I think emacs has a perfectly fine control scheme, and is certainly a very powerful program, but I prefer the simplicity of vim’s modal input. Many vim users, however, seem unaware of many things that vim can do. (Many emacs users, similarly, seem to think vim isn’t capable of emacs-like tricks.)

But I am here to dispel these false notions! Here’s a list of many possibly-lesser-known vim tricks, many of which I use on a daily basis. They aren’t complicated, so you don’t need to know much about vim to figure them out. If you have any other tips, leave them in a comment. If you don’t know how to use vim but want to find out, type “vimtutor” at a command line and read the instructions.

Multipliers: Prefixing any command with a number causes that command to occur that many times. If you want to write the character ‘x’ 80 times, just type 80 i x [ESC] and the character x will be inserted 80 times. This works for all sorts of things.

Actions and motions: You might be unaware, but actions like delete (d), change (c), and yank (y) take “motions” as an argument of sorts. Motions include end of file (G), start of line (^), end of line ($), next word (w), and matching brace (%). Executing any of these motions as a command causes the cursor to immediately jump to that location. (If the cursor is on a parenthesis, square bracket, or curly brace, % jumps you to the matching one, whether before or after.) Their true power comes when you combine them with actions, however: d% deletes the matching braces and everything in between; cG deletes the rest of the file and places you into insert mode; yw will copy the next word. In Scheme, I’ve found d% and y% indispensable for moving around S-expressions.

Repeat last command: The . command repeats the last thing you did. If your last action was to delete the current line, . will delete the new current line. That’s all.

Navigating directories/archives: If you open a directory or archive in vim, it will show a file listing. You can move up and down the list as usual, but if you hit Enter, it will open the file/directory that your cursor is on. If you’re in an archive, it extracts a copy into a temporary directory.

Make: The command :mak or :make runs “make” in the current directory, showing you the result. If there are any compile errors, vim will automagically jump your cursor to the first one, even if it’s in a different file. The command :cn gets you to the next error, and :cN to the previous. This drastically reduces the time it takes to compile and fix errors, and is definitely the most useful tip here for any C programmers.

TOhtml: Your files are syntax highlighted when you view them in vim. But what if you want a syntax-highlighted copy of your file to distribute to others? Use the command :TOhtml, which opens a new edit buffer with an HTML syntax-highlighted version of your file, ready to save. If you prefer CSS instead of <font> tags, which you should, then first use the command :let html_use_css=1 or, better yet, put it in your .vimrc file so it’s always set.

Searching: To find a string in a file, use / followed by that string. For instance, /char [Enter] will jump you to the next occurrence of “char.” The setting :set hlsearch (best placed in your .vimrc) causes vim to highlight every matching search occurrence in the file. Simply typing / [Enter] will get you to the next one. :noh turns off the highlighting.

Indentation: If you want to indent with spaces instead of tabs, set expandtab in your .vimrc. Either way, set both shiftwidth and tabstop to the desired size of your tabs. To reindent the current line, use the command ==. A single = is an action, so =G will reindent every line from the current one to the end of the file. gg=G will therefore reindent your entire file. (gg moves to the beginning of the file.)

Splitting: You can edit multiple files side-by-side in a single instance of vim. Type in :sp for a horizontal split, or :vs for a vertical split, followed by the filename you want to open. You can further split either of these new windows. ctrl-W followed by an arrow key (or h, j, k, l) moves your cursor between files. Commands like :e, :w, and :q only affect the active split. (For instance, to close one file and continue editing the others, type in :q while that window is active.

This is just a small subset of vim’s features. I try to learn something new every week. By slowly adding more features to your repertoire, you’ll get increasingly efficient with your editing, without any additional thinking. And after all, that’s the goal with any text editor.

This is an issue that has been at the front of my mind for a long time, and one which, despite first appearances, is surprisingly storied and even controversial.

Math is usually perceived as a completely rigorous field concerned with finding “correct” answers, and verifying the “correctness” of theorems. This is true in a very limited sense — math is indeed an axiomatic system under which certain conclusions follow from the premises, and others do not.

Of course, an axiomatic system can encompass any set of “truths” — after all, one can take any set of (non-contradictory) statements as a foundation. Thus, merely choosing axioms is an aesthetic decision not strictly based on any sort of reality.

