The most well-read cities in the United States

From Berkeleyside: Berkeley is the third most well-read city in the US, according to amazon.com data.

This is among cities with population 100,000 or greater. Number 1 is Cambridge, Massachusetts (105K people); number 2 is Alexandria, Virginia (140K); number 3 is Berkeley, California (112K); number 4 is Ann Arbor, Michigan (114K); number 5 is Boulder, Colorado (100K). There are 275 cities of population greater than 100,000 in the US; Alexandria, the most populous of these five, is ranked 177.

My first thought upon seeing this is that these are all small cities, and of course you expect to see more extreme results in small cities than in large cities. Small cities are perhaps more likely to be homogenous. (This seems especially likely to be true for small cities that are part of larger metropolitan areas.) Actually, my quick analysis of the top five doesn't hold up for the top twenty; the average rank of the top twenty cities listed at amazon is 127.1, which is LOWER (although not significantly different) from the 138.5 you'd expect if being on this top-twenty list was independent of size. But it's certainly possible that, say, some 100,000-person section of the city of San Francisco actually has higher amazon.com sales than Berkeley. (There are a surprisingly large number of bookstores in the Mission.)

Also, people in college towns tend to read a lot -- that's no surprise (although one does hear that students don't read any more a lot these days). Four of the top five (all but Alexandria) are college towns; also in the top 20 are Gainesville (Florida), Knoxville (Tennessee), and Columbia (South Carolina). And in case you're wondering, Alexandria is not named after the city in Egypt with the Great Library.

The NIST Handbook of Mathematical Functions

The National Institute of Standards and Technology has released what you might call a "trailer" for the revised edition of Abramowitz and Stegun's Handbook of Mathematical Functions. The original version is available online (it's public domain).

The print version is called the NIST Handbook of Mathematical Functions, and is available in hardcover and paperback.

There is also, not surprisingly, an online version, the Digital Library of Mathematical Functions, which takes advantage of new technology: three-dimensional graphics, color, etc. Think MathWorld, but less idiosyncratic. It jsut went public today.

And it includes Stanley's Twelvefold Way, which makes me smile.

However, some small part of the original Handbook's primacy as a reference comes from the fact that in a list of papers which are alphabetical by last name of the first author, it usually comes first. The first editor of the new book is Frank Olver, so it won't have that advantage.

Street-Fighting Mathematics is a book

From the Chronicle of Higher Education: The Gospel of Well-Educated Guessing, on Sanjoy Mahajan's Street-Fighting Mathematics. (Previously: here, and here.) It's now a real book!


Here's a calculation I hadn't heard of before, and don't actually know the details of:

They were both right, in a sense: some of the calculations he pulls off have a hint of Houdini. For instance, he can start with two paper cones, to find the relation between drag force and velocity, and—believe it or not—arrive at the cost of a round-trip plane ticket from New York to Los Angeles. He works out the problem in a blur of equations, remarking that a gram of gasoline and a gram of fat contain the same amount of energy, that drag force is proportional to velocity squared, and so on. The number he arrives at ($700) isn't the cheapest deal out there, but it's roughly right.

I've recently priced PHL-(SFO/OAK) flights, and this is roughly right. (And this uses chemistry, which is awesome because I was a chemist in a former life. Gasoline and fat are both basically long chains of carbon atoms.) The article tells of other similar party tricks. It would be nice to see some details, but the Chronicle seems to pitch itself at a humanities-ish audience.

Calculus made awesome

Calculus Made Awesome, by Silvanus P. Thompson, is available online. (Okay, so it's actually called Calculus Made Easy, but I like my alternate title better.)

Furthermore, unlike the modern calculus texts, it is nowhere near large enough to use as a weapon, even if you buy the print version, a 1998 edition fixed up by Martin Gardner Next time I teach calculus I must make sure to tell my students this book exists. And it's in the public-domain and free online, so it's not like I'd be recommending another expensive book. How much has calculus really changed in a century, anyway?

Thanks to Sam Shahfor reminding me of it. See also Ivars Peterson's review of the 1998 reissue. John Baez likes it but doesn't like that the new edition is longer than the old one.

List of free mathematics books

Possibly of interest: free mathematics books online. Many seem to be either quite recent (since, say, around 2000) or more than a few decades old, although I haven't systematically checked this statement; this isn't surprising, as the very old books tend to be in the public domain and the very new books tend to have been produced in a period when computers were more universal than they were in the past.

