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.

Noah Feldman

In looking at the brewing civil war between the two groups in Iraq, it's easy to assume that the cause is ancient hatred. Nothing could be further from the truth. For the overwhelming majority of Iraqi history, Sunnis and Shi'ites have lived peacefully side by side, and numerous Iraqis are the children of mixed marriages. Instead we are witnessing in Iraq what occurs when government collapses and there is no state around capable of guaranteeing personal security.

What do you do when your family is in peril and you cannot turn to the government for protection? The answer is that you will take security wherever you can get it. You need to find some group that will be capable of keeping you safe, and that group had better be one that can count on your loyalty just as you can count on its protection. If you are a member of my ethnic, racial or religious group, then we share at least some basic bond, which may be enough to ensure our loyalty to one another. I need some assurance that you will have my back, and identity is better than nothing.

Sunnis and Shi'ites may find themselves joining militias or supporting denomination-based political parties even if they are not particularly pious and would much prefer not to. Something similar happened in the former Yugoslavia when its government collapsed with the fall of communism and nothing replaced it. Ethnic activists - call them identity entrepreneurs - will always form the core of the new militia. These radicals will emphasize symbols, like al-Askari mosque that was blown up last week in Iraq, and hope that followers will react by strengthening their commitments to the group itself.

Is it possible to break the cycle of violence that gets under way when identity groups move toward civil war? One answer is for an outside force to impose a solution. The killing did not stop in Bosnia or Kosovo until Western powers showed they were willing to bomb. But this approach is not viable in Iraq, where U.S. bombs came first and civil strife has followed. Instead the only way out of the violence is for Iraqis to realize that they have more to gain by negotiating a settlement between their groups than they do by allowing a full-blown brothers' war to break out.

Vali Nasr Author The Shia Revival (forthcoming)

What lies at the heart of the sectarian violence in Iraq is not so much religious dispute as it is a very secular competition for power and prominence in the new Iraq. Iraq is not all that different from Northern Ireland or Bosnia, where religion paraded as ethnicity and became a vehicle for communal rivalries. In the vacuum of power left by the fall of Saddam Hussein, the game of numbers has favored Shi'as, who are 60% of the population. It is for this reason that they wholeheartedly embraced democracy. Disgruntled Sunnis, on the other hand, vested their fortunes in boycott and violence, hoping that as spoilers, they would gain leverage in negotiating over the future.

Few in the West recognized the depth of either the Shi'a anger at the Saddam regime or the Sunni rage born of loss of power. There is a strong sense of Iraqi identity among both Shi'as and Sunnis, but as strong allegiance to sect and ethnicity in every election has shown, a shared notion of what Iraqi identity means and how each community sees the future of Iraq is fast disappearing. As happened in Bosnia, in Iraq mixed marriages and shared memory of coexistence will not be enough to stop internecine violence.

Shi'as embraced the political process that the U.S. set in place in 2003 in the hope that it would guarantee their security and serve their interests. There is indication now that many Shi'as are having second thoughts. Already overstretched in facing the Sunni insurgency, the U.S. can hardly afford losing the Shi'a as well. If tensions escalate to a full-blown civil war, Iran, Saudi Arabia, Jordan and Syria may all join the war to protect their co-sectarians and to scramble for pieces of a failed Iraq.

Pulling Iraq back from the brink will be difficult. Building a strong central government and an effective security force will help. The challenge is to get them up and running before events on the ground pass a point of no return.

James D. Fearon

By any reasonable definition, there has been a civil war in progress in Iraq at least since the Coalition Provisional Authority formally handed over authority to the Iraqis in 2004. A civil war is a violent conflict within a country fought between organized groups seeking to compel a major change in government policies or to take control of the center or a region. The insurgents in Iraq target the U.S. military, but they are also fighting against the Shi'ite-dominated Iraqi government and killing large numbers of Iraqis. There is little reason to think that if the U.S. suddenly withdrew, the insurgents would not continue to fight to control or shape the government.

When we hear talk about incipient civil war in Iraq, the fear is of an escalation of the current insurgency into a much bigger war. Analysts may have in mind something like the U.S. Civil War, with Sunni and Shi'ite armies fighting each other across well-defined fronts. Or they may imagine a sudden spasm of massive communal conflict and ethnic cleansing along the lines of Bosnia or Rwanda. Neither scenario is all that likely, although bouts of violent ethnic cleansing are certainly possible in a few parts of the country, especially Kirkuk.

My guess would be that as the insurgency continues to create insecurity, sectarian militias will continue to grow in power and influence. They will increasingly supply local security, but in the form of protection rackets that extort as they protect. They will clash with each other over territory and control of revenue sources. Since the Sunnis remain highly disorganized, some of these local fights may initially be intra-Shi'ite. But in the absence of effective political incorporation and protection from national police and army units - which are heavily infiltrated by Shi'ite militias - Sunnis will gradually form a patchwork of militias. Neighborhood-by-neighborhood conflict and violence will increase. Think Lebanon.

Juan Cole

If you look at the ethnic conflicts and street demonstrations during Iraq's modern history, it is remarkable how few have involved Shi'ites fighting Sunnis. During the colonial era, Iraqis were united by their opposition to the British occupation. Sunni and Shi'ite tribes cooperated in rebelling against British rule, and were only put down with a bombing campaign in 1920 that killed 9,000. In 1941 mobs targeted Iraq's small Jewish population; Jews had been a valued part of the Iraqi national fabric but were accused, unfairly, of being pro-colonial. After World War II, much of the violence in Iraq was fueled by issues of class. In 1948 slum dwellers and railway and oil workers revolted against a government treaty with Britain. In 1959, Arab nationalists assassinated Communist Party members, while mobs in Mosul and Kirkuk attacked and killed rich businessmen and landowners.

