Does the conflict in Iraq amount to a civil war? In many ways, the public debate over this question is largely political. Calling Iraq a "civil war" implies yet another failure for the Bush administration and adds force to the question of whether U.S. troops still have a constructive role to play.

Politics aside, however, the definition of civil war is not arbitrary. For some -- and perhaps especially Americans -- the term brings to mind all-out historical conflicts along the lines of the U.S. or Spanish civil wars. According to this notion, there will not be civil war in Iraq until we see mass mobilization of sectarian communities behind more or less conventional armies.

But a more standard definition is common today:

1) Civil war refers to a violent conflict between organized groups within a country that are fighting over control of the government, one side's separatist goals, or some divisive government policy.

By this measure, the war in Iraq has been a civil war not simply since the escalation of internecine killings following the bombing of a Shiite shrine in Samarra in February, but at least since the United States handed over formal control to an interim Iraqi government in June 2004.

Here's why: Although the insurgents target the U.S. military, they are also fighting the Shiite-dominated Iraqi government and killing large numbers of Iraqis. There is little reason to believe that if the United States were suddenly to withdraw its forces, they would not continue their battle to control or shape the government.

Political scientists who study civil war have proposed various refinements to this rough definition to deal with borderline cases. One issue concerns how much killing has to occur -- and at what rate.

2) For a conflict to qualify as a civil war, most academics use the threshold of 1,000 dead, which leads to the inclusion of a good number of low-intensity rural insurgencies.

Current estimates suggest that more than 25,000 Iraqis have been killed in fighting since the U.S.-led invasion in March 2003 -- a level and rate of killing that is comparable to numerous other conflicts that are commonly described as civil wars, such as those in Lebanon (1975-1990) and Sri Lanka (beginning in 1983).

The organization -- or rather, disorganization -- of the warring communities in Iraq means that a large-scale conventional conflict along the lines of the U.S. Civil War is unlikely to develop. More probable is a gradual escalation of the current "dirty war" between neighborhood militias that have loose ties to national political factions and are fighting almost as much within sectarian lines as across them.

This is roughly what happened in Lebanon and at a lower level in Turkish cities in the late 1970s. Ethnic cleansing will occur not as a systematic, centrally directed campaign (as in Bosnia), but as a result of people moving to escape danger.

And there's another twist to the terminology:

3) If the conflict in Iraq becomes purely a matter of violence between Sunni and Shiite communities driven by revenge and hatred rather than by political goals, many political scientists would say that it is something other than civil war.

Almost no one, for example, calls the Hindu-Muslim violence in India a civil war.

A civil war has to involve attempts to grab power at the center of government or in a given region, or to use violence to change some major government policy.

In Iraq's case, however, the vacuum of power at the center means that communal violence will inevitably be tied to struggles for political power and control.

A final complication concerns the nature of international involvement. Some argue, for example, that the war in Bosnia should be seen as an interstate war rather than a civil war, since the Bosnian Serb forces were armed and directed largely by Belgrade. Post-Mobutu violence in Congo is often termed a civil war, even though fighters have been closely tied to armies from neighboring states.

4) A conflict may be both a civil and an interstate war at the same time.

The Vietnam War, for instance, clearly comprised both a civil war in the South and an interstate war involving the North, the South and the United States.

Iraq may be moving in this direction. The United States and Britain are already openly involved, and such neighboring countries as Iran and Syria are more covertly involved. Not that it matters to the people dying there, but the next debate here may turn on whether what is already a civil war in Iraq should be viewed as an interstate war as well.

In his State of the Union address in January, President Bush stressed the importance of improving math education. He proposed to "train 70,000 high school teachers to lead advanced placement courses in math and science, bring 30,000 math and science professionals to teach in classrooms, and give early help to students who struggle with math."

But where will these teachers come from? And will the training of teachers be sufficient to increase the number of students choosing math and science careers? And why does all this matter?

Because mathematics is the foundation of the natural sciences. It is no coincidence that Isaac Newton, the man who formulated the law of gravitational attraction that revolutionized our understanding of the universe, was also the man who popularized the calculus. And the natural sciences, however pure, are what give us airplanes, cable TV and the Internet.

