The Struggle Against Army Math

This essay is reproduced here as it appeared in the print edition of the original Science for the People magazine. These web-formatted archives are preserved complete with typographical errors and available for reference and educational and activist use. Scanned PDFs of the back issues can be browsed by headline at the website for the 2014 SftP conference held at UMass-Amherst. For more information or to support the project, email

The Struggle Against Army Math

by Madison SftP

‘Science for the People’ Vol. 6, No. 1, January 1974, p. 24 – 35




In February 1971, Karleton Armstrong was apprehended by Canadian police in Toronto, arrested, and jailed pending extradition proceedings. Imprisoned for the following 21 months, Karl was subjected to an extraordinary series of legal machinations by the judicial apparatus of Canada and the U.S. Finally, a hearing in Toronto concluded that the bombing had no “political connotations”, thus permitting Karl’s extradition under Canadian law.1 This was followed by a secret flight to Madison in the governor’s personal airplane, a $450,000 bail which is the highest in Wisconsin’s history, and months of pre-trial motions in a court room protected by closed circuit TV cameras, guards with metal detectors, and scores of redtape for anyone hoping to enter. 

Events climaxed surprisingly on September 28, 1973 with the announcement of a bargain between Armstrong’s lawyers and the prosecutors. Karl pleaded guilty to four counts of arson for his role in the August 1970 bombing of the Army Mathematics Research Center in which physics researcher Robert Fassnacht was accidentally killed. Armstrong also pleaded guilty to assorted lesser charges, including six bombings directed at ROTC offices, a Selective Service office and an ammunition plant. In exchange, the prosecution dropped the murder charges from first to second degree, recommended that the sentences be arranged to cover a maximum of 25 years in prison, and allowed Karl a two week hearing to persuade the judge to reduce the sentence. 

The mitigation hearing was a crucial part of the bargain as explained by William Kunstler, one of Karl’s lawyers: 

“Karl agonized over his decision. On one hand, he didn’t want to let down his many friends and supporters, who saw in his trial an opportunity to expose the insidious connection between the Army Math Research Center and the war in Southeast Asia. On the other hand, he wanted to be assured that he would be able to get such information before as much of the American public as he could reach. When his lawyers told him that it was extremely doubtful that such evidence would be admissible in a criminal trial, he elected to plead guilty and present it in full at a mitigation hearing. It was a hard choice for anyone to make and I am proud of Karl for the courageous way in which he resolved it. 

Karl’s decision was particularly courageous because he had a good chance of being acquitted of first-degree murder in a jury trial. The evidence indicated that stringent precautions had been taken to ensure that no one would be injured in the detonation. Madison police ignored an anonymous warning to evacuate the building. Karl might have escaped a lengthy sentence after a long legal battle-which might have buried the politics of his act. 

The serious question for us …
is not the question of our own welfare
but the moral trajectory of our act—
that it fly undeflected, to the heart of the matter,
which is
the infamy of the widening war
the grief torture dislocation death
rape murder terrorism
inflicted by our government
upon the innocent.

— from “The Passion of Dietrich Bonhoeffer”
by Daniel Berrigan

The Mitigation Hearing 

The hearing began on October 15, 1973, a triple anniversary: Karl’s birthday, the first moratorium against the Vietnam War, and the commencement of the war crimes trials at Nuremberg. Karl had already voiced the themes for his defense when he pleaded guilty: 

The acts with which I have been credited were undertaken with the purpose of crippling the efforts of the American government to wage an illegal, criminal and aggressive war against the Indochinese peoples, to prevent further loss of life, devastation and suffering. I have acted out of a sense of moral responsibility and felt for me, not to have taken action against this war would have been criminally irresponsible. I am not happy about the death of a human being and the injuries suffered by others as a result of these actions, but I do not apologize for having taken these actions.. These actions were intended as an affirmation of life and great precautions were taken to prevent injury to human life. 

According to attorney William Kunstler, Karl’s unprecedented use of a purely political defense is “historic.” 

In the two weeks of mitigation pleadings Karl’s defence was elucidated by the testimony of witnesses such as Vietnam veterans, Philip Berrigan, Anthony Russo, Daniel Ellsberg, biologist Egbert Pfeiffer and historian Gabriel Kolko. Many of the Vietnam veterans clearly found Karl’s act of resistance preferable to their own war crimes which they graphically explained to the court. After telling of his years in Vietnam, one marine, Sam Schoor, concluded, “I’ve killed many more people than Karl Armstrong and would gladly serve next to him in jail.” 

International law expert Richard Falk reviewed the U.S. government’s many crimes in Vietnam such as the indiscriminate killing of civilians by the electronic battlefield and automated air war, and then discussed the legal basis for resistance: 

“To stop the commission of great crimes, one may have to commit lesser crimes. What is illegal or criminal has to be understood in the context of the larger notion of the illegality and criminality of the war …. In the light of the Nuremberg tradition and the absence of constitutional redress, the sense of the right and the duty of the individual to take the law into his own hands is reinforced.” 

