Good Math, Bad Math

Moving on

Mon, 02 Aug 2010 10:41:49 -0500

Finally, at long last, I can tell you what I've been up to with finding a new home for this blog. I've created a new, community-based science blogging site, called Scientopia. With the help of many wonderful people, we're ready. We launched this morning. So to continue following GM/BM - along with the work of many other wonderful bloggers, like Scicurious, Grrlscientist, Mike Dunford, Dr. Skyskull, and lots of others, come on over to Scientopia, the new home of Good Math/Bad Math"/a>.

Goodbye, Scienceblogs

Wed, 07 Jul 2010 22:01:49 -0500

So my decision is made. I'm closing up around here. I'm in the process of working out exactly where I'm going to go. With any luck, Seed will leave this blog here long enough for me to post an update with the new location. But I'm through with Seed and ScienceBlogs.

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Seed, Conflicts of Interest, and Sleaze

Tue, 06 Jul 2010 20:47:04 -0500

As my friend Pal wrote about, Seed Media Group, the corporate overlords of the ScienceBlogs network that this blog belongs to, have apparently decided that blog space in these parts is now up for sale to advertisers.

We've been advertiser supported since I joined up with SB. I've never minded that before. Providing a platform and bandwidth takes money, which has to come from somewhere. The way that ads have been handled before has been no problem: the ads are clearly distinguished from the content. There's no way that you're going to mix up one of my posts with a paid advertisement.

Until now.

Seed has, in its corporate wisdom, decided to let Pepsico buy its way into a blog on ScienceBlogs. Pepsi writes SMG a nice check, and suddenly their content gets mixed in to the ScienceBlog RSS feeds, the ScienceBlog feed to Google News, etc., exactly the way that my blog posts do.

This is not acceptable.

For now, I'm suspending my blog for a few days. If Seed decides to back out of this spectacular stupidity, then I'll start posting here again. If not, then I'll go looking for a new home for GM/BM. The money that I've made from the ads that Seed has sold has been nice - but it's not worth my integrity.

If Blogs here are for sale, then I'm gone.

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Searching for Topics

Mon, 28 Jun 2010 11:02:24 -0500

As regular readers have no doubt noticed by now, posting on the blog has been slow lately. I've been trying to come back up to speed, but so far, that's been mainly in the form of bad math posts. I'd like to get back to the good stuff. Unfortunately, the chaos theory stuff that I was posting just isn't good for my schedule right now. Once you get past the definitions of chaos, and understanding what it means, actually analyzing chaotic systems is something that doesn't come easily to me - which means that it takes a lot of time to put together a post. And my work schedule right now means that I just don't have that amount of time.

So, dear readers, what mathematical topics would you be particularly interested in reading about? Since I'm a computer scientist, my background obviously runs towards the discrete math side of the world - so, for the most part, the easiest topics for me to write about are from that side. But don't let that limit you: tell me what you want to know about, and I'll take the suggestions into consideration, and figure out which one(s) I have the time to study and write about.

I don't want to limit you by making suggestions. I've tried that in the past, and the requests inevitably end up circling around the things I suggested. But I really want to know just what you want to know more about. So - fire away!

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Saturday Recipe: Ginger Scallion Sauce

Sat, 26 Jun 2010 17:19:41 -0500

Today's recipe is something I made this week for the first time, and trying it was like a revelation. It's simple to make, it's got an absolutely spectacularly wonderful flavor - light and fresh - and it's incredibly versatile. It's damned near perfect. It's scallion ginger sauce, and once you try it, it will become a staple. To quote David Chang, whose cookbook I learned this from: if you've got ginger scallion sauce in the fridge, you'll never be hungry.

There are two main variations of this: there's a cooked version, and a raw version. Mine is the raw version. I love the freshness of flavor, and while cooking it will intensify some of the flavors, it will also detract from that delightful freshness.

Ingredients

Fresh ginger - roughly one inch, peeled.
A bunch of fresh scallions.
A teaspoon, give or take, of coarse salt.
1 tablespoon of soy sauce.
1 tablespoon rice vinegar.
1/4 cup oil - peanut oil, canola oil, or something other neutral oil.
A dash of sesame oil.