In the 2003 Program for International Student Assessment, a test that measures math literacy, American 15-year-olds performed worse than their peers in 23 countries, as well as those in Hong Kong. It's not hard to see why. According to the National Council of Teachers of Mathematics, 40 percent of the nation's middle school math teachers do not have the equivalent of an undergraduate minor in math. The average starting salary of a teacher is only $30,000, whereas the average starting salary for a recent college graduate in computer science or engineering is $50,000.

Short of following the British, who have proposed paying experienced math teachers more than $100,000, with a guaranteed minimum of $70,000, where will we find a way to attract the thousands of teachers George Bush wants?

New York State initiated an innovative program to bring teachers from Jamaica for two or four years to teach in New York schools. Jamaica, a developing nation where one U.S. dollar equals 65 Jamaican dollars, is nonetheless a stable, English-speaking nation with an unbroken democratic tradition; it stands poised to beat the United States in establishing the world's first Institute for Mathematical Methods in Counterterrorism. When teachers for the New York program were recruited on the campus of the University of the West Indies, recruiters found more experienced math and science teachers than they ever dreamed they would.

But you can have all the teachers in the world and still not inspire kids to learn math. My friend Autumn e-mailed me about her nephew, Joshua: "He's upset because he's asked several of the math teachers why math is important or what are certain formulas used for -- there has to be a use, correct?"

Autumn told her nephew about my work in counterterrorism and for the television crime drama "Numb3rs." Autumn reported, "He's told his math teachers about you as well, and about the show 'Numb3rs.' He's informing them that through something called lattice theory you are managing to fight terrorists -- all with math."

Mathematics is art, and should be appreciated for its beauty, not simply for its utility. But we cannot expect 11 year-olds to cherish totally order-disconnected topological spaces as much as professional mathematicians do.

As I first proposed in January 2005, television shows like "Numb3rs" (or "Medium") -- where the main characters are mathematicians -- could work with the National Council of Teachers of Mathematics to show kids how math is really used; the council and Texas Instruments are now working together to use "Numb3rs" to promote math literacy in schools.

Another way to inspire kids is to relate mathematics to something they see every day. In order to excite students and draw funding to his school, school superintendent Ronald Ross of Roosevelt, N.Y., has begun looking into the idea of creating a curriculum involving math and counterterrorism. What kinds of topics would students learn?

The opening line of the Oscar-winning movie "A Beautiful Mind" is "Mathematicians won the war." During World War II, the mathematics underlying cryptography played an important role in military planning. Winston Churchill admired Alan Turing, the mathematician who had mastered the German codes, recognizing him as the man who had perhaps made the single greatest individual contribution to defeating Hitler.

At Los Alamos, the lab that built the atomic bomb, Cliff Joslyn uses lattice theory to mine data drawn from thousands of reports of terrorist-related activity to discover patterns and relationships that were previously in shadow.

Lattice theoretical methods developed at MIT tell us the probability that we have disabled a terrorist cell, based on how many men we have taken out and what rank they hold in the organization. Lauren McGough, a Massachusetts high school student, tested the accuracy of this model by getting her classmates to pretend they were terrorists, passing orders down a fictitious chain of command, essentially confirming what the theory predicts.

High school students could learn algebra, trigonometry, calculus and logic while also learning concrete applications involving homeland security. No longer would students yawn and ask, "What is math good for?" Beauty could defeat both terror and boredom.

Whatever you may think of the State of the Union address, when it comes to supporting math education, we should all see pi to pi. President Bush is correct when he says that mathematics education in America must improve if the United States is to stay economically competitive, but the stakes are much higher than that. During the Cold War, the United States would not have tolerated a military gap between itself and its adversaries. Yet today, with 61 percent of all U.S. doctorates in math going to foreigners (15 percent to Chinese), we readily accept a "math gap."

Dollar for dollar, the best defense against our adversaries' weapons of mass destruction may be our allies in the Americas, armed with weapons of math instruction.

Improving math education is not merely a smart idea. It is a matter of national security. Algebra is one revolutionary Islamic concept we cannot afford to neglect or ignore.

Before attacking Iraq in March 2003, the United States told the world that Saddam Hussein had reconstituted his nuclear weapons program in defiance of the United Nations. That claim, used to justify the war, was based on assessments provided by the United States intelligence community. But as everyone now knows, those assessments were wrong. So Jeffrey T. Richelson's history of American nuclear intelligence, including our attempts to learn about Iraq's nuclear program, could hardly be more timely.

