Tardy announcement: I’ll be giving a talk tomorrow at Harvard. Details. If you’re in the neighborhood, please look in.

Update: Harvard has posted video of the talk; please have a look, if you’re interested, and you’re equipped to play RealMedia files. I’m not equipped, so I haven’t seen it myself. But I’m sure it’s wonderful (thanks to the AV expertise of Bill Countie and Geoff Maness).

Slides are here (PDF), though they won’t make much sense without the narration.

The mail has brought me an early copy of The Princeton Companion to Mathematics, a 1,034-page compendium edited by Timothy Gowers (as well as June Barrow-Green and Imre Leader, associate editors), with contributions from more than 130 other authors. I’ve only just begun to browse through its pages, but already I’m completely charmed. This is one of those books that makes you wish you had a desert island to be marooned on.

In the preface, Gowers is at pains to establish that his book is a companion, not an encyclopedia. What that means, in part, is that authors are allowed to exhibit attitude and personality. Gowers wastes no time in doing so himself. The preface begins by citing a definition of “pure mathematics” written by Bertrand Russell in 1903:

Pure Mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants….

Russell is allowed to go on in this vein for another eight lines, and then Gowers remarks: “The Princeton Companion to Mathematics could be said to be about everything that Russell’s definition leaves out.”

Since I haven’t yet read more than 5 percent of the book, I’m in no position to review it here, but I think I can say a little more about the nature of what’s in it. Two big sections are essentially reference material: 99 short articles on mathematical concepts (arranged alphabetically) and 96 biographies of mathematicians (arranged chronologically, excluding living persons; the median birthdate is 1822). A third section gives brief accounts of 35 “theorems and problems,” many of them either open or recently solved (the Riemann hypothesis, the Mordell conjecture, Fermat’s Last Theorem) but also including a few classics (the three-body problem, the insolubility of the quintic).

Lots of good stuff in all of those sections, but not a lot of surprises. What I’m really warming up to are the parts of the book where authors are given freer rein to follow their own particular instincts or obsessions and where they express more distinctively personal views.

Gowers himself wrote a 76-page introduction that undertakes to explain what mathematics is all about, not only as a body of knowledge but also as a cultural phenomenon and as a way of thinking about the world. (The last section of the chapter is titled “What do you find in a mathematical paper?” At one point, Gowers begins to sound a little like Russell: “The object of a typical paper is to establish mathematical statements.”)

Seven more essays take another stab at introducing mathematics, this time working from a historical perspective. For the most part the sequence of topics follows an uncontroversial trajectory through the past two millennia: numbers, geometry, algebra, analysis, proof. But the editors have also decided to put algorithms on an equal footing with these subjects, a choice that would have been unlikely 50 years ago. On the other hand, the historical progression culminates in “The Crisis in the Foundation of Mathematics.” The crisis in question is that of the intuitionist rebellion and Gödel’s incompleteness results, events of the 1920s and 30s. It’s rather like a political history of the world that ends with the conflict between communism and fascism. But I suppose that the rest of the book could be taken as an effort to fill in the record of what’s happened since then.

The best bits of all come at the end of this weighty volume. Although the Companion claims to focus on “pure” mathematics, 14 chapters on “The Influence of Mathematics” show a definite leaning toward applications. We get views of mathematics in chemistry, biology, economics, statistics, music and art. And there are a few more narrowly focused essays, on topics such as wavelets, traffic and cryptography.

The book’s last section, titled “Final Perspectives,” is where I would recommend beginning. Here are the contents:

• The Art of Problem Solving, by A. Gardiner.
• “Why Mathematics?” You Might Ask, by Michael Harris.
• The Ubiquity of Mathematics, by T. W. Körner.
• Numeracy, by Eleanor Robson.
• Mathematics: An Experimental Science, by Herbert S. Wilf.
• Advice to a Young Mathematician, with contributions from Sir Michael Atiyah, Béla Bollobás, Alain Connes, Dusa McDuff and Peter Sarnak.
• A Chronology of Mathematical Events, by Adrian Rice.

Details: Princeton University Press. ISBN: 978-0-691-11880-2. Price: $99. The book’s web page has PDFs of a few chapters, as well as an interview with Gowers. Gowers also has a blog with a few entries pertaining to the book.