Instructions

Mince the ginger. Toss the minced ginger into a food processor.
Cut the roots off of the scallions, cut them coarsely, and add them to the food processor.
Add the rest of the ingredients to the food processor.
Run the food processor until everything is finely ground into a smooth sauce.

That's it. Ginger scallion sauce. Taste it - make sure it's got enough salt. Don't add any soy sauce - just use plain salt if it needs any.

So what can you do with it? Just about anything. A few great ideas:

Ramen noodles. Just cook up a batch of ramen, and toss it with a tablespoon of the sauce. You can also add some stir fried meat and veggies to make it a bit more filling.
Grilled meats. Use a bit of the sauce as a marinade, then grill it, and dress it with a bit of the sauce when it's done.
Use it instead of mayo on a sandwich.
Add a bit more vinegar, and use it as a vinaigrette over a salad.
Sautee some shrimp, and toss some ginger-scallion sauce in just before they're done.
Get a nice whole fish, steam it cantonese style with just a bit of salt, soy, and sake. Spoon a bit of the sauce over it when it's done.

If you wanted to try to cooked version, you take the ginger, scallions, and salt, and puree them in the food processor. Then put them into a large pot. In a different pot, heat the oil up until it just starts to smoke, and then pour it over the ginger/scallion/salt mixture. When it cools, whisk in the rest of the ingredients.

But like I said - I think it's best to just stick with it raw.

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The Surprises Never Eend: The Ulam Spiral of Primes

Tue, 22 Jun 2010 11:58:52 -0500

One of the things that's endlessly fascinating to me about math and science is the way that, no matter how much we know, we're constantly discovering more things that we don't know. Even in simple, fundamental areas, there's always a surprise waiting just around the corner.

A great example of this is something called the Ulam spiral, named after Stanislaw Ulam, who first noticed it. Take a sheet of graph paper. Put "1" in some square. Then, spiral out from there, putting one number in each square. Then circle each of the prime numbers. Like the following:

If you do that for a while - and zoom out, so that you can't see the numbers, but just dots for each circled number, what you'll get will look something like this:

That's the Ulam spiral filling a 200x200 grid. Look at how many diagonal line segments you get! And look how many diagonal line segments occur along the same lines! Why do the prime numbers tend to occur in clusters along the diagonals of this spiral? I don't have a clue. Nor, to my knowledge, does anyone else!

And it gets even a bit more surprising: you don't need to start the spiral with one. You can start it with one hundred, or seventeen thousand. If you draw the spiral, you'll find primes along diagonals.

Intuitions about it are almost certainly wrong. For example, when I first thought about it, I tried to find a numerical pattern around the diagonals. There are lots of patterns. For example, one of the simplest ones is that an awful lot of primes occur along the set of lines f(n) = 4n²+n+c, for a variety of values of b and c. But what does that tell you? Alas, not much. Why do so many primes occur along those families of lines?

You can make the effect even more prominent by making the spiral a bit more regular. Instead of graph paper, draw an archimedean spiral - that is, the classic circular spiral path. Each revolution around the circle, evenly distribute the numbers up to the next perfect square. So the first spiral will have 2, 3, 4; the next will have 5, 6, 7, 8, 9. And so on. What you'll wind up with is called the Sack's spiral, which looks like this:

This has been cited by some religious folks as being a proof of the existence of God. Personally, I think that that's silly; my personal belief is that even a deity can't change the way the numbers work: the nature of the numbers and how they behave in inescapable. Even a deity who could create the universe couldn't make 4 a prime number.

Even just working with simple integers, and as simple a concept of the prime numbers, there are still surprises waiting for us.

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Metaphorical Crankery: a bad metaphor is like a steaming pile of ...

Thu, 17 Jun 2010 13:45:15 -0500

So, another bit of Cantor stuff. This time, it really isn't Cantor crankery, so much as it is just Cantor muddling. The post that provoked this is not, I think, crankery of any kind - but it demonstrates a common problem that drives me crazy; to steal a nifty phrase from youaredumb.net, people who can't count to meta-three really shouldn't try to use metaphors.

