## Kernighan's history of Unix offers lessons for the future

Kernighan's history of Unix is a first person account of how the operating system came to be, and how it came to be so ubiquitous. It's also a great overview of Bell Labs culture, and why it was so productive. The book is interesting, readable, and informative. A fairly quick read that's still full of useful knowledge.

Each chapter is a part of the Unix development history, followed by a short biography of someone involved. This worked really well for me. The history part of the chapter connected what I knew about the OS with people who worked on it and the design decisions they struggled with. The biography section of the chapter would then go into more detail about one of the people involved in those design decisions, often showing some interesting background about why they would have had the thoughts they did.

There are a lot of anecdotes about how and why things were built the way they were. Some of these come off as simple reminiscences (and the book is part memoir). Other anecdotes really clarify the technical reasoning behind some of Unix's features. I'm used to thinking of Unix through the lens of Linux, which is (currently) enormous and complicated. That's pretty different than Unix during it's development, when the source code could be printed in a single textbook alongside with line-by-line annotation. The book gave me the impression that Unix was created by some highly educated people that were just seeing what they could do, while also trying to make their own jobs easier. Each step of Unix's creation was a small improvement over what someone was already doing, followed by a bit of cleaning up to make it all work nicely together.

A particular string of anecdotes really changed my understanding of the OS and it's associated command line tools. When I use command line utilities like grep or sed or Make, it's often in a cookbook format to accomplish a very constrained goal. I viewed them as tools that have specific configurations I can tweak, like I drill that I can change the torque on. That's pretty distinct from how a lot of the Unix authors viewed things. They were very much working from the idea of languages. For Kernighan, sed and grep aren't tools. They're parsers for a general language. Make isn't a configuration file, it's a language that gets parsed. This is one reason that they were able to make so many powerful tools: they treated every problem like a translation problem, and wrote languages that would help. This change in mentality from a "tool" to a "language" feels large to me.

In addition to gaining insight into the decision process that drove Unix development, the book also spends a fair amount of time on the culture of Bell Labs. Bell Labs has a kind of mythic status, appearing in many history of technology books (including at least one that focuses on Bell Labs specifically). This was the best book I've read that described the organization of Bell Labs from the perspective of someone who was there.

## Bell Labs Funded People

Bell Labs was a great example of the "fund people" method of research, as opposed to pretty much everything today. This idea, which is also discussed explicitly in the book, is that the modern scientific enterprise is too focused on giving money only to people with concrete problems they're trying to solve. By only giving funding to short term concrete projects, our society is limiting the process of discovery and improvement.

The Bell Labs approach to funding research was:

step 1) hire people at the cutting edge of their field
step 2) there is no step 2. If you're doing something here, you're doing it wrong.

Kernighan describes how he started working at Bell Labs, and was never given a project. He just kind of wandered around the halls talking to people until he found some interesting things to work on. Everybody at Bell Labs was like this. Nobody had an official project, they just did whatever they felt like and published or patented what made sense after the fact.

Some people sometimes had concrete projects. Apparently when Ken Thompson started at Bell Labs, he was asked to help write the Bell Labs component of Minix. Minix was an operating system project being run out of MIT that had very lofty goals and a very complicated design philosophy. Bell Labs eventually pulled out of the project for reasons that aren't clearly explained in the book. Something something it wasn't working out, it's me not you.

After Thompson's Minix project got scrapped, he just worked on whatever he felt like. He made an early space-explorer video game, he worked on a hard disk driver. Then he realized he was "three weeks from an operating system" if he reused some of the software he'd just written. So he made Unix. No grant process, no management review, no scheduling it around other obligations.

After that, several other people in Thompson's department got interested in it. They all worked together to add tools and features and clean it up. None of them had to clear their work with management or get a new job-code put on their timecards. They just worked on the software.

That's not to say that they were just playing around. Kernighan describes a lot of specific concrete problems that they were trying to solve. Things like document preparation, automated legal brief generation, and resource sharing. Everyone who worked on those problems did so in a way that was generalizable, so they were adding tools and techniques to Unix as they worked. Ken Thomson and Dennis Ritchie, the two main Unix inventors, were at the cutting edge of operating systems and language design, and they just kept trying new things and increasing what was possible.

There are a few specific takeaways here that are relevant to attempts to fix modern science funding.

### Who do you fund?

Kernighan has several pages on how Bell Labs selected people to hire. It seems that Bell Labs would have certain engineers build relationships with specific universities. Kernighan himself worked with CMU for over a decade. These Bell engineers would visit their university several times a year to talk to professors and meet their grad students. Promising PhD candidates would go out and interview with Bell Labs engineers in a process that reminded me a lot of the Amazon and Google in-depth interviews I've done.

### Teams

The secret of Bell Labs' success, at least according to Kernighan, lies in the network effects of their office. They hired all of these people who were at the cutting edge of their field, and then put them all on a single campus. There were no remote workers.

This really kills my plan to get myself fully funded and then do math research in my underwear from home.

The Bell Labs network offered two things. The first was obviously access to solutions. If you didn't know how to do something, corporate culture demanded that you go ask the leading expert two doors down from you. The culture was a "doors open" policy, and people would often drop whatever they were doing when someone came to them with questions.

The other benefit of the Bell Labs network is less obvious: it's the problems. All of the people at the cutting edge of their fields were trying to cut a little farther. They were all trying to solve their fields problems. That meant that everyone had interesting problems to work on. This ended up driving a lot of Unix development. Someone would show up in Kernighan's office needing help with some random thing, and Bell Labs would end up with another ground-breaking piece of software several months later.

It's possible that the internet could replace some of this. Stack exchange and Quora, in particular, seem ripe for this kind of experts-helping-experts exchange. That said, I think the in-person nature of Bell Labs lowered barriers to collaboration.

There was also the knowledge that anyone you helped was an expert in their own field, which is not the case with online tools now. I think that would change the dynamic significantly.

### Bringing back the greatness

If you wanted to bring back the greatness of Bell Labs in its heyday, you'd fund people instead of projects. But that slogan admits a reading that's fairly isolationist. I don't think you could match Bell Labs' output by funding a bunch of individuals around the country. I don't even think you could match it by funding a bunch of small teams around the country. I think you need the big research center energy. Fund people, but make them work near the other people you're funding.

Jonathan Edwards has an interesting post about stagnation in software development. He argues that the Great Stagnation in economic development is also impacting software, and his evidence is that we're still using a lot of the tools that are discussed in Kernighan's history. Unix and C/C++ are still used all over the place. Edwards would like to see better operating systems, better languages, better filesystems, better everything. He argues we don't have better everything because "we've lost the will to improve" since the internet lets software developers make easy money.