Economics has always mystified me, but watching the death spiral on Wall Street this past week has left me even more baffled than usual. I don’t pretend to understand what’s happening. But this morning I’ve begun to wonder if maybe—just maybe—there are some aspects of this debacle that I fail to understand not just because I’m thick but because they actually don’t make sense.

The part that’s puzzling me right now is an accounting rule known as “mark to market.” If I understand correctly, the gist of the mark-to-market principle is that the value of a thing is whatever you could get for it if you were to sell it right now. “Fair value” = “current exit price.” For some kinds of assets there’s a longstanding tradition of assigning value in this way, but in the past year the rule has been applied much more broadly. (See Rule 157 of the Financial Accounting Standards Board, which took effect November 7, 2007.)

Is it really a good idea to equate price and worth quite this rigidly? I’m not trying to overturn all that adamsmithian capitalist orthodoxy about a willing buyer and a willing seller and an invisible hand. And I understand that mark-to-market was introduced as an improvement over the mark-to-wishful-thinking kind of accounting that led to the previous round of scandals, such as Enron. Still, could it be that we’ve gone too far?

Consider the guy who sells umbrellas on the sidewalk around the corner from the New York Stock Exchange. Under mark-to-market rules, his inventory is assigned a much higher value on rainy days and becomes nearly worthless when the sun shines. I suspect that the umbrella guy has a very clear understanding of how the weather affects his business, and yet he doesn’t just throw away all of his stock whenever the sky turns blue. He knows that under those conditions he can’t sell an umbrella at a profit, but he still considers the umbrella to have a certain intrinsic worth. He hangs on to it. He carries it on his books as an asset. This concept of inherent value is apparently too sophisticated for the traders inside the exchange. In that world, if the Emir of Dubai doesn’t need an umbrella on Monday, then all umbrellas are “toxic.” But if the Secretary of the Treasury asks for an umbrella on Friday, then everybody invests in umbrellas.

Surely I’m missing something. It can’t be this stupid.

A couple of weeks ago I had the pleasure of chatting with Colin Marshall, whose radio show “The Marketplace of Ideas” is broadcast and webcast from KCSB in Santa Barbara. The main subject of our talk was Group Theory in the Bedroom, but the conversation also wandered into topics like the nature of publishing in the Internet age. The interview is available in streaming audio or as an MP3 file or as a podcast distributed via the iTunes store.

And if you grow weary of my voice, you might browse some of the other interviews—maybe Denis Dutton of “Arts and Letters Daily,” Michael Shermer of The Skeptic, or Steve Wozniak.

This is an update to “Shut up and calculate!” (which was posted here three weeks ago, 2008-08-12).

Many thanks to the readers who have suggested programming languages or environments for inquisitive computing. There are a dozen or so recommendations in the comments to the earlier post, and I’ve received even more by private correspondence. Here is a summary of the suggestions, in no particular order. (Numbers in parentheses indicate the number of times each system was mentioned.)

• R (2). The open-source version of the S language for statistics. (Also accessible through Sage.)
• Sage (1). Open-source mathematical software.
• Haskell (2). Lazy functional programming language.
• Python (2). Scripting and programming language.
• MATLAB (2) and Mathematica (1). $$$ mathematical software.
• Programmable calculators (3). The specific machines mentioned were the HP-15C, the TI-83 and the TI-V200.
• Yorick (1). An open-source scripting language that emphasizes scientific computation.
• Octave (2). Open-source mathematics software similar to MATLAB.
• SuperCollider (1). Language for audio and music synthesis.
• Fathom and Tinkerplots (1). Data-analysis and graphics software from Key Curriculum Press.
• DERIVE (1).Successor to muMath; now discontinued.
• Excel and other spreadsheets (2).
• APL (1), C (1), Forth (2), Fortran (2). Old favorites.
• UBASIC (1). BASIC with bignums and rationals; last release seems to be 1998; MS-DOS only.
• Scala (1). Recent open-source language that uses Java runtime facilities or .NET.
• “Roll your own” (2). Two readers politely suggested that if I think I know how a programming environment ought to work, then I ought to build one myself.

I’m impressed and surprised by the wide spectrum of responses. It’s not just that there were so many different answers but also that they come from some very distant corners of the computing universe. Several of the systems mentioned are new to me, and I plan to give them a look.

From all of the above it appears that we have some happy campers out there—people who have found programming tools that suit their needs. Others share at least some aspects of my discontent. But given the vastly differing preferences expressed here, it seems unlikely that any one solution could please everyone.