In a comment on my earlier note about Wolfram Alpha, Daniel Asimov takes me to task for failing to explain “what Wolfram Alpha is.” I’ll accept the criticism, but I have to add that the question he raises is a real toughie. What, indeed, is Wolfram Alpha? Much of the prerelease hype (e.g., CIO, ZDNet, Telegraph) suggested it was to be some kind of search engine—a “Google killer,” or else, as Steven Levy wrote, “more like the anti-Google.” Another common theme (Infotoday, Guardian) suggests that Alpha is a manifestation of the semantic web, “that thing that Sir Tim Berners-Lee has been banging on about.”

There have been lots of other attempts to answer the ontological question. Jonathan Zittrain (via the New York Times) calls Alpha a “computable almanac.” Larry Greenemeier, writing for Scientific American, says: “Think of it as Ask Jeeves with [a] PhD.” Stephen Wolfram, the creator of Alpha, tells Rudy Rucker: “If anything, you might call it a platonic search engine, unearthing eternal truths that may never have been written down before.” Yuri Alkin, in a blog called Connections, had the wit to present the question to Alpha itself: “Who are you?” he asked. The polite reply, worthy of HAL or Commander Data, was “I am a computational knowledge engine.”

After a few weeks of sporadic poking around with Alpha, I’m finally ready to take my own shot at answering the big question. If you ask me, Wolfram Alpha is an oracle. Not an oracle in the computer-science sense—a hypothetical black box that simplifies complexity analysis by always giving the correct answer for queries of a specific form. I mean an oracle in the Greek-mythology sense—a sybil in a cave or a temple, whose responses to questions are often helpful but tend to be enigmatic and require careful interpretation. Sometimes you get just the answer you were looking for. Sometimes you get no answer at all. Sometimes the answer leaves you more perplexed than when you began.

All in all, perhaps it’s better to set aside the question of what Alpha is and ask what it can do.

It can do your homework (or your students’ homework):

Query: Limit (x^p)^(1/p) as p->0

Answer: \(\lim_{p \to 0}(x^p)^{1/p} = x\)

It can graph a function:

Query: plot sin(x)/x from -10 to 10

Answer:

It makes a handy desk calculator:

Query: 12 choose 3

Answer: 220

Query: factor 8549176323

Answer: 3 × 127 × 22438783 (3 distinct factors)

It provides access to a rich trove of “curated” data:

Query: molecular weight vanadium dioxide

Answer: 82.9403 (grams per mole)

It offers links to “live” data, updated in real time, on topics such as the weather and financial markets:

Query: weather Buenos Aires

Answer:

But the big payoff of a service like this lies in combining factual queries with mathematical or algorithmic analysis. Surely that’s what a “computational knowledge engine” should be good at, no? I’ve been trying hard to make Alpha perform in this way. So far I’ve found the process pretty frustrating.

Here’s a case study. Remembering an old story about Kansas being flatter than a pancake, I submitted the query, “flattest state in the U.S.” The response was another question: “Did you mean ‘fastest state in the U.S.?’”

Well, no, I didn’t mean that, but out of curiosity I clicked the link to learn which is the fastest state in the U.S. The reply, in its entirety, was this:

The oracle was in deep enigma mode. I decided to go back to the “flattest” question. Let me add that I hadn’t really expected my first query to work; a ranking of states by flatness is not something you’d find in an almanac (computable or otherwise), and indeed the concept of flatness has various possible definitions. I thought I could give Alpha some help by being more explicit.

Query: All US states maximum elevation - minimum elevation

Answer: Did you mean: US states maximum elevation minimum elevation

I wasn’t quite sure how to respond here, but it doesn’t cost anything to try, so I accepted Alpha’s rephrasing of the query. What I got back was not the answer I was looking for, but it was not entirely without interest:

Query: US states maximum elevation minimum elevation

Answer:

The scatterplot of highest and lowest elevations by states tipped me off that the data I’m looking for are in the system somewhere. Indeed, one of those dots in the lower left corner, with both lowest and highest elevations near zero, is probably the answer to my question (at least if we define flatness as the difference between maximum and minimum elevation). But how to identify the dot? Or, for that matter, how to identify the conspicuous outlier—the one state with a minimum elevation well above 1,000 feet?

I allowed myself to be distracted by the latter question. There are a couple of obvious guesses for the state with the highest lowest elevation, so I tried one:

Query: Colorado minimum elevation

Answer: 3314 feet

Hmm. That’s not the outlier in the scatterplot; 3300 feet is well off the chart. That means at least one state was clipped from the graph. Another query makes this more obvious:

Query: US states minimum elevation

Answer:

It appears that ranks 1 through 4 lie somewhere above the top edge of the graph. Is there some way to force Alpha to plot the complete data set, without arbitrary cropping? For some kinds of plotting, I’ve figured out how to control the range of the independent variable (see the command “plot sin(x)/x from -10 to 10″ above), but in this context I’ve not discovered the key, if there is one. And, as far as I can tell, there is no warning given when a plot is chopped.