There are a couple hundred books listed here, which is not anywhere near the number of free mathematics books available (legally) online. Various other lists exist, with varying degrees of overlap. Sometimes I flirt with the idea of attempting a more complete list but I realize it would become out of date quite quickly.

The large sieve and its applications

You should vote at thebookseller.com for Emmanuel Kowalski's The Large Sieve and its Applications, which has been shortlisted for the "Diagram Prize for Oddest Book Title of the Year". (Apparently "large sieve" sounds like it has something to do with cooking, if you're not a mathematician.)

Brian Hayes on the continental divide

Brian Hayes recently made a fascinating post about determining the location of the Continental Divide. Apparently that's what those pictures on the cover of this month's Notices were. This is to go with David Austin's review of Hayes' Group Theory in the Bedroom, and Other Mathematical Diversions. The book consists, apparently, of reworked versions of Hayes' essays for The Sciences and American Scientist, essays which in general I highly recommend; the post by Hayes that I linked to is his reflections on redoing this map with the data that's available now, which is better than the data he had in 2000 when he wrote the original column. (Not surprisingly, the major discrepancy between the divides obtained then and now is in Wyoming, where there are some pathological areas that don't drain to either ocean.)

Neal Stephenson's Anathem

I haven't really digested it yet (in fact, I'm only a third of the way through it), but Anathem by Neal Stephenson is making a strong run at the title of My Favorite Book. Mostly because there's math hidden everywhere inside it. And how could I resist this, which almost feels like something out of The Glass Bead Game:

Three fraas and two suurs sang a five-part motet while twelve others milled around in front of them. Actually they weren't milling; it just looked that way from where we sat. Each one of them represented an upper or lower index in a theorical equation inolving certain tensors and a metric. As they moved to and fro, crossing over one another's paths and exchanging places wile traversing in front of the high table, they were acting out a calculation on the curvature of a four-dimensional manifold, involving various steps of symmetrization, antisymmetrization, and raising and lowering of indices. Seen from above by someone who didn't know any theorics, it would have looked like a country dance. The music was lovely even if it was interrupted every few seconds by the warbling of jeejahs.

Please, Internet, deliver me video of this mathematical dancing. Somewhat more seriously, though, moving pictures often float in my mind (and I suppose the minds of others) as I attempt to understand various mathematical structures.

Stephenson gave an interesting
talk/Q-and-A at Google about the book, if you've got an hour to kill. I think if you liked Cryptonomicon
you'll like this one; on the other hand I was disappointed by the Baroque Cycle, which lots of people seem to have liked. I suspect this has to do with the times in which they're set; the Baroque Cycle takes place a few centuries ago, Cryptonomicon during World War Two, and Anathem on another planet entirely, but one in which the secular world is roughly comparable to present-day Earth. (Perhaps a bit too comparable; Earth intellectual history and the intellectual history inside Anathem are essentially the same thing with different names.) Except in Anathem, the mathematicians live in what are essentially monasteries cut off from the outside world. I don't think I could handle that. I suppose some would argue that universities aren't the Real World, though...

The Know-It-All

I'm reading The Know-It-All: One Man's Humble Quest to Become the Smartest Person in the World, by A. J. Jacobs -- I remembered hearing good reviews, and they're for the most part right. It's an amusing book to read, because it's part trivia and part A. J. Jacobs, who's an editor at Esquire, read the entire Encyclopedia Britannica in a year or so, which is long enough that while he was reading it amusing stories unfolded in his life -- most of which have some trivia worked in. Is it great literature? No. But it's fun. And there's the inevitable moment when you know something that Jacobs doesn't mention he knows; that makes you feel a little smarter when you're reading any book, but especially one with this book's subtitle.

It's amusing and interesting to be reading this during my less ambitious attempt to read the The Princeton Companion to Mathematics
. But I'm mostly making this post because I couldn't resist sharing this, from when Jacobs joins Mensa:

Of course, I'm terrified that I'll be rejected. In fact, I'm pretty sure that they'll send me a letter thanking me for my interest, then have a nice hearty laugh and go back to their algebraic topology and Heidegger texts and Battlestar Galactica reruns.

