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Good Math, Bad Math

Finding the fun in good math; Shredding bad math and squashing the crackpots who espouse it.

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Mark Chu-Carroll (aka MarkCC) is a PhD Computer Scientist, who works for Google as a Software Engineer. My professional interests center on programming languages and tools, and how to improve the languages and tools that are used for building complex software systems.

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August 7, 2008

Encryption, Privacy, and You

Category: Encryption

As you've probably heard, the US customs service has, recently, asserted the right to confiscate any and all computers and/or digital storage carried by anyone crossing the US border. They further assert the right to demand all passwords, encryption keys, etc., from the owners. They even further assert the right to keep or make copies of any data that they find, and to share it without limit with anyone they choose.

I don't think I really need to stress how insane this is. Back when I worked for IBM, I frequently travelled to Canada, because I worked with development labs in Toronto and Ottawa. When I did that, I carried a computer full of stuff that IBM considered to be highly confidential and highly sensitive. (I've even still got a wall-plaque from IBM thanking for me work on a project, where I'm not allowed to ever tell anyone what I did to earn it!) What this policy says is that the border service would have the right to turn that information over to anyone they wanted, without informing me or IBM that they had done so. Further, some of the information on that laptop was encrypted, and I did not have the key. They were encrypted with a system that would only allow them to be opened if the computer could contact a particular IBM server from inside the IBM firewall. So not only could the border service have confiscated the computer and passed on confidential or private information - but they could have arrested me for refusing to decrypt the information on the computer - even though I couldn't decrypt it.

This isn't new news. They've been doing this for a while, and we know they've been doing it - they've made absolutely no attempt to hide it.

The reason that I'm writing about it now is because I just read something on Salon about how an allegedly knowledgeable and tech-savvy person recommends coping with this, and I can't possible disagree more strongly. On the Salon Machinist blog, Denise Caruso wrote:

Swire notes that agents at the border are going further than just taking image copies of people's hard drives. They're actually demanding passwords and encryption keys so they can examine the contents.

Of course, they promise to destroy the copies and the keys as soon as they're done -- as long as they don't find anything illegal, like a downloaded song you didn't pay for -- so no security worries there, right? There's no such thing as a crooked customs or Border Patrol agent.

This gives government agents access to information they would never get by opening up your suitcase. In addition to e-mail, spreadsheets, documents and personal financial information like credit card receipts and photos, nowadays they can also listen to your stored Skype calls and voice mails.

Not to mention that just having encrypted data on your hard drive causes suspicion, or at least throws down the gauntlet. If you were looking for illegal stuff and you ran into a file that looked like this,

qANQR1DBwU4D/TlT68XXuiUQCADfj2o4b4aFYBcWumA7hR1Wvz9rbv2BR6WbEUsy
ZBIEFtjyqCd96qF38sp9IQiJIKlNaZfx2GLRWikPZwchUXxB+AA5+lqsG/ELBvRa
c9XefaYpbbAZ6z6LkOQ+eE0XASe7aEEPfdxvZZT37dVyiyxuBBRYNLN8Bphdr2zv z/9Ak4
/OLnLiJRk05/2UNE5Z0a+3lcvITMmfGajvRhkXqocavPOKiin3hv7+Vx88

wouldn't you immediately need to know what it said? It could be a conspiracy! It could be a list of child pornographers! It could be a copyrighted magazine article! It could be a bootleg Led Zepplin video!

Urgh.

So I figure the best solution is to encode your files rather than encrypt them, so that you could hide your stuff in plain sight. If agents don't know something is encrypted and it looks innocuous, they won't compel you to give them the key. "Here's your laptop, ma'am. Sorry for the inconvenience."

That's the wrong answer. The solution isn't to try to hide the fact that you're taking your own/your employer's privace seriously. The answer is to make encryption so absolutely routine that (A) finding encrypted files on a computer is so common and routine that it can't be used as a distinguishing characteristic to allow them to justify confiscating your computer, and (B) to make it so incredibly painful and laborious for them to get any data off of a computer that they give up.

The first part of instructions for how to do this are below.

August 6, 2008

Bad Fundies! Chabad takes on Science.

