MATHEMATICS AND WAR 30 November 2001

We consider here mathematics as a bellicose art. We address:

In 1939, the British crystallographer and science historian John Desmond Bernal wrote: "Science and warfare have always been most closely linked; in fact, except for a certain portion of the nineteenth century, it may be fairly claimed that the majority of significant technical and scientific advances owe their origin directly to military or naval requirements."

This is to some extent also true for mathematics, though the mutual relation between military demands and emerging mathematical concepts is intricate. A few examples:

Babylonian clay tablets from about 1800 B.C. deal with siege computation: the number of bricks needed for siege ramps, the volume of earth to be dug and how much workforce was required. But the same calculations were used when building a temple or digging an irrigation canal. Moreover, many "real-world" mathematical problems on the thousands of preserved tablets show (for instance in the choice of unknowns for the quadratic equations) that they were meant to puzzle and train the student rather than to contribute to the solution of problems in real situations.

In spite of its declared praxis-remote motivation, Euclidean geometry has for two millennia trained geometric judgement and modelling capacity of immediate military relevance. The graecist John Onians traces the origin of Greek mathematics to "the importance of absolute order in the military sphere which gave mathematics a dominant role in all Greek culture". An overlooked aspect of the present decline of Euclidean geometry in school curricula are the diminishing demands for distributed independent geometric calculations in the electronic battlefield.

Archimedes (ca. 287-212 B.C.) demonstrated during the siege of Syracuse his command of the full geometry of three-dimensional translations and rotations and of precise volume and weight estimates for plates, bodies and tools of a most delicate shape. His engineering calculations were admired, the artifacts copied and the proportions disseminated through all the Hellenistic, late Roman, and Arab times, and through the Middle Ages.

The military writer Aeneas Tacticus (around 360 B.C.) published a whole chapter with ideas about cryptology (for instance sending a book with one letter thinly marked in each line - an early instant of steganography). Subsequently Polybios, Caesar, and Augustus introduced true bilateral substitutions, i.e. permutations (of vowels and consonants) for military communication. The Islamic Middle Ages took over this complex of applied mathematics and (occasionally military inspired) techniques, expanded and moulded it into arithmetic and algorithmic form – like moving squadrons and armies - and handed it over to the early Modern period.

The Renaissance had a high appreciation of the possibilities of mathematics in every practice. Specific military needs (cartography, artillery and ballistics) partly preceded, partly met specific civilian needs (the theory of the central perspective, bookkeeping, merchants' calculation and algebra). Keeping it a strict military secret, the fifteenth-century Portuguese court took up a systematic development of navigational mathematics which led them to distinguish between great circle arcs (= geodetic curves) and loxodromes (curves of constant angles like the parallels). When Niccolò Tartaglia (1500?-1557) tried to 'give rules for the art of the bombardier' (Nova scientia, 1537), he abandoned Aristotle's concept of a piecewise linear trajectory and created the modern concept of a function with a smoothly curved graph (though not yet the correct parabola orbit, later derived by Galileo). The Flemish mathematician Simon Stevin (1548-1620), later quartermaster general of the army under Maurice of Nassau, engineered a system of sluices to flood certain areas in defence of besieged cities and thus founded the modern statics and hydrostatics (De Beghinselen der Weeghconst, 1586).

Towards modern times. All these germs were systematically investigated in the following centuries, sometimes directly on military order like Harriot's practical and theoretical work on cartography (around 1585); Hooke's study of elasticity, ordered by the English Royal Society on behalf of the Navy which wanted to cut down the consumption of wood in shipbuilding; the construction of accurate chronometers and, alternatively, the creation of a precise theory of the lunar movement for precise empirical determination of geographical latitude; the fortification mathematics of Sébastien le Prestre, Marquis de Vauban, optimising the complex polygonal fortification structures with Monge's descriptive geometry as spin-off; the introduction of infinitesimal mathematics into the military curriculum to consider the effect of air resistance on the velocity of a projectile (then an impossible task). Forts and Howitzers were on the frontispieces of almost any mathematics book of the 16^th to 18^th century, even when they dealt solely with pure mathematics.

The practice of war, however, had no important influence on the development of mathematics: practical military or civilian preoccupations of that time proved fruitful for the development of mathematics and mathematics based physics and engineering only when they were related to the network of current scientific knowledge and theory. This is illustrated by the life, work, and lasting influence of the mathematical heroes of that time, Huygens, Fermat, Galileo, Newton, Leibniz, the Bernoullis, Euler, d'Alembert, Laplace, Gauss, Cauchy, Riemann, Poincaré, Hilbert, none of them very eager in military matters.