Falk then recalled cases during World War II when this principle was used by President Roosevelt and others to encourage the violence of the resistance movements against the Nazis. 

When asked about the legal responsibility of AMRC mathematicians for their work on the electronic battlefield, Falk cited the Nuremberg trials of scientists who developed Zyklon-B gas for the Nazis’ death chambers. Because these German scientists knew the use intended for their research, the Nuremberg tribunal found them guilty of crimes against humanity. 

It was during the time of the Sputnik that I decided to become a scientist because the country needed scientists. It was later when I entered college that I saw that science is studied for the benefit of the few, that it oppressed people. 

The bombing was a statement of moral as well as political protest. The depth of Karl’s moral outrage over the Vietnam War was pronounced in the testimony of Armstrong, his parents and the people who knew him in the movement. In his father’s words, “He couldn’t take it just because it was 10,000 miles away and he knew that what was happening there was just as important as if it were happening to my best friend right here at home.” 

As the hearing reached its end Karl recounted the process which prompted the son of a Madison worker to set bombs. First, as a nuclear engineering student. His struggle then became focused on the Vietnam War, and he went through a history familiar to many: dozens of demonstrations starting with draft protests in 1965, campaigning for Eugene McCarthy, marching in Chicago in 1968 and Washington in 1969. 

Concluding that peaceful means had been exhausted, he testified: 

I debated the use of violence. But I had a horror and revulsion against the war. I am a very non-violent person. I don’t like to use violence. I don’t feel comfortable with violence. After fire-bombing ROTC and the Badger Ordinance Plant, I felt very alienated from the violence I was using. I was wishing that there was some other way to stop the war. I alternated between pacificism and violence. 

But events led him to plan the bombing of Army Math: “I thought if the bombing of AMRC saved the life of one Indochinese, the destruction of professors’ research—all that destruction versus one life—to me, would be worth it.” 

The hearings were a compelling reaffirmation of Karl’s moral and political commitment. As Gabriel Kolko testified, “He will be respected and honored, and his action will be a milestone in the history of the resistance to the Vietnam War.” But the moral worth of Karl’s actions did not weigh as heavily as the political pressure which induced Judge William C. Sachtjen to hand down a 23 year jail sentence (two years being exempted from the maximum sentence for the time which Karl has already spent in jail). 

Army Math on Trial

The political import of the bombing of Army Math was not as widely grasped as its moral underpinnings. 

Too few people in Madison at that time understood the nature of AMRC and U.S. imperialism to follow up on Karl’s act of violence. The mitigation hearing, however, helped to build consciousness about Army Math here in Madison, although a media blackout kept the story from the rest of the nation.2

Along with The AMRC Papers, the record of the hearing is a powerful indictment of the center’s work. Steve Hawkins, a veteran of the Air Force in Southeast Asia, testified about the employment of AMRC research in the war. His list included the electronic battlefield, electronic and computer components of which had been produced at Project MICHIGAN with AMRC’s assistance, vehicles and methods of transporting weapons in the jungle (the work of the Waterways Experiment Station, a regular AMRC customer), and the dispersal of various chemicals in aerosol mists (a frequent element in the AMRC consultations with the Army’s chemical and biological warfare experts). In one of the most chilling moments of the hearing, Hawkins explained how the Air Force would lay down a cloud of CS tear gas over the Vietnamese countryside, and then drop napalm to cause a chemical reaction producing lethal hydrogen cyanide. Although the idea for the chemical reaction (obviously a war crime) was probably generated elsewhere in the Army’s research system, AMRC reports clearly mention research on the aerosol technology needed to control the cloud of gas. 

In cooperation with Armstrong’s defense, Science for the People also put AMRC’s acting director R. Creighton Buck on the stand at the mitigation hearing. The defense had subpoenaed records of AMRC’s reports which were delivered by Army Math’s director. After a technical examination on the nature of the documents, the prosecutor subjected Buck to a lengthy cross-examination in which all the center’s rationalizations for its work were once again enunciated. 

Today the Army Mathematics Research Center is still in operation. Fighting, hardship and injustice in Indochina are not over. American imperialism wields its puissant forces elsewhere. Graphically “bringing home” the horrors of Vietnam, the Armstrong trial was a summons to revitalize our commitment to the anti-imperialist struggle. It is our responsibility to Karl, to the spirit that engaged him, that we pursue that fight-because the “real murderers are still at large.”