In "Spying on the Bomb," Richelson, the author of several books on American intelligence, has brought together a huge amount of information about Washington's efforts to track the nuclear weapons projects of other countries. He examines the nuclear projects of Nazi Germany, the Soviet Union, China, France, Israel, India, South Africa, Taiwan, Libya, Pakistan, Iran and North Korea, as well as Iraq. Through interviews and declassified documents as well as secondary works, he sets out briefly what we currently know about those projects and compares that with assessments of the time.

This may sound like heavy going, but Richelson writes with admirable clarity. And along the way he has fascinating stories to tell: about plans to assassinate the German physicist Werner Heisenberg during World War II; about discussions in the Kennedy and Johnson administrations on the possibility of attacking Chinese nuclear installations; about Indian measures to evade the gaze of American reconnaissance satellites; and about the bureaucratic infighting over the estimates on Iraq.

The United States has put an enormous effort into gathering information about the nuclear projects of other countries. After World War II it equipped aircraft with special filters to pick up radioactive debris from nuclear tests for isotopic analysis. It created a network of stations around the world to register the seismic effects of nuclear explosions. Most important, in 1960 it began to launch reconnaissance satellites that could take detailed photographs of nuclear sites in the Soviet Union and China. Richelson occasionally speculates about the role of communications intercepts and of spies, but these appear from his account to have been much less important than the other methods of collecting information.

Through these means the United States has gathered a vast quantity of data, sometimes to surprising effect. Intelligence played a crucial role in the cold war, for instance, by reducing uncertainty about Soviet nuclear forces. Alongside such successes, however, there have been failures. One notable example concerned the first Soviet test, which took place in August 1949, much sooner than the C.I.A. had predicted. Another was the failure to detect Indian preparations for tests in May 1998, even though at an earlier time the United States, with the help of satellite intelligence, had managed to learn about preparations the Indians were making and to head off their tests.

But the most serious failure of all was in Iraq in 2003, because in no other case did the intelligence assessments serve as justification for the use of military force. The information needed for avoiding political surprise is one thing. That needed for preventive war is quite another, if only because of the consequences of making a mistake.

Beyond making the uncontroversial recommendation that "aggressive and inventive intelligence collection and analysis" should continue, Richelson draws no general conclusions. That is a pity, because his rich material points to issues that cry out for further analysis. He suggests in one or two cases that failures sprang from the mind-set of the intelligence community, but he does not elaborate on this point. He has little to say about relations between policy makers and the intelligence community, even though the quality of intelligence and the use made of it depend heavily on that relationship.

His focus is no less narrow in his discussion of foreign nuclear projects. He concentrates on the programs themselves, paying very little attention to their political context. Does that reflect a technological bias in nuclear intelligence? Would, for example, the prewar assessment of Iraqi nuclear capabilities have been more accurate if it had paid more attention to the broader political and economic circumstances of Hussein's regime?

The task of intelligence has become more complex than it was during the cold war. A single dominant nuclear opponent has now been replaced by a number of nuclear states, along with states and stateless terrorists that are aiming to get their hands on nuclear weapons. Meanwhile, the technology needed for producing nuclear weapons has become easier to acquire.

Many critics believe the recent performance of the intelligence community shows it has not responded adequately to this new situation. Richelson does not have much to say on this question; nor does he discuss the likely impact of the current reforms, initiated in response to the Iraq war, on the quality of intelligence. His reticence may imply that he does not think reform is necessary. Still, it is disappointing that he does not draw on his historical survey to discuss whether new approaches are needed for dealing with nuclear threats, and, if so, what those new approaches might be.

George Habash, a militant and former secretary-general of the Popular Front for the Liberation of Palestine, once characterized terrorism as a "thinking man's game." Using mathematics, researchers at Stanford University's Center for International Security and Cooperation (CISAC) have made fighting terrorism a thinking man's game as well.

CISAC affiliate Lawrence M. Wein of the Graduate School of Business and CISAC Science Fellow Jonathan Farley are both applying mathematical models to homeland security problems, such as preventing a nuclear detonation in a major U.S. city and determining whether terrorist cells have likely been disrupted.