Nevertheless, I was able to identify the four missing states. Accompanying the rank-order graph above was a helpful list, whose first entries were: Colorado 3314, Wyoming 3100, New Mexico 2844, and Utah 2001. The highest visible dot in the graph represents the fifth state in the sequence—Montana, with a minimum elevation of 1801 feet. The list gave the first five states and the last five in the ranking. This looked promising. If I could get a complete list of minimum elevations for all the states, and then the corresponding list of maximum elevations, perhaps Alpha could also give me the differences. I would ask it to alphabetize both lists, then subtract them element by element, and finally take the minimum of the result, or else sort again according to magnitude.

A button next to the truncated list of states promised “More.” I pressed it. Now I had the first 10 and the last 10 states, but I was still missing the 30 in the middle. Something else had changed as well: All the numbers were different, with the list of elevations beginning 1010, 945, 867. After a moment’s perplexity, I realized that Alpha had decided to shift from feet to meters. No matter. Three more presses of the “More” button finally got me a complete list of minimum state elevations (in meters). And the same rigmarole soon produced the analogous list of maximum elevations (again in meters).

But now I was stumped. How do I sort the list alphabetically? Can I subtract one list from another? Can I do anything to transform the output of a command? Is there any way to compose commands, so that the output of one routine becomes the input of another? Not a clue.

But perhaps I could do it the other way, slicing the salami crossways instead of longitudinally. Instead of compiling a list of maxes and a list of mins and then subtracting, I could subtract lowest point from highest point state by state and then list the results. Searching through various help files and lists of examples, I eventually came to a page on “Elevation Data,” with a subcategory “Minimum and Maximum Elevations.” And there, at the bottom of the page, was this suggested query: “Montana maximum elevation - minimum elevation.” Clicking on it gave me the result “11,007 feet.” So I could get the elevation range for a single state. All that remained was to persuade Alpha to map the same computation over all the states….

But wait. That’s where this story began, with the query “All US states maximum elevation - minimum elevation.” It didn’t work when I tried it before, and it still doesn’t work now.

I tried some minor variations in phrasing and punctuation, such as this one:

Query: (US states maximum elevation) - (US states minimum elevation)

Answer: 4341 feet

What does the number 4341 mean? A “Show Details” button led to the explanation:

Instead of subtracting the vectors element by element, the program is taking the median of each elevation list and then subtracting. (If I had wanted to do that, I wouldn’t have known how to ask for it.)

Finally, shown below in full detail is what came back after one further attempt to formulate the “flattest state” query:

Who asked about Albanian currency? I guess this is what the sybil says when she’s tired of listening to all of my questions.

*   *   *

Wolfram Alpha is an ambitious project, as its makers would be the first to proclaim. Here’s what the “About” page tells us:

Wolfram|Alpha’s long-term goal is to make all systematic knowledge immediately computable and accessible to everyone. We aim to collect and curate all objective data; implement every known model, method, and algorithm; and make it possible to compute whatever can be computed about anything.

It’s hard to resist making fun of these lofty and all-encompassing aims, especially when a fairly simple geographic query returns a result expressed in units of Albanian Lek-feet. All the same, I still applaud the attempt to create such a service, and I hope that Stephen Wolfram and his colleagues achieve some reasonable fraction of their goals.

The main sticking point, it seems pretty obvious, is not in collecting and curating data or in formulating models, methods and algorithms. It’s the access part. How am I to communicate with the system? How am I to specify which bits of systematic knowledge I’d like to retrieve, and how do I tell Alpha which models, methods and algorithms to apply? For more than 50 years the answer to this question has generally been a programming language of some kind. The designers of Wolfram Alpha have deliberately turned their back on that option, in favor of a natural-language interface. I’m sure they made this choice with the best of motives, in order to reach out to a wider audience that might be intimidated by formal notation. Unfortunately, the natural-language interface is so limited that we’re effectively left with no notation at all.

In a way, talking to Wolfram Alpha is rather like communicating in a natural language—a foreign language you don’t happen to speak. With grunts and gestures and a few stray nouns you may be able to get across the most rudimentary touristic needs—”Where toilet?” or “How much?”—but if you want to carry on a real conversation, you need more vocabulary and, most of all, you need grammar. I’m skeptical that Wolfram Alpha will ever be of much use without such a linguistic structure.

I have a new column out in American Scientist, on “compressive sensing”:

When you take a photograph with a digital camera, the sensor behind the lens has just a few milliseconds to gather in a huge array of data. A 10-megapixel camera captures some 30 megabytes—one byte each for the red, green and blue channels in each of the 10 million pixels. Yet the image you download from the camera is often only about 3 megabytes. A compression algorithm based on the JPEG standard squeezes the file down to a tenth of its original size. This saving of storage space is welcome, but it provokes a question: Why go to the trouble of capturing 30 megabytes of data if you’re going to throw away 90 percent of it before anyone even sees the picture? Why not design the sensor to select and retain just the 3 megabytes that are worth keeping?