While this may seem like a pedantic point, choosing axioms in math is actually a somewhat controversial issue. Apropos Godelian incompleteness, there is an infinite set of independent axioms to accept or reject. A few prominent examples are the axiom of choice and the continuum hypothesis in set theory, and a generalized Fubini’s theorem in analysis.

Historical axiomatization.

But has math always been a purely axiomatic system? Not really, as it turns out. Euclid‘s Elements is perhaps the greatest (or at least, most famous) specimen of axiomatization in mathematics. By taking several (reasonable) geometric truths as self-evident, Euclid was able to develop many non-obvious results in geometry.

(I must, of course, pause to note that one of his axioms is significantly less self-evident than the other. In fact, altering that axiom — the parallel postulate — yields a different kind of geometry, akin to conducting your business on the surface of a sphere instead of a plane.)

However, the lack of a concise shorthand for algebraic notions appears to have held back the rigorous treatment of non-geometric concepts. For instance, Newton’s second law, , was expressed by him as “The alteration of motion is ever proportional to the motive force impress’d.” While it sounds pompous now, that’s simply how equations were discussed back in the day. Imagine solving a set of equations written that verbosely!

In fact, great mathematicians like Euler often stated results somewhat informally, and without proof. Even by the middle of the nineteenth century, Galois had to invent many terms in group theory to explain his highly-technical theory linking fields and groups.

Nicolas Bourbaki.

The true breakthrough in axiomatization began in the 1930s by a group of French mathematicians operating under the pseudonym Nicolas Bourbaki.

Bourbaki aimed to produce a coherent treatment of modern mathematics, publishing nine volumes covering a large portion of the field. While opinions on Bourbaki vary drastically, I think it’s evident that they accomplished their goal admirably, and in the process, very heavily influenced the way mathematics is performed.

Bourbaki took a very rigorous approach to mathematics, systematically building up concepts from set theory to algebra, topology, analysis, and beyond. The development is very rigorous and dry; no actual problems or applications are discussed, and virtually no diagrams are included.

At the time, Bourbaki’s books were undoubtedly the best references available; it is not surprising, then, that their new approach had a profound effect on mathematicians (particularly nascent ones). Even their vocabulary has stuck, such as the empty set symbol and the words injective, surjective, and bijective.

Since then, mathematicians have essentially agreed to conduct mathematics more or less in the manner of Bourbaki. (It may seem surprising that math was at one point much less rigorous, but this merely reflects the huge influence of Bourbaki.)

Rigor considered harmful?

In my view, the most important question at this point is how beneficial (or detrimental) rigor is to mathematics. For reasons I will explain in the next installment, it seems evident (though initially counterintuitive) that rigor often helps clarify the situation. At the same time, I believe there is an alarming dearth of non-rigorous treatments of math.

Please comment on this if you have anything to add or ask; I plan on writing at least several more posts about mathematics, and I would like to focus on whatever points everyone finds most interesting.

Pause for a conversation about genres. “Metal” encompasses a hugely diverse group of sounds, from the fast, aggressive riffs in thrash metal, to the melodic, punk-inspired metalcore, and even the stupidly violent, distorted…sounds…of grindcore (ugh). Progressive metal, a highly complex, technical genre, often doesn’t sound like metal at all. Dream Theater is one of the oldest, most successful bands in the genre, and in my opinion, the best.

Apparently, they’re one of those bands everyone either loves or hates. All the album reviews I have found are either too raving, or–ack! they just don’t get it at all!–somewhat confused and unimpressed. There doesn’t seem to be anyone who listens to them in serious moderation. (As a side note, while I was looking through the last.fm API, I discovered that many examples are about Dream Theater; I was confused until I realized that Audioscrobbler’s founder is a huge fan.)

Octavarium is probably my favorite album of theirs, though I had a hard time deciding whether to write about it or Six Degrees of Inner Turbulence, a double album ending in a namesake 40+ minute eight-movement epic. (Note to Dream Theater fans: I’m sorry; I just don’t get why everyone thinks Metropolis Pt. 2 is so good!)