The problem is: You use a metaphor to describe some concept. The metaphor isn't the thing you describe - it's just a tool that you use. But someone takes the metaphor, and runs with it, making arguments that are built entirely on metaphor, but which bear no relation to the real underlying concept. And they believe that whatever conclusions they draw from the metaphor must, therefore, apply to the original concept.

In the context of Cantor, I've seen this a lot of times. The post that inspired me to write this isn't, I think, really making this mistake. I think that the author is actually trying to argue that this is a lousy metaphor to use for Cantor, and proposing an alternative. But I've seen exactly this reasoning used, many times, by Cantor cranks as a purported disproof. The cranky claim is: Cantor's proof is wrong, because it cheats.

Of course, if you look at Cantor's proof as a mathematical construct, it's a perfectly valid, logical, and even beautiful proof by contradiction. There's no cheating. So where do the "cheat" claims come from?

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The Unfalsifiable Theory Of Everything from viXra

Fri, 11 Jun 2010 14:29:53 -0500

Today is another bit of rubbish from viXra! In the comment thread from the last post, someone (I presume the author of this paper) challenged me to address this. And it's such a perfect example of one of my mantras that I can't resist.

What's the first rule of GM/BM? The worst math is no math.

And what a whopping example of that we have here. It's titled "Spacetime Deformation Theory", by one Jacek Safuta. I'll quote the abstract in its entirety, to give you the flavor.

The spacetime deformations theory unifies general relativity with quantum mechanics i.e. unifies all interactions, answers the questions: why particles have mass and what they are, answers the question: what is energy, unifies force fields and matter, implies new theories like spacetime deformations evolution.

This is a theory of principle (universal theory delivering description of nature) and not constructive theory (describing particular phenomenon using specific equations).

The theory is falsifiable, background independent (space has no fixed geometry), not generating singularities or boundaries.

This is hard to believe but a belief has nothing to with it. The real intellectual challenge is to falsify the theory.

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Gravity, Shmavity. It's the heat, dammit!

Tue, 08 Jun 2010 11:47:36 -0500

Sorry for the ridiculously slow pace around here lately; I've been ridiculously busy. I'm changing projects at work; it's the end of the school year for my kids; and I'm getting close to the end-game for my book. Between all of those, I just haven't had much time for blogging lately.

Anyway... I came across this lovely gem, and I couldn't resist commenting on it. (Before I get to it, I have to point out that it's on "viXra.org". viXra is "ViXra.org is an e-print archive set up as an alternative to the popular arXiv.org service owned by Cornell University. It has been founded by scientists who find they are unable to submit their articles to arXiv.org because of Cornell University's policy of endorsements and moderation designed to filter out e-prints that they consider inappropriate.". In other words, it's a site for cranks who can't even post their stuff on arXiv. Considering some of the dreck that's been posted an arXiv, that's pretty damned sad.)

In my experience, when crackpots look at physics, they go after one of two things. Either they pick some piece of modern physics that makes them uncomfortable - like relativity or quantum mechanics - and they try to force some argument that their discomfort with it must mean that it's wrong. The other big one is free energy - whether it's perpetual motion, or vacuum energy, or browns gas - the crackpots claim that they've found some wonderful magical process that defies the laws of thermodynamics in order to make limitless free energy. The cranks rarely (not never, but rarely) go after the kinds of physics that we experience every day.

Well, this is something different. This guy basically wants to claim that gravity doesn't really exist. And along the way, he claims to have solved the problems of dark matter and dark energy. See, we've all got it totally wrong about gravity! Gravity isn't a force where matter attracts other matter. It's a force where warm things attract other warm things! Gravity is actually a force created when things radiate heat.

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Big Number Bogosity from a Christian College Kid

Tue, 04 May 2010 20:34:19 -0500

I know that I just posted a link to a stupid religious argument, but I was sent a link to another one, which I can't resist mocking.

As I've written about quite often, we humans really stink at understanding big numbers, and how things scale. This is an example of that. We've got a jerk who's about to graduate from a dinky christian college, who believes that there must be something special about the moral atmosphere at his college, because in his four years at the school, there hasn't been a single murder.