The answer to this question leads to some ingenious mathematics and technology—but I’ll leave all that for the column itself. Here, regrettably, I need to try to patch up a few soft spots in my telling of the story. To make the best of the situation, perhaps I can find something interesting to say about my mistakes.

Here’s the first trouble spot. I wrote:

If we have N equations in N unknowns (and if the equations are all distinct) the system is certain to have a unique solution.

The equations in question are linear, and they have coefficients restricted to the set {0,1}. In my first draft, the parenthesis read “(and if the equations are all linearly independent),” but I wanted to avoid that bit of jargon, and I persuaded myself that because of the restriction on the values of the coefficients, distinct equations would necessarily be linearly independent. That’s wrong. Take the system:

\[ \begin{split}
0x + 0y + 1z& = 2 \\
1x + 1y + 0z& = 3 \\
1x + 1y + 1z& = 5 \\
\end{split}\]

We have three distinct equations in three unknowns with all coefficients in {0,1}, but the system is not linearly independent because the third equation is the sum of the first two. And the system has infinitely many solutions (namely all those that satisfy \(x + y = 3).\)

So I’m left wondering: Is there some way I could have explained the point correctly without a long digression on linear independence, the rank of a matrix and other niceties of linear algebra? Peter Renz, who pointed out my error in the first place, suggested this phrasing: “What is needed is that each equation adds information to the whole, so that no equation can be derived from the others.” I like that way of putting it, but on the other hand it doesn’t tell you how to look at a system of equations and determine whether each one adds information. Still, maybe it’s the best we can do.

The second expository pothole reads as follows:

For each vector element \(x\), the least-squares rule calculates \(x^2\) and then sums all the results. The search for a sparsest vector can be framed in similar terms, the only change being that \(x^2\) is replaced by \(x^0\). The zeroth power of \(0\) is \(0\), but for any other value of \(x\), \(x^0\) is equal to \(1\).

A reader noted that he had been taught—and had gone on to teach others—that \(0^0\) is an “indeterminate form,” without a definite value. And indeed it’s true: \(0^0 = 1\) if we take the expession as the limit of \(x^0\) as \(x\) approaches \(0\), but \(0^0 = 0\) if we view it as the limit of \(0^x\) as \(x\) goes to \(0\) from above. Given this discontinuity, I should not have declared so glibly that \(0^0 = 0\), as if there could be no controversy about it. Instead I could have phrased it along the lines of, “In this context it’s convenient to adopt the convention that \(0^0\) is equal to \(0\).”

But, on looking at the situation a little more closely, the whole business seems really murky. Here’s a passage from Concrete Mathematics, by Graham, Knuth and Patashnik:

Some textbooks leave the quantity \(0^0\) undefined, because the functions \(x^0\) and \(0^x\) have different limiting values when \(x\) decreases to \(0\). But this is a mistake. We must define

\(x^0 = 1\) for all \(x\),

if the binomial theorem is to be valid when \(x=0, y=0\), and/or \(x=-y\). The theorem is too important to be arbitrarily restricted! By contrast, the function \(0^x\) is quite unimportant.

In spite of this authoritative pronouncement, however, workers in certain other fields adopt the opposite convention. The tangled sentences that got me into this pickle were written an attempt to explain—again without introducing some hard-core terminology—the difference between the \(L_2\) and the \(L_0\) vector norms. And it appears that the usual definition of the \(L_0\) norm just doesn’t work unless \(0^0 = 0\). But I had never bothered to think clearly about this until my reader raised the question.

Suppose we’re given the four-dimensional vector \(\mathbf{x} = (2,0,1,5)\), and we’re asked to define its length \(\|\mathbf{x}\|\). The most familiar definition of length is the Euclidean “square root of the sum of the squares” formula, which corresponds to the \(L_2\) norm:

\[\|\mathbf{x}\|_2 = (2^2 + 0^2 + 1^2 + 5^2)^\frac{1}{2} \approx 5.477.\]

The \(L_1\) norm is defined in the same way but with first powers rather than second powers:

\[\|\mathbf{x}\|_1 = (2^1 + 0^1 + 1^1 + 5^1)^1 = 8.\]

The \(L_1\) norm is really just the simple sum of the vector components. In the same way that the \(L_2\) norm measures Euclidean distance, the \(L_1\) norm implements the Manhattan, or taxicab, metric.

Following the same scheme, we can invent lots of higher norms. Here’s the length of our example vector in the \(L_5\) norm:

\[\|\mathbf{x}\|_5 = (2^5 + 0^5 + 1^5 + 5^5)^\frac{1}{5} \approx 5.011.\]

We can generalize this idea in a formula that defines the \(L_p\) norm for any positive integer \(p\):

\[\|\mathbf{x}\|_p = \left( \sum_{i=1}^n |x_i|^p \right)^\frac{1}{p}.\]

As \(p\) increases, the \(L_p\) calculation gives greater and greater weight to the larger components of the vector. By taking the limit as \(p \to \infty\), we can even define the \(L_\infty\) norm, which is essentially a max operation—only the largest element contributes. For our example vector, \(\|(2,0,1,5)\|_\infty = 5\).