Iraqi Muslims have not all along been severely divided by religious sect. There have been many instances of strong cooperation between Sunnis and Shi'ites. Other social divides have led to mob violence in the past, but Iraqis have overcome them to re-establish national unity. It remains to be seen whether they can accomplish this feat again.

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.

You burst into the room. Sitting on a chair, blindfolded, his hands tied behind his back, is your prisoner. The room is dark, except for a lonely naked bulb hanging from the ceiling. He is sweating. He is afraid.

"Tell me where it is!" you scream. "Now!" You know there is little time left. Somewhere in your city, a time bomb is ticking. Whether it spits serin into the air, uranium into the water or atomic fire into the heavens, you do not know.

He does. But he is not talking. Involuntarily, you raise your hand as if to strike. What you are about to do violates the law and your conscience. And yet...

In peacetime, torture ranks next to murder as a primal sin. But during war, the debate begins over whether this evil can ever be justified to combat the seemingly greater evil of the enemy. Harvard law Professor Alan Dershowitz has said torture should be legalized.

In early October, the U.S. Senate voted 90-9 to ban it. Although Secretary of State Condoleezza Rice and President Bush have both recently asserted that "We do not torture," five U.S. Army Rangers were charged in November for punching and kicking detainees in Iraq, secret U.S. prisons have caused anxiety in Europe, and Vice President Dick Cheney has battled to win the CIA an exemption from the torture ban. As late as December, the U.S. House of Representatives stood poised to defeat the White House.

Few of us will ever be asked to torture. But, indirectly, all of us have to make a choice: to support, as citizens, those politicians who back torture, or those who seek its prohibition.

The decision of an individual to support, or reject, torture seems at first to be a purely moral question. But what would be the long-term consequences to society if we were to make this radical break with the past?

One cannot do experiments with societies, or predict the future, but, it turns out, one can attempt to address this issue using the cold, hard tools of mathematics and logic. This story begins in 1963.

The United States and the Soviet Union are on the perpetual brink of war, balanced like two sides of an equation. On the American side are "game theorists" like Thomas Schelling, recently awarded the Nobel Prize for his work on the strategy of conflict. On the Soviet side, there is the solitary mathematical psychologist Vladimir Lefebvre.

Just as mathematics could be used to describe logical reasoning, Lefebvre saw that mathematics could be used to describe ethical reasoning. If something was good -- for example, "church," "democracy," "prosperity," "kindness" -- it had value "1."

If something was evil -- "earthquake," "famine," "military defeat," "murder" -- it had value "0." But rarely were ethical situations so simple. For instance, "killing" is bad (0) but protecting one's country is good (1) -- so is war 1 or 0?

Lefebvre saw that, at the crudest level, there were essentially two types of ethical systems. Those that held that employing evil means to attain just ends was good, and those that saw that employing evil means to attain good ends was wrong.

There were also, crudely put, two types of relations between individuals: those entailing compromise (or cooperation) and those entailing confrontation.

Of course, evil people rarely see themselves as evil. So Lefebvre had to incorporate in his model of human nature the capacity of human beings to judge -- correctly or incorrectly -- the goodness or evil of their own acts, and to reflect upon their own judgments, and others'. "Reflexive Theory" was born.

It quickly became a paradigm within the Soviet defense establishment, with the publication of books such as "Mathematics and Armed Conflict." Nothing like it was known in the West.

With very simple assumptions -- for instance, that an individual who correctly sees his actions to be good when they are good, and evil if they are evil, is more highly regarded by society than an individual who incorrectly sees himself -- Lefebvre showed that in a society that accepted the compromise of good with evil, individuals would more often seek the path of confrontation with each other.

Lefebvre's insights were called upon by the State Department during negotiations with Mikhail Gorbachev in Reykjavik, Iceland. (And perhaps Lefebvre's model could be re-enlisted to help U.S. officials understand and negotiate with Arab and Muslim heads of state, who must also negotiate with their people.)

In support of Lefebvre's revolutionary new theory, a survey of Soviet émigrés and Americans was conducted in the 1970s. They were asked questions like, "Should a doctor conceal from a patient that he has cancer in order to diminish his suffering?" Overwhelmingly, the Americans would say no, and overwhelmingly, the Soviets yes. The Soviets accepted the compromise of good with evil; the Americans rejected it.

What does this mean? If Americans begin to accept the use of torture, American society might turn into a society of individuals in conflict.

Not uniformly, thanks to something called free will, but generally, with harmful consequences for society: Imagine two roads, with a stream of cars moving along each one. Each driver wants to reach his destination as quickly as possible; on occasion, drivers will impede each other.

On the first road, drivers rise in their own, and in other drivers', estimation if they yield. Drivers on the second road lose face when they yield. It is clear that traffic will move faster on the first road than on the second.

It can be argued that repressive states like Saudi Arabia, which bred most of the Sept. 11 hijackers, are on the second road. If the United States moved to accept torture, it could veer toward the second road, too -- the road of the Soviet Union.

And we know where that road ends. The Soviet Union no longer exists.