In arguing that the United States has achieved nuclear primacy and thereby made mutual assured destruction obsolete, Lieber and Press rely on some questionable assumptions about the status of Russia's strategic forces.

To make their case that the Russian strategic nuclear arsenal "has sharply deteriorated," Lieber and Press quote statistics showing that Russia today has "39 percent fewer long-range bombers, 58 percent fewer ICBMs, and 80 percent fewer SSBNs [ballistic-missile-launching submarines] than the Soviet Union fielded during its last days." These numbers are generally correct, but a similarly one-sided examination of U.S. forces would have painted a similarly dire portrait; after all, the U.S. nuclear arsenal today has 66 percent fewer strategic bombers, 50 percent fewer ICBMs, and more than 50 percent fewer ballistic missile submarines than it possessed during the Cold War.

Lieber and Press claim that Russia's strategic bombers "rarely conduct training exercises," implying that Russia's military is neglecting strategic aviation. Yet Russia's strategic air forces participated in four major exercises in 2005 alone. Similarly, Lieber and Press refer to a number of failures during launches of submarine-launched ballistic missiles (SLBMs) in 2004 to illustrate the decline of the Russian navy. But they do not mention the much larger number of successful launches or the Bulava SLBM development program, which has been quite successful so far.

It may be true, as Lieber and Press write, that "over 80 percent of Russia's silo-based ICBMs have exceeded their original service lives," but regular and successful tests of ICBMs that have been kept in silos for 25 years or longer suggest that this has not affected the missiles' reliability. Russia is decommissioning its ICBMs not because they are unreliable but because it does not need them. Lieber and Press claim that the development of the next-generation Russian ICBM, the Topol-M, has been "stymied by failed tests." But of 15 flight tests conducted to date, only one has failed -- a remarkable achievement for any missile-development program. It is true that the production rate of new Topol-Ms is low, but that may be because Russia has decided to concentrate on producing the mobile version of the missile, which it will begin deploying this year.

Lieber and Press are right to state that Russia may end up having as few as 150 land-based missiles by the end of the decade. But about half of those ICBMs would probably be road-mobile Topols and Topol-Ms, which, if operated properly, would have a good chance of surviving a first strike. Lieber and Press dismiss Russia's mobile missiles by saying that they "rarely patrol." In reality, very little is known about Russia's mobile-missile patrol rates, and although it is quite plausible that they are low, it is a stretch to assume that they are zero.

Lieber and Press describe Russia's early warning system as "a mess." In fact, although the system is past its prime, it has lost surprisingly little of its effectiveness. It may seem counterintuitive, but Russia would gain very little were its early warning system to be deployed to the fullest extent. Adding the capability to detect SLBM launches would not dramatically increase the time available to the Russian leadership for assessing attacks. The much-discussed "gaping hole" in the radar coverage east of Russia also should be put in context. Missiles launched from the Pacific could not cover the entire range of Russian targets that would need to be destroyed in a first strike. The scenario that Lieber and Press postulate, in which "Russian leaders probably would not know of the attack until the warheads detonated" because of flaws in their early warning system, is simply impossible.

The National Security Agency is a riddle wrapped in a mystery inside a code. Secrets lie within. Located in Fort Meade, Md., it dwarfs the CIA. Its budget is black, unknown. And, most disturbing of all, it is the world's largest employer of mathematicians.

One of its secrets, recently revealed, is that it's monitoring millions of phone calls to learn just who was calling whom. (Technically, only telephone numbers are being recorded, but you don't have to be Q from James Bond to get a name from a number.) This information was being used to determine who might be a terrorist.

Legal or not, the spying program isn't worth violating our civil liberties for. The information one can glean will hardly help us win the war on terror.

With the NSA data, you can draw a picture with nodes or dots representing individuals, and lines between nodes if one person has called another. Mathematicians who work with pictures like this are called graph theorists. The field of social network analysis deals with trying to determine information about a group from such a graph, such as who the key players are or who the cell leaders might be.

But even when you know everyone in the graph is a terrorist, graphs don't contain information about the order or hierarchy of the cell. Researchers look instead for graph features like centrality: They try to identify nodes that are connected to a lot of other nodes, like spokes around the hub of a bicycle wheel. Monterey Naval Postgraduate School researcher Ted Lewis, in his textbook "Critical Infrastructure Protection," defines a critical node to be such a central hub.