What about going in the other direction, to \(p = 0\)? In a fuzzy-headed, hand-waving way, it’s clear that what we want in this case is the sum of the zeroth-powers of the vector components. In other words, we simply want to count the components, ignoring differences in their magnitudes. Unfortunately, though, we can’t just plug \(p=0\) into the formula. That would give:

\[\|\mathbf{x}\|_0 = \left( \sum_{i=1}^n |x_i|^0 \right)^\frac{1}{0},\]

which just won’t do. The most serious problem, of course, is the \(\frac{1}{0}\) exponent, but there’s also the question of what meaning to assign to \(0^0\). As I understand it, the communities of workers who care about using the \(L_0\) norm solve these problems by introducing a special-case definition such as this:

\[\|\mathbf{x}\|_0 = \sum_{i=1}^n |x_i|^0,\]

along with the further proviso that \(0^0 = 0\). Accordingly, \(\|\mathbf{x}\|_0\) is a count of the nonzero components of the vector \(\mathbf{x}\). For our running example,

\[\|\mathbf{x}\|_0 = (2^0 + 0^0 + 1^0 + 5^0) = (1 + 0 + 1 + 1) = 3.\]

I guess this works, but if we need a special case anyway, wouldn’t it be easier just to define the \(L_0\) norm directly as the count of the nonzero elements? In formal terms this could be done using some sort of delta function instead of \(x^0\). Taking this approach, the \(0^0\) issue would never enter the case—and maybe I’d have a better chance of staying out of trouble.

I’ve let the tropical mathematics bandwagon pass me by. It’s not that I’ve been unaware. I noticed the stream of preprints, the special session at a joint mathematics meeting a few years back, workshops in Palo Alto and Moscow, upcoming events in Montreal and Berkeley. I’ve noticed all this, but I’ve been resisting it. Maybe it was the term “tropical” that put me off—a little too whimsical, with too-deliberate designs on my curiosity. But now the tropical craze has made it to the cover of Mathematics Magazine, with a lead article by David Speyer and Bernd Sturmfels (preprint version available here). I guess it’s finally time for me to figure out what the fuss is all about. The notes below—a work in progress—trace my efforts to do so.

For starters, what is that word “tropical” supposed to mean? Speyer and Sturmfels explain: “The adjective tropical was coined by French mathematicians, including Jean-Eric Pin, in honor of their Brazilian colleague Imre Simon.” Pin, in a 1998 paper (.pdf), deflects the credit to another French mathematician, Dominique Perrin, again noting that the name honors “the pioneering work of our brazilian colleague and friend Imre Simon.” Simon himself, in a 1988 paper (.ps), attributes the term to yet a third French mathematician, Christian Choffrut. Apparently, no one wants to lay claim to the word, and I can’t entirely blame them. Speyer and Sturmfels go on: “There is no deeper meaning in the adjective ‘tropical’. It simply stands for the French view of Brazil.”

The fundamental objects in the tropical world are the real numbers \(\mathcal{R}\) augmented with \(\infty\), and two operations on those numbers, tropical addition, denoted \(\oplus\), and tropical multiplication, \(\otimes\). Tropical addition is simply the minimum operation; that is, a \(\oplus\) b is equal to the lesser of a and b. For example:

\[ \begin{split}
8 \oplus 3& = 3 \\
-8 \oplus 3& = -8 \\
x \oplus \infty& = x
\end{split}\]

Tropical multiplication is equivalent to addition in conventional (non-tropical) arithmetic: a \(\otimes\) b = a + b. Some examples:

\[ \begin{split}
8 \otimes 3& = 11\\
-8 \otimes 3& = -5\\
x \otimes \infty& = \infty .
\end{split}\]

So that’s the key idea in a nutshell: plus is min and times is plus. This looks like a fairly strange and arbitrary reshuffling of definitions. First we get rid of addition, then we bring it back but call it multiplication. Why not just leave addition alone and redefine multiplication as min? There is a reason for doing it the other way around. In tropical arithmetic, the min operation \(\oplus\) plays the same role that addition plays in ordinary arithmetic, and \(\otimes\) has the same role as multiplication. In particular, consider the distributive law:

\[ a \otimes (b \oplus c) = (a \otimes b) \oplus (a \otimes c) .\]

This rule holds correctly when \(\oplus\) is min and \(\otimes\) is +, but it wouldn’t work if the definitions were swapped. The commutative and associative laws are also valid in tropical arithmetic. Speyer and Sturmfels point out that the system even fulfills “the Freshman’s Dream”:

\[ (a \oplus b)^{3} = a^{3} \oplus b^{3} .\]

On the other hand, there’s trouble with subtraction. No value of \(x\) can satisfy an equation such as \(x \oplus 2 = 5\). Because of this asymmetry, the algebraic structure of tropical arithmetic is classified as a semiring (whereas ordinary arithmetic on \(\mathcal{R}\) is a ring). Here’s another bit of jargon for jargon’s sake: the tropical semiring is idempotent. This just means that \(a \oplus a \oplus a \oplus … \oplus a = a\).