Octavarium isn’t their best example of extreme technical proficiency, and isn’t particularly metal, but it’s just so rich. It’s absolutely their most progressive album, in its diversity of sounds and abundance of self-reference. Even if you don’t like metal, I think the title track at least is required listening for everyone reading this post. It’s just a totally different way of doing music.

Their eighth studio album, Octavarium is a sort of concept album about cycles, filled with the numbers eight and five in surprising places. (Here’s a ten-page summary of little details people have found in the album.) In particular, the dominating cycle in the album is that the first track starts on F, the next on G, and so forth until the eighth and final track, which starts again on F and ends with the same piano note which begins the album.

“The Root of All Evil” starts out the album, centered around a driving, heavy guitar riff. Written by their drummer about his past alcoholism, it’s actually the third song in his five-song “Twelve-Step Suite” mirroring the steps of Alcoholics Anonymous, and is full of references to the previous two songs.

“The Answer Lies Within” is, in stark contrast, a slow, melodic ballad dominated by piano and strings. This track perhaps best showcases James LaBrie’s soft, soaring vocals. This is one of the two songs on the album which almost sounds like it could be played on the radio.

“These Walls” is a mellow, atmospheric song which slowly builds to a climax that just washes over the listener. Ensemble doesn’t get much better than this.

“I Walk Beside You” was Dream Theater’s attempt at a single on this album; it’s rare to see a DT song clocking under five minutes (or often, even eight). It’s catchy and interesting, and I do wonder why it wasn’t actually released as one.

“Panic Attack” is the virtuosic performance on the album, with a strange guitar riff and constantly morphing, complex rhythms. It only gets stranger (especially starting around 4:39) but somehow captures the breathlessness of a panic attack.

“Never Enough” is a rant about ungrateful fans, a pretty unique song in DT’s catalog if only for how much it sounds like Muse.

“Sacrificed Sons” is about 9/11, a dark, largely instrumental track with a long introduction and gothic-sounding vocals. It’s definitely my least favorite track on the album: it’s a bit too slow for me, and the instrumental break at 4:15 is, while great, rather incongruous.

“Octavarium” is, in my opinion, Dream Theater’s best song. I really don’t want to give much of it away, so I won’t say too much here. It starts with a fantastically expansive four-minute solo on the Haken Continuum, before a flute melody leads to the first of five sections. The first two sections are downtempo and soft; the third, “Full Circle,” (13:48) is an upbeat wordplay tribute to many of the bands which influenced Dream Theater. But the fourth section, “Intervals,” (18:30) is my favorite part. It references each of the tracks on the album, in order, while distorted samples from the tracks play in the background; crescendoing masterfully from a low growl to a hair-raising climax (“Trapped inside this Octavarium”).

Seriously, listen to this album. At least the last track. YouTube quality doesn’t really do it justice, so try to find a better-quality version somewhere (or just ask me). And do comment here if you listen to it; I’m interested to hear what other people think.

I’m glad all of you made it out here tonight. One great thing about math is how diverse it is–not just of mathematicians, but of the entire field itself. Math ranges from the purest, most useless subjects–the ones I like–to applied fields, like engineering; at the same time, some fields like computer science straddle the dichotomy, strangely both applied and pure at the same time.

As some of you know, I’ve recently come to like computer science a lot. I’m still learning the ropes, but it’s already pretty easy for me to find the computer scientists at a breakfast buffet. They’re usually in a queue by the hash table.

See, the choice between math and related fields is a tradeoff between purity and utility. The more applied fields actually get some important results.

For example, in computer science, they recently deduced the best way to walk: the logic gait. Of course, if you do too many logical ands, your ampersands may get inflamed, a serious condition called conjunctivitis.

In electrical engineering, they’re working on making even better semiconductors. I think adding gallium to silicon is totally dope.

In physics, they recently discovered a new salsa a thousand times better than pico de gallo. It’s called nano de gallo. And they excavated a derivative of the velociraptor: the acceleraptor.

Seriously, though. You probably saw that Wall Street Journal article about how mathematician is the best profession, since it pays well, and sitting in a chair doesn’t really have any occupational hazards.

Still, some mathematicians have gotten into entrepreneurial endeavors. I’ve been eating out lately at Markov’s chain of restaurants. They’re kind of artsy; you don’t actually get to order off a menu. They just choose your dish randomly based on the probability that you’ll like it. I highly recommend their Chinese, though. I quite enjoy their random wok.