Yeah, seriously. He really believes that his school is special, because it's gone four whole years without a murder:

Considering that the USA Today calculated 857 college student deaths from 2000 to 2005, how does one school manage to escape unscathed? It's certainly not chance or luck. For Patrick Henry College, it's in our Christian culture.

Critics mock us for our strict rules - like no dancing or drinking on campus, no members of the opposite sex permitted in your dorm room, nightly curfew hours - and the lack of a social atmosphere it creates. We have been the subject of books (God's Harvard), television shows, op-eds, and countless blogs who rant against our brand of overbearing right-wing Christianity that poisons society's freedom.

Yet, what is the cost of students being able to "express" themselves? Is that freedom worth the cost of drunk driving deaths, drug related violence, and love affairs turned fatal?

There were 857 college student deaths in the five-year period from 2000 to 2005! Therefore, any college where there weren't any murders in that period must be something really special. That christian culture must be making a really big difference, right?

Well, no.

According to Google Answers, the US Census Department reports that there are 2363 four year colleges in the US. So, assuming the widest possible distribution of student deaths, there were 1506 colleges with no student deaths in a five-year period. Or, put another way, more than 60% of colleges in the US went that five-year period without any violent student deaths.

Or, let's try looking at it another way. According to the census, there are 15.9 million people currently enrolled in college. The school that, according to the author, is so remarkable for going without any murders in the last four years? It has 325 students. Not 325 per class - 325 total.

In other words, among a group making up less than 2/1000ths of one percent of the college population, there were no murders. Assuming that the distribution of violent deaths is perfectly uniform (which it obviously isn't; but let's just keep things simple), given that there were 857 violent deaths in the student population as a whole, how many violent deaths would you expect among the student body at his dinky christian college?

That would be a big, fat zero.

The fact that there were no violent deaths at his school isn't remarkable, not at all. But to a twit who's incapable of actually understanding what numbers mean, that's not the conclusion to be drawn. It's also not that the violent death among college students is actually remarkably rare. Nor is it that most college students will go through college without any violent deaths on campus. No - according to a twit, with 857 violent campus deaths over five years, the only reasonable conclusion is that there must be something special about the ridiculous religious rules at his college that prevented the great rampaging plague of violence from touching the students at his school.

I actually spent five years as an undergraduate at Rutgers University in NJ. During that time, there were no violent student deaths. (There was one death by alchohol poisoning; and there was one drunk driving accident that killed four students.) But zero violent deaths. Gosh, Rutgers must have been an absolutely amazingly moral university! And gosh, we had all of those horrible sinful things, like dancing, and co-ed dorms! How did we manage to go all that time with no violence?

It must have been the prayers of the very nice Rabbi at the Chabad house on campus. Yeah, that must be it! Couldn't just be random chance, right?

Ok, now let me stop being quite so pettily snide for a moment.

What's going on here is really simple. We hear a whole lot about violence on campus. And when you hear about eight-hundred and some-odd violent deaths on campus, it sounds like a lot. So, intuitively, it sure seems like there must be a whole lot of violence on campus, and it must be really common. So if you can go through your whole time in college without having any violence occur on campus, it seems like it must be unusual.

That's because, as usual, we really suck at understanding big numbers and scale. 800 sounds like a lot. The idea that there are nearly sixteen million college students is just not something that we understand on an intuitive level. The idea that nearly a thousand deaths could be a tiny drop in the bucket - that it really amounts to just one death per 100,000 students per year - it just doesn't make sense to us. A number like 800 is, just barely, intuitively meaningful to us. One million isn't. Fifteen million isn't. And a ratio with a number that we can't really grasp intuitively on the bottom? That's not going to be meaningful either.

Bozo-boy is making an extremely common mistake. He's just simply failing to comprehend how numbers scale; he's not understanding what big numbers really mean.

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The Danger When You Don't Know What You Don't Know

Mon, 03 May 2010 19:35:52 -0500

A little bit of knowledge is a dangerous thing.

There's no shortage of stupidity in the world. And, alas, it comes in many, many different kinds. Among the ones that bug me, pretty much the worst is the stupidity that comes from believing that you know something that you don't.

