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Des mathématiciens et des guerres

Antonin Durand
Laurent Mazliak
Rossana Tazzioli

The mathematical management of uncertainty in Britain and America in the Second World War and after

William Thomas

Texte intégral

1This paper discusses mathematical work done primarily by two groups. The first group is the Air Warfare Analysis Section, or AWAS, a small group of mathematicians, founded in 1940, which worked under the British Ministry of Aircraft Production. The second group is the Applied Mathematics Panel, or AMP, which was founded in late 1942, and worked under the Office of Scientific Research and Development, an American civilian organization. Most attention will be paid to one of the AMP’s subsidiary contract organizations, the Statistical Research Group, housed at Columbia University.

2What I want to emphasize is that a major feature of the work of both these groups was the use of mathematical analysis to articulate some of the assumptions governing the design of equipment, of military tactics, and industrial procedures; and to analyze the logical consequences of those assumptions. Where these assumptions were based on admitted uncertainties, such analysis could ascertain whether or not those uncertainties actually made a difference when deciding between two courses of action. If indeed they did make a difference, such analysis could indicate what sorts of data needed to be obtained, and what tests needed to be conducted to choose intelligently between decisions.

3I want to emphasize that these mathematicians’ models were not thought to yield portraits of things like combat that were, in a strict sense, “true”, or that could, on their own, give an imprimatur of authority in favor of one decision over another. Rather, I must emphasize, these models were intended to make more explicit on what basis decisions were, in fact, being made, and to provide a framework that would allow decisions to be made that were better informed by comparison.

4So, we begin with a figure who, to my knowledge, is entirely unknown to the history of mathematics: Leslie Bennett Craigie Cunningham, usually referred to as L. B. C. Cunningham. Cunningham spent his career up until World War II as an instructor of ballistics in the British Air Ministry’s No. 1 Air Armament School at Eastchurch on the coast east of London. He had studied mathematics at the University of Edinburgh, and ultimately, while working for the Air Ministry, earned a PhD under the Edinburgh mathematician E. T. Whittaker for theoretical work on bomb ballistics. Whittaker, by the way, may also be known to some of you as a historian of theories of electricity and the ether.

  • 2 Director of Armament Development to Chief Superintendent, Royal Aircraft Establishment, 11/9/1937 (...)

5In November 1937, Cunningham sent a paper to the Air Ministry’s Director of Armament Development, outlining a quantitative model of combat. Cunningham noted that his work departed from older theories of gunnery “in that it takes account of the simultaneous exchange of fire between two opposing forces instead of considering the performance of only one side at a time, firing against a passive target.”2 So, what an ordinary equipment test would do is gather data about the specifications of a weapon: its accuracy and precision, the rapidity of fire, the damage its ammunition can cause to opposing targets, and so forth. On the basis of these specifications, designers must make choices between weapons, or, for example, when designing an airplane, they must choose different armament configurations.

6The difficulty in these situations is that one weapon, or one configuration, is not necessarily obviously better than another. The rapidity of fire in one weapon might be outweighed by the devastating effects of another, for example. What Cunningham’s theory did was estimate the effects of different weapons or configurations on the actual outcome of combat, since these outcomes were ultimately the only meaningful way of saying whether one design was superior to another. It accomplished this by incorporating weapons specifications into mathematized assumptions about equipment damage and “stated tactical methods either of approach of withdrawal” in air combat. Then, given the probabilistic assumptions about these things that one already held, one could come up with what one, by implication, thinks is the preferred configuration, by virtue of it being a logical consequence of those assumptions.

  • 3 Cornford E.-C., “Dr. L.B.C. Cunningham”, Nature, 158, 1946, p. 408-409.

7Cunningham further developed his “mathematical theory of combat” with other Air Ministry researchers during the years just prior to the war, and in the first year of the war. Then, in 1940, he was named the head of a new Air Warfare Analysis Section. Although AWAS had responsibility over a number of quantitative matters, Cunningham’s theoretical work formed a major part of the activities of AWAS. Two of his papers on the mathematical theory of combat formed the first two reports AWAS produced. During the war, close association with field-based operational research groups, staffed largely by scientists, even provided a means by which theoreticians could obtain reasonably reliable data on actual combat conditions, so that assumptions underlying their theories could be modified as need be. According to Cunningham’s obituary, he was in line for a “high appointment” at the Royal Aircraft Establishment following the war. However, he died in 1946 at the age of 51.3

  • 4 Mindell D., Between Human and Machine: Feedback, Control, and Computing before Cybernetics, Baltim (...)