Even the Ghostbusters have gone into math lately, studying Hilbert spaces. They’ve already made a lot of breakthroughs in spectral theory.

But Rick Astley hasn’t had as much success in game theory. He’s studying the prisoner’s dilemma, but he just can’t get past his moral hangups about ratting out the other prisoner to lessen your own sentence. Even if he knows the other prisoner won’t rat him out, Rick’s still never gonna give him up.

And ever since the economy tanked, cartesian products of rings have become more popular, since grad students can only afford to study free modules.

The military pulled out of some math research, too. At the NSA they can’t even generalize any more; they’ve been reduced to lieutenantizing. They’ve also been reconsidering some policies at their complex prisons. The problem with conjugal visits is that sometimes they result in those elements multiplying, which becomes a real problem of square magnitude.

But really, now. I’ve been making it sound like math isn’t good for anything, but that couldn’t be further from the truth.

I use calculus in the grocery store all the time. For example, how do you differentiate between cuts of beef? Prime rib.

And algebra comes in handy on the dance floor. You know the robot, right? It’s composed only of rigid motions, so it turns out it’s actually a subgroup of isometries.

And analysis is indispensable in the kitchen. The other day, I made a sequence of sandwiches for myself: first a p-naught butter and jelly sandwich, then a p1 butter and jelly sandwich, then a p2 butter and jelly sandwich… I tried to eat the whole series, but I realized the sandwiches didn’t get smaller, so I diverged from that plan.

To me, at least, math is really exciting. Seriously. I just get really excited whenever I’m doing math. Like, the other day, I had the surface integral of the curl of a vector field, and I was really stoked to turn it into a path integral instead.

One problem I’ve noticed in math classes is that there’s so much material in each class, but it isn’t compact enough to cover finitely. Good professors know that it’s easier once you add on a point at infinity; then you can always cover it finitely.

Anyway. Probably a lot of you have AT&T Wireless plans, right? They changed their name one and a half years ago or so; that’s because they realized how important linear algebra is, and they wanted to be invertible. So now they’re no longer singular.

A recent medical study found that Viagra works on some ring members with zero powers. After the trial, they were no longer nilpotent.

Anyway, I think it’s time for the next act, so I’m going to leave you with a little physics problem… (A demo ensues.)

Now, I don’t disagree that a “tiered Internet” is a bad thing, but I think that calling for legislation necessitating complete net neutrality is an overreaction, and is very likely in fact a bad idea. There are reasons why some degree of non-neutrality might be necessary or even preferable, but these points seem to get lost in a big “you vs. The Man” war in which, if you don’t stop The Man, he might severely restrict your Internet!

But maybe we should educate ourselves before we engage in this debate. The concept of net neutrality seems straightforward–essentially, that all Internet communications should be “treated equally”–but gets complicated once we start discussing what exactly that entails. So I figured that I would devote a few posts in here to explaining what net neutrality is, why people think it’s good, and why other people think it’s bad. It’s not a simple debate, but it’s irresponsible to support net neutrality as a savior of the Internet without realizing how harmful it might actually be.

“As we move to a broadband environment and eliminate century-old non-discrimination requirements, a lightweight but enforceable neutrality rule is needed to ensure that the Internet continues to thrive.” -Vint Cerf, co-creator of the TCP/IP protocol underlying the Internet, and net neutrality proponent

“I am totally opposed to mandating that nothing interesting can happen inside the net.” -Bob Kahn, co-creator of TCP/IP and net neutrality opponent

(Hey, by the way, you definitely ought to leave comments on my blog all the time!)

Then, in November, I bought The Orange Box, and played Portal. Portal is an excellent though short first-person puzzle game that forces players to think hard about game physics — that’s a first. It features a “portal gun” which allows the player to teleport. I really enjoyed Portal, as did most people.

But what I’m writing about now is a game released in 2000, called Deus Ex. Developed by the generally disappointing group Ion Storm, Deus Ex is considered one of the best PC games of all time. I (not so) recently completed it, and I must wholeheartedly agree.