From britannica.com, it looks like the EB does not have an article titled "algebraic topology". But there are articles titled "topology" and "mathematics" with sections named "algebraic topology", and they do seem to have serious treatments of mathematics in there. Jacobs admits (p. 338) that "the math sections... are my bête noire.

In which I buy the Princeton Companion to Mathematics

I cashed in 5.793 kilograms of assorted coins today at the bank. That's $115.19, for a density of $19.88 per kilogram; I know the weight because the bank's coin counting machine gave a receipt that had the numbers of each type of coin I received. I then immediately proceeded to the bookstore and purchased the The Princeton Companion to Mathematics (That was $106 with sales tax; perhaps I should have bought it from Amazon.com, where it's cheaper. Oh well, it's too late now. But at ten cents a page it's still a good deal.) I told myself I wouldn't buy it, but I'm doing some private tutoring on the side, and so there's a bit of extra money floating around, and I couldn't help myself...

I want to compliment the design of it; the side of the book which faces out on the shelf has "The Princeton Companion to" in small letters and then "Mathematics" in large letters, perhaps giving the impression that the book contains all of mathematics. This is not true, but from the portions of it I've seen online and the part I've read so far, it seems to admirably solve the optimization problem of squishing down mathematics to an object that only weighs five pounds. That's less than the coins I hauled to the bank to get the cash I paid for it! For links to various reviews, see this entry from Tim Gowers' blog. Gowers is the editor, and also wrote substantial parts of the book, although it has many other contributors; you can play the party game "how many of these names do I recognize?" I was interested to see that I recognize many more than I would have when I started grad school.

Yes, I'm actually trying to read it cover-to-cover. Like many mathematicians, there are a bunch of things that people seem to assume I'm familiar with, but I only heard about very briefly in some course in my first year of grad school in which I was holding on by the skin of my teeth. Indeed, this is one of the uses that's recommended in the introduction! I will resist the urge to turn this blog into a Companion to the Princeton Companion to Mathematics.

A couple of links

1. Jordan Ellenberg's review of Andrew Hodges' book One To Nine. Read the review, if only because it uses the word "mathiness". Ellenberg's review seems to imply that the book has similar content to most popular math books; sometimes I wonder how the publishing industry manages to keep churning out these books, but then I remember that the same thing is true in most other subjects and I'm just more conscious of it in mathematics.

2. Open Problem Garden, which is a user-editable (?) repository of open problems in mathematics. Thanks to Charles Siegel, my fellow Penn mathblogger, for pointing this out. The majority of the problems given there are in graph theory; that seems to be because Matt Devos, one of the most prolific contributors, is a graph theorist.

But I have to say that "garden" feels like the wrong word here; gardens are calm and peaceful and full of well-organized plants, which doesn't seem like a good way to describe problems that haven't been solved yet. "Forest" seems like a better metaphor to me -- certainly when I'm working on a problem that's not solved, it feels like hacking my way through a forest, not walking around a garden. Also, the use of "forest" enables bad graph theory jokes -- the problem of "negative assocation in uniform forests", due to Robin Pemantle, in particular sounds like it could be about sketchy people you meet in the woods.

(I gave a talk back in February where I mentioned this problem. I'm glad I didn't think of that joke then, because it's really bad and I would have just embarrassed myself.)

Link to review of The Drunkard's Walk

Here's a review of Leonard Mlodinow's book The Drunkard's Walk. I will resist the temptation to review the review. (Although I've been meaning to write down what I think of the book. I might do so.)

For some difficult-to-explain reason it is illustrated with a couple images from Jessica Hagy's indexed.

Richard Stallman: not a mathematician

Something I didn't know: Richard Stallman ("rms") of free software fame took Harvard's Math 55, often called "the hardest math class in the country". (I think I've linked to the Math 55 article before, but I can't find it.) He then took more math courses, and turned out to be quite good at it, but for some reason -- his biographer speculates it was because he shied away from the competitive aspects of mathematics -- he gravitated to computer science.

David Harbater (professor of mathematics at Penn) said of that class:

"only 10 really knew what they were doing." Of that 10, 8 would go on to become future mathematics professors, 1 would go on to teach physics.
"The other one," emphasizes Harbater, "was Richard Stallman."