This is particularly dangerous for people like me, who write blogs like this one where we try to explain math and science to non-mathemicians/non-scientists. Part of what we do, when we're writing our blogs, is try to take complicated ideas, and explain them in ways that make them at least somewhat comprehensible to non-experts.

There are, arising from this, two dangers that face a math or science blogger.

There is the danger of screwing up ourselves. I've demonstrated this plenty of times. I'm not an expert in all of the things that I've tried to write about, and I've made some pretty glaring errors. I do my best to acknowledge and correct those errors, but it's all too easy to deceive myself into thinking that I understand something better than I actually do. I'm embarrassed every time that I do that.
There is the danger of doing a good enough job that our readers believe that they really understand something on the basis of our incomplete explanation. When you're writing for a popular audience, you don't generally get into every detail of the subject. You do your best to just find a way of explaining it in a way that gives people some intuitive handle on the idea. It's not perfect, but that's life. I've read a couple of books on relativity, and I don't pretend to really fully understand it. I can't quite wrap my head around all of the math. That's after reading several entire books aimed at a popular audience. Even at that length, you can't explain all of the details if you're writing for non-experts. And if you can't do it in a three-hundred page book, then you certainly can't do it in a single blog post! But sometimes, a reader will see a simplified popular explanation, and believe that because they understand that, that they've gotten the whole thing. In my experience, relativity is one of the most common examples of this phenomenon.

Todays post is an example of how terribly wrong you can go by taking an intuitive explanation of something, believing that you understand the whole thing from that intuitive explanation, and running with it, headfirst, right into a brick wall.

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Iterative Hockey Stick Analysis? Gimme a break!

Thu, 29 Apr 2010 21:46:23 -0500

This past weekend, my friend Orac sent me a link to an interesting piece of bad math. One of Orac's big interest is vaccination and anti-vaccinationists. The piece is a newsletter by a group calling itself the "Sound Choice Pharmaceutical Institute" (SCPI), which purports to show a link between vaccinations and autism. But instead of the usual anti-vac rubbish about thimerosol, they claim that "residual human DNA contamintants from aborted human fetal cells" causes autism.

Among others, Orac already covered the nonsense of that from a biological/medical perspective. What he didn't do, and why he forwarded this newsletter to me, is because the basis of their argument is that they discovered key change points in the autism rate that correlate perfectly with the introduction of various vaccines.

In fact, they claim to have discovered three different inflection points:

1979, the year that the MMR 2 vaccine was approved in the US;
1988, the year that a 2nd dose of the MMR 2 was added to the recommended vaccination schedule; and
1995, the year that the chickenpox vaccine was approved in the US.

They claim to have discovered these inflection points using "iterative hockey stick analysis".

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Finger Trees Done Right (I hope)

Mon, 26 Apr 2010 10:47:53 -0500

A while ago, I wrote a couple of posts that claimed to talk about finger trees. Unfortunately, I really botched it. I'd read a bunch of data structure papers, and managed to get myself thoroughly scrambled. What I wrote about was distantly related to finger trees, and it was useful to help understand how fingertrees work - but it was not, in any way, shape, or form, actually a description of fingertrees. Since then, I've been meaning to write a proper post explaining finger trees - but with the work on my book, and with chaos at work, I just haven't had the time. This time, in order to do my damnedest to make sure that I don't screw it up again, I'm basically go to describe finger trees over a couple of posts by walking through the best finger-tree paper that I could find. The paper is "Finger Trees: a simple general-purpose data structure", by Ralf Hinze and Ross Patterson. This might by the paper that introduced the structure, but I'm not sure.

The point of finger trees is pretty simple. It's very similar to the point of zippers. Programming in functional languages is terrific. As I've described before, there are a lot of advantages to writing functional code. But there are also a lot of places where a naive implementation of an algorithm using a functional data structure is dreadfully inefficient. Functional code may be prettier, more maintainable, and more reusable - but imperative code is frequently much more efficient. When you're doing an operation that, conceptually, modifies a piece of a complex data structure, then functional code can really suck. Finger trees give you a way around that - for many common updatabale data structures, you can build finger-tree versions that are very close to or fully as good as imperative, updating structures.