Gradually, mathematics as an academic discipline itself emerged in the sense of systematic research attached to universities, institutes of technology, and other institutions of higher learning. Only now could the reached maturation of mathematical knowledge lead to direct industrial applicability and by this to enhancing the military strength of a country. Prerequisites for this maturation were, in addition to an immense quantitative growth, a complete reorganization of knowledge and a far-ranging division of intellectual labour. Physics and engineering sciences were organised independently. Pure and applied mathematics were separated from each other conceptually and institutionally.

Military traces in 20. century mathematics. On the eve of and during World War I there were various attempts at organising technical development on a large-scale scientific basis. The general picture, however, was different: the French Supreme Command did not find a better use of its young scientific and technical talents than sending them into the trenches, while the German Supreme Command widely granted leave of absence for continuing research, also basic research. The intention was to save scientific manpower and thus preserve the basis for post-war technical development.

The Second World War brought a new alliance between mathematicians and the military, leaving a triad of complex artifacts behind: the atomic bomb with its stationary model in service of energy production; the jet propulsion for rockets, intercontinental bombers and charter travel; and the computer. Less visible was the creation of the new mathematical sciences of operations research (P.M.S. Blackett, G. B. Dantzig), coding (A. Turing), statistical quality control (A. Wald), signal processing and time series analysis (N. Wiener, C.E. Shannon, A. Kolmogorov), simulation and scientific computing (S. Ulam), numerical weather prediction (J. v. Neumann).

The complexity of artifacts, of production processes and of globalized social relations had their break-through in the aforementioned triad of deadly products, which the new alliance between mathematicians and warlords left behind after World War II. A new faith in the omnipotence of computer supported modelling emerged. Description, prediction and prescription (design) became based on a universal belief in the conversion of complex situations and problems into simplified and seemingly controllable perceptions.

According to Carl von Clausewitz, the brilliant war historian and war theoretician, such conversion is intimately related to the essence of war. But it is quite opposite to traditional mathematics taking pride in emphasising differences between the perceived and real complexity of a problem.

At the heart of it all is the electronic computer, which supports the operation of processes beyond the grasp of human minds using vast collections of software whose magnitude and complexity put it beyond the reach of reliability. Consider for instance the steering of nuclear power plants or of organisations of a size which need world markets for their sustenance. This blurs the borderlines between what we do and do not understand, what we can make and what we can control.

The successful co-operation between mathematicians and the military during World War II and the subsequent arms race let mathematics expand grossly as a science devoted to complexity and simplification. In that process, our profession was and is so much success oriented that essential quality marks of traditional mathematics, the discussion of the range, validity and consequences of our calculations and derivations, are often sacrificed.

Through history there are a few instances where we can point to a single mathematical calculation, concept or a mathematics based construction of immediate military relevance.

Polybios judged it necessary to imprint Roman commanders that the hypotenuse is substantially longer than even the longest other side in a rectangular triangle and hence ladders for storming a wall should be substantially longer than the shear height of the wall.

Archimedes' mechanical calculations could not prevent the fall of Syracuse but supported the excellence of siege machinery in Roman warfare.

The application of ordnance (canons, howitzers) became efficient only with quantitative relations between the slope of the pipe and the range (though with empirical tables more reliable than theoretical deductions up till the 1940s).

We are used to attaching Galileo's work in geometrical optics for better telescopes to the confirmation of the new planetary view by the discovery of the Jupiter moons, but that was not what Galileo demonstrated to the Venice Seniors from the top of the tower of St. Marc. What he demonstrated was the military relevant capacity of seeing ships and their details long before encountering them.

Elaborated astronomical calculations of astrology decided the opening of quite a number of battles up till Wallenstein in the Thirty Years War in Germany.

A single calculation almost "on the back of an envelope", Peierl's and Frisch's 1940 estimation for the critical mass of U_235 to be less than 10 kg to maintain the fission process, convinced first the British and then the US government of the feasibility of the atomic bomb - contrary to Heisenberg, who did not do this calculation and, according to various testimony even after August 6, 1945 remained convinced that rather a whole metric ton of U_235 was needed for one single atomic bomb.

Another calculation (of 1943), again by Peierls and Frisch, who had then moved to Los Alamos, showed the feasibility of the Plutonium bomb, namely how conventional explosives had to be proportioned and placed around a sub-critical Pu ball to generate those shock waves which bring the Pu together into a critical mass - instead of blowing it up (details are still not disclosed).

According to rumours even more advanced calculations (mostly numerical simulations) were performed by Ulam and v. Neumann for nuclear branching processes and shock waves of explosives - now also involving fission bombs - as a triggering device for thermonuclear fusion, the hydrogen bomb.

A legendary case of mathematical ingenuity was Turing's contribution to the design of the "Colossi" computers and the breaking of the German code Enigma, by the Germans believed un-breakable until the end of the war.

The list of war-relevant mathematical ingenuity on the eve of and during World War II could be largely expanded. However, the question remains: were any mathematical efforts decisive for war?

Historians of World