8Cunningham’s theories were welcomed with enthusiasm by a rather more famous mathematician across the Atlantic, Warren Weaver. During peacetime, Weaver was head of the program within the Rockefeller Foundation that funded research in the natural sciences. During the war, he was placed at the head of the section of the Office of Scientific Research and Development responsible for developing new fire control technologies. “Fire control” is the branch of engineering responsible for aiming an artillery gun, an anti-aircraft gun, or a turret gun mounted on an airplane, and controlling the timing of its firing in such a way that it maximizes the probability of scoring a hit on a target being tracked by the gun’s operators. Notably, Norbert Wiener began developing his work in cybernetics on contract under Weaver for work on fire control, but his contract was not renewed because the work was not deemed sufficiently practical.4 Most work done under Weaver involved collecting piecemeal information on combat results and equipment designs, which could result in modifications to device designs, or in the techniques used to operate them.

9As to the task of integrating this piecemeal data into a coherent picture of the fire control problem, Weaver was impressed by the potential in Cunningham’s methods. He wrote in his final wartime report that he “showed and praised” AWAS’s first two papers on combat theory “to various officers of the Army Air Forces and of the Naval Bureau of Aeronautics until, in self defense, they suggested that we try to digest and simplify these papers, interpreting them in terms not so formidably mathematical.” In the summer of 1942, Weaver approached Columbia University statistician and economist Harold Hotelling to recruit a team that could help him with the task. Hotelling referred Weaver to his former students, Allen Wallis from Stanford University and Jacob Wolfowitz, who was at that time teaching in New York Public Schools. Wallis and Wolfowitz formed the core of a new Statistical Research Group (SRG) at Columbia University, and set to work interpreting and developing Cunningham’s theory.

10The first new application of Cunningham’s theory that the SRG considered was whether configurations of eight. 50-caliber guns or of four 20 mm guns were preferable for a fighter making a tail attack on a bomber. The first task was to set down what kinds of information, of estimates, or of guesses were needed “before the answer would be forthcoming” Weaver recalled.

11These questions (nature of combat, bomber and fighter armament, value and variations of accuracies, ammunition, vulnerabilities, etc.), were then discussed at very considerable length with experienced officers. As a result estimates were arrived at which everyone agreed were almost certain to bracket the true values, although in many instances the true values were admittedly unknown.

12The questions could then be explored mathematically to ascertain whether or not the uncertainties in certain factors bore upon the final decision between the two configurations. For four out of five assumptions of vulnerability, the analysis favored the. 50-caliber gun configuration; for the fifth assumption, the 20 mm guns had a 20% advantage in securing a favorable outcome of the duel.

  • 5 All quotes from Warren Weaver, “Comments on a General Theory of Air Warfare”, AMP Note no 27, Janu (...)

13Weaver recalled, “The study was thus necessarily inconclusive. It did, however, make clear just what sort of information was necessary to obtain a conclusive answer, and it furnished the necessary analytical methods”. Or, as one of the SRG’s reports on Cunningham’s theory pointed out, “in conjunction with statistical theory, [the Mathematical Theory of Combat] can suggest ways to get the most information with the least experimenting, which is important when the planes, pilots, scientists, and time for experimenting are scarce”.5

14Now, it is at this point that I would like to leave Cunningham’s mathematical theory of combat behind for a little while, and turn to a better-known wartime development: sequential analysis. As the economist and historian Judy Klein has shown in some detail, sequential analysis derived from the development of statistical methods in industrial quality control. In industrial quality control, one takes from a particular production lot a sample of a certain size. Based upon whatever standards one wants to ensure in an industrial product, one can test this sample, and if a certain percentage of the sample fails to live up to the standard, the entire lot will have to be discarded.

