I only recently got around to counting up the results, and of those 48, I grouped them into six essentially different responses:

• pound (or pound sign, or “pound that shit bro”),
• hash (or hatch),
• number (or number sign),
• sharp,
• octothorpe,
• and a few other responses (tic-tac-toe board, and 0×23).

Many respondents seemed to have a primary answer, followed by caveats (e.g., “Pound, except in music, where it’s a sharp.”) so I differentiated between the primary and subsequent responses for each person. The following graph shows the results, where dark blue indicates primary responses, and light blue indicates all subsequent responses.

If you want the raw responses for some reason, and you don’t want to cull it yourself from my Twitter and Facebook, I can send you a spreadsheet. Graph is courtesy of the Google Charts API, which is a pretty sweet way to make easily-modified charts. (Check out the image URL if you’re curious how it works. I wrote a trivial Perl script to make it easier to incrementally edit the parameters and view the results.)

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.

In mathematics, a proof is no more than a convincing argument of a statement’s validity. But as justification for a theorem, a proof alone is wanting. Why state one theorem over another?

The metric of real-world utility seems not to apply. After all, the very objects of study are, in some sense, artificial.

But the allure of mathematics–to me, at least–is how well those objects fit together: the degree to which mathematics unifies apparently different concepts is staggering. Perhaps nobody said it better than G.H. Hardy in A Mathematician’s Apology: “The ‘seriousness’ of a mathematical theorem lies in…the significance of the mathematical ideas it connects.”

If research mathematics is about connecting ideas, surely math education ought to convey that interconnectedness. On the contrary; until reaching senior-level courses, mathematics appears an amalgam of essentially disparate concepts.

I’ll get a tracking number from, say, Amazon, before the package information is entered into their system. Fair enough. Other services (USPS, and I think FedEx as well) will return a page stating that the tracking number has been issued but they don’t have the package yet. But this is what I get from UPS (with the tracking number blacked out):

UPS tracking screenshot

The shipping date reflects my package’s shipping date, but the status, delivery date, signature, location, destination, service, and weight all correspond to a package which (presumably) previously used this same tracking number? (To wit, the delivery date is over a year before the shipping date.) Clicking on some links on that page gets me further tracking information about that previous package.

It seems as though they don’t actually delete tracking data–they merely overwrite it with new data as it comes in. They currently have only the shipping date of my package, so that’s the only part of that page which has been updated with my data. (Once they receive the package, all the remaining data is updated to reflect my package.)

This doesn’t seem like the right thing to do–they should clear out all the previous data once the tracking number is reissued. It doesn’t really matter, but I do get some information about some arbitrary package from about a year ago, and it could be potentially confusing. More importantly, it’s just kind of weird, and it’s happened to me several times recently.

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.

• yggdrasil: old headless Compaq box procured from Jonah; main entry point in LAN (port 22 forwarded here)
• midgard: my primary computer; micro-ATX desktop I built in March
• ratatoskr: Windows 7 VM on midgard; needed a separate name for networking (samba, mostly?) purposes
• nidhoggr: my Dell Latitude D620 laptop; recently rehabilitated since I’m carrying it around campus next year
• huginn: cell phone’s Bluetooth device name
• muninn: iPod touch (8 GB, 1st gen) for music and internet in my pocket

I think that’s about it. All computers are running Ubuntu; midgard and nidhoggr have Vista and XP installs, respectively, in case I need Windows for something.

I’m very satisfied with them, especially because the only earbuds I had were the iPod ones, which are basically terrible. I have a pair of headphones that I’m pretty satisfied with, but lately they’ve been sitting on my keyboard, and I’ve been using the earbuds for my computer and iPod.

(Yesterday I also bought a pair of cheapo speakers at the IU surplus warehouse for $10. They’re okay; the sound isn’t very clear, but they’re not for music anyway.)

Anyway, some people have asked me what I think of the earbuds. Since MS Paint is worth a thousand words, here’s my review:

They’re pretty much as good as my headphones, which is an accomplishment for cheap-ish earbuds, which are typically quite lousy. At any rate, I do recommend them, assuming you’re interested in buying earbuds for less than $100. According to the Internet, Skullcandy’s cheaper earbuds aren’t as good as the Titans, but I haven’t tried any others. In audio equipment, though, more money does translate into better sound (up to a certain point) so I wouldn’t try skimping.

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.

Much of my initial exposure to music was of the classical variety, and perhaps as a result, I have always conceived of music as something larger than a sequence of singles. I don’t like listening to the radio, because the selection is so manic (and repetitive). I prefer to listen to artists one at a time, and in particular, albums from start to finish.

So when I think of my favorite music, I think primarily in terms of albums. Which albums do I love in their entirety? Which albums do I keep turning back to, and rarely skip a track? I started forming a list, and realized that these are albums I have quite a bit to say about, and want to evangelize as simply great albums.

The first one I’m going to mention is the Foo Fighters‘ The Colour and the Shape.

The Foo Fighters are an “alternative rock” (whatever that means) band fronted and formed by Dave Grohl (previously Nirvana‘s drummer) shortly after Kurt Cobain’s death. Their eponymous first album was entirely written and performed by Grohl himself. CatS was their second album, and first with a band besides Grohl. Of course, Grohl was dissatisfied with their drummer, William Goldsmith, and ended up re-recording almost all the drum parts without telling Goldsmith, who subsequently left the band.

The album loosely follows the arc of a relationship, but I wouldn’t really consider it a concept album. It’s probably Nick’s favorite, and definitely one of mine. I don’t need to comment extensively on it, since theoretically you will go listen to it, but here are my liner notes:

“Doll” is a short, quiet prelude to the album, immediately followed by the fast, loud “Monkey Wrench.” One of the popular songs off the album, and one of my favorite, it uses the loud chorus, quiet verse structure that the Foo Fighters follow in many of their songs. (I think this song, in particular, shows ’80s metal’s influence on Grohl, a genre to which he later paid tribute with his solo metal side project, Probot.)

The next few tracks build from a calm happiness to a sort of loud elation that seems to climax at the end of “Up in Arms.” Following that is “My Hero,” a song that received a lot of radio play but which I honestly think is the weakest track on the album. I really don’t understand why people like it so much, besides its vague subject matter.

A few tracks later comes “February Stars,” a quietly resigned song that seems to signal something went wrong. After that is “Everlong,” without question my favorite Foo Fighters song, and one of my favorite songs ever. It’s the first track on the album since “Monkey Wrench” that I’d call driven, and it’s pretty much the opposite of “Monkey Wrench”‘s “good riddance.”

To me, “Everlong” perfectly captures the point just past the apex of a relationship, when everything’s too perfect and you feel an impending fall. It’s a sort of blissful desperation to keep things exactly the way they are. (“And I wonder / If everything could ever feel this real forever / If anything could ever be this good again.”) It’s a catchy song, almost therapeutic in its recklessness, as you remember what it is to feel that way. (Or maybe it’s just about sex, but is that reading really any different? SURPRISE CLAIM)

“Walking After You” follows through on the breakup, with a whispered, sad hope that things can be back that way. (“If you walk out on me, I’m walking after you”) But the album ends with hope, in what’s maybe my second favorite track on the album, “New Way Home.” It’s a persistently hopeful track that ends in the repeated insistence that “I’m not scared.”

I’ll also mention that one B-side to this album is a great cover of Gerry Rafferty’s “Baker Street.” You should check it out.

Anyway, I’m about out of things to say, so I hope you’ll check out this album, and love it as much as I (and Nick) do. If nothing else, listen to Everlong.

NEXT UP: Dream Theater’s Octavarium, an immense concept album made of total awesome.