And now you ask: What’s it all good for?

As I understand it, Imre Simon and those French mathematicians who take a tropical view of Brazil were interested in the tropical semiring mainly in connection with automata theory. The idea goes something like this. A finite-state automaton can be represented as a directed graph, where the nodes of the graph are states of the machine and the edges are transitions between states. Suppose the edges are assigned numeric labels, and a path from an initial state to a final state accumulates the sum of all the edges traversed along the way. There may be multiple paths that connect the same initial and final nodes, and so it makes sense to ask which of these paths has the minimal sum. These questions are conveniently addressed in the algebra of tropical arithmetic, where the summation of the labels is the tropical product \(\otimes\) and the taking of a minimum is the tropical sum \(\oplus\). (Expressing the rules in a formal algebra rather than an ad hoc convention helps with certain proofs about the automata.)

Automata theory may be where it all began, but the current interest in tropical mathematics seems to have a different focus. The emphasis has shifted to algebraic geometry—an area of mathematics that I look upon with equal parts of fascination and fright. The main aim in this realm is to understand the solution sets of polynomial equations.

If you plot the values of an ordinary polynomial, you get a smooth curve. The graph below at right shows the simple quadratic function \(y=2x^2+x-1\) with its two real roots at \(x=-1\) and \(x=1/2\). What is the equivalent graph for a tropical polynomial? To construct such a beast, we first have to decide what \(2x^2\) might mean in a tropical context. In ordinary arithmetic, \(2x^2\) is shorthand for \(2 \times x \times x\), and so the obvious translation is \(2 \otimes x \otimes x\); translating back to conventional arithmetic, this expression becomes \(2 + x + x\), or in other words \(2x + 2\). Thus the quadratic term \(2x^2\) becomes a linear function, and the same transformation applies to higher powers also: In general \(kx^{n}\) is converted into \(nx + k\). Thus the exponent in the original equation becomes the integer coefficient that determines the slope of a line. The graph of the tropical polynomial is shown at left. The green line traces the complete function \(y=2x^2 \oplus x \oplus -1\); it is the union of segments and rays drawn from the three lines \(y=2x+2\), \(y=x\) and \(y=-1\). For each value of \(x\), the function takes on the least value of \(y\) that lies on one of these lines (the “lower envelope” of the lines). Graphs of all tropical polynomials share this same basic appearance: They are piecewise linear functions and they are always concave downward.

Just as the ordinary polynomial \(2x^2+x-1\) can be factored into the form \((x+1)(2x-1)\), the tropical polynomial \(y=2x^2 \oplus x \oplus -1\) has the unique factorization

\[2 \otimes (x \oplus -2) (x \oplus -1).\]

The constants –2 and –1 appearing in these factors correspond to the “bend points” in the graph of the function, and they play a role analogous to the roots of the ordinary polynomial. They are values of the variable \(x\) where two or more of the linear constraints are simultaneously satisfied. The locus of all such points is the solution set, the object of particular interest to the algebraic geometers.

For a polynomial in one variable, the solution set generally consists of isolated points. The situation gets hairier in higher dimensions, where the solution sets become algebraic “curves”—though they sure don’t look very curvaceous. Take the two-variable tropical polynomial function \(z(x,y) = x^2 \oplus y^2 \oplus 1\). The three terms of the polynomial define three planes in \(\mathcal{R}^3\), and the value of the function is the lower envelope of these planes:

The “bend points” are now “bend lines” where the planes intersect. Projected onto the \(x,y\) plane, this tropical “curve” has a distinctive Y-shaped form. At right, the rays in red represent the solution set. They divide the plane into three regions, in each of which a different linear relation has minimal height; the rays themselves define the locus of points where two (or, at the origin, all three) of the terms are minimal. This trident motif is preserved in all tropical curves, not just this simple example. The illustration on the cover of Mathematics Magazine shows a more complicated graph, part of a proof of the tropical version of Bézout’s theorem. (The theorem states, with various caveats, that two algebraic curves of degree m and n intersect in mn points. If I understand correctly, it carries over fully from “temperate” to tropical mathematics.)

The mere idea of reconstructing all of arithmetic with different operations, and then building higher-level structures (algebra, geometry) on top of this new foundation—all that’s enough fun to justify the project. But it seems there are also practical applications. Speyer and Sturmfels discuss a biological problem: Given genetic sequences that define pairwise distances between species, construct a phylogenetic tree showing the evolutionary history of the species. Tropical arithmetic turns out to be useful in finding a tree where all the distances are metrically consistent—that is, they obey the triangle inequality.

