether a number is even or odd in an even base. The digital root test works for this purpose in an odd base.
Outstanding question: the digital root test "doesn't work" for certain digits (in general, most digits), in the sense that it can and sometimes will give wrong answers for those digits. (It's not guaranteed to give a wrong answer every time -- if it were, it would still be a perfectly accurate test, you'd just have to reverse the interpretation of the results, much in the sense that someone who can actually get a 0% on a multiple choice exam or a Rhine ESP test, while still answering the questions, has something strange going on.) The point is that it gives answers which are not directly connected to what the test is supposed to be testing for: we would like to say that the test result and the actual divisibility property are "uncorrelated". But is this true?
I'd like to know whether the test gives you any information at all in this case. (One example: adding the digits in a base 10 number to test for divisibility by 6. We know that this "doesn't work", but is it right more often than 1/6 of the time, wrong more often than 5/6, or is it right exactly 1/6 of the time? Can you get any information at all by applying this faulty test, or is it completely useless?)
I wrote a program to experiment with this. It's not clear quite yet what the result is.
I'm trying to write these proofs and explanations of them up nicely in LaTeX. The slight problem is that I don't know a whole lot of LaTeX. I guess this is a great way to learn more.
0 is divisible by everything, and everything is divisible by 1. Sounds like blood types, doesn't it?
What cryptographic systems lack in subtlety, they make up for in malice...(Whitfield Diffie and Mary Fischer, explaining the difference between cryptographic systems and God, "Deciphment Versus Cryptanalysis", in Richard Parkinson, Cracking Codes: The Rosetta Stone and Decipherment)
Sumana mentions a piece by Annalee Newitz (whose writing I've sometimes read in the past; she likes to write about free software reasonably often). This one, on the other hand, is about love and sex, and what they have to do with one another. Newitz mentions the "scarcity economy of love" and suggests that it's a problem that people either think that sex must be connected with love, or that it must not. This reminds me a lot of the divisibility problem: there, if the divisor is a divisor of the highest possible digit, the divisibility of the sum of the digits in a number must be the same as the divisibility of the number itself. Either they are both divisible, or they are both not divisibile. On the other hand, if the divisor is not a divisor of the highest possible digit, then the divisibility of the sum of the digits in a number will certainly sometimes be the same as and certainly sometimes be different from the sum of the digits of that number, depending on what number you choose. So I guess Newitz is telling us that sex and love are like the divisibility of a number and the divisibility of the sum of digits in the number, where the highest digit in the base is not itself divisible by the divisor of interest: they're sometimes the same and sometimes different, sometimes both present, sometimes neither, and sometimes either one by itself in the absence of the other, and if we assume that they're one way or the other all the time, we're sure to be mistaken eventually.
Of course, I am one of these people who are part of the problem, per Newitz's view, on account of having a traditional (in two senses) theory about the situation.
Yeah, I'm really far from being able to relate to her attitude. I go far beyond what she criticizes as unreasonable.
And, separately, alas for us if her conclusion is right:
[O]ne often hears the truism "communication is the key." The idea is that we can bridge that gulf of relationship misunderstandings if we're just "honest," and tell our sex partners up front what we expect from them. But communication and honesty can't possibly be solutions to a problem whose roots are self-delusion and plain old uncertainty. If few of us truly know what sex means to us, or what we want out of our dates, how can we be honest about our feelings unless we say something like, "Duh, I don't know"? That's the sort of honesty we could all do without.We long to use words like "honesty" when it comes to love and sex not because we are confident about our intentions but because we want to ward off the disorienting ambiguity of desire.
I remember quoting, and I'll quote again, the lines from Aeneid VI:
O tandem magnis pelagi defuncte periclis!
Sed terrae graviora manent. In regna Lavini
Dardanidae venient; mitte hanc de pectore curam;
sed non et venisse volent. Bella, horrida bella,
et Thybrim multo spumantem sanguine cerno.
Non Simois tibi, nec Xanthus, nec Dorica castra
defuerint; alius Latio iam partus Achilles,
natus et ipse dea; nec Teucris addita Iuno
usquam aberit; cum tu supplex in rebus egenis
quas gentes Italum aut quas non oraveris urbes!