Daniel Chess also makes an interesting point, about a "brilliant" proof that Stallman came up with his sophomore year:

"That's the thing about mathematics," says Chess. "You don't have to be a first-rank mathematician to recognize first-rate mathematical talent. I could tell I was up there, but I could also tell I wasn't at the first rank. If Richard had chosen to be a mathematician, he would have been a first-rank mathematician."

It's interesting that someone would even say this. Nobody would say that you don't have to be a great chef to recognize great cooking, because it goes without saying.

I learned this from Free as in Freedom: Richard Stallman's Crusade for Free Software by Sam Williams, which is available online for free.

Shoup: A computational introduction to number theory and algebra

A Computational Introduction to Number Theory and Algebra, online book, by Victor Shoup. (There's also a print edition, but the online PDF-book will remain freely available.) It is what it sounds like; "computational" doesn't mean "non-mathematical" but rather means that a lot of the applications are chosen with regard to their usability in computer science, specifically cryptography.

From reddit, where various commenters have pointed out things like "the author says that book is elementary but it really isn't." Of course, this is fairly common. The author actually says

The mathematical prerequisites are minimal: no particular mathematical concepts beyond what is taught in a typical undergraduate calculus sequence are assumed.
The computer science prerequisites are also quite minimal: it is assumed that the reader is proficient in programming, and has had some exposure to the analysis of algorithms, essentially at the level of an undergraduate course on algorithms and data structures.

As usual, "elementary" means something like "a smart, well-trained undergrad could read and understand it"; it's a term of art like anything else. But in some provinces of the Internet there seems to be an idea that everything should be immediately understandable to all readers at first glance. (I use Reddit for the links it gives me to interesting news, not for the comments.)

Bollobas and Riordan's "Percolation" -- cheap at Amazon

Amazon.com is selling Bollobás and Riordan's Percolation for $40.28 (marked down from $79).

Percolation isn't a subject that I'm interested enough in to buy the book (especially since our library has it), but I have talked about percolation here before and gotten some interesting responses, so I thought there might be someone out there who'd appreciate knowing.

The back of the universe

Brent Yorgey is working on a book intended to introduce high-school-level students to mathematics through hands-on problem solving. (His working title is The Math Less Traveled.) In his preface, he writes:

If you're having trouble with a problem, don't look at the answer -- put it down and come back later. This is how 'real' mathematics works: there's no 'Back of the Universe' where mathematicians can go look up the answer when they can't solve a problem!

If there were a "Back of the Universe", of course, that still wouldn't be practical, because it would be really far away.

Some might argue that the "Back of the Universe" is in fact the universe itself, at least in problems that have some interpretation in terms of real-world things. But just like the back of the textbook, physical experiment only gives answers, not solutions. If you throw a ball into the air and see that its trajectory is parabolic, that's something -- but experiment can't generally tell you why something is true.

The other wonderful thing about "real mathematics" is that, unlike for homework, you don't have to do all the problems; I believe I recently came across something stating that if a research mathematician solves one out of every one hundred problems they look at, they're doing well. This counterbalances the fact that real problems are much harder than textbook problems. (I wish I could cite this claim; it's on one of the many "advice for graduate students in math" web pages out there.)

Things I found from browsing David Aldous' web page

David Aldous has a list of non-technical books related to probability, complete with short reviews, and a much shorter list of technical books. This was constructed for his undergraduate seminar From Undergraduate Probability Theory to the Real World.

Also found from Aldous' page: Current and Emerging Research Opportunities in Probability, a report dating from 2002. This document summarizes views on that subject by the probabilists at a workshop with a similar name. Very short summary: probability is important -- here are a bunch of examples -- and we should make sure more people learn it. Since I talked about pretty pictures yesterday, I should point out that this report contains many pretty pictures.

A random fact about Aldous: very often when reading papers related to things he's done (which therefore cite him), the first reference in the bibliography is to one of his papers. This is, of course, because his name falls very early in the alphabet. This is true often enough that if I see [1] in the right sort of paper, and I just want to know the author of the work in question, I don't bother flipping back to the bibliography. It's very important here that he's at the beginning of the alphabet; the same phenomenon doesn't happen with a researcher at the end of the alphabet, because bibliographies are not all the same length. Also, papers with multiple authors tend to end up early in an alphabetized bibliography because there is something of a convention for collaborators to put their names in alphabetical order. (See, for example, the most recent submissions to the arXiv in mathematics. As that list stands right now, you'll have to go back a few pages to find a collaborative paper that's not alphabetized -- I was actually starting to think the arXiv might automatically alphabetize author names.)