That’s about as far as I’ve been able to get in my explorations of the tropical latitudes. But in closing I’d like to say another word about the naming of things in science and mathematics. My sober, fuddy-duddy opinion is that we’d be better off in the long run giving things plain, descriptive names, without too much metaphorical spin on them. Physicists once had a lot of fun with quarks and gluons, charm and strangeness, infrared slavery and ultraviolet freedom; but eventually the joke wears thin. I gather that biologists are now regretting the game of giving genes names like Sonic Hedgehog. Tropical mathematics may also get tired of explaining over and over again about “the French view of Brazil.” On the other hand, I have to admit that if all those papers and workshops had talked about min-plus algebra, they never would have caught my eye.

WolframAlpha, the new whatchamacallit from Stephen Wolfram, is up and running. It’s also stumbling: I’ve gotten a fair number of error messages. But now is not the time for nitpicky griping. I just want to say bravo to the whole idea.

I applaud because Alpha invites us to use a computer for actual computing. Did you know that this machine that brings you YouTube and Facebook can also evaluate integrals and calculate correlation coefficients? What a thought!

I have ranted elsewhere about the sad decline of casual, inquisitive programming, and the lack of tools that will draw more people into that mode of exploring their world. WolframAlpha is not exactly the answer to my plea. It is a powerful computational tool cleverly disguised as a mild-mannered search engine. You’re not writing programs; you’re just “searching” for the roots of x² + x – 1 or the limit of (1 + 1/n)ⁿ as n goes to infinity or the sine of 10⁵⁰. I have a feeling that this search-engine metaphor may wear thin pretty quickly, but it may also serve its obvious purpose—breaking down barriers for those who wouldn’t dream of writing a line of Python or Lisp but who are masters of Google.

WolframAlpha is not the first thing of its kind. (My Sage friends will be quick to point out that they have had an online notebook interface up and running for a couple of years.) I’ve only just begun to explore Alpha, and I’m sure there are both treasures and pitfalls I haven’t glimpsed yet. Still, for the moment, I just want to cheer it on.

In recent weeks my junk-mail folder has received well over 100 copies of the folksy image reproduced above. I don’t read Russian, but I know just enough of the alphabet to sound out the second word of the headline. My русскоговорящего friend Dana MacKenzie kindly translated the rest of the message:

Village Spammer

The era of posters and announcements has passed. We will distribute your ad on the Internet.

As Dana points out, this is spam for spammers: metaspam. There’s a lot of that going around. After all, the entire online economy is supported by advertisements for companies supported by advertisements for companies supported by….

Incidentally, the painting that the spammers defaced is “Village Postman,” 1959, by Pyotr Krokhonyatkin. I identified it by guessing the title and running a query through Simple Image Search, which led me here.

While I’m on the subject of spam, here’s an update to my running tally of monthly receipts:

For March and April combined, 57 percent of my spam was in Russian (or some other language that uses the Cyrillic alphabet).

The tattered magazine shown above was on newsstands 60 years ago. You could have bought a copy for 50 cents—which wasn’t cheap at the time. The article featured on the cover, “Mathematical Machines,” surveys the whole topic of computational technology, from a primer on binary arithmetic to breathless reports on the hottest new machines—the Edvac, the Univac, Project Whirlwind. Other articles in the same issue include “Submarine Canyons,” “Greek Astronomy,” “Titanium: A New Metal” and “The Evolution of Sex.”

The creators of this ambitious magazine were Dennis Flanagan and Gerard Piel, two young writers who had become friends while working at Life magazine. In 1947 they set out on their own, planning to launch a wholly new magazine about science. When they learned that Scientific American was about to fold up after more than a century of publication, they bought it and made it over. By the time of that April 1949 issue, the new magazine had already found the formula and the format that would define it for a generation of readers.

One important part of the formula was a bit of an accident. Flanagan and Piel had expected to hire professional writers to produce the main articles, but they couldn’t find enough of them. So they began inviting scientists to tell their own story, in collaboration with an editor. This shortcut would soon become a key element of the magazine’s identity and its claim to credibility.

I grew up reading Scientific American, though I was not a subscriber. I discovered a disreputable shop on Race Street in Philadelphia that sold out-of-date issues for a dime each. These were copies with their covers torn off; years later I learned why. Newsstands are entitled to return unsold copies of magazines for full credit, but shipping all that paper back to the publisher was expensive and wasteful. So the dealers sent back only the covers and promised to pulp the rest of the magazine. Some of the discarded copies never made it to the recycling yard.

My adolescent reading of all those gray-market magazines was probably my main qualification when I joined the staff of Scientific American in 1973. I spent a decade there. I’ve done lots of other stuff since then, but the years with Flanagan and Piel were the great formative experience of my working life; in my own mind, I’ll always be a former editor of Scientific American. And I still have a deep affection for the magazine, even though I dislike what’s been done to it in recent years—so much so that I find it painful to read.