In Madison, the struggle against U.S. imperialism has refocused on the Army Mathematics Research Center (AMRC) at the University of Wisconsin. Two important events in the fight to abolish AMRC have taken place this fall: Karleton Armstrong was tried and sentenced for his part in the 1970 bombing of the center, and a book called The AMRC Papers was published by the Science for the People collective in Madison. The book and the Armstrong trial exposed Army Math’s work to people throughout Wisconsin, and brought increased support for the two demands raised by Science for the People: close AMRC, and replace it with a People’s Mathematics Research Center (PMRC). 

A History of Struggle 

The background of the movement against Army Math was described in the article “Calculus for Conquest” in the March 1973 Science for the People. In Madison, the trail of expose, protest and demonstrations built up to a massive bombing on August 24, 1970, which destroyed the center’s building and accidentally killed a researcher. The Army Math center was seriously damaged by the bombing, but it soon resumed its work in another building on the fringes of the campus. In contrast, the movement against AMRC was dissipated by the shock over the death and by the repression that followed from the FBI, grand juries, and the local police. 

Fresh resistance to AMRC welled up in 1972 from two sources. First, Karl Armstrong, one of the four men accused of bombing AMRC, was captured in Toronto, Canada. As the legal systems of Canada and the United States put Karl on trial, the Left slowly came to his defense. The people who defended Armstrong had different attitudes towards the bombing of AMRC and the resulting death, but everyone was united by the idea that the American government, the murderer of more than a million in Indochina, had no right to try Armstrong for a single death. Under the slogan, “The real murderers are still at large,” the Armstrong defense also became an attack on U.S. imperialism and on AMRC in particular. 

The second attack on Army Math came from the Science for the People collective in Madison. Beginning in June 1972, SFTP demonstrated against the symposia which AMRC holds twice yearly on the Madison campus. During these actions, we condemned the center’s clear perversion of science for destructive ends as well as its role in preparing the U.S. for the Vietnam war and future wars against Third World movements. It quickly became clear to us that this condemnation by itself was not enough. We needed more concrete information on the workings of AMRC to provide a basis for our arguments. We also needed to present an alternative to AMRC to juxtapose what is and what could be. In this light, we developed the idea of a People’s Mathematics Research Center (PMRC), a center which would apply AMRC-type mathematical tools to the problems of working people. By contrasting PMRC with the reality of AMRC, we hope to build awareness of the misuse of U.S. science. 

The AMRC Papers

While we were picketing the AMRC symposia, we had ample opportunity to debate the nature of AMRC with the participants who crossed our picket lines. In these discussions, Army Math’s apologists argued primarily that the center did only “pure” research. We realized that the only public evidence to the contrary was circumstantial fragments such as travel vouchers for the center’s trips to Army bases. Even such basic documents as the AMRC contract were not available to the public. 

Consequently, in the Fall of 1972, members of the collective began a project of research on Army Math. While searching for the original Army-AMRC contract in the University Archives, Paul Still came across the papers of former U.W. president Fred Harvey Harrington. In Harrington’s files, Paul found not only AMRC’s contract, but a series of reports which the center regularly sent to the Army. These Quarterly and Semi-Annual Reports included lists of contacts between AMRC scientists and the Army. This was a major discovery. These reports spelled out in great detail the services which AMRC provided for the Army. The University Administration attempted to withhold these reports; but they are required to release most official documents to any member of the public by Wisconsin’s “Open Information Law.” The entire collective was soon working on the AMRC papers—digging out missing reports, collating the information and interpreting the facts. 

In our research, we concentrated on AMRC’s consultations with eleven Army bases which were producing weapons and strategies for counter-insurgency warfare, chemical and biological weapons, missiles and conventional weapons. We also found the center’s contract and budget, a list of the staffs highly inflated salaries, and documents on the maneuvers which brought AMRC to Wisconsin. 

Material of this extent and complexity could be published only as a book, and the entire collective started the writing in early 1973. After considerable discussion, we settled on the kind of book which a non-scientist could read and a scientist would believe. The chapters written by scientists were therefore reviewed by non-scientists. We also realized that all of our details about Army Math’s work made sense only in their political context—the U.S. Army’s role in guerrilla wars past and future. In long discussions, we clarified our thinking on U.S. imperialism and the People’s Mathematics Research Center, and included chapters on both these subjects in the book. 

We also decided to publish the book ourselves. We did all the work of typesetting and laying out the pages ourselves with equipment and advice of the local underground newspaper, Takeover. The people in the collective had little previous experience with publishing, so we learned each step in the process through trial and error. Some parts of the work turned out pleasantly: finding graphics and photographs, drawing the eagle on the cover out of equations from AMRC reports and seeing a good-looking page come together in our hands. However, publishing a 130-page book was a large burden for the fourteen people in the Madison collective, even with two of us working full-time on the project. 

Launching the Book

While The AMRC Papers was being printed, we began organizing a movement against AMRC. According to our analysis, Army Math could be removed3 by Wisconsin officials if they decided that the center were a political liability. Thus, our strategy is to mobilize public sentiment against AMRC throughout Wisconsin. 