Wein, who teaches operations classes about different business processes used to deliver goods and services, has focused his research on bioterrorism and border issues. He has performed, he says, the first mathematical analyses of hypothetical botulism poisoning, anthrax outbreaks and smallpox infections.

"One overriding theme of my work is that all these homeland security problems are operations problems," said Wein, the Paul E. Holden Professor of Management Science. "Just as McDonald's needs to get hamburgers out in a rapid and defect-free manner, so too does the government have to get vaccines and antibiotics out and test the borders for nuclear weapons or terrorists in a rapid and defect-free manner."

In collaboration with Stephen Flynn of the Council on Foreign Relations, a nonpartisan research center, Wein recently has conducted research to improve security at U.S. borders and ports. Port security has received significant attention recently owing to the furor over Dubai Ports World's bid to manage six terminals at major U.S. harbors. The aim of Wein and Flynn's work is to prevent terrorists from bringing into the country a nuclear weaponbe it an atomic bomb or a so-called "dirty bomb," or conventional explosive packed with radioactive waste.

"Of all the problems I've studied, this is the most important because the worst-case terrorist scenario is a nuclear weapon going off in a major U.S. city and also it is the one the government has dropped the ball on the most," Wein said. "They have done a very poor job."

Instead of using the existing approach, where U.S. Customs actively inspects a minority of containers based on information from a specialized tracking system designed to identify suspicious containers, Wein and Flynn have recommended the government use a multi-layer, passive screening system for every container entering the country. Under their system, Customs would photograph a shipping container's exterior, screen for radioactive material and collect gamma-ray images of the container's contents. If terrorists shielded a bomb with a heavy metal such as lead to hide it from radiation detectors, gamma-ray imaging would allow inspectors to see the shielding and flag the container for inspection. Wein and Flynn believe this whole process would cost about $7 per container.

"Right now about maybe 6 percent of the containers are deemed suspicious and they will go through some testing and the other 94 percent of the containers just waltz right into the country without an inspector laying an eye on them," Wein said. "What we're proposing to do is 100 percent passive testing."

Wein's earlier work addressed a different threat: bioterrorism. In 2005, Wein revealed the nation's milk supply was vulnerable--a terrorist could potentially poison 100,000 gallons of milk by sneaking a few grams of botulinum into a milk tanker. Although the government and dairy industry have collaborated to intensify the heat pasteurization formula for milk, Wein is still pushing for additional botulinum testing, which he says would cost less than 1 percent of the cost of milk.

Wein also has used math to study smallpox outbreaks, the U.S. fingerprint identification system and U.S.-Mexico border security issues. Wein's congressional testimony on the fingerprint identification system in 2004 led to a switch from a two-finger system to a 10-finger system. His 2003 research on anthrax attacks resulted in a Washington, D.C., pilot program to use the U.S. Postal Service to distribute antibiotics throughout the capital after an outbreak. Seattle is now testing a similar program.

"In Washington, D.C., now, if there is a large-scale anthrax attack, postal workers will be the first to get their Cipro and, on a voluntary basis, they will go door-to-door distributing antibiotics," Wein said.

He said the common thread throughout his research is queuing theory, or the mathematical study of waiting lines, but he also draws upon mathematical epidemiology for his smallpox studies; air dispersion models for the anthrax model; supply chain management for the milk study; probability theory for the fingerprint identification system; and models for nuclear transport and detection for his work with containers.

From tainted lactose to lattice structures

While Wein is working on improving the government's counterterrorism systems, Jonathan Farley is working to figure out when terrorist organizations have been effectively disrupted. His mathematical model is designed to help law enforcement decide how to act once they have captured or killed a terrorist or a number of terrorists in a cell.

A professor at the University of the West Indies who will chair the Department of Mathematics and Computer Science there next year, Farley is on a one-year science fellowship at CISAC. In 2003, he co-founded Phoenix Mathematical Systems Modeling Inc., a company that develops mathematical solutions to homeland security problems.

He is using lattice theory--a branch of mathematics that deals with ordered sets--to determine the probability a terrorist cell has been disrupted once some of its members have been captured or killed.

"Law enforcement has to make decisions about what resources they should allocate to target different cells," Farley said. "The model should provide them with a more rational basis for allocating their scarce resources. ... It will inform you when you're making decisions about how much time and effort and how much money you're going to spend going after a particular cell."