15In a traditional quality control test, one defines ahead of time what the sample size is, and what the number of items that fail to meet the standard is that will cause the production lot to be rejected. So – to use an artificial example – if one has a production lot of 50,000 artillery shells, one might select 1,000 of them to test, and if 5 of those 1,000 fail to explode, then the entire lot of 50,000 shells will be destroyed. Naturally, if one reaches that number before the entire sample has been tested – in our example, if 5 shells fail after only testing, say, 4,000 – the inspector may truncate the test and reject the lot. However, inspectors also felt that if a large number of items did not meet the standard (or if a very high percentage did meet the standard), that they were justified in truncating the test before the predetermined number of failures had been reached. In general, but especially during the war, one had an incentive to truncate tests as early as possible, since many products had to be tested, and time and resources for testing were scarce. So, the Navy proposed the problem to the Statistical Research Group at Columbia University to determine whether early truncations were legitimate, and, if so, exactly what rules determined when the test should be truncated.

  • 6 Klein J. L., “Economics for a Client: The Case of Statistical Quality Control and Sequential Analy (...)

16The problem was given to Allen Wallis and another young economist, Milton Friedman, who, of course, would go on to fame as a champion of the economics of the free market at the University of Chicago. However, they passed it along to their senior colleague, statistician Abraham Wald, who worked out the underlying statistical theory to incorporate information gathered while the testing process is in progress. Now, as I remarked, this story is already rather well-known.6 However, the development of sequential analysis becomes more striking when one considers the conceptual links between Cunningham’s theory and sequential analysis.

  • 7 Wallis W. A., “The Statistical Research Group, 1942-1945”, Journal of the American Statistical Ass (...)

17Most obviously, there is the emphasis on using mathematical theory in order to focus data gathering processes in situations where the opportunity for data gathering is limited. This objective even found its way into the so-called “mathematical recreations” pursued by the SRG on its lunch breaks: “Given twelve coins, all of the same weight except for one, using only a two-pan balance, find the odd coin and determine whether it is heavier of lighter than the others, making only three weighings.”7 If you work out the problem, you can see that it involves designing a highly efficient sequence of weighings, and decisions must be made during the sequence about how it will unfold based on a very efficient use of information gathered during the testing process. And this information is not of a concretely defined character; the meaning of the information obtained in one weighing, such as whether the odd coin is in one or the other pan, can only become clear based upon the results of future weighings.

18Beyond efficiency in experimentation, these projects also place an emphasis on making explicit something I like to call meta-calculations. If a calculation arrives at a solution to a stated problem, a meta-calculation is the consideration of some factor outside the calculation as stated, which bears upon the solution’s validity to some larger, and partially implicit goal. So, Cunningham’s mathematical theory of combat incorporated meta-calculations concerning the anticipated effect of weapon-design choices on the outcome of combat. Sequential analysis, likewise, incorporated meta-calculations incorporating information gathered during the testing process to revise the structure of the testing process mid-test, which was not a possibility allowed for in ordinary testing procedures.

19Now, I would like to propose a somewhat more radical – but I believe defensible – suggestion concerning the mentality underlying these mathematical projects. This is: under Warren Weaver, mathematicians working for the Applied Mathematics Panel understood the selection of calculating projects to work on, and their strategy in pursuing these projects, as itself a kind of meta-calculative problem. The main difference between this suggestion and the cases of Cunningham’s mathematical theory of combat and sequential analysis is that this process of selecting and managing problems was never actually turned into an explicit calculation; it was merely treated in the same spirit. Thus, AMP chose projects based upon their implicitly calculated assessment of their ability to make some constructive, practical contribution to the war effort. They pursued and prioritized those projects based upon their ability not to arrive at a mathematically elegant solution, but based upon their ability to use their mathematical knowledge to make some contribution to a problem given constraints on time, data availability, and the availability of qualified personnel. Thus, for example, the failure of Norbert Wiener’s work under the Panel.

20To illustrate the depth with which this spirit was felt, we can look to the case of mathematician Edwin Paxson’s introduction to Warren Weaver, while Paxson was consulting with the proving ground at Eglin Field in Florida. Paxson chided Weaver and other members of the Panel for not doing any work on a certain kind of sight. Weaver, taken aback, replied that the panel had nothing to do with the sight. According to Weaver’s notes on the meeting, Paxson, in turn, pointed out that “certain tactical problems have to be solved quickly and on the spot in the various theatres of war. These are presumably to be handled by the Operational Research Group. Then there are certain basic long-range developments […] which the NDRC is particularly well equipped to handle.” However, there were also “a lot of stop-gap problems”. These could be addressed with “methods or pieces of equipment which are admittedly not perfect, but which are nevertheless very much better than nothing”. Included in this category was the sight in question: this intermediary category of technology was something on which AMP should be working as a matter of course.