Our first task was launching the book with enough publicity so that people would read it. To ensure that the media would not ignore us, we arranged a press conference in the office of Madison’s left-liberal mayor, Paul Soglin. In preparing for the press conference, we spent several weeks seeking support from reporters, editors and politicians. The result was good press coverage in Wisconsin papers and even in the New York Times

Having built this wave of publicity, our problem is to transform the press coverage into a general consciousness of the nature of AMRC, and to work with people to channel this consciousness into political action. To develop this consciousness, we are going out into the community—to clubs, classrooms, labor unions and city ward organizations—speaking about Army Math. To generate actions, we organized a demonstration protesting AMRC’s Fall symposium and introduced a resolution in the Madison City Council calling for the cancellation of the center’s contract. 


The following sections are excerpted from Madison SftP’s recent book (copies available), The AMRC Papers: An Indictment of the Army Mathematics Research Center. These excerpts are meant to convey the thorough documentation the book provides of the objectives and activities of the Center, and the critique it offers of the misuse of mathematics exemplified by the Army Mathematics Research Center (AMRC). 

The Army has realized from the outset that a university setting is essential for the kind of mathematics research center it needs, where close scientific contact between Army research and development personnel and other scientists, primarily academic ones, is possible. Additionally, only by providing a stimulating university environment can the Army draw the top researchers in the desired fields of applied mathematics to such a center. These researchers would not work in the more controlled environment of an Army base or laboratory where the options, and the publishing so important for the esteem of their scientific peers, are more restricted. These researchers, however, eagerly come to the University of Wisconsin to do the same research for the Army. What the Army expects from its partnership with the University is outlined in the objectives of the Army-University contract in which the University agrees to fulfill the objectives and scope, utilizing its best efforts, personnel, and facilities.

The contract states that the objectives are: 

  1. To provide a group of highly qualified mathematicians which will conduct mathematical research in the areas cited in (1)-(5) of paragraph A below. The emphasis in this research is to be on long range investigations with the intention of discovering mathematical techniques that may have application to the scientific and technical needs of the Army. The research is to supplement (not replace) that of existing Army facilities. 
  2. To provide for the Army a source of advice and assistance on mathematical techniques, mathematical programs and mathematical problems. 
  3. To provide a center for stimulating scientific contact and cooperation between Army scientific personnel and other scientists. 
  4. To increase the reservoir of mathematicians that may be called upon by the government for assistance in the event of national emergency. by acquainting mathematicians with problem areas relevant to Army needs. 

Contract DA-31-124-ARO-D-462
Modification P010, June 1973  

What AMRC Does 

The programs carried out by AMRC fall into several categories: research, consulting training, providing technical services, and working with academic scientists. These are each described in detail in The ARMC Papers where extensive information from letters, proposals, and reports is given as evidence of AMRC activities. This chapter not only demonstrates how these different activities fulfill AMRC’s contractual obligations, but how they are essential aspects of military research—the design and testing of weapons, and the formulation of military and political strategy. 

The mathematical papers published by AMRC provide AMRC spokesmen with the excuse that no secret work is done, in accordance with a University regulation, and that the Center’s total work is found in the “open literature.” But what is omitted from publication are the ways in which this so-called “pure research” is in reality directly applied towards solving the Army’s mathematical problems. The clearest method of this application is through the permanent staff’s consultations with Army base mathematicians, involving lectures, symposia, and orientation sessions with large groups of Army personnel, and other times the advising of smaller groups of Army mathematicians on specific problems… 

As the trail of Army-AMRC consulting was traced out by Madison SftP, the Army’s growing dependence on mathematical models became an obvious fact. Through mathematical modeling AMRC has helped the Army in three important areas. First, it has helped design new weapons and the technological components of new weapons systems. Second, it has aided in the testing of weapons. Third, AMRC has helped analyze and plan strategies for future warfare systems. Again, the real situation is simulated as a game in mathematical terms. The player of the game is the Army strategist, who tries out various strategies to determine which best attain the Army’s goal. The assumption is then made that the strategy working best in the game will work when the situation is faced in actual combat.

The Army transforms AMRC’s mathematical tools into military hardware and strategy at a number of research bases. These bases are a crucial step in the process which pipes “pure” University research into the American military machine. Gathered there are the scientists and engineers who apply AMRC’s work to strategies and weaponry. Providing these bases with the latest mathematical techniques has been AMRC’s primary purpose since its birth. 


The following excerpt is a discussion of AMRC’s work with STAG (Strategy and Tactics Analysis Group)—one of the ten Army Bases for which Madison SftP was able to obtain the most information on ARMC consulting. It is an example of the in-depth research contained in The AMRC PAPERS. 