Is Edwards right? Will it be useless to Fund People, Not Projects because developers have lost their way? Edwards want another Unix project, another breakout operating system that fixes all of Unix's mistakes. Kernighan doesn't think it'll happen.

The thing is, OS development now is not even in the same reference class as OS development when Unix was invented. In the 60s, all operating systems sucked. The people using them were all experts by necessity, and they were always ready to jump to the next big thing in case it was more usable. By contrast, operating systems now are adequate and used by the non-technical public. People may whine a lot about their OS, but they mostly do what people need them to do. Any newer/fancier OS would require people to learn a new skillset without offering them many benefits. No Unix replacement can spread the way that Unix did.

Unlike Edwards, Kernighan isn't disheartened. Kernighan just thinks that the next major developments will be in different fields. And indeed, we have seen major improvements in things like machine learning over the past decades. Machine learning is in exactly the reference class of early Unix: it currently sucks, everyone who uses it is an expert, and they're all ready to jump to the next thing in case it solves their problems better. ML is exactly the kind of place we'd expect to see another breakout hit like Unix, and it's also an area that Edwards explicitly says doesn't count.

Software developers are people. They enjoy solving problems, but want to solve problems that actually matter. I'm sure Edwards has a long list of improvements that can be made to any particular human programming software paradigm, but in our current place in the timeline those improvements have low expected value relative to other areas.

Software developers have not lost the will to improve. That will is still as strong as ever, but it's pointed in a slightly different direction now. Edwards may not like that direction, but I think it's a direction that will have a better impact on everyone. Now we just need to get all the bureaucracy out of the way, so the developers can actually push forward the state of the art.

## The most underrated scene in The Expanse

I've been catching up on the Expanse lately, and just finished season 3. One of the earlier episodes in the season had the most sci-fi scene I've watched so far. It wasn't the magic rocket engines or the arm-band cancer cures that impressed me. It was the retrieval of the Nauvoo.

The Nauvoo is an enormous ship debuted back in season 1. Intended as a generation ship, it's incomplete when the intrepid heroes try to use it to batter an asteroid out of the way. The Nauvoo misses the asteroid, and ends up in a difficult to reach orbit without power or propellant.

So Drummer goes off to retrieve it. There's a lot of in-space rendezvous in the Expanse, but this one really brings home how advanced their spaceflight hardware and software really is. To retrieve the Nauvoo, they use a hundred or so autonomous engines that fly out to it, clamp on, and then redirect the orbit.

Every part of this is impressive compared to what we can do now. Let's break it down:

1. Coordination: There are dozens, maybe a hundred spacecraft all working autonomously and in concert to move the Nauvoo. We don't know how to do this now (though it is one of NASA's research focuses). Coordination is a hard problem because you have to let all the spacecraft communicate with each other, tell each other where they are, and decide which spacecraft is going to do what. Swarm robotics is hard. Imagine how hard normal robots are, and multiply that difficulty by the number of robots in the swarm to get a sense for how impressive it is.
2. Localization: Each of these autonomous rocket engines navigates to the proper place on the Nauvoo before clamping on. Navigating with respect to a large craft is difficult because most human crafts are pretty featureless. If you just look at the side of a battleship, it's really hard to get a sense for where on the battleship you are. You have to do a combination of computer vision and odometry to figure it out. In this case the robots might be able to localize against each other to figure out where they are next to the Nauvoo, but that's also something we haven't quite figured out how to do yet.
3. Attachment: In space operations, you'll sometimes hear the term RPOD for rendezvous, proximity operations, and docking. RPOD activities are very difficult, because you want to make sure the spacecraft don't collide instead of docking. Part of this is the localization problem above, but part of it is also just how do you attach to the spacecraft? When the two spacecraft are designed to dock, this is kind of easy. We do it all the time when a new craft docks with the ISS. When the craft you're docking with wasn't designed for that, it's much harder. We've only successfully done this once, when a Northrup Grumman craft docked with a satellite earlier this year.
4. Control: Finally, once the swarm of robots is attached to the Nauvoo in all the right places, they fire up their engines to reorient the thing. The control problem here is staggering. The Nauvoo is enormous. It's probably filled with some leftover propellant, some water, and a lot of gases. That means it's going to slosh as it moves, so the dynamics of the Nauvoo will change over time. Each robot needs to fire its engines at precisely the right time, for precisely the right duration, in order to get the Nauvoo going where it needs to go. They need to coordinate that firing, while also learning in real time how the Nauvoo actually responds to the thrust.

The reason I find all of this so impressive in the show is that we can almost do all of it. We're actively getting closer to being able to do this, but every part of it is still outside our grasp.

When I see the med-tech that they have, my brain just fills it in as magic and moves on. I don't know enough about biology or medicine to know how likely it is that we can cure cancer or hypoxic brain injuries using some arm-cuff. It's cool, but it's not inspiring since it feels like I can't have it.

The space robotics in the Expanse is both cool and inspiring. We can get there, and it's going to be amazing.

## A mediocre history of Bayes' rule

The Theory that Would Not Die starts out strong. This history of Bayesian thought by Sharon Bertsch McGrayne was recommended to me after I finished reading a biography of Claude Shannon, and I was pretty excited to read about how Bayesian thought developed before and after Shannon.

The first few chapters were a great introduction to the Reverend Thomas Bayes, to Pierre-Simon Laplace, and to some of the controversies of the early 1800s. Going into this book, I thought that I understood the origin of Bayes' rule, and just had to learn about how it became popular now. My preconception was exactly backwards.

The myth of Bayes rule is that Bayes himself created it to address a question from David Hume about the existence of god. Hume claimed that our experience of the world says there can't be miracles. Since we've never seen a miracle, if someone reports one we should always believe they're mistaken (or lying). Bayes supposedly created his formula to prescribe exactly how much hearing about a miracle should increase our belief in god.

It turns out that there's very little evidence that Bayes was thinking about that when he created his rule. He wrote a single paper about his rule, in the form of a probabilistic thought experiment involving tossing balls onto a table. That paper was published after Bayes died, along with some religious interpretations of it written by Bayes friend who found that paper in Bayes' effects.

At this point, everyone forgets about it. Pierre-Simon Laplace, facing some very complicated data analysis problems in astronomy and biology, re-invents something very similar to Bayes rule a few decades later. It was Laplace, and not Bayes, who really popularized the idea of "the probability of causes" for the first time. He used it extensively for many of the problems that he faced, and only learned about Bayes' paper by chance. Apparently Bayes' prior probabilities (always equal odds) were new to Laplace, and Laplace then incorporated that idea into Laplace's formulation.

After Laplace died, few people used his methods. There seems to have been some form of smear campaign against Laplace, with people actively avoiding methods he'd created. It wasn't until the world wars that people started using the probability of causes again.