There are two problems. First, the central player might not be as important as the hub metaphor suggests. Jafar Adibi of the University of Southern California looked at e-mail traffic between Enron employees before Enron collapsed, and drew the graph. He found that if you naively analyzed the graph, you could mistakenly conclude that one of the central players was CEO Ken Lay's ... secretary. But that wasn't the person who ran the company into the ground.

Second, as the journal Studies in Conflict and Terrorism reported in 2003, you can kill all the central players in a terrorist cell and still leave the cell with a complete chain of command -- still capable of carrying out a devastating attack.

Expert Kathleen Carley of Carnegie Mellon was able to correctly predict -- twice -- who would take over Hamas when its leaders were assassinated, and her analysis used detailed information about the individuals in the organization, not just what anonymous nodes were linked with what. The moral is that the graph theory approach is inadequate. For useful results, it's important to utilize the lattice theory approach, which takes into account order and hierarchy.

The other questionable aspect of the NSA spying program is that it seeks to work out who might be a terrorist based on their calling patterns. While I agree that anyone calling 1 (800) AL-QAEDA is probably a terrorist, guilt by association is not just bad law, it's bad mathematics, for three reasons.

The simplest reason is that we're all connected. Not in the Haight-Ashbury/Timothy Leary/late-period-Beatles kind of way, but in the sense of the Kevin Bacon game. Sociologist Stanley Milgram took individuals unknown to each other, separated by a continent, and asked one person to send a package to the other -- but only by sending the package to an individual he or she knew, who could then only send the package to someone he knew, and so on. While Milgram's interpretation of the results has since been questioned, the conclusion that emerged is that it took only six mailings, on average, for each package to reach its intended destination.

For example, President Bush is only three steps away from Osama bin Laden. And terrorist hermits like the Unabomber might be connected only to very few people. So much for guilt by association.

The second reason the NSA methodology is flawed is the concept "strength of weak ties," made famous by Stanford sociologist Mark Granovetter. Robert Spulak of the Joint Special Operations University puts it this way: You might not see your college roommate for 10 years, but if he were to call you up and ask to stay in your apartment, you'd let him. This is the principle under which sleeper cells operate: There is no communication for years. The links between nodes that the NSA is looking for simply might not exist for the real threats.

Formal concept analysis, a branch of lattice theory, helps rectify this situation. Individuals who share many of the same characteristics are grouped together as one node, and links between nodes in this picture, called a concept lattice, indicate that all the members of a certain subgroup, with certain attributes, must also have other attributes. For instance, you might group together people based on what cafés, bookstores and mosques they attend, and then find out that all the people who go to a certain cafe also attend the same mosque (but maybe not vice versa). While this tool has in fact been used by researchers at Los Alamos National Laboratory to sift through hundreds of terrorism-related reports, it's still dangerous to rely on the math.

The NSA data mining is flawed because, as Kennedy and Lincoln buffs know, two people can be a lot alike without being the same person. Even if there is only a 1 in 150 million chance that someone might share the profile of a terrorist suspect, it still means that, in a country the size of the United States, two people might share that profile. One is just minding his own business. The other is Cat Stevens.

This isn't to say mathematicians are useless. In September 2004 -- 10 months before the July 7 bombing of the London Underground -- mathematician Gordon Woo warned that London was a hotbed of jihadist radicalism. But Woo, who works for the Bay Area company Risk Management Solutions, didn't anticipate the bombings using math. He used his knowledge and experience of London, especially the Wood Green area. That's what law enforcement officials should be doing.

As for tracking terrorist financing, it may already be too late. The terrorism of the future, according to mathematician Stefan Schmidt of the Technical University in Dresden, Germany, may be the terrorism of the futures -- when bombs explode, the stock market drops. Schmidt wants to quantify the impact on the market of a terrorist incident. The only people who know when a bomb will explode are the terrorists. By playing the market, they may already have obtained as much money as they need -- in perfectly legitimate ways -- thus stifling Treasury Department efforts to cut off the source of their funding.