STAG is an Army group which plans battle tactics and strategies at all levels. One of the main tools used in such planning is war games. These “games” are sets of mathematical equations representing a combat situation. The military strategist makes a decision; the decision is then represented by placing values in the equations. The answers, which are often found by computer, predict what would be the combat results in a real war given the strategist’s decisions. AMRC helped STAG prepare equations which will simulate, as correctly as possible, what really happens in combat. These are called “models” of war.

AMRC’s help began in 1960 and has continued through 1972, according to the Center’s latest written report, and may well be continuing today. Assistance has focused on increasingly sophisticated combat situations, including guerrilla warfare, as the Army’s work has progressed over the years. The mathematical problems in finding equations to represent the complexities of combat are very intricate and require AMRC’s expertise. 

To give a sense of what is required in such models of war, a basic model, the Lanchester Model, is described in the box on pg. 33. AMRC has been involved in attempts to improve this model over the last several years… 

Contacts between AMRC and STAG have taken place periodically from 1960 to the end of 1972. These consultations have been primarily with. one STAG person, Dr. R. Howes, …who works on computer models of guerrilla warfare, including the Lanchester model. The first consultation is described in AMRC’s 1968 Annual Report: 


March 21, 1968. In response to a detailed request for assistance with a problem concerning measures of effectiveness which was received from Dr. David R. Howes, U.S. Army Strategy and Tactics Analysis Group (STAG), Bethesda, Maryland on March 6, 1968 Professor J.B. Rosser wrote ‘to Dr. Howes to suggest a meeting between STAG personnel and Professor Bernard Harris. A preliminary study of the problem indicated that a solution technique could be provided. 

April 4, 1968, et seq. An exchange of correspondence began, resulting in an appointment for Dr. Howes to visit MRC on April 26, 1968. 

April 26, 1968. Dr. Howes consulted with an MRC group consisting of Professors M. Fox, B. Harris, G. Kimeldorf. and J.B. Rosser regarding the estimation of a parameter measuring relative combat effectiveness in a computer simulation of war games. The solution to this problem for the special case considered which had been worked out at MRC in the interim was presented by the MRC group. It seemed that it dealt adequately with the problem. 

“Effectiveness” in Howes’ work is probably the factor of combat effectiveness used in models such as the Lanchester equations of combat. 

In 1969, Howes wrote Rosser: 

Dear Mr. Rosser: 

Following our telephone conversation on 23 December, I discussed your Center’s draft of orientation lectures on mathematical programming with Colonel Carpenter, our Commanding Officer. Colonel Carpenter was most encouraged to hear that this series is close to realization. He hopes that STAG can avail its personnel of these lectures at the earliest date, since the maintenance and operation of large programming models has become a STAG responsibility. 

While STAG can look to other sources for instruction in various technical aspects of the operation of computerized programming models, it is only by means of a series such as yours that STAG personnel can be brought to appreciate the concepts and theories which underlie the computer models. 

I hope that you will be able to give early attention to your draft. 

Howes’ only paper in the Defense Department’s indexes appeared in 1971. It was entitled: ”GUEVARA, A Computerized Guerrilla Warfare Model” (AD-863 983L); the “L” attached to the code means that access to this paper is limited to those with the proper security clearance. 

In 1971, Howes’ consultations with AMRC continued when “Prof. J.B. Rosser furnished Dr. David R. Howes of STAG an approximating function for a certain integral” (27 April 1971 Semi-Annual Report)… 

Howes consulted AMRC again in 1972. According to the 20 October 1972 Semi-Annual Report: 

On 2 April 1972 Professor Louis B. Rail, Associate Director, returned material sent to him by Dr. David R. Howes, U.S. Army Strategy and Tactics Analysis Group, Bethesda, Maryland, concerning dynamic Lanchester equations. Since the problem of obtaining oscillations in a Lanchester model seemed to be fairly difficult, Professor Rail suggested using a Volterra model for the attrition rates and cited two references that might be of interest in this connection. 

This work on the Lanchester warfare models described above is an attempt to make the models apply to the more complicated situation where the probabilities of various outcomes of the combat change according to the progress of the fighting. The “Volterra model” referred to is an alternative type of equation that might be used in the Lanchester theory. It is interesting to note that John Nohel of the UW Mathematics Department has now, and has had for some time, a Defense Department grant to study these Volterra equations in their abstract form. Nobel, it should be noted, works occasionally for AMRC. 