What went wrong? I think the genre of this story is tragedy: a noble enterprise undone by its own success. By the 1980s the magazine was prosperous enough to attract the attention of predatory investors. Their reasoning went something like this: If those eggheads can sell half a million copies and a hundred pages of ads with a magazine crammed full of physics and mathematics and molecular biology, just imagine how well we can do if we get rid of all that boring science. In 1986, under pressure from stockholders, the company was sold to Verlagsgruppe Georg von Holtzbrinck—by no means the worst of the suitors, but the terms of the purchase left the magazine desperate for new revenue. And the collateral damage was even sadder: Flanagan and Piel parted ways and never spoke again.

Over this past weekend I learned that Scientific American is going through another rough patch and will likely see further changes. (Sources: The New York Times, Folio, Portfolio.) Ad pages are down 18 percent. John Rennie, the editor since 1994, has “taken the opportunity… to find other engaging things to do.” At least 20 staff members will lose their jobs. The magazine will be uprooted from its own offices and will move in with the parent organization, Nature Publishing Group (another Holtzbrinck acquisition). Rumor has it that the new editorial direction will be even more consumer-oriented, focusing on science that’s “useful in everyday life.” The old commitment to articles “written by the scientists who did the work described” is under question.

I’ve been stewing over all this for a few days—or maybe it’s been a few decades. In any case, I’m tired of stewing; it’s time to move on.

At a personal level, I commiserate with those who are going to be hurt most—the staff members (including some old friends) whose jobs are in jeopardy. But in journalism and publishing these days we’re all sailing in leaky tubs and bailing as fast as we can.

When I take a step back and look at the larger cultural significance of these events, I think first of how important Scientific American was in my own upbringing and education. It’s my reflex to ask: How will kids like me get turned on to science without access to those coverless bootleg magazines at a dime a piece? But what a ridiculous question! The world has changed. The resources available today to a curious 14-year-old are far superior to anything I could have imagined back in the sixties. The little shop on Race Street was a treasure chest in its time, but it can’t compare with Wikipedia and the arXiv and PLoS and Weisstein’s World of Mathematics and Sloane’s Sequence Server and all the other bounty that’s now just a click away. Indeed, that’s a big part of the problem that faces Scientific American, and other publications.

We did good work at Scientific American, back in the old days. I’m proud to have been a part of it. But what we were building was not a monument to be preserved for the perpetual admiration of future generations. It was a channel of communication, a way of linking scientists with a broader audience. I believe it’s still important to keep those channels open, but the best way of doing it may be different in 2009 than it was in 1949.

On page 1 of the April 1949 issue of Scientific American is an advertisement from the Radio Corporation of America announcing—with considerable fanfare—the company’s latest innovation for audiophiles: the 45-rpm vinyl record. Today, the replacement of the replacement of the replacement of that recording medium is teetering on the edge of obsolescence. And yet music goes on! I want to believe that science journalism can also make a transition to a new medium, and perhaps come out of it better than ever.

My “Computing Science” column in the new issue of American Scientist looks at economics—including the current malaise—through the lens of control theory. This is not a new idea. The Keynesian prescription for smoothing out cycles of boom and bust is essentially an application of feedback control, although Keynes himself never described it that way. By the late 1940s the connection became explicit, as economists and control engineers worked together to refine tools for taming the business cycle. The two disciplines had another flirtation in the 1970s. Economists today seem less enthusiastic about methods borrowed from control theory, but they have certainly not given up on the aim of bringing the economy under control. In the past six months governments and central banks have intervened on a scale never seen before, all in an effort to “stimulate” economic recovery.

I’ll let the column itself tell the rest of this story; here I would just like to add a a couple of personal reminiscences about control engineering as seen by an interested outside observer.

Over the years I’ve had the opportunity to visit a lot of control rooms—at oil refineries, power plants, steel mills, railroad switch yards, sewage-treatment plants, particle accelerators, observatories, Internet operations centers. They’re not all exactly alike, but they do seem to share a common architecture, and also aspects of a common culture. For example, standard control-room iconography shows normally running machinery in red, whereas stopped or faulty equipment is green—the opposite of what you might guess after a lifetime of exposure to traffic lights. Years ago most control rooms had a “mimic board”—a long wall covered with a schematic diagram of the plant, festooned with gauges and switches and indicator lights. Now the operators are more likely to sit at ordinary computer screens. (I do hope they’re not watching YouTube videos while the reactor melts down.)

Two control rooms at the same plant—a paper mill in Catawba, South Carolina—serve to frame my impressions.