Math is just a tool. Used wisely, math can indeed help win the Battle of Britain (by breaking the German codes). But used unwisely -- as seems to be the case in the NSA telephone caper -- your approval rating might just hit an all-time low.

It's been said that World War I was the chemists' war and World War II the physicists' war, but that World War III is destined to be the mathematicians' war.

With the gravest threat to U.S. security now posed by rogue terrorists who simultaneously hold a grudge and have access to weapons of mass destruction, some of this country's prime human calculators are on the case.

Combatting terrorism not with mortars and missiles but with mathematical models, they intend to prove this theorem: There literally is safety in numbers.

In their vanguard is an amiable Stanford University professor who, by devoting himself to the application of math principles to doomsday scenarios, is beginning to acquire the nickname "Dr. Doom." The attacks of Sept. 11 inspired mathematician Lawrence Wein to channel his expertise into some of the most compelling questions of our time.

He has ciphered the risks of our "woefully inadequate" inspection of container ships, assessed the effectiveness of border-control fingerprint checks to spot terrorists, and performed what may well be the first math analyses of hypothetical botulism, anthrax and smallpox contaminations.

Recall the agitation last year over warnings that the nation's milk supply was vulnerable, based on the calculation that one terrorist with a few grams of botulinum could contaminate a tanker and potentially poison 100,000 gallons of milk? That's just one of many tidings of comfort and joy brought to you by Wein.

Another such upper is that just about 6 percent of containers shipped into U.S. ports will be categorized as suspicious and subjected to tests for a nuclear device, based on a system that relies largely on reporting by the shipper. The rest, Wein noted, "just waltz right into the country without an inspector laying an eye on them."

At Stanford's Graduate School of Business, he teaches a core course in operations, and he says the parallels are strikingly similar: Just as McDonald's needs a well-designed distribution system to get its hamburgers out quickly, so the government needs a well-designed distribution system to get vaccines and antibiotics to citizens who might sicken or die from a bioterror attack. In math lingo, some of these computations rest on "queueing theory" -- the notion that a lot of needy people must line up behind a limited number of distributors.

And what do we do after crunching millions of numbers to arrive at specific prescriptions? Wein has taken on the role of necessary nag to a torpid federal bureaucracy and recalcitrant industries.

In fact, the feds scrambled to try to suppress publication of his damning milk study as a threat to national security -- an argument the journal Proceedings of the National Academy of Sciences rejected before publishing the work.

These days, Wein regularly testifies at congressional hearings, addresses scientific forums and pens op-ed pieces in national newspapers about precisely what the government should be doing -- and where it is falling short.

"I believe my work demonstrates that numbers really matter, and we need to pay attention to what they tell us," said Wein, a critic of the federal government's homeland security failures in key areas, particularly at the ports. "I really don't bring my own personal political views into this -- I would be just as critical of the federal bureaucracy if a Democrat were in the White House.

"Bureaucracy just isn't designed to respond nimbly."

On the other hand, politicians are eager to do something, anything, that might thwart a terror attack or save lives. But in an information vacuum, that makes them easy prey for entrepreneurs hawking all kinds of safety gadgets.

So, much of Wein's math analysis goes beyond documenting threats to assess the most effective, cost-efficient remedies.

Take the milk scare, for example. Wein urged the government to mandate that milk tanks and trucks be locked, that two people be present when it is transferred along each part of the supply chain, and that milk truck drivers use a 15 minute test to detect any toxins. The estimated cost of all this: 2 cents per gallon.

Some other experts disputed Wein's assumptions, and dairy industry groups insisted they already had taken extra steps such as raising pasteurization temperatures to secure the milk supply. They dismissed the scenarios Wein described as "highly unlikely or impossible."

To avoid being victimized by a radioactive device aboard a tanker -- otherwise known as a "poor man's missile" -- he and colleague Stephen Flynn of the Council on Foreign Relations advocate using techniques such as gamma-ray imaging to screen 100 percent of containers. They calculate the cost at $7 per container.

Some of Wein's calculations have translated into real reform.

In 2004, he presented to the White House his findings that the system of collecting only two fingerprints from incoming foreign visitors ran a high risk of missing terrorists because so many prints turned out to be of poor quality. His solution was simple: Take prints from all 10 digits. The government adopted it.