The 20 October 1972 AMRC Semi-Annual Report further states that ”On 5 April 1972 MRC TSR Nos. 1140, 1142 and 1158 were mailed to Dr. David R. Howes, US Army Strategy and Tactics Analysis Group, Bethesda, Maryland, at his request:’ No. 1140, “Computational Solution of Ratio Games by Iterative Linear Programming,” was written by Stephen Robinson. The motivation for this study is spelled out in the 3 May 1972 Semi-Annual Report: 

Dr. Stephen M. Robinson consulted with Mr. David R. Howes at STAG, Md. on March 13, 1972. He presented a new method for formulating and wiving an optimal delaying action problem discussed by representatives of US-STAG at the 1971 Army Numerical Analysis Conference. He also pointed out possible applications of a ratio-game model to an optimal weapons allocation system Work on the latter subject is continuing. 

No. 1158 was written by Fred Brauer of the UW Mathematics Department on the “Predator-Prey problem.” The predator-prey type of equation offers a way of predicting when and with how much effort a predator can destroy a prey. Howes would be interested in using this type of equation in place of those in his current warfare model to see if it gets better results. The prey would be the guerrillas, and the predator the US. 

The predator-prey problem occurs in ecology as well as in Howes’ studies on guerrilla warfare. For this reason, the Technical Summary Report written on the predator-prey equations includes a note that it will be published in the “open literature”; the other two reports sent to Howes contain no such remark. The ecologists however will have to wait about two years for the AMRC report to appear in the journals, while STAG received a copy immediately. 

The third paper sent to Howes, No. 1142, was also written by Stephen Robinson. This paper concerns the Von Neumann economic model, a mathematical description of the functioning of an economic system. The fact that Howes requested information on economic models suggests that the economy of a country in which a guerrilla war occurs is a factor in the ·warfare models currently being constructed. In other words, the US is determined to manipulate the economy of whole countries in order to defeat guerrilla movements. This concern with economic modeling is a new’ direction in AMRC’s research, as we describe in the last section. 



The idea of mathematically representing combat to aid in making strategic decisions was first suggested by F.W. Lanchester around the time of the First World War. Lanchester formulated a battle between two forces in terms of equations which described the rate at which each side’s strength decreased. He was especially concerned with air combat. Today, Army researchers have modified Lanchester’s equations and assumptions to use them as a means of predicting the outcomes of other combat situations such as ambushes or guerrilla warfare. Primary efforts have been devoted to the problem of using the equations to determine, from the initial condition of the armies, which side will ultimately win. 

Lanchester himself dealt with two situations. In the first instance, each side knows only the general location of the opposition forces; and as units on either side are destroyed, the remaining forces then distribute their fire uniformly over the whole battlefield. In the second situation, each side knows the exact location of each opposing unit; and as units on either side are destroyed, the fire is concentrated on the surviving units. 

Actual Battles

These simple cases worked on by Lanchester have now been expanded to make the modeling more realistic. He assumed that opposing units had the same firepower, and to modify the equations, introduced different weapons such as tanks versus rifles.

But the outcomes of actual battles are determined by factors more complex than the numbers of troops and weapons. Model builders today incorporate human variables which introduce elements of certainty, that one side will defeat the other. 

Troop morale and training is an important factor. Whereas Lanchester’s equations described attrition resulting from hostile fire only, today’s equations are formulated to include desertion and surrender which also cause attrition. Retreat and advancement of forces could be predicted on the basis of the attrition they suffered; observed casualty rates and theoretically “acceptable” rates would be compared to determine whether a force should advance or retreat. 

Equations could be modified by allowing each side to add reinforcements during the fighting. Lanchester assumed that the size of the forces was fixed during the battle. 

An additional factor to be considered is the role of the military decision-maker in influencing the course of events. Essentially, depending on the factors taken into consideration, a war game played with opposing forces of the same relative strengths could have an infinite number of outcomes. 

Counter-Insurgency Planning 

Serious counter-insurgency modeling began in the early sixties by S.J. Deitchman, and was continued by M. B. Schaffer of the RAND Corporation. Their mathematical models were based on Mao Tse-Tung’s three phases of guerrilla warfare, which are summarized by Schaffer: 

The first two phases of insurgency are characterized by small-force ground-yielding operations by the insurgents but overall military superiority on the part of the counterinsurgents. In phase II the insurgent operations become increasingly military; however, they continue to be basically small-force guerrilla activity which cause the defense to fragment and the engagements to be localized and relatively isolated. In phase III the insurgents take the strategic offensive and operate with larger, more conventional forces. 

— Lanchester’s Models of Guerrilla Engagements 

To model guerrilla warfare, Schaffer extended the basic Lanchester equations to include the effects of battlefield desertions, capture of prisoners, supporting weapons, and changes in weapon efficiency over time (as could be caused by rusting or extended use). Schaffer’s equations represent three kinds of combat—skirmish, ambush and siege—which occur in “phase II” of Mao’s strategy. 

Skirmish is a battle in which surprise is not a factor. An ambush involves an element of surprise and, because of this, a smaller force could defeat a larger one. Siege involves an attack on a fortified position such as a strategic hamlet in Indochina. Here, timing the use of supporting weapons such as artillery or aircraft is critical. If a preliminary “softening-up” is undertaken, then the element of surprise is lost. 