  • 8 Diary of Warren Weaver, 24/2/1944, and Letter from Warren Weaver to E. W. Paxson, 28/7/1944, both (...)

21Paxson had chosen the correct words. Inelegant but innovative ad hoc solutions to diverse problems were AMP’s bread and butter. Weaver replied that AMP’s leadership had “been preaching exactly this same philosophy on every possible occasion during the last twelve or eighteen months”. Returning to New York, he felt that the Panel should do everything in its power to help Paxson. By July 1944 Weaver decided that he wanted Paxson to work for AMP as the Panel’s resident expert on aerial gunnery, and, after securing the permission of Gardner and Bowles, Paxson moved the next month.8

22As Paxson joined AMP, it was undertaking a major project to evaluate potential tactical uses of the B-29 Bomber, under a contract known as AC-92. The B-29 was a remarkable product of the aeronautical engineering of the time, having remarkable speed and range, and it quickly became the primary weapon that the United States deployed against mainland Japan, with devastating effects. However, the airplane had been rushed through design and production, meaning that it not only had a number of technical flaws, there was also no established tactical doctrine indicating how it should be used in combat.

23To address the problem at all adequately, Weaver and AMP felt that an enormous set of controlled studies had to be conducted, which included a large number of experimental flight tests to be undertaken at the Army Air Forces testing range near Alamogordo, New Mexico, under a contract with the University of New Mexico in Albuquerque and in collaboration with the Second Air Force. A further set of gunnery model experiments were done at the Mount Wilson Solar Observatory in Pasadena, California in which special lights were affixed to model bombers in place of guns, and the resulting light patterns were measured by photocells to determine the firing characteristics of bomber formations with various armament configurations. In addition, the Panel felt it was necessary to conduct tests on the central fire control system of the bomber, gun camera tests at Eglin Field in Florida, collaborative studies with General Electric, psychological studies on subjects such as gunners’ perceptions and reaction times with the NDRC’s Applied Psychology Panel, analytical studies of special problems such as aircraft vulnerability and flak, and, to tie all the other work together, “broad analytical studies, such as those which relate to bombing effectiveness, general ‘economic’ theory of bombing, etc.” to be done under contract with Princeton University.

24Throughout the course of the AC-92 contract, Weaver had “steadily and stubbornly” insisted that all aspects of the problem needed to be worked out simultaneously in order to interrelate component test designs properly and to understand the significance of results achieved. However, the Army Air Forces leadership did not take the same perspective as AMP, and only took a real interest in the experimental aspects of the project being worked on in New Mexico. They eventually came around to support the Pasadena model tests as well, but were colder toward the others, and, according to Weaver’s end-of-the-war report, were “totally (and in most cases emphatically) uninterested” in the integrative Princeton studies.

25Disappointment with the AC-92 project led Weaver to reflect in his end-of-the-war report on his early impressions of Cunningham’s mathematical theory of combat, and his experience with AC-92. He also reflected on the nature of decision-making in war, which was reflected in the title of the report, “Comments on a General Theory of Air Warfare”. He illustrated the central problem using a scenario that might be encountered by a hypothetical colonel charged with selecting a new radar-guided bombsight for development. The colonel’s choice meant weighing differences in the weight and performance of different bombsights, how complicated they were to use, and the time it would take to develop them, each of which had to be considered in light of various other factors. Weaver observed.

26He cannot possibly himself know all the necessary things, but somehow he should certainly bring to bear on these questions a wide and precise knowledge of the probabilities of bombing accuracies; the logistics of the theatres in which these sights are to be used; the nature of the enemy targets; all the vast field of terminal ballistics; the importance of the time factor (which means war plans, among other things); the psychology and physiology of operation of bombsights, selection, and training of bombardiers; accessibility of qualified personnel; the basic cost of accomplishing the same objectives otherwise; the present and potential future effectiveness of the enemy’s fighter attack against our bombers; etc.

27Clearly the choice was monstrously complex, and the hypothetical colonel could not be expected to make a perfectly informed decision even if all the necessary data were actually available. To consider how such problems could be most fruitfully addressed, Weaver imagined a utopian solution: a “Tactical-Strategic Computer (TSC)”.