A Mathematician's Apology available online

Hardy's A Mathematician's Apology is available online. (It's in the public domain in Canada, although not in the US.)

I'll refrain from commenting -- obviously I have thoughts, but I have things I really should be doing instead of trying to articulate them (like preparing for the class I'm teaching tomorrow morning!) -- and just let Hardy's work, one of the classic answers to questions such as "What is mathematics?" and "Why do mathematics?", stand on its own.

They say nobody reads anymore

John Armstrong is begging me in a comment at at Concurring Opinions to comment on this article from the Associated Press, which makes the following claims:

• One in four adults say they read no books at all in the past year


• "The typical person claimed to have read four books in the last year -- half read more and half read fewer." -- so although they don't want to use the word "median", they're saying the median number of books read is four.


• "Excluding those who hadn't read any, the usual number read was seven." Assuming that "usual number" means "median", this is saying that five-eights of people (the quarter who read no books, plus half of the rest) read seven or less books in the last year.

So what do we know about the distribution? One-quarter of people read no books; one-quarter read between one and four; one-eighth read between four and seven; three-eighths read more.


They claim a 3% margin of error, as well, which is standard for polls involving a thousand people (as this one was), but that margin of error only applies to the survey as a whole. The article includes a lot of claims of the form "Xs read more than Ys", but the number of Xs or Ys that were polled is less than a thousand, so the margin of error is greater.

It seems hard to get these numbers, though. The article claims that "In 2004, a National Endowment for the Arts report titled "Reading at Risk" found only 57 percent of American adults had read a book in 2002, a four percentage point drop in a decade. The study faulted television, movies and the Internet." (Emphasis mine.) If you interpret both of these claims at face value, 18 percent of non-readers have been converted to readers in the last five years. This seems unlikely.

I keep a list of the books I read; there are approximately one hundred and seventeen books on it. Yes, you read that right. That number's a bit inflated by the fact that about a third of those books were books I was rereading. But it's also a bit deflated because I have a tendency not to count textbooks, research monographs, and so on. (There's a pile of books up to my knee -- mostly library books, which is why they're in a pile, so I remember to return them -- which I read in the last year but aren't on this list.) I've also probably read a dozen or so books while sitting at the bookstore because I was too cheap to buy them, and some book-length online works...

Of course, I'm not typical.

A "book" doesn't seem like the right unit here, though. For one thing, some books are much longer than others. To take the two books on my shelf that I suspect have the most and fewest words, I estimate that Victor Hugo's Les Misérables has about 700,000 words, and Susanna Kaysen's Girl, Interrupted has about 40,000. For another thing, the person who reads a lot of books is going to engage a lot more deeply with some of them than others. There have been books that I've read in two hours and never gone back to; there have been books that I've returned to over and over again and find new insights every time. Most, of course, fall somewhere in between. And what if you don't finish a book? Does it still count?

I think the more useful metric would be "how much time have you spent reading books in the past year?" Because you can't ask people that, I suspect the Right Thing to do is to call people up and say "how much time have you spent reading books in the last week?", doing so at various times of year to control for the fact that reading is probably higher at some times of year than others, and averaging the results. But no one cares that much. (Actually, I suspect people in the publishing industry care very much, but they're not releasing their findings if they have any.) Or perhaps "how many words have you read in the past year?", but counting this seems almost impossible. (It wouldn't surprise me to learn that this number is rising, as a lot of online content is in written form.)

And as "Katie" commenting at Concurring Opinions points out, anyone at the high end of the spectrum probably underestimates. I don't know if this is true. I would have estimated I read "about two books a week" in the last year, for 104 books in the year; when I went and looked at the list, the actual count was 117. This might not matter all that much, because the pollster was asking about the median, and the people who are going to have trouble are the ones that are above the median. But I suspect that this poll is a lot like the recent polls about the number of sexual partners; people feel that they should lie about the number of books they read. I'm not sure in which direction they're likely to lie, though; I suspect it's correlated with education, and also with how many books one thinks one's friends read.