From WWI up to the present day, the military seems to have been a great user of Bayes' rule. While statisticians and other academics were debating the merits of Bayes' equal prior and finding it groundless, the militaries of the world were using Bayesian updating in everything from aiming guns to cracking codes. The military used it because it worked, the academics rejected it because they couldn't see how it could make sense.

There were some notable academics who embraced Bayes' rule around this time, especially Turing and Shannon. The book gives a pretty good overview of what Turing accomplished. I was left even more impressed with Turing after this, and even more upset at his treatment by Britain after the war. The book unfortunately didn't really go into detail on Shannon's use of Bayes' rule.

During WWII, breaking German and Japanese codes was crucial to the war effort. The British didn't really understand that cryptography had advanced since the 1800s, and had to be given the solution to early Enigma cyphers. Polish mathematicians had managed to crack it before the war began. Britain then hired Turing to expand on the Poles' work, and he basically created modern computing in order to do it. A large part of his work was generating likely priors for different messages. Then using Bayesian updating to determine the rest of the message. After the war, all of his work was made confidential and Turing sworn to secrecy. He was later hounded into suicide by the British government.

After this period, a lot of arguing happened. That's my best summary of the rest of the book. Many of the Post-WWII chapters were just chronicles of the arguments between people I'd never heard of before.

Each chapter was nominally about some use of Bayes' rule in history. For the period after WWII, these chapters were arranged moderately chronologically so you could see how Bayes' rule was rejected and embraced in various times and places. That overall structuring makes sense, so it's unfortunate that the book didn't work at all.

The book really played up the personalities of the people involved, and ignored the actual math to a large extent. McGrayne avoided in depth discussions of the math. There was maybe a handful of equations in the entire book. For a history of math, that's pretty unhelpful. I was hoping to understand how the theory was developed, how new pieces that made Bayes' workable in the modern world actually worked. Instead I got to read pages and pages about how various people were all jerks to each other.

The emphasis on what Bayes' rule was used on, and the people who used it, also caused the book to feel very chaotic. The chapters were nominally in historical order, but the later chapters especially jumped around a lot in the careers of various people. I ended up reading about someone in one chapter for three pages, forgetting about him for 50 pages, and then seeing him again in another chapter with the assumption that I would still remember why he was important. I went into this book looking for a high level overview of a theory's development, and instead of got the knitty-gritty back and forth between dozens of people over the course of decades. For me, this level of detail obscured the higher level points I was reading for.

I would have loved to see more discussion of the math. More simplified examples of what the people were actually working on, and how they were trying to do it. More equations in my math book, in other words.

I was also pretty surprised by what the book focused on in later chapters. It discussed Kalman filters only tangentially, in spite of the fact that they are an incredibly useful and common application of Bayes' rule. I was waiting for the Kalman filter chapter with baited breath, and it never came. Instead I just got one sentence about how Kalman himself claimed that his filter wasn't Bayesian at all.

This is such a missed opportunity! The book spends pages and pages talking about how hard it was for Bayesians to get recognized, and the little arguments between Bayesians and frequentists. Then it can't spend a single page talking about the Kalman filter or why Kalman didn't think it was Bayesian? The Kalman filter led the development of a huge family of Bayesian filters and models that are used in all of aviation today. Instead we get an entire chapter about some rando who got sent to Europe to look for a submarine. That's cool and all, but what about the entire space program?

The lack of discussion on Kalman filters, throwing mathematical terms around without helping the reader understand them, the lengthy digressions into tiny spats between statisticians. All of this makes me question how the topics the book focused on were chosen. I'm left wondering whether I actually got a useful history of Bayesian thought, or just the tidbits that this author was particularly interested in.

This was disappointing, as the first few chapters were so good. My recommendation: read through to the end of WWII, then find a different book.

## The truth about my kids

A friend on facebook recently asked for examples where people believed something wrong on purpose, just to improve their lives in some way.

While the idea seemed strange to me at first, I've slowly come to realize that there's something similar that I'm doing in my thinking about my kids. And I think that thing is actually very important to raising my kids well.

Unlike the original request for examples, I'm not trying to believe something I know in my heart is untrue. Instead, I'm actively trying to avoid learning certain things about my kids. Or perhaps it would be better to say that I'm actively avoiding trying to build certain models about my kids.

## Self-Fulfilling Models

My kids are very young, only two and a half right now. That means that there's an enormous amount about who they are (or will be) as people that is unknown right now. Will they be interested in arts or sciences (or both)? Will they be introverts or extroverts? Will they like science fiction or mysteries?

Our kids aren't super messy when they eat. The main exception to that is yogurt. When we feed them yogurt, it gets all over their faces, their arms, their shirts. The yogurt gets everywhere. After they've finished eating it, I'll come in with a cloth to clean them up. Doing this makes it really obvious that they're using one hand more than the other. One hand is usually pretty clean, and the other is so covered in yogurt it looks like melted wax.

I actively try not to pay attention to which hand it is. I try to avoid putting together patterns about whether the messy hand changes each time they eat yogurt or not. I try to avoid pointing it out to them.

My goal is to let them develop their handedness, left or right, independently of my own preconceptions. I don't want to push them to use a hand that they're not interested in using, or to prematurely settle on using only one hand.

Especially now, during this time of covid-induced hibernation, my kids get the majority of their understanding of the world and what's good from me. How I think about them influences how I treat them, and how I treat them has an outsized impact on how they grow. I don't want my own peculiarities to unduly shape who they could be as people.

## The importance of letting kids be themselves

Whether our kids are left handed or right handed isn't a huge deal (but it was historically). On the other hand, there are some personal traits that will have a major influence on my children's lives. I'm trying to avoid learning those or modeling them as well.

The biggest example of this is intelligence. There's a whole argument about what "intelligence" means for kids, how correlated intelligence in one domain is with another domain, how it changes with age, and how easy it is to determine. I want to sidestep all of that by appealing to a twin study.

Specifically, a study of my twins. They're fraternal twins, so genetically different but same environment. They look different and they act different. They also figure things out at different rates. One of my kids is much more verbal than the other, much faster in answering questions, and much better at remembering (or at least repeating) song lyrics.

In spite of how I try not to notice that disparity, it still surprised me when the disparity was disrupted. I taught the kids how to play "I spy…" the other day. The kid who is faster at answering questions and filling in song lyrics really struggled with understanding how the game is played or what the point of it was. The other kid, who is generally quieter and slower to answer questions, understood how the game is played immediately and joined in.

I've heard some parents talk about their kids as being "smart" or "slow". I don't think either of those models are very helpful to actually raising kids. To me, what seems most important is what would challenge a kid at any given moment. Regardless of what a kid is capable of now, the goal is still to raise the most competent, compassionate, and grounded kids that I can.