In 2052, you are JC Denton, a member of UNATCO, the United Nations Anti-Terrorist Coalition. An early test subject for nano-augmentation, you have superhuman abilities due to modifications made to you at birth. The world is in a downward spiral, as the Gray Death pandemic is killing the lower classes; while a vaccine, Ambrosia, exists, it is in short supply.

You have just been assigned your first UNATCO mission. A terrorist group, the NSF, has captured a shipment of Ambrosia on Liberty Island and you are tasked with recovering it. As you continue to pursue the NSF, you end up at LaGuardia Airport where the Ambrosia is being kept…and you find that a UNATCO ally very close to you is actually working for the NSF.

Uncovering the mystery reveals sinister connections with FEMA, Majestic 12, and the Illuminati as you travel between New York, Hong Kong, and Paris. The final battle takes place at Area 51, where the future of the world is placed in your hands. Do you want to merge yourself with the global communications network and become benevolent dictator of the world? Do you want to destroy the network and plunge the world into a second Dark Age? Or would you prefer to return the Illuminati to power, guiding the world’s governments with an invisible hand?

The massive amount of freedom afforded the player by Deus Ex is incredible. Every area is designed to allow many ways to accomplish each task. It is actually possible to beat the game without killing anybody — and even without going to such extremes, players’ strategies can range from stealth to all-out violence. There’s no sense of the “right way” to do anything in Deus Ex — any way that works is great. The plot is highly fluid, and while players end up in the same places at the same times, small actions from hours ago influence who survives and who is friendly.

All in all, I thought Deus Ex was a great game with a great story. I’m playing Halo right now, which kinda bores me with its focus on massive battles. Can anybody suggest more great games?

Wow, there are some great acts here. In particular, I think the basis we just heard was great. His music spanned our three-space quite nicely. Anyway, I was going to bring some predatory birds here, but then I realized it wasn’t a talon show.

Okay, I’d like to make a request of you before I start my act. Please laugh very loudly at everything I say, because nobody might actually find it funny.

So, math comedy. When I told my friends I was going to do a math stand-up act, one of them replied, “Chuck Norris knows the tangent of pi over two!” Well…okay. I’m not sure how to respond to that.

Math comedy is certainly a niche audience, though. Even among mathematicians. If you ask a statistician if they’ve heard a joke before, they say “Probably.”

Anyway, there are a lot of oldies-but-goodies. There’s the joke about the mathematician who gives a talk about 13-dimensional space. Afterwards, an engineer comes up to him and says, “Wow, how could you possibly visualize 13-dimensional space?” The mathematician responds simply, “That’s easy, I just visualize n-dimensional space, and set n equal to 13.”

Of course, many sub-disciplines come with their own occupational hazards. They say topologists can’t tell the difference between a doughnut and a coffee mug, them being homeomorphic and all. I’m not sure if that’s true; I’ll ask Kent after the show.

And then physicists get their own brand of flak from mathematicians. Physicists, you see, use a special brand of mathematics. The really fuzzy type…that’s usually wrong, but somehow comes up with the right answers all the time. I think one thing in particular illustrates physicist math. Those of you who know some physics may know that electric and magnetic waves propagate as orthogonal sinusoidal waves. The direction in which they are pointing, the vector representing the energy flux of the wave, that’s called the Poynting vector. I don’t know about you, but I never make distinctions about which of my vectors are pointing. They all are!

Anyway, the other day I was going to a geometry conference, and I was speaking on constructible diagrams. I was flying out of the airport, but I was stopped at security because of my straightedge and compass. They found my weapons of math construction. I ended up missing my plane. But it’s okay; luckily I had three points in my pocket, so I defined my own plane and got there on time.

You know, we mathematicians are always trying to prove to everyone that there’s math everywhere. In particular, there’s a lot of math in the Bible; did you know that? For example, a lost story from the gospels. One day, Jesus said, “The kingdom of heaven is like x squared plus 3x plus 5!” Somebody went up to Matthew and asked him, “What is Jesus talking about?” “Don’t worry,” responded Matthew, “that’s just another one of his parabolas.”

Then there’s also the story in Genesis with Noah’s Ark. After the ark landed, Noah told all the animals to go forth and repopulate the world. Two snakes stayed behind, and told him, “We can’t do that until you build us a wooden desk.” So, whatever, he built it, and lo and behold, they started to reproduce. He asked them what the problem was, and they said, “Well, we’re adders. We need log tables to multiply.”