28Without going into too many details, the TSC was imagined to be able to take a variety of variables bearing on the outcome of a mission, and then to determine, on the basis of prevailing assumptions about how missions worked, how particular choices bore on mission outcomes. It might even be set to work through all possible combinations of choices to arrive at the best one. The point here, consistent with my prior discussion, is not that mathematics could guarantee an optimal mission result. Rather, the TSC was intended to illustrate that mathematics could help work through the consequences of existing decisions in light of larger contexts of equipment choices, mission planning, and even the strategy of the war. It was also intended, like Cunningham’s mathematical theory of combat, to pinpoint places where investigation and testing was likely to have a major impact on the decisions that were ultimately made.

  • 9 Weaver, “Comments”, op. cit., p. 4.

29Weaver’s argument was predicated on the reality that these sorts of choices would inevitably be made in amalgamating opinions – that analysis and planning, whether “excellent, good, mediocre, or disastrous,” were simply facts of life. So, Weaver insisted, the comparison of the worth of various options “has to be made either by analysis, or magic, or blind guess.” Protests concerning the complexity of the problem and appeals to more abstract or intuitive forms of judgment or common sense represented no true epistemological alternative for him: they were nothing more than “disorganized and feebly intuitive shadows of a real analysis.” For his part, he was offering no epistemological alternative: “I am simply arguing for facing the complexity and the facts, and pushing analysis to its usable limit.”9

30Now, as we track this attitude into the postwar period, this general attitude toward using mathematics, not as a way of suggesting deterministic laws for deeply variable and uncertain things like combat, but as a way of keeping track of the consequences of uncertainties encountered in the planning of these things. Archival evidence shows that the development of systems analysis at the RAND Corporation – often described as an attempt to create a “science of warfare” – was envisioned as following in this vein.

  • 10 Mirowski P., Machine Dreams: Economics Becomes a Cyborg Science, New York, Cambridge University Pr (...)

31What is less clear is how this wartime experience bore upon new postwar liaisons between mathematics and economics. The idea that mathematics could be used to describe behaviors that decision-makers under existential pressure – be it a war, or merely market competition – would find and adhere to, regardless of whether or not they were rational calculators is a subject of much interest. It is not, for example, a large leap from seeing mathematics formalize quality control procedures already practiced by experienced inspectors, to seeing economic actors making optimal plans using the best available information. This question has already been broached by historians such as Philip Mirowski10. However, the subtleties of this process, and conclusions regarding what sort of epistemological or authoritative status these models were supposed to have seem to remain open questions.


2 Director of Armament Development to Chief Superintendent, Royal Aircraft Establishment, 11/9/1937 and associated documents, The National Archives of the UK (TNA): Public Record Office (PRO) AIR 13/879.

3 Cornford E.-C., “Dr. L.B.C. Cunningham”, Nature, 158, 1946, p. 408-409.

4 Mindell D., Between Human and Machine: Feedback, Control, and Computing before Cybernetics, Baltimore, Johns Hopkins University Press, 2002, p. 276-282.

5 All quotes from Warren Weaver, “Comments on a General Theory of Air Warfare”, AMP Note no 27, January 1946. A copy is available at TNA: PRO AIR 52/106.

6 Klein J. L., “Economics for a Client: The Case of Statistical Quality Control and Sequential Analysis”, in Toward a History of Applied Economics, edited by Roger E. Backhouse and Jeff Biddle, Durham, Duke University Press, 2000, p. 27-69.

7 Wallis W. A., “The Statistical Research Group, 1942-1945”, Journal of the American Statistical Association, 75, 1980, p. 320-330.

8 Diary of Warren Weaver, 24/2/1944, and Letter from Warren Weaver to E. W. Paxson, 28/7/1944, both in “Paxson, E. W. Correspondence” folder, Box 8, Records of the Office of Scientific Research and Development, Record Group 227; National Archives at College Park, College Park, MD.

9 Weaver, “Comments”, op. cit., p. 4.

10 Mirowski P., Machine Dreams: Economics Becomes a Cyborg Science, New York, Cambridge University Press, 2002.


Junior Research Fellow au Centre for the History of Science, Technology and Medicine de l’Imperial College (Londres). Il est spécialiste d’histoire de la recherche opérationnelle et de ses interactions avec les questions militaires.

© CNRS Éditions, 2013

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