For me as a parent, it doesn't matter if one kid is learning something faster than another. What matters is how I can help both of them learn and grow as effectively as possible. That's going to be different for each kid, but my overall job is the same.

If I get wrapped up in comparing my kids to each other, or to some arbitrary "standard" development track, that makes it harder for me to do my real job of raising them.

## On modeling humans

None of this is to say that I don't think the idea of intelligence is useful (or similar summary measures like "compassionate" or "grounded", for that matter). I actually do think summarizing a person as being "smart" can really help when you're trying to figure out whether to hire person A or person B for a job. Or when you're trying to decide what leader to elect. Or when you're thinking about who you trust to have good information about something.

If I were a perfectly fair person, this would be less of an issue. If I were actually able to use my models of the world only when they were applicable, and ignore them when they weren't applicable, then this wouldn't matter. I could go ahead and think of one of my kids as left-handed, or estimate their intelligence right now, and not let it impact how I raised them.

But I am not perfectly fair. If I start modeling a person, my model of them will influence every thought I have about them. I can't even avoid modeling them. When I notice someone answering a question quickly, that automatically leads me to thinking certain things about them. When I notice them usually using their left hand, I automatically label them left-handed.

This is why I actively try to avoid making certain observations about my kids. It's why I actively try to avoid letting certain observations coalesce into models of them. I know my job, and it's not judging them. It's nurturing them. As a human myself, that's easiest to do when I haven't already decided what they'll grow into.

## PS

Just to be clear, I also think my kids are both very smart. And caring, creative, athletic, and cute. I'm so proud of them for all of that, but those adjectives aren't useful when figuring out what games to play or what to teach them next.

## Supposed Spartan Superiority

A while ago I stumbled on the blog Acoup, which is mostly military history written by an Assistant Professor of history at NCSU. As a binged through the blog's backlog, I stumbled on a series about Sparta that totally surprised me. Here's a post I made about it on Facebook.

"When I read about real Spartan history recently, I was pretty surprised that they were only mediocre as individual warriors, they were terrible at warfare in general, and were incredibly brutal to their absolutely huge numbers of slaves."

This started a fascinating argument about Greek history. It seems like everyone pretty much agrees that Sparta was pretty brutal to their lower classes (though I had had no idea about that until I read the blog posts). What people disagree about is whether they were good at war. This makes sense, given that it's such a staple of popular culture. Some of the people in that FB discussion know way more history than I do, so I ended up unsure how to weight their statements against those of Acoup.

A friend recently posted a fact-check of some of Acoup's Sparta claims. That gave me more trust in Acoup's other claims.

In the end, I remain pretty convinced by Acoup's claims about Sparta's (lack of ) prowess at naval and siege warfare, as well as logistics. Where I was left most confused was in the claims about hoplite warfare. As one of my friends mentioned in that FB argument:

"[W]e're talking about putting down a lot of skilled contemporary analysis. The idea that [historical sources like Xenophon] were just suckered and there's nothing to it is almost shockingly arrogant, given the scope of their capabilities and accomplishments. As far as I can tell essentially no contemporary sources are like 'actually, the Spartans are bad and unimpressive'."

# The crux of the argument

everybody: Spartans were the best warriors in the world
Acoup: Spartans society was horrible to live in and highly immoral by modern standards. Also they didn't win very many wars.
Me (on FB): Seems like Spartans were terrible at warfare
FB friends: they were actually great warriors, and everybody in antiquity knew it

It's easy to get side-tracked by these types of arguments. The overall idea of Spartans in pop culture is definitely that they're peak warriors. When I argued with people who defend Spartan military acumen, I found that they fell back on hoplite battle as what they meant by that.

This feels a bit like a motte and bailey argument to me, where a push back on overall military competence gets rebutted with a much smaller claim. Maybe the people who defend the bailey (Spartans were awesome at war) are separate from those that defend the motte (Spartans were good at hoplite battles), but it's hard to say.

In any case, I want to be clear about what question I'm trying to answer. I just want to know if Spartans won more battles than they lost.

If Spartans won most of the battles they fought, then I have to admit that they were better than their contemporaries. If they lost most of their battles, then they weren't. If it was about 50/50, then maybe they were only average.

There's a lot of minutiae that goes into this, because battles are never clean competitions between comparable forces. The Spartans that are most renowned were those who went through the agoge schooling (called Spartiates), but many in Spartan forces were helots who hadn't had that training. How do we gauge those differences?

Similarly, Sparta often went to war alongside allies. Depending on time period, they allied with the Thebans, the Athenians, the Persians, etc. If we want to know how good the Spartiates were, we should discount battles where most of the soldiers on the Spartan side were from allied forces.

We should also look at how the Spartans actually won their battles. If they won pitched battles against a numerically superior opponent, then that's good evidence that they were strong warriors. If they won against a surprised force that was much smaller than them, I'd take that as not arguing for their skill at arms.

Spartan military might waxed and waned over time. To do this right, we should give the most weight to the battles when they were strongest.

But all of that sounds like a lot of work. So I'm going to take a list of Spartan battles (this one from wikipedia), and just look at win/loss record. I think that'll give a good first order approximation to their abilities. I'll look at a few individual battles after that to get a sense for the more specific questions.

# Overall Battle Performance

I made a table from Wikipedia's list of Spartan battles, and you can find it at the end of this post.

I'm not a historian, and I don't have the decade+ to become one to answer a question that is pure curiosity on my part. It seems pretty likely that the Wikipedia list isn't comprehensive, but my naive estimation is that it probably contains the more well known and impactful of Sparta's battles. If you know of a better list, let me know and I might update my analysis later (time permitting).

There were 35 battles on that list. Of those, they had 16 wins, 16 losses, and 3 ties. To me, that seems to indicate that they were mostly fighting people who were about as good at war as they were. By this measure, they certainly weren't terrible at war (as I maybe mis-interpreted Acoup as saying). On the other hand, they aren't the gods of war that pop-culture makes them out to me.

## Individual battles

Having looked at how the Spartans did overall, it will also be instructive to look at a few individual battles and see why the Spartans won or lost. Here are a few important ones:

### Battle of the 300 Champions

I find the battle of the 300 champions very useful for this argument. Sparta and Argos were going to war, but they didn't want to waste all their armies. They decided they'd have 300 men of each side fight, and that would determine who won the battle. The rest of the militaries withdrew to prevent interference, so it was really just these two groups of 300 people each.

I'm not sure who was chosen to be in the Spartan group of 300, but it seems a safe assumption that it was primarily Spartiates. Those Spartans who had been through the Agoge and were the best warriors Sparta had to offer.