Equations aid the planner in balancing some advantages against others. But as Schaffer points out, these equations cannot predict the outcome in guerrilla warfare because they do not take into account political, sociological, economic or moral factors. They do help in estimating casualty rates in both sides, demonstrated by the emphasis on “enemy body-counts” during the Indochina War. 

Recent modifications in these equations now take into account intelligence about the opposing force, command efficiency, and search and reconnaissance to pin-point the enemy’s location.

Effect of Funding on AMRC 

I don’t believe that the type of research program we have and the areas in which we work would be any different if, for example, the entire contract were to be assumed by the National Science Foundation or the Department of Health, Education & Welfare or anything else. 

Stephen Robinson, 27 March 1973
interview with People’s Video

This long-standing claim that AMRC is unaffected by its Army funds was completely refuted by staff member J. Ben Rosen in an interview which he gave the Daily Cardinal (11 May 1971) on leaving the Center for a new job: 

The influence of the source of funds is felt by the selection of people appointed to the MRC. They are chosen keeping in mind research for military application and not for, say, ecology. The research in both these fields may be the same, but not generally. 

The Army funds have been such a great stigma for the Center that, in 1971, an application was made to the National Science Foundation (NSF) for sufficient funds to cover AMRC’s entire operation. In a 1972 interview, Robinson admitted that the NSF application was made “to get the radicals off our backs.” 

Eventually, Army Math’s proposition was turned down, presumably because NSF could not afford the Center’s $1,400,000 annual budget. Nonetheless, the Center’s willingness to apply for NSF funds has occasionally been taken as proof of AMRC’s independence from the Army. An inspection of its proposal to NSF shows otherwise. 

First, the funding request left the AMRC contract with the Army intact, with the same principles of Army assistance and the same coordination by the Army Research Office through the Army Mathematics Steering Committee. Further, the Permanent Staff would have remained unchanged under the proposed NSF grant, and thus their partiality for Army work would not be hindered. 

So the request itself was simply a political cosmetic, designed with no purpose beyond deception, and the continued protection of the Army’s work force in academia. 


The Army Mathematics Research Center is expanding the scope of its research to include investigations in broader areas. This shift parallels the increasing interest of the military in political and social sciences. The motivation behind this interest was outlined by former Chairman of the Joint Chiefs of Staff Maxwell D. Taylor, in an interview with Bill Moyers shown on Public Television in Madison, 17 March 1973: 

We ought to take time out now and take a deep breath and look at what we learned in Vietnam, and then try to project our present threats and problems into the future and ask ourselves what kind of threat we are likely to face which might require the use of military strength… The problems that might cause a nuclear or major war are still there but I would say diminished in intensity, whereas I see endless increase in the field of limited problems [for example limited war] arising from all sorts of things to include population growth, which happens to be one of my hobbies at this time. 

According to Taylor, the burdens of excess population cause weak governments to collapse, create discontent in populations, endanger .. democratic” government, and intensify 

the competition that’s going to arise between the industrial nations fighting for the diminishing supplies and raw materials on which they depend. The easiest example of the economic pressures which could lead to military operations is in the case of oil . . . If indeed access to oil, for example, would be shut off, that would be a situation which might very readily lead to military operations. Meanwhile, many minerals are going to become scarce in the coming decade, so that this whole globe is going to be grasping for solutions to the depletions of these stocks. 

The military is very interested in anticipating possible future conflict situations and maintaining control in those areas in which it is already involved. As Taylor confirmed in the interview, military commitment follows the flag and wherever the flag is put, the commitment will escalate.


The National Science Foundation, in an attempt to respond to the “mounting interest throughout our society in the ethical and human value implications of science and technology” has appointed the onetime counterinsurgency task force chief for Southeast Asia under the Kennedy and Johnston Administrations, Charles Maechling, Jr., to head up a new program in ethics. Maechling, a lawyer, served as the State Department’s director for internal defense from 1961 until 1963 and was the chairman of a National Security Council task force on counterinsurgency from 1961 until 1966. The Ethical and Human Value Implications of Science and Technology program, which will be run jointly with the National Endowment for the Humanities, will have access to various kitties held by the Director. The amount of these monies varies during the year, but this year they totaled the considerable sum of $2 million. 