Category: Debunking Creationism

Someone sent me another stupid Jewish article. It's still not the wonderful relativity denial that I lost, but it's pretty delicious as stupidity goes. This time it comes from Chabad. For those who aren't familiar with it, Chabad is a Chasidic organization, which originally formed around people following a very famous Rabbi from the town of Lubav after he emigrated to the US. Chabad grew into a very large fundamentalist organization that is very devoted to what they call outreach. (I call it proselytization.)

Anyway - on to the article: "Are Science and Religion a Contradiction?". Large swaths of it are just rehashes of standard fundamentalist crap - indistinguishable from the kinds of rubbish we routinely hear from the various Christian fundies, but with a bit of low-budget hebrew mixed in. For example:

In the 19th Century it was the prevailing view of scientists and modernists that human reason was infallible in "scientific" deductions and that sciences such as physics, chemistry, mathematics etc., were absolute truth, that is to say, not merely accepted truths but absolute. Speaking in Jewish terms this meant the establishment of a new idolatry, not of wood and stone, but the worship of the contemporary sciences and philosophies.

In fact, in the face of dogmatic and deterministic views of science prevailing at that time, a whole apologetic literature was created by well-meaning religious advocates and certain rabbis who saw no other way of preserving Torah heritage in their "enlightened" communities except through tenuous and spurious reinterpretations of certain passages in the Torah in order to accommodate them to the prevailing world outlook. No doubt they knew inwardly that they were suggesting interpretations in Torah which were at variance with Torat Emet, but at least they felt they had no alternative.


See, it's just another version of the old "science is a religion" shtick. No better than the dreck you'll find on, say, James Dobson's website.

Of course, if all that they did was rehash the same-old christian dreck, there'd be no good reason to waste my time writing about this meshugas. But they've got a few unique touches that are worth a moment or two.

August 1, 2008

Friday Random Ten, August 1st

Category: Music

  1. Kansas, "Byzantium": an example of why Kansas fans waited so long for Kerry Lofgren to return to the band. The guy's a brilliant songwriter. Even with Walsh's voice clearly aging and suffering from abuse, this is fantanstic stuff.
  2. Isis, "Wrists of Kings": Fairly hard post-rock. I like Isis a lot, but one thing about them that takes some getting used to is the "lead singer". In general, Isis has a sound a lot like Mogwai, but they do use vocals. And their vocals consist of a guy screaming hoarsely in the background. This track doesn't have the really awful vocals; in fact, the singing here is pretty reasonable.
  3. Genesis, "Squonk": I've been going back and listening to old Genesis lately. And it's really amazing to listen to. You can see why so many of the neo-progressives start out by trying to sound like Genesis - it's such an amazing, unique, engaging sound. This is off of the first Genesis album after Peter Gabriel left, so it's a pretty dramatic change from what preceeded it. But it's still distinctly Genesis - listening to "The Lamb Lays Down on Broadway" followed by this, it's clearly the same band.
  4. Marillion, "How Can it Hurt?": a bit off of the worst album Marillion ever made. Considering the album that it came off of, this isn't a bad song; but for Marillion, this is crap.
  5. Sonic Youth, "Rats": typical Sonic Youth. That means that it's fantastic, strange, full of odd tonalities and controlled noise. It somehow manages to be smooth and mellow and noisy and dark all at the same time.
  6. Godspeed You! Black Emperor, "Static: Terrible Canyons of Static": The deities of post-rock. No one can do this kind of music like Godspeed. I dearly wish they'd get back into the studio and do some more music. Their spinoff "A Silver Mt. Zion" is really good, but it's not Godspeed.
  7. King Crimson, "Indiscipline": Brilliantly goofy song from King Crimson. This consists of wonderfully improvised dark noisy guitar and spectacular drumming, interspersed with Adrian Belew talking about some thing, without ever saying what the thing is.
  8. Kruzenshtern and Parahod, "March": progressive klezmer. Wow. This is strange, and wonderful, and amazing, and really strange. I'd kill to be able to play my clarinet like that!
  9. Naftule's Dream, "The Aimless Path": In a wonderful coincidence of iTunes shuffle randomness, another bit of progressive klezmer. Naftule's Dream is a spinoff of the Shirim Klezmer orchestra. Shirim is fantastic, mostly traditional Klezmer. Naftule's Dream is the stuff that really pushes the boundaries of the genre. Highly recommended, and a lot easier to find than K&P.
  10. The Flower Kings, "Man Overboard": The Flower Kings are, probably, my favorite band. They started off as a blatant Genesis ripoff (see my comment above on "Squonk"), but they've evolved into a really amazing band with a real distinctive sound. You can't mistake the Flower Kings for anyone else - just a couple of seconds of any of their songs, and you know that it's them. This is one of my favorite of their shorter tracks - only 3 1/2 minutes! But it's got the most amazing twisted rhythm and chords in the chorus, connected by a highly contrasting smooth melodic bridge. It's amazing.