This gives one of the most pure comparison points available for Sparta's military prowess (at least as of 546BC). It's a pitched battle between equal numbers of people, so obviously whoever wins is better at battle.

But it infamously came down to a tie. Technically two Argos soldiers survived and one Spartan soldier survived. This argues pretty strongly against the Spartan exceptionalism theory.

On the other hand, since the outcome of the battle of 300 was so in doubt, Sparta and Argos went ahead with the full battle that they'd tried to avoid earlier. Spartans defeated the Argos in this larger battle.

Another interesting tidbit is that Argos challenged Sparta to a rematch 100 years later, which Sparta declined.

### Battle of Sepeia

The battle of Sepeia, also between Spartans and Argives, was a total victory for the Spartans. The Spartans completely devastated the Argive military.

Did the Spartans win through superior skill at arms? No, the Spartans ambushed the Argives while they were all eating lunch.

I don't want to knock this tactic. All's fair in war, as they say. But it doesn't seem to be strong evidence that the Spartans were great hoplite soldiers, as it's much easier to destroy an opposing force when their hands are full of food instead of weapons.

### The Athenian Sicilian Expedition

During the Peloponnesian War, the Athenians invaded Sicily. Over the course of the expedition (which involved many battles), the Spartans and their allies completely destroyed the Athenians' expedition.

Athens lost 10000 hoplites and 30000 oarsmen. That was a huge blow to their military. As Wikipedia says, " the defeat of the Sicilian expedition was essentially the beginning of the end for Athens". After this defeat, many previously neutral parties allied with Sparta.

In a counterfactual world where Athens hadn't invaded Sicily, they may have won the Peloponnesian War. Even if they had invaded, if they'd withdrawn when they realized they were losing they could have saved some of their soldiers to fight in later battles. My (naive) reading of this battle is that Sparta would have had a much tougher time winning the war if Athens hadn't been defeated here.

Should we give Sparta the credit for this? There were definitely Spartans involved in the Sicilian fight against Athens. It's hard to get a sense for numbers here, but my take on the Wikipedia article is that it was mainly Syracusans who fought off the Athenians, and they were just assisted by the Spartans. In other words, the Syracusans may have been one of the main reasons that the Spartans won the war.

# The Spartan Mythos

Based on that list of battles, I have to revise my original assumptions. Spartans weren't terrible at war. They also weren't obviously superior at it. If I had to summon a warrior through time to organize my assault against a great evil, it's not clear I should choose a Spartan over an Argive (I would obviously choose Alexander the Great, a Macedonian).

Why then is Sparta held up as a core of military mastery? I honestly think it's because they liked war so much. They had a whole school devoted to teaching their kids to be warriors. It might not have helped them to conquer and hold their neighbors (which they obviously wanted to do). It did impress all of their neighbors though, and it made it clear what Sparta valued.

People in middle class America don't fight hoplite battles. When we go to war, we have the best equipment and the most people. Our soldiers don't necessarily need to emulate great generals or ancient soldiers, they just need to have values that work well with military discipline. Our civilians don't even need that much, they mostly just want to feel connected to a sense of physical pride and motivation. The Spartan mythos provides these things, even if it doesn't have much to do with Sparta itself.

I also think the Spartan mythos was pretty useful to Sparta itself. It seems pretty clear that Sparta was drinking it's own koolaid. A friend asks the very reasonable question:

"The Spartans were, by all accounts, backwards, agrarian and few in number. But they seem to have had an outsized influence in the geopolitics of the day."

I think recent American politics have shown that you don't necessarily have to be skilled and exceptional to have a big impact on something. What really worked for Trump was just a willingness to push for what he wanted, and keep pushing regardless of what other people said or who might have been a better fit. It seems pretty clear the Spartans would have supported that mindset.

## On nuclear weapons policy, Biden beats Trump

There are a lot of uncertainties about the result of a nuclear war, but one thing seems clear: it would be bad. How bad depends on things like who we go to war with, number of nuclear weapons used, weather patterns, etc. Wikipedia documents some estimates that the US government produced during the cold war as saying that nuclear war with the Soviet Union could lead to the death of 70% of all Americans. My wife and I have two kids. If 70% of us die, that's three out of the four people in our family. My kids, my wife, gone.

Since the cold war, the number of nuclear weapons stockpiled has been reduced by about 85%. That said, there are over ten thousand nuclear weapons in the world, over half operated by other countries. Things would still be very bad if we got into a nuclear war.

In a lot of ways, I see nuclear war deterrence as one of the more important responsibilities of a US president. It doesn't matter what other policies they put in place if they get us involved in a war that kills 70% of Americans. I want a president that will continue drawdown of current nuclear stockpiles, prevent other countries from continuing their nuclear weapons programs, and provide stability to the international environment. That's one part of why I'm going to vote for Biden.

As President, Donald Trump has spent the last four years making our country much less safe from nuclear war. It can be hard to evaluate some of the nuclear decisions his administration has made, given the international landscape. Unilateral disarmament seems likely to make the US less safe, and modernizing our nuclear arsenal improves reliability and safety. Because of that, we have to look at all of the administration's decisions as a whole to determine if they're improving the security of Americans.

• actively called for development of new nuclear weapons (both in type and in quantity).
• focused on increasing capabilities to match Russia and China, instead of negotiating for more international drawdown
• broadened the definition of "extreme circumstances" under which nuclear weapons could be used
• pulled out of the INF. This was done because Russia wasn't meeting targets, but the INF governed more than just the US and Russia. Pulling out of the INF reduces leverage on other nuclear powers governed by the treaty, as well as those not officially in the treaty but who abided by it (like Germany and Slovakia).
• introduced development of low-yield nuclear weapons
• looks likely to not extend the New START treaty, which places limits on the number of nuclear weapons Russia and the US can deploy. Russia is currently complying with that treaty.

In addition to leaving nuclear non-proliferation treaties and calling for development of new nuclear weapons, the Trump administration has also completely failed to reign in North Korea and Iran.

North Korea currently has around 30 nuclear weapons and can produce something like 6 more weapons each year. Several years ago, North Korea offered to start reducing nuclear capabilities in exchange for lifting of sanctions, but Trump walked away. A couple months ago, North Korea said that negotiating with Trump had been "a nightmare" and that they were going to increase their nuclear weapons stockpile.

Iran's nuclear weapons development had, prior to Trump's election, been governed by the JCPOA. The JCPOA was an agreement between Iran, the US, and several other nations. It governed how Iran was to eliminate its stockpile of enriched uranium. Iran would still be able to build nuclear power plants, but not to enrich enough to build nuclear weapons. In 2018, Trump's administration withdrew from that agreement over the protest of every other member (including Iran). After the US withdrew, Iran and the other countries in JCPOA attempted to continue abiding by the agreement. The US's withdrawal from JCPOA led to series of skirmishes which culminated in the US killing an Iranian major general. In response to that killing, Iran has said that they won't abide by the JCPOA at all. By their actions, the Trump administration increased the likelihood of Iran getting nuclear weapons.