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Friday Random Ten, 4/23/2010

Fri, 23 Apr 2010 19:30:21 -0500

Stellardrive, Inlandsix: Reasonably good instrumental prog. They're not particularly exceptional, but they're decent.
Gong, "The Octave Doctors and the Crystal Machine": Gong is a perfect example of one of the differences between the great prog bands, and a lot of the neo-progressive stuff. I can't quite describe exactly what it is - but you listen to a band like Gong, and you never get bored. You can listen to it over, and over - and it's always interesting. Even though the individual features of the music are similar to what a lot of less brilliant bands do, they manage to put them together in a different way. I can listen to a neo-prog band like Jadis or Frost once or twice a month; if I listen to them more than that, they start to bug me. But I can listen to Gong twice a day, and never lose interest.
Parallel or 90 Degrees, "Backup": One of the really great neo-progressives. Po90 is Andy Tillison's other band, and they are brilliant. Not as brilliant as groups like Gong, but pretty damned amazing.
Jadis, "All You've Ever Known": Here's exactly what I'm talking about. The beginning of this Jadis track is actually sort-of like the Gong track above. But somehow, it's dull when Jadis does it. Listening to them right after Gong and Po90, they frankly sound terrible. I really like Jadis, but they can't hold a candle to the prog greats.
And So I Watch You From Afar, "If it Ain't Broke, Break It": Really good post-rock. ASIWYFA is on the louder end of post-rock, and they're really good at it. They're one of my most recent post-rock discoveries, after being recommended to me by a reader of the blog, and I'm really enjoying them.
Genesis, "Your Own Special Way": And now, my favorite band of all time. I love Genesis. Even after Peter Gabriel left, they still wrote some of the best prog rock of all time. There's a reason why so many neo-prog bands were inspired by them. Even when they're doing a song like this, which is basically a silly sappy ballad, they make it into something really special.
Jacob Hoffman with Kandel's Orchestra, "Doina and Hora": an incredibly old recording of traditional klezmer, led by probably the greatest Klezmer xylophone player ever. If you have any appreciation for Klezmer, this will absolutely knock your socks off.
The Flower Kings, "Soul Vortex": Ah, the Flower Kings. The only neo-progressive band that I've found that's really as good as the original prog guys. Whatever that elusive "it" that the great bands had that made them endlessly listenable was, Roine Stolt and the Flower Kings have it.
Transatlantic, "The Return of the Giant Hogweed": On their latest album, Translatlantic added a disk of covers of their influences. Naturally, no group made up of members of the best neo-progressive bands could possibly not include a classic Genesis track. It's a very faithful cover, and it works really well.
Marillion, "Forgotten Sons": An old favorite of mine: one of the lesser known tracks from Marillion's very first full album. From the very start, Marillion was really something special.

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Book Update

Wed, 21 Apr 2010 15:04:23 -0500

Very quick post here: the third beta of my AppEngine book "Code in the Cloud" was released this morning. If you've bought a copy of the beta, you can go to your pragmatic account, and download a fresh copy with all of the fixes and new material.

If you haven't bought a copy... Well, if you're interested in cloud programming, I'd like to think that this book is a pretty good overview of the subject. It's about Google AppEngine, but I've done my best to write it so that it discusses the nature of cloud programming in general, using AppEngine as a specific example of a cloud platform. Buying a copy supports your friendly math blogger and makes me happy; and when I'm happy, I'm more likely to write more posts for the blog :-).

The main reason that I'm mentioning it is that a few people have asked me to provide a forum on the blog for questions. I'm happy to answer questions, and I'd love to hear feedback from anyone who's read it - both positive and negative. (And, to be honest, the negative feedback is generally more useful, so I'm very serious when I say that constructive negative feedback is welcome. Anything that you can tell me now, before it's printed, is something that I can fix!)

So if you've got any questions or comments about the book, please go ahead and put them in the comments here. On the other hand, if you find any errors, you're welcome to put them here, but it would help me more if you could file an erratum at the book's pragmatic press page, linked above.

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