His 2003 research into the most effective response to an anthrax attack prompted pilot programs that are now testing his finding that the best way to widely distribute antibiotics would be by mail carriers, who already go door to door.

Albert Einstein once observed: "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

And sure enough, one of the impediments to fighting terrorists with mathematics is the paucity of hard data.

An anthrax scenario illustrates the point. Estimates of fatalities per kilogram of poisonous agent released vary from a low of zero to a high of 660,000. That's because of variables: the dose required to cause infection, the percentage of people who survive infection, the degree of aerosol dispersion, the density of the population, the environmental stability of the agent and the effectiveness of a public health system response.

As Wein acknowledges, "It's not like we can do massive clinical trials on this."

The trail of what-ifs is so convoluted that in 2002 the National Academy of Sciences cited "an irreducible uncertainly of several orders of magnitude in the number of people who will be infected in an open-air release."

This makes some experts suspicious of anybody's calculations, even from a researcher with a pedigree like Wein's.

"With so much uncertainly surrounding the outcome of a bioweapons attack, it does not make sense to plan extensive biodefense programs when more-certain public threats, particularly those involving nuclear weapons, require attention," argued Allison Macfarlane, research associate in the Science, Technology and Global Security Working Group at MIT.

But Wein is undeterred, and is busy on new assessments. Yet he insists his work has not impaired his ability to sleep soundly.

"People really are much more likely to suffer from cancer than a terrorist attack, but that doesn't merit the same attention," he said. "We should do what makes sense to protect ourselves from both."

One day perhaps Marina Agaltsova will join the diplomatic corps at a foreign embassy, or help write policy positions for the Russian government. Coit Blacker hopes that the lessons from her Stanford-sponsored distance-learning course will stick.

Agaltsova was among a group of Russian students brought to the provincial city of Yaroslavl in late May for an academic conference that capped this year's five distance-learning courses offered at nine universities across Russia by the Initiative on Distance Learning at the Freeman Spogli Institute for International Studies.

Through videotaped lectures, web readings and online chat sessions with senior research scholar Kathryn Stoner-Weiss and 14 other Stanford instructors, students in Democracy, Development and Rule of Law explored democratic ideals and practices, studying examples in Latin America, Asia and the former Soviet Union. "The course taught me that there is a black side to the reforms" that followed perestroika in Russia, Agaltsova says. "I learned more about Russian history [in the course] than I had learned in school."

That's the idea, says FSI director Blacker, who wants to re-establish the teaching of critical analysis, lost under decades of Communist rule, in Russian universities. "The social sciences were disemboweled," he says. He wants to develop future generations of diplomats and policy makers whose worldview is shaped "by how they think, not what they're told to think."

This year, to cap off the courses, 40 students came to Yaroslavl to participate in a mock United Nations Security Council session addressing Iran's nuclear program. They traveled from the farthest reaches of the Russian hinterlands, like Amur State University in Blagoveschensk, 4,800 miles from Moscow.

The arms control simulation is a teaching tool developed for the Stanford undergraduate class International Security in a Changing World, taught by Blacker and Scott Sagan, a political science professor and director of the Center for International Security and Cooperation within FSI. Sagan has exported the simulation to several universities in the United States where his former graduate students now teach--UC-Berkeley, Dartmouth, Columbia, Duke--but this was the first one he has conducted overseas.

This year's scenario was the International Atomic Energy Agency's referral of Iran to the U.N. Security Council for failure to fully disclose its nuclear activities. During the simulation, students submitted proposals to their heads of state, played by Blacker, Sagan and Russian faculty members. By the end of the two-day session, delegates had overcome seemingly intractable differences during four intensive sessions led by Stanford third-year law student Matthew Rojansky, acting as U.N. undersecretary-general for legal affairs. The council's resolution gave Iran three months to comply with the IAEA's requests and provided for Iran to obtain nuclear fuel from Russia, with the production and waste disposal to occur on Russian soil under IAEA controls.

After the session closed, students set aside their delegate roles to reflect on what they had learned. Narina Tadevosian, a student from Yakutsk State in far eastern Siberia, said she was surprised at "how strict Russia was" in taking a leading role in the session.

"If only it were so in real life," she added.