Let's contrast with Biden. Biden wants to extend the New START treaty. He wants to drawdown nuclear stockpiles. We have evidence from his decades in politics of him working to reduce the likelihood of nuclear war. Biden has also released a letter describing his approach to nuclear disarmement treaties in the past.

This quote in particular is why I think Biden will manage our nuclear policy well:

"Despite what some extreme voices argued at the time, the arms control agreements we hammered out with the Soviets were not concessions to an enemy or signs of weakness in the United States.
They were a carefully constructed barrier between the American people and total annihilation."

Joe Biden

## Bad Math: Tax Plan Tweet Edition

It's pretty common these days to talk about how much Jeff Bezos could buy for us. Facebook and twitter are both full of people saying things like this:

The idea being that if Jeff Bezos just paid his share of our country's upkeep, we could have a utopia.

There's a problem with this idea, but it's not a political problem. It's a math problem. It's a problem with what people mean when they say "wealth" or "taxes".

## He doesn't have the cash

First thing's first: Bezos has somewhere around $200 Billion now (a quarter of his and MacKenzie Scott's wealth went to her in the divorce). That wealth is not cash in a bank account. It's Amazon stock that he owns. There's no way for him to spend that much money, because he doesn't have it as money. If we wanted to take our 4.7% of his wealth, we'd have to start by selling about$10B worth of his Amazon stock. Selling that much stock would have an impact on Amazon's stock price, so we'd likely have to sell more shares than you'd expect to reach that amount.

Bezos spends around $1Billion a year on his new space company, Blue Origin. He makes a big deal about that, I think, in large part because he has to justify selling that much stock every year. If Bezos just started selling Amazon stock for no reason, especially that amount, the price of the stock would plummet and he'd lose a lot of his wealth. People would assume that he knew something bad about Amazon's business. Now if he were selling the stock to pay for College For All, that would be a pretty good signal to the market that Amazon was still a stable company. The market price for the stock likely wouldn't drop much from our$10B sale due to a lack of confidence.

I was originally going to write something here about $10B being a lot to sell on the stock market, but it turns out that's not true. The NASDAQ (where Amazon is listed) clears over$100B per day. Bezos could probably find someone to buy his $10B of stock pretty easily. ## Normal US Taxes don't work like that Let's say that we decide it's a good idea. That$10B that people are paying Bezos for his stock wasn't doing us any good before (it was probably just wrapped up in some other big tech stock). We're going to have Bezos fund our College.

The thing is, we can't do this by just taxing Bezos normally. The US has an income tax. That's a tax on income. Bezos actually doesn't have much income, he just has assets that are worth a lot. This means that if President Biden (I hope that's who we have next year!) says that 2021 will have a super high tax on everyone named Jeff Bezos, we wouldn't actually get much money.

Jeff Bezos doesn't pay any taxes on his stock until he sells it, and then he only pays taxes on the appreciation. Though given that he got the stock when it was likely worth a single dollar, the tax will apply to pretty much the entire amount he's selling.

So if Bezos did sell $10B in stock next year, then we'd only be taxing that$10B. The current capital gains tax would be 20% for him, so we'd only get $2B from that sale. In order to fix this, we'd need to tax wealth, not income. We could absolutely do that, and Piketty has advocated for that kind of tax to help deal with some of the social problems that we're facing now. But I don't really think that Jack Califano, from our original tweet up above, is thinking about it like that. Obviously I don't know what his understanding of our current tax situation is, but if he were proposing something as radical as a wealth tax I'd have expected him to play it up. ## Math vs. Politics Enough of this tax bracket, stocks-vs-cash nonsense. We want our College For All; let's get Bezos to pay for it. We're going to pass a law that says Bezos has to sell enough stock every year to provide 4.7% of his wealth to the US government, each year, effective immediately. Tomorrow, Bezos sells enough stock to get a grand total of$10B from it (remember that his current wealth is around $200B, so 4.7% is slightly less than$10B). Now can we all have free college?

No, no we can't. Because Bernie Sanders says that he needs $48B per year to pay for his College For All plan. I have no idea where Califano got the 4.7% number, but it makes no sense. Bezos could only pay for College For All for four years, even if we gave him a wealth tax of$50B/yr.

Bezos has a lot of money, but he doesn't have that much.

## Why

I'm sympathetic to Califano's argument. It would be really nice for Bezos to pay to improve America. Hell, it would be nice for him to pay to improve Seattle (more directly than he's doing by building nice buildings and bringing in employees who spend money). People have been trying to get Amazon to pay more Seattle taxes for a long time, and I hope that they've finally managed it.

I'm a bit more reluctant about the idea of a wealth tax, but I could be convinced by the right arguments and experiments.

I want free college (and free healthcare, and affordable housing) for everyone in America. I want it so bad that I'm willing to pay attention to reality to find ways to get it. This is why I get so frustrated when people start talking about having Bezos pay for everything.

Bezos has a lot of money, but he doesn't have that much. Simple "tax the rich" schemes also generally won't access most of his money, so saying that we should tax Bezos without specifying that you want to totally change the way taxes are done in America seems disingenuous.

I want people to work towards a better world together. To do that, we all need to know what direction a better world is. If we just ignore the math, I'm afraid of where we'll actually end up.

## Life Events Are Community Events

I read Achtung Baby, by Sara Zaske, when my kids were about six months old. The book is part travelog, part parenting advice, by a woman who moved her family from the US to Germany. She raised kids for five years there, came back to the states, and wrote a book on the difference in parenting cultures.

One of the biggest things that stuck with me about this book was the discussion about Einschulung, and life milestones parties more generally. An Einschulung is a party thrown for a kid when they first go off to school, and it is a big deal. Everyone is invited, family, extended family, friends, neighbors. Everyone comes out to celebrate and acknowledge this major milestone for the kid.

The Einschulung is compared with two other major parties that Germans have in their lifetimes: their Jugendweihe celebrating entrance into young adulthood and their wedding. These are the three parties that define the arc of many a German's life, and they help tie the person into their community.

## Community

What struck me most about the Einschulung, Jugendweihe, and wedding is how they were seen in Germany. Or perhaps how Sara Zaske saw them. These were parties that were in honor of a given person, but not always for them. Often the parties were for the community that person belonged to as much as they were for the person themselves.

This really appealed to me when I was reading it as a new parent. I still had fresh memories of my own wedding two years before, and how my wedding had changed my views on weddings overall. Prior to my own wedding, I'd often viewed wedding invites with distaste. When some friends got married, I'd feel obligated to go in order to show my support, but I often felt pretty isolated at weddings. I didn't know how to interact with the event, or the people at the event.