Science, vol. 180, June 1, 1973, p. 939 

Economic Modeling 

Today, AMRC still works for the Army. Its changes in policy correspond to the world political and economic situation. AMRC’s Assistant Director Stephen Robinson, in a recent interview with People’s Video on 27 March 1973, talked about diminishing resources and the future: 

We’re using up a lot of resources. Some of these are non-renewable resources. I don’t think we’ve done very much thinking about the consequences of this. Now part of the effort here [at AMRC] is to take a look at what might happen in the future, when we keep on with this growth pattern that we’re in now. We keep on using up these resources-what’s going to happen? The preliminary studies that were done at MIT tend to indicate that some rather unpleasant things might happen. We’ll like to find out if that’s true. And if so, are there policies we can follow that will tend to avoid this?4

An economic model is a system of equations which attempts to describe the relationships between factors in a country’s economy such as the availability of raw materials, industrial and agricultural production, trade and foreign investment. Such models have not been successful because of the extremely complex relations among the factors. The rewards from developing the simple models which now exist, and so increasing their predictive abilities for our military policy makers, makes a large investment in research worthwhile. 

The effort Robinson talks about is a new program in economic modeling, coordinated by himself and H.R. Day of the UW Economics Department, who was also a member of Robinson’s Ph.D. thesis committee. The 20 October 1972 Semi-Annual Report briefly describes the 1972 summer cast of characters: B.E. Easton and Lynn McLinden; J.P. Aubin of Paris, interested in competitive equilibrium, B.P. Stignum of Northwestern University, in dynamic stochastic processes; and D.G. Tarr of Ohio State University, interested in oligopoly models. All but Tarr will be back for the 1973 summer program. 

Population Dynamics

Held in June 1972, at the same time as the summer program in economic modeling, was AMRC’s Symposium on “Population Dynamics.” As Maxwell Taylor noted in his interview, this is an important subject for military planners. In line with this, AMRC consultations with the military planners of the Strategy and Tactics Analysis Group (STAG) have been on the upswing during the past few years (see Chapter 1). An example of STAG’s work is the paper written in 1971 by David R. Howes, entitled: “GUEVARA, A Computerized Guerrilla Warfare Model.” This work which AMRC and STAG collaborated on clearly falls into the area which might be called social science modeling. 

In order to obtain the information needed for these more complex social models, AMRC is attempting to expand its influence into other University departments. We have already mentioned that Economics and Demography faculty have cooperated with AMRC. In addition, there were held during the 1973 spring semester joint seminars of personnel from AMRC and various departments, including Mathematics, Computer Sciences, and the Social Systems Research Institute. The joint work undertaken by AMRC and these additional academic departments tends to give AMRC a more respectable appearance. Increased respectability enables AMRC to entice more academicians to contribute towards AMRC’s work for the Army, under the guise of continuing normal scholarly research. 

These inter-departmental seminars are merely local versions of the yearly symposia and seminars which AMRC holds. Their purpose is the same: to collect as much information as possible, in the hope that some of it might be useful to the Army. 

As the Army Math Research Center expands its work into seemingly more abstract and less technical weapons research, spokesmen will claim that the Center is doing work which is of benefit to all citizens rather than solely helping the Army. AMRC staff will claim that their discoveries have “good uses” as well as Army applications. While there may be some truth in their theoretical statements, in practice it will not be true. The work will be tailored to the Army’s needs. 

In the future, we can expect an increasing diversity in AMRC’s research activities, as United States’ foreign policy and military needs grow, and require the designing of systems for social control beyond the development of new weaponry. AMRC can be expected to try to sell this new research as beneficial to all since it deals with “social problems.” But as long as this research is directed toward the needs of the military instead of the needs of the people, it cannot be said that AMRC is serving the public. AMRC’s newest research for social and economic manipulation can only be stopped by political action from people opposed to the imperialists’ use of science. 


>> Back to Vol. 6, No. 1 <<



  1. Technically speaking, Canada guarantees to political fugitives protection from extradition or deportation to the country fled from if they are charged with politically motivated crimes. Armstrong’s extradition hearing is a remarkable example of international ruling class solidarity in defiance of the facts of the matter and their own laws. For a description of the hearing see “Karl Armstrong: After the Bomb” by David Wagner, obtainable from Takeover, Box 706, Madison, Wisc. 53701.
  2. The best story of the hearing which was distributed around the country appeared in The Nation, 26 November 1973. Hopefully, a publisher can be found to make a book out of the hearing’s transcript.
  3. AMRC could not exist without several favors granted to the center by the University. For example, AMRC’s permanent staff were granted permanent leave by the Math department, in order to receive salaries far above the University maximums from the Wisconsin Alumni Research Foundation (WARF). In May 1973, the Math faculty considered withdrawing these permanent leaves, thus breaking AMRC’s contract, but the motion lost by a 4 to 1 margin.
  4. The “MIT study” refers to The Limits to Growth, published in 1972, describing a global model designed by the MIT Project Team headed by Dennis L. Meadows. Present trends in world population industrialization, pollution, food production and resource depletion are analyzed, with the conclusion that unless enormous changes occur, sudden and uncontrollable declines in population and industrial capacity will result.