The other day I was proving a theorem. It was a long theorem, with a lot of significant intermediate stages. I got to one of those stages, and I said to myself, “Do I have to finish? Lemma stop here.”

Medicine has made great strides recently. When right triangles get old, they sometimes start to sag, their right angle turns into 89 degrees, 88 degrees… Anyway, they made this injection, you can just apply it to the triangle, and the angle will snap back up to a right angle. It’s called Pythagorean serum.

The other day I was at the concession stand. I wanted a medium order of Fibonachos, and my friend wanted a small order. But then I realized that a small plus a medium cost the same as a large.

I usually eat more healthily. I found a grape that could commute, it’s called an abelian grape.

I thought up a great anagram for Banach-Tarski. Ready? It’s… “Banach-Tarski Banach-Tarski.”

Some people have wondered why Newton didn’t contribute to group theory. It’s because he wasn’t Abel.

Have you heard? A former vice president recently released some rap tapes to teach computer science. It’s called “Al Gore Rhythms.”

Even mermaids like math. They wear algae bras.

Okay, just one more and I’ll leave you guys alone. So, as you know, lately, the military has been having issues with how its officers are perceived. Some kernels have expressed concern at their rather zero-dimensional images.

Format wars are nothing new to the movies. A bit before our time, VHS and Betamax had a showdown to be the cassette tape format, and VHS definitively won. I would speculate myself on the reasons for VHS’s triumph, but this topic has been exhaustively discussed by people far more qualified than I. Google it.

Likewise, DVD+R and DVD-R now (mostly) coexist peacefully, as most people have +/- drives that can read and write both.

What’s the difference? + and – have different ways of storing data. The differences are largely esoteric, and have to do with both the physical configuration of pits and the logical groupings of data on the disc itself. That said, those esoteric differences are actually manifested in the performance of the discs, especially with regard to error correction. Both are highly usable, and because they are largely compatible, neither has stomped out the other.

Unfortunately, it seems that 4.7 GB is not enough for people nowadays. Even dual-layer DVDs, which store about 8.5 GB, are somehow inadequate for everyone’s oh-so-sensitive eyes. (I don’t get it. I’m relatively happy with VCD-quality 700 MB movies, and don’t understand why people need such high-definition movies. On the flip side, a lot of people are happy with 128 kbps MP3s, but I myself want at least 192 kbps, and even higher for classical music.)

Anyway, two new optical disc formats have been invented. HD DVD, championed by the DVD Forum consortium, stores about 15 GB per layer. Blu-ray, pushed by Sony, stores a ridiculous 25 GB per layer. Both owe their existence to still-expensive blue lasers which can write data more compactly than the lasers used by DVDs.

Both BD and HD players are rather expensive at the moment; the cheapest HD DVD player retails for $150, and most Blu-ray Disc players are over $300.  If Blu-ray players are more expensive, what’s the advantage, besides higher capacity?

Sony’s PlayStation 3 console, despite its poor showing against the Xbox 360 and Wii, can play BD. While it is expensive, the PS3 is the only example of convergent technology yet to hit high-density optical disc players. No other player can do anything else.

Another, and in my opinion, even bigger issue, is that consumers are unclear about the difference. In particular, in a market saturated with High Definition: HD-upscaling DVD players, HD broadcasts, HD-ready televisions…I think that “HD” is too generic a moniker. “HD DVD” sounds simply like a better sort of DVD, not a completely different format. Blu-ray has a distinctive name and, being backed by Sony rather than a poorly-defined group of companies, just has a sort of presence that HD DVD doesn’t.

Recently, Warner announced that it would start releasing high definition films exclusively in Blu-ray format. They are now the fifth studio to exclusively support Blu-ray, among such giants as MGM, Disney, and Twentieth Century Fox. HD, on the other hand, is supported only by Universal and Paramount.

Blu-ray and HD DVD are too expensive to coexist peacefully, at least for the near future. Each requires a several hundred dollar investment. I am pleased that it looks like one will die out, because coexisting formats just result in a nightmare for uninformed consumers. Good for you, Sony. Ever since Betamax failed, we knew you’d rule video formats again some day.