July 31, 2008

Nonsense Pretending: Probability as a Disguise

Category: Debunking Creationism

Once again, you, my readers, have come through with some really high-grade crackpottery. This one was actually sent to me by its author, but I didn't really look at it until several readers sent me the same link because they thought it was my kind of material. With your recommendations, I took a look, and was rewarded. In a moment of hubris, the author titled it A Possible Proof of God's Existence from Multiverse Assumptions.

This article is basically a version of the classic big-numbers probabilistic argument for God. What makes this different is that it doesn't line up a bunch of fake numbers and saying "Presto! Look at that great big probability: that means that it's impossible for the universe/life/everything to exist without God!". Instead, it takes a more scientific looking approach. It dresses the probability argument up using lots of terms and ideas from modern physics, and presents it as "If we knew the values of these variables, we could compute the probability" - with a clear bias towards the idea that the unvalued variables must have values that produced the desired result of this being a created universe.

Aside from being an indirect version of the big-numbers argument, this is also a nice example of what I call obfuscatory mathematics. See, you want to make some argument. You're dead sure that it's right. But it doesn't sound convincing. So you dress it up. Don't just assume your axioms - make up explanations for them in terms of math, so that it sounds all formal and mathy. Then your crappy assumptions will look convincing!

With that said, on to his argument!

July 30, 2008

Solving Tic-Tac-Toe: Game Tree Basics

Category: Game Theory

tic-tac-toe.png

Moving on from simple zero-sum games, there are a bunch of directions in which we can go. So far, the games we've looked at are very restrictive. Beyond the zero-sum property, they're built on a set of fundamental properties which ultimately reduce to the idea that no player ever has an information advantage over any other player: the complete payoff matrix is known by all players; no player gets to see the other players strategy before selecting their own; and so on.

Moving on to more interesting games, we can relax those assumptions, and allow information to be hidden. Perhaps each player can see a different part of the payoff matrix. Perhaps they take turns, so that one player gets to see the others strategy before selecting his own. Perhaps the game isn't zero-sum.

Non-zero sum games turn out to be disappointing from a game theory point of view. Given a suitable set of restrictions, you can convert a non-zero-sum game to a zero-sum game with an additional player. In the cases where you can't do that, you're pretty much stuck - the mathematical tools that work well for analyzing zero-sum games often simply don't work once you relax the zero-sum requirement.

The more interesting ways of exploring different realms of games comes when you allow things to get more complex. This comes about when you allow a players strategy selection to alter the game. This general takes place in a turn-taking game, where each players strategy selection alters the game for the other player. A simple example of this is the game of tic-tac-toe. The set of strategies of the game for a given player at any point in time is the set of open squares on the board. Each time a player makes a move, the game is altered for the other player.

This makes things much more interesting. The easiest way to think of it is that now, instead of a simple matrix for the game, we end up with a tree. Each move that a player can make creates a new game for the other player. By making each game position a tree node, and adding children nodes for each position that can follow it, you can build a tree describing the complete set of possible game positions, and thus the complete set of ways that the game could play out.

July 25, 2008

Teaching Multiplication: Is it repeated addition?

Category: Math Education

I've been getting peppered with requests to comment on a recent argument that's been going on about math education, particularly with respect to multiplication. We've got a fairly prominent guy named Keith Devlin ranting that "multiplication is not repeated addition". I've been getting mail from both sides of this - from people who basically say "This guy's an idiot - of course it's repeated addition", and from people who say "Look how stupid these people are that they don't understand that multiplication isn't repeated addition".

In general, I'm mostly inclined to agree with him, with some major caveats. But since he sidesteps the real fundamental issue here, I'm rather annoyed with him.

July 24, 2008

Numeric Pareidolia and God in Π

Category: numerology

piproof.png

There's one kind of semi-mathematical crackpottery that people frequently send to me, but which i generally don't write about. Given my background, I call it gematria - but it covers a much wider range than what's really technically meant by that term. Another good name for it would be numeric pareidolia. It's been a long time since I've written about this kind of stuff, and someone just sent me a pretty typical example, so what the hell. It revolves around a mess that he put together as an image, which is pretty much a classic example of obsessive silliness.