When planning our wedding, my wife and I wanted to really emphasize the family and community aspects. We ended up doing a lot of non-standard wedding things, but the closer we got to the wedding, the more those seemed to matter less to me than having people there to witness our love. I think several of our plans for our wedding were amazing, and a couple fell kind of flat, but the thing that was most meaningful to me was just having so much family and so many friends there with us as we said we loved each other.

I really understood then that our wedding wasn't just for us, it was for our community as well.

This was a pretty new experience for me. I was a lonely kid, and had a lot of trouble making friends. It wasn't until I was in my mid-twenties that I understood that it was possible to have a social interaction with someone new that wasn't emotionally painful. My wedding was the final push I needed to see community in a different way.

Reading about Einschulung in Achtung Baby got me wondering if I could have had a much more socially comfortable adolescence if I were raised in a culture that emphasized community in a more formal way. It's really too bad the US is the way that it is, but maybe I could do something like that for my kids anyway.

## Other states

I was pretty surprised to discover that, at least in one US state, there are parties like Einschulung. Several Michiganders that I know recently told me about the tradition of high school graduation parties there.

In Michigan, everyone apparently throws a high school graduation party. You invite your family, your friends, your friends' families, your neighbors, your parents coworkers. It's a regular community ritual. It's such a big deal that high schoolers carefully schedule their graduation parties to not overlap with their classmates, so that everyone can go to everyone else's parties.

I was immediately excited about this and wondering if Michiganders felt less isolated and more tied in to their community than I had as a high schooler.

## Not Always Good

It turns out that one of the people who was telling me about this tradition had hated their graduation party. They felt that it was a party for their parents, not for them, and that they were being forced to go. In fact, the way they described it reminded me a lot of how I felt about other people's weddings before I'd had my own.

This raises some interesting questions. Why hadn't I felt like I was a part of a community when I was going to other people's weddings before I had my own? Why did my friend wish they hadn't had a high school graduation party, instead of feeling like it was their community supporting them?

It's hard to speculate on my own about what someone else was feeling at some party, but I think my own lack of community feeling was related to a sense of support going one way only. I went to weddings of friends and families because I wanted to be there for them, which is actually a large part of what I consider to be a good community. But I didn't feel communal about it, I felt like it was an obligation. I was doing it to avoid being cast out of my community, not because I wanted to build a stronger community. I don't think I even had a sense that community could benefit me in any way.

This is not to say that community never benefited me before I got married (though the amount of community support I received after my wedding was mind boggling). Looking back, I see many times when my community was supporting me throughout childhood and young adulthood. My experience of that support at the time was confused though. People would do something nice for me, and I wouldn't understand why. I'd feel like I had to pay them back immediately, or that they were condescending to me. It also wasn't clear to me if someone was "in my community" or not (a question not at all helped by the introduction of Facebook).

This makes me wonder if my wedding was a turning point because it really shoved my face in the idea that other people were helping me because they wanted me to be happy. Planning and throwing a wedding is not an easy thing, and there's often a ton of family stress on top of that. We never would have been able to throw the wedding that we did without the help of a huge number of people.

I think this is why the other two parties described in Achtung Baby sounded so good to me. If I'd had the idea that other people might genuinely want to help me when I was a kid, or that they might actively want me to be a part of their community, then I would have had a much happier childhood. The Einschulung seems like a very stark demonstration of that fact to a kid when they could most use it: right before joining a huge group of people they've never met before to do something totally new.

## Celebrating With My Kids

This idea of the reason behind community parties offers up some ideas for how to do similar things for my kids as they age:

1. throw parties at milestones that are a Schelling Point for the community that you're in
2. let people help with party set up in a way that's visible to others
3. invite everyone that's in your community, but not everyone you know
4. make sure the party will actually be enjoyable for the kids

I think point 4 is pretty important. I really enjoyed my wedding, and I suspect I'd be feeling differently about community if I hadn't (even if my community had been exactly as helpful). I also suspect that this was what went wrong for my friend who hated their own graduation party. If the party had been structured more to their liking, then they could have recognized the community aspects of it more easily.

I also think point 1, about choose milestones that make sense for your community, is important. This makes me think that birthdays are more important than I had been thinking before this, and going forward I plan to place more importance on my kids birthdays and on my own birthday.

## Knowledge Bootstrapping Experiment 2

Last month, I started experimenting with AcesoUnderGlass's Knowledge Bootstrapping method. I started out with a small project learning some facts about radiation and electronics. That worked well, so I then went to learn about something a little less straightforward: GPT-3's likely impact on AI safety.

I have to be honest, selecting this topic may have been a bit of a mistake. I was seeing a lot headlines and posts about GPT-3, and I had a pretty immediate emotional reaction of "GPT-3 isn't a big deal and people don't know what they're talking about."

In a lot of ways, this is great. I learned a lot about the current state of AGI research, and some of the current major players in AI safety. Deciding (before doing any research) to write a post about the topic is what gave me the motivation to actually read all those articles, and then read the background, and then read even more background. I haven't really kept up with these things for the past three years, so things had changed a lot since I had last looked into it. This project gave me the push I needed to finally learn how the transformer architecture really worked, as well as uncovering some of what DeepMind has been doing. I hadn't even known that MuZero existed before starting on this project.

## Motivation

All of this leaves me still excited about the knowledge bootstrap method, but I'm also noticing that keeping my motivation for a research project up is hard. When I have a blog post that I'm excited about writing, it's easy to put in effort to learn and write. When someone is wrong on the internet, of course I'll be burning to write about it. The more I wrote my post, the more clear it became that I was the one wrong on the internet.

That started sapping my motivation to write, even though the things that I was writing changed enough that I still stand by their accuracy.

As I closed in on answering most of the questions that I had come up with in my original question decomposition, I had such a different understanding of the topic that I realized I had an enormous number of new questions. I answered those questions, and then the questions that followed from that. Eventually, I came to the point where I thought I had a decent stance on the original safety question I had. At that point, I also realized how much detail there was to making a decent prediction about GPT-3's implications on future safe AI. And much of that detail was (and is) still unknown to me.

As I began to realize how much I'd have to research in order to do the topic justice, I could feel my excitement fade. Given that I've had a very stuttering relationship with this blog over the past decade or so, I could recognize that if I let my excitement about the topic drive me into perfectionism I wouldn't post anything. I also recognized that if I didn't post that blog entry, I'd feel like a failure and there would be a long drought in me posting anything at all.

I decided that I had enough for a high level post and wrote it, but I ended up writing a more milquetoast thesis than I had originally intended.