While at Stanford, Farley hopes to unearth the perfect structure, mathematically speaking, for a terrorist cell--or in other words, a cell structure that is most resistant to the loss of members.

"If it's possible to determine the structure of an ideal terrorist cell, you can focus on a much smaller number of possibilities, because it makes more sense to assume the adversary is going to be smart rather than stupid," Farley said.

Farley has suggested it is possible Al-Qaida and other terrorist organizations already may have figured out the perfect structure for a terror cell by trial and error.

"I don't expect Osama bin Laden to be reading lattice theory in his caves in Afghanistan," said Farley. "But if it follows from the mathematics, perhaps heuristically, the terrorists will have come to the same conclusion--that this is the best way to structure a terrorist cell."

Although Farley acknowledges his model is not a panacea for terrorism, he hopes it will help reduce guesswork that might be involved in pursuing terrorists.

"It's not that I think mathematics can solve all of these problems," Farley said. "Because it can't. But it's better to use rational means to make decisions rather than guesswork."

John B. Stafford is a science-writing intern at Stanford News Service.

Since the controversy over cartoons of the Prophet Muhammad erupted, Europe's leaders have shown remarkable--and uncharacteristic--courage under fire. Refusing to apologize for the alleged slight to religious Muslims, a chorus of Continental voices has instead risen to the cartoons' defense, citing freedom of expression as the very essence of liberty, democracy and the European Way.

Unfortunately, free speech is about the weakest card in Europe's hand these days. An Austrian court's conviction and sentencing of the British historian David Irving to three years imprisonment for Holocaust denial is merely the most recent footnote to European hypocrisy on freedom of expression over the past decade.

The European Convention on Human Rights, which legally binds all EU states and supersedes domestic law, explicitly guarantees "the right to freedom of expression" including "the freedom to hold opinions and to receive and impart information and ideas without interference by public authority."

This provision is in keeping not only with the U.S. Bill of Rights, but with the central instruments of international human rights law to which Europe and America claim adherence. Yet Europe's interpretation of free expression has diverged markedly from America's broad deference to First Amendment freedoms of speech, assembly and religion.

American courts have upheld the publication of false, even racist materials, the right of neo-Nazis to rally in Jewish neighborhoods, and the objections of some citizens to the Pledge of Allegiance and to school dress codes on religious grounds.

European governments, on the other hand, have consistently trampled analogous rights, outlawing publication of hate speech, trade in Nazi paraphernalia, and the wearing of distinctive religious clothing, to name but a few recent examples.

According to the Austrian court that convicted him on Monday, David Irving's offense was to have "denied, grossly played down, approved, or tried to excuse" the Holocaust in print or other media, in violation of a 1992 statute. Although he has not been tried at home in Britain, Irving was convicted and fined in Germany in 1995 for "inciting race hatred."

At best, Irving is a monumentally terrible historian, who, only after publishing dozens of books on World War II, read the notes of the Holocaust mastermind Adolf Eichmann and came around to admitting that the Nazi genocide might actually have occurred. At worst, he is an artless but unrepentant bigot, on the model of America's David Duke or Austria's own Jörg Haider, but without any independent political power.

Why, then, is Irving's Holocaust denial, like other minority and extremist views in European society, of such great concern to lawmakers? If European governments want to guard against the repetition of genocide, they should actively educate their citizens in tolerance and respect for different cultures and beliefs, not gag those who express conflicting ideas.

Europe's suppression of free speech is guaranteed to spawn and incubate precisely the kind of bigotry and sectarian violence it is intended to prevent. Hounded for the unthinkable crime of publishing false history, David Irving appears almost heroic as he stands up to censorship, fines and imprisonment, making him a kind of martyr for neo-fascist groups.

Likewise, suppression of young Muslims' rights to dress or worship as their religion requires lends government sanction to already widespread anti- Muslim attitudes. This official xenophobia in turn breeds simmering resentment that has already exploded into mass violence and been manipulated by radical Islamists to recruit willing terrorist agents from within European society.

While European leaders should be praised for their belated conversion to the cause of free speech, outraged Muslims around the world are right to allege a double standard. Until Europe consistently respects its own guarantees of free expression, and actively promotes tolerance instead of clumsily stifling dissent, its brave rhetoric will ring disappointingly false.