One of those rooms was in charge of a papermaking machine, a stupendous contraption the size of a football stadium, which takes in a slush of digested trees at one end and delivers huge reels of pristine white paper at the other end, with the sheet of paper-to-be threading between banks of steel rollers at a rate of more than 60 kilometers per hour. The control room for the paper machine was a calm and laid-back place. I sat with an engineer for an hour or so, talking about the thundering machinery outside the soundproofed booth, and all the while keeping an eye on the display screen in front of us. The indicators hardly budged the whole time. One key set of sensors in the papermaking process monitors the thickness of the moving sheet; if the thickness varies from the set point, a correction is made by blowing either warm or cool air onto one of the steel rolls. The temperature change causes the roll to expand or contract very slightly, and thus to squeeze the sheet a little more or less tightly. This is a subtle adjustment. Here’s what I wrote about it at the time:

It should be noted that the overall control strategy in papermaking is to operate the machine in a stable, steady-state condition. Blowing hot air on a steel roll cannot effect a large or a rapid change in the roll’s diameter; in the vocabulary of control theory, the actuators for this system have only limited authority. The limitation is intentional. The paper machine must run continuously for long periods, and changes in its state are designed to be gradual and confined to narrow bounds.

The other memorable control room at the Catawba mill was in the powerhouse, where boilers generate steam needed throughout the plant, as well as electricity. The main fuel for the boilers is “black liquor,” a strange watery brew created in the digesters that leach lignin and other kinds of organic gunk out of the wood. Black liquor is not a particularly good fuel, but the plant has to get rid of the stuff anyway, so burning it makes sense. But because of wide variations in the caloric value of the liquor, the boilers are hard to control. The first thing I noticed when I sat down with an operator in the powerhouse control room was that all the finish had been rubbed off one area of the console, surrounding a button whose label had also been rubbed away. I soon learned the function of that button: It silenced the alarm buzzer that sounds whenever some parameter wanders out of bounds. On the day of my visit, the buzzer was going off every minute or two. The operator displayed a well-developed Pavlovian reflex, slapping the silence button in milliseconds every time the buzzer sounded. Then he would turn to the screen, find the cause of the problem, and make some adjustment to fix it. Thus the powerhouse control room was a much busier place, and the mood was tense. If I had been doing that job, I’d have been ready for a beer at the end of my shift.

Somewhere in the bowels of the Treasury Department or the Federal Reserve (or maybe the World Bank), there must be a control room where operators at glowing screens watch the ebb and flow of money and make adjustments when some economic parameter strays from the set point. I wonder about the mood of that room. Is it one of those placid, hushed working environments, where controllers gently nudge the system back toward equilibrium? Or are they batting at buzzers all day?

Browsing through some notes I jotted down sometime in 1988, I come upon this sentence:

Mathematicians feel about computers much as the Nepalese feel about jet aircraft.

Did I read that somewhere, or did I make it up? My notes give no clue to the provenance of the thought, and Google is unhelpful.

Sometime in the early 1980s my friend Greg Chaitin decided to go digital. He wanted to have all of his own writings, along with some other documents he particularly valued, ready at hand in machine-readable form. This was long before Postscript or PDF; scanners and OCR were exotic technology. So Greg hired a typist. I know about all this because he sent me a printout of his digitized oeuvre—an impressively thick sheaf of fanfold paper, with sprocket strips still attached. (The irony of this mode of distribution was not lost on either of us.)

I’m finally trying to catch up with Greg. Instead of hiring a typist, though, I’ve bought a cute little document scanner, which has been busily transforming heaps of moldering cellulose into shiny new bits. The scanner produces PDFs, which then run through an OCR program to build an index, making the page images searchable. (Bring on the Googlebot!) The result falls short of ideal—files are obese, graphics are low-res, there’s no opportunity to edit or reformat—but still it’s better than rummaging through cobwebby filing cabinets in my basement. And I don’t have to FedEx cartons of fanfold to my friends.

I’ll be adding links to my publications list as I upload the files. Here are a few teasers:

The HP-41C: A Literate Calculator? Byte, January 1981, pages 118-138. PDF (3.3 MB)

40,000 Points of Light. Pixel, Vol. 1, No. 2, May-June 1990, pages 36-41. [On a graphics language in which an image is defined as a locus of points.] PDF (2.4 MB)

Rank-and-File Thinking. Lotus, June 1985, pages 73-77. PDF (1.8 MB)

Theory & Practice: On the bathtub algorithm for dot-matrix holograms. Computer Language, October 1986, pages 21-32. [On patterns generated by severely oversampled signals.] PDF (4.6 MB)

The Information Age: The Numbering Crisis in World Zone 1. The Sciences, Vol. 32, No. 6, November-December 1992, pages 12-15. [On the impending shortage of telephone numbers.] PDF (1.8 MB)

I note for the record that every one of the publications mentioned in the list above is now defunct. I wonder if there’s any significance in that?

For those of you who have been waiting patiently for the paperback edition of Group Theory in the Bedroom, and Other Mathematical Diversions: It’s finally here. Today is the official release date.

For those of you waiting for the movie version, well, so am I.