The most important thing for me in any kind of learning project is keeping up motivation. For work related topics, there's enough external motivation that I can power my way to a solution one way or another. For personal projects, even personal projects that could help me out at work, I need to stay interested throughout the process to have any hope of success.

My first experience of Knowledge Bootstrapping showed me that an emphasis on questions could help me keep my motivation up. By keeping my thoughts close to my original questions, it was easy to remember why I was doing the thing. This second experience of the process showed me that the blog output itself is still a big part of my motivation, and I'll need to plan around that in future projects.

## Question Decomposition

I still view question decomposition as one of the more important components of Knowledge Bootstrapping. My original project had a very straightforward set of questions, and after I decomposed them it was easy to pull answers out of the sources I found. The hardest part of my Radiation+Electronics mini-project was finding sources that went deep enough to truly answer my questions.

The GPT-3/AI-safety mini-project was much different. When I first started decomposing questions about (before I started doing much reading) I had a ton of trouble figuring out what my primary question even was. Then I had trouble breaking that down into questions that reading books/papers could answer. I did my best to decompose the questions, then went and tried to answer them. That helped me orient myself to the field again, and when I came back to try answering my original questions I could clearly see some better question decompositions.

I ended up iterating this process several times, and I think for difficult or new topics this is probably crucial.

Elizabeth says that if you're not sure what notes to take when you're reading a source, you should go look at your questions again. That isn't great advice if you're having trouble with the decomposition step itself. I tried to address this by emphasizing the difference between what I was reading and what I already thought, and writing that down. That also helped me to figure out what my questions were, as I would sometimes realize I disagreed with something but be uncertain why.

Elizabeth emphasizes doing a brain dump of what you think about any given source before you really start reading it. I didn't do this very much in my first mini-project, but I did it for every source in this project.

I now think that my radiation+electronics mini-project didn't need much of the brain-dump step because I'd been thinking about the topic on and off for several years. I pretty much knew what I already knew. I also had a mindset that was focused on fact acquisition and model building, but I didn't have to worry much about conflicting information or exaggeration.

With GPT-3 and AI safety, there's no settled science about the topic. Everything is new, so people are all very excited. That meant that I had to be more careful with what sources I was using. I also didn't have a good handle on what questions I was trying to answer at the beginning, which meant that it was harder for me to notice what was important about each source's content.

This is where the pre-read brain-dump really shines. Before I did an in-depth read of any source, I'd free-write for a while about what I expected the source to say. I'd also write about what I personally thought about the expected content of the source. Then when I went to read the source, it was easy for me to notice myself being surprised. That surprise (or disagreement, or whatever) was the trailhead for the questions that I should have been asking at the beginning.

Interestingly, this seems to be the exact opposite of the reason that Elizabeth does it. She talks about how, if she didn't get her brain dump on paper, those thoughts would be floating around her head interrupting her reading process.

When I don't do the brain dump, I don't have any of those thoughts floating around my head as I read. That makes it really hard for what I read to latch on to what I already know. I'll sometimes read something and feel like I understand it, but then be unable to recall it even ten minutes (or ten pages) later. By brain-dumping, I prime my mind with all those thoughts so that I'm actually engaging with and thinking about the content in the source.

(Though Elizabeth also talks about this a bit here, where she says breaking the flow of a book is a sign of engagement).

In the past I've tried to address this with Anki. When I was reading textbooks cover to cover, I'd create flash cards of the major things I learned. This was generally very effective, but I've ended up with a truly enormous number of cards. I haven't kept up on my Anki training for the past couple weeks, and I now have hundred of cards in my backlog. It's also pretty slow to do this, and really takes me out of the flow of reading.

A good future workflow might be something more like:

1. question decomposition
2. source selection
3. brain-dump
5. post-process notes and write blog post
6. generate anki cards that are more focused

## Tools

One of the things that held me back during my first Knowledge Bootstrapping mini-project was being unfamiliar with some of the markdown features that Elizabeth makes common use of. Because of that, my writing project was slower and more awkward than I think is Elizabeth's experience.

I took some time (really just ten minutes or so) to look up some of the markdown features that I had wanted to use in my first project. Using those made this second project a lot easier. I was a lot more comfortable drafting the post and referring to each source. I'm beginning to see how the process itself could become more natural and get in the way less.

I still feel pretty curious about Elizabeth's actual workflow during note-taking and synthesis though. She described it at a high level in her post, but I'm more interested in the nitty-gritty at this point. What does she make a tag, and why? How does she manage her tags? Does she really actually use that many of them?

## Math Puzzle: 2D planes in N-D spaces

I was playing around with robot localization the other day, and realized that the angular degrees of freedom a robot has follow an interesting pattern. A robot that can just move around a floor has only one degree of angular freedom; it can rotate to the left or right. A flying robot, on the other hand, has three angular degrees of freedom: it can pitch, roll, or yaw.

That made me curious how the number of angular degrees of freedom is related to the number of spatial degrees of freedom. If we could build a robot that could move in the 4th spatial dimension, how would it rotate?

A robot that can only move along one line is the degenerate case. This one dimensional bot has a single binary degree of freedom in rotation. It can point either forward or backwards.

A robot that can move in a single plane, like a roomba, has a single angular degree of freedom in rotation. It can rotate however it wants as long as it remains parallel to the floor.

A robot in arbitrary three space has the traditional angular degrees of freedom of roll, pitch, and yaw.

But what about a robot in arbitrary spatial dimensions? What would a robot's degrees of freedom look like in 4-D space, or n-D space?

As normal, the linear degrees of freedom equal the dimensions of the space. So in n-D space there are n linear degrees of freedom.

At first, I naively thought that the number of angles the robot could rotate around would be the number of axes also. After all, in three spatial dimensions there's one orthogonal axis for each plane the robot rotates in. But for higher dimensional cases this doesn't quite follow. The plane that's defined by being orthogonal to one axis is actually a hyperplane. It's made up of all (n-1) dimensions at a right angle to the axis of rotation. If our robot sensors are still just 2D or 1D devices, then we probably want to be more precise about what plane they're rotating in.

What we're interested in for robot rotations is (I think) the number of 2D planes that the robot could rotate around. Just by happenstance, the number of 2D planes in 3-space is the same as the number of axes. But for higher dimensional spaces we actually have to define the 2D plane by choosing two orthogonal vectors and finding their span. We can find a sufficient set of planes by choosing orthogonal vectors that are all aligned with an axis of the space.

So the number of 2D planes in an nD space is , or . Here's a list of the number of 2D planes of rotation for a few different numbers of spatial dimensions:

• 2D = 1 plane of rotation
• 3D = 3 planes of rotation
• 4D = 6 planes of rotation
• 5D = 10 planes of rotation
• and so on