The general idea of this kind of silliness is finding some kind of numeric pattern, and convincing yourself that there's some deep, profound truth behind that pattern. There are a couple of typical kinds of this: number/letter correspondence (classical gematria, which uses the fact that the hebrew characters are used both as letters and numbers, so a word can be interepreted as a number, and vice versa), distance coding (like the infamous "torah codes", where you find words "hidden" in a text by picking out characters according to some pattern and using them to form words), and simple numeric patterning (where you take numbers - generally some sort of constant - and find some sort of pattern supposedly hidden in its digits). Todays crackpottery is the third kind - it's written by a guy who believes that there are mystic secrets encoded into π and the square root of two that were put there by God, and that the existence of those patterns are proof of the existence of God.

This little bundle of rubbish - like all of the kinds of things I described above - are examples of pareidolia involving numbers. As I've written about before, we humans are amazingly good at finding patterns. We've got a strong natural talent for looking at things, and finding structures and patterns. That ability serves us well in many of our ordinary endeavors. The problem with it is that there are apparent patterns in lots of things. In fact, if you look at things mathematically, the odds of any text or constant not containing interesting patterns is effectively nil. If you're willing to consider all sorts of patterns, then you can find patterns in absolutely everything. The question that you need to ask is whether or not the pattern is simple the result of our ability to find patterns in noise, or whether it's something deliberate.

July 21, 2008

Utility Functions

Category: Game Theory

Before we move beyond zero-sum games, it's worth taking a deeper look at the idea of utilities. As I mentioned before, in a game, the scores in the matrix are given by something called a utility function.

Utility is an idea for how to mathematically describe preferences in terms of a game or lottery. For a game to be valid (that is, for a game to have a meaningful analysis and solution), there must be a valid utility function that describes the players' preferences.

But what do we have to do to make a valid utility function? It's simple, but as usual, we'll make it all formal and explicit.

July 18, 2008

Friday Random Ten, July 18

Category: Music

  1. The Flower Kings, "Underdog": a neo-progressive track with the lead played by a bagpipe and a steel guitar. How can you not love that?
  2. Broken Social Scene, "Ibi Dreams of a Pavement": A post-rock track with vocals. Very good stuff - very dense. Like I said it's got vocals, but they're not the dominant part of it - they're actually almost in the background.
  3. Marillion, "Heart of Lothian": a track off of my favorite Fish-era Marillion album. It's hard to take this in isolation - the whole album is really one continuous piece of music - with recurring themes, lyrical motifs, etc. This is really just a continuation of what came before it - it's not a standalone. But it's amazing - the kind of music that can give you chills even the hundredth time you've listened to it.
  4. Naftule's Dream, "Speed Klez": Rollocking progressive Klezmer, played by a mix of instruments including clarinet, trombone, electric guitar, and who knows what else. Amazing, dazzling stuff.

  5. Magma, "Ork Alarm": progressive rock, which really staddles the line between modern classical and progressive rock. Very complicated stuff. Not the easiest listen - it's very strange, and takes a few listens before you really understand it enough to enjoy it. But like modern classical music, it's worth the effort. This group is really one of the most amazing ensembles in progressive rock.
  6. Happy the Man, "Stepping Through Time": a piece off of the reunion album of the great american progressive band. I'm a huge fan of HtMs old work. This new album isn't bad - it's interesting, complex music, with nice melodies, time changes, and amazing musicianship. But it's strangely lacking something. It's soulless. It just feels very mechanical.
  7. Isis, "All Out of Time, All Into Space": more post-rock. Very atmospheric, dark.
  8. Lunasa, "Island Paddy": quite a transition from the last one; this is a straightforward traditional Irish showoff piece. Bouncy fun that makes you want to get up and dance.
  9. Sonic Youth, "Queen Anne Chair": a wonderful little snippet. This is off of Sonic Youth's album "The Destroyed Room", which is assembled from experimental studio clips. It sounds like Sonic Youth playing post-rock.
  10. Hawkwind, "You Shouldn't Do That": Amazing strange but wonderful stuff from early Hawkwind. Depending who you ask, this is either early progressive or psychedelia.

July 16, 2008

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