I'll try a quick informal translation:
Oh you who passed such dangers on the sea!
But graver ones remain on land: into the kingdom
of Lavinius you Trojans will come, so send that fear away.
But you won't wish that you had come! I forsee wars,
horrible wars, the Tiber flowing red with blood.
The rivers of Troy and the enemy camps will
come back again for you; another already in Latium
Achilles waits for you, his mother, too, a goddess. And Juno
who hates you won't leave you alone: when you in dire straits
go as a beggar, what cities won't you ask for help
in Italy?
I told Zack that the best figurative translation of "Non Simois tibi ... defuerint" for Americans might be "It will be another Vietnam for you".
One reason this passage is so disturbing is its context. Aeneas has survived a long war -- in which his country was destroyed -- and then wandered for years at sea and nearly been killed there, too. He's learned that his destiny is to go to Italy and to found a new civilization there. So he dutifully heads for Italy, never having been there before, and not entirely sure he'll make it. But in Italy, he believes, "the fates offer us peaceful seats" ("Latium, sedes ubi fata quietas / ostendunt").
So now, through much effort (the theme of the first half of the poem), Aeneas has almost made it to Italy at last, and he stops by to ask this prophetess to give him more advice about his destiny. In some sense, he believes that the poem is about to end, for he's on the verge of reaching Italy, and reaching Italy was what he was supposed to do, wasn't it? And then she says this! She tells him that, not only will Italy not be peaceful, but that he'll practically have to fight the Trojan war all over again from scratch when he gets there. (Indeed, the entire second half of the Aeneid is devoted to the war Aeneas does, in fact, have to fight when he finally makes it to Italy; the Sybil wasn't joking around, and the river Tiber does run run with blood.)
The great Dryden translation has
Escap'd the dangers of the wat'ry reign,
Yet more and greater ills by land remain.
The coast, so long desir'd (nor doubt th' event),
Thy troops shall reach, but, having reach'd, repent.
Wars, horrid wars, I view -- a field of blood,
And Tiber rolling with a purple flood.
Simois nor Xanthus shall be wanting there:
A new Achilles shall in arms appear,
And he, too, goddess-born. Fierce Juno's hate,
Added to hostile force, shall urge thy fate.
To what strange nations shalt not thou resort,
Driv'n to solicit aid at ev'ry court!
I believe that the line "Sed non et venisse volent" ("But they will not also wish to have come", or Dryden's "but having reach'd, repent") must have been the most shocking to Aeneas of everything he hears in the whole poem. It's true that Aeneas gives a speech in reply in which he denies being afraid of anything, but immediately beforehand he's just expressed his state of mind and clearly shown that he was relying on reaching Italy to be the end of the story. And, directly in reply, the Sybil says no, there is a second half of the Aeneid, books seven through twelve, and in them you're going to fight your war again, another Vietnam...
Annalee Newitz is the Cumaean Sybil.
Speaking of having a theory, I wrote more of "Existence and Uniqueness". This is starting to remind me of my collage (which I started about a year ago), a private creative project which I always mention in my diary that I'm continuing to work on.
Slightly edited from a post by Daniel Wang to peacefire-technical:
Did we also mention PanGo and GlobalTrack's customer tracking system?In fact, over 2000 malls in the US have installed their marketing and sales lead system that will allow them to detect and track transponder signals from cell phones, PDAs, and anything that is enabled with WAP, 802.11b, Bluetooth, CDMA, or GDMA.
Guess what? They're already working on getting the station to read phone and PDA serial numbers too, so they can track where YOU specifically go every single time you visit the mall.
Oh look, by the year 2010, it's estimated that almost everyone will have some sort of wireless device, whether it's a Proximity card or a cell phone or a satellite dish. Welcome to real-world persistent solid-state client identifiers, or in other words, real-world cookies. Bum bum bum!
... there are some downsides to devices which easily and automatically interoperate with other devices, aren't there?
I sent a letter to Linuxcare about my severance agreement. I went to the chiropractor.
Tomorrow, I was going to go to HSC, but now I'm not.
Contact: Seth David Schoen