And what's so great about books, anyway? Why do we assume that reading books is automatically better than reading any other source of the written word? I suppose the argument is that a 100,000-word book requires more intellectual effort to read than, say, one hundred 1,000-word newspaper or magazine articles, because there is more interrelation among the ideas. But books come, for the most part, predigested. A lot of the real intellectual work is done in taking those clippings from various sources and making a book out of them. But this isn't a study of how intellectuals read, it's a study of how the person in the street reads. And even "study" is a bit too strong. They called up a thousand people and asked them some desultory questions. The AP did the poll itself. Let's face it, they're just trying to sell newspapers.

books, companions to books, and varying levels of detail

When I was taking my first graduate algebra class, I was desperate for any sort of supplementary material. One of our textbooks was Lang (we also used Hungerford) and so naturally I tried to find anything on the Internet that would explain what Lang was saying; Lang's textbook can be terse at points. (Incidentally, there's a note on the back of my copy of Hungerford, quoting from a review of the text, which says something like "the book is sufficiently understandable that an averagely prepared graduate student can read it and understand most of it." I found this to be true, at least at those times when I was actually willing to sit down with the book instead of desperately flipping through it to find the theorem that would magically make my homework work out, and I remember being quite amused by the fact that this was sufficient praise that Springer would actually print it on the book.

At one point during my searching I found George Bergman's A Companion to Lang's Algebra, which is basically what it sounds like -- a couple hundred pages of supplementary material, which Bergman wrote in his attempts at teaching the basic graduate algebra course at Berkeley. This material includes proofs of things that Lang doesn't bother giving a proof of, notes about other things that one might like to think about when reading a certain passage, alternate notations, some supplementary exercises, and so on. (For example, at the beginning of the fourth section there's an explanation of where matrix multiplication comes from; no graduate textbook would include this, because it's pretty much standard, but it's good to see how it fits into the new framework one is learning.)

In general mathematicians have a habit of leaving out some details when writing -- some of which are crucial, and some of which are not. This is usually explained by saying that the reader should be able to fill in the details. While ideally this is true, sometimes one just doesn't have the time to fill in those details, or -- more importantly -- sometimes one just doesn't have the ability, and seeing some sort of example of a proof or a calculation would make it easier to fill in similar details later. Now, it would be unnecessarily pedantic to fill in all the details of every calculation. But I suspect that in the case of a book like Lang which is widely used, all the details have been worked out by someone. All the interpretive notes that one might want are in someone's mind. Wouldn't it be nice if, when you came to a sentence that said "it is easy to see" and you don't see it, you could just click and see an explanation -- either written by Lang, or by someone else who has either struggled with the same point or watched their students struggle with it? (And yes, I know Lang is dead, so we can't ask him to do this.) In general, having a web site full of various explanations of such classic texts would be useful. A lot of this material is already out there on the Internet; people have written it up for the use of their classes. But having it all in one place would be useful. These various lecture notes, supplemental handouts, and so on could be threaded together into a sort of virtual textbook. Indeed, one could probably learn from the notes people have written on the Internet on any classic textbook without having the textbook itself.

And not just for books; research papers, too, could benefit from this. It's often written in a paper that "the reader can see that"; but you know that the author of the paper has actually done the calculation, and wouldn't it be nice if you could just click on the paper and have it appear? Sometimes we shoot too much for terseness, at the expense of clarity. And sometimes terseness is good; one doesn't want to be disrupted by the unnecessary details.

This is different from the idea of, say, a textbook which is in the form of a Wiki, which anyone can edit. What I'm proposing is that the original text is inviolable, not least because in many of these cases there may be a lot of printed copies out there, but that anybody who wants to -- or at least some reasonably large set of people -- could write comments on the text, instead of everyone sitting around and sweating through the details on their own. Something like having a paper which lots of people have scribbled notes in the margin with lots of different colors of pen, except you could actually read the notes, you could decide that you only wanted to read certain people's notes (the ones who seemed to struggle at the same points you did, or who were particularly good at explaining things, and so on). I'm not sure to what extent the mathematical culture would like this, because we don't place much value on mathematical exposition, and so exposition of exposition might be thought even less valuable. But it's interesting to think about.

(Although I think that this is a new idea -- I don't recall seeing anything like this before, which combines multiple people's notes on a text -- I am not under the delusion that nobody's thought of this before; I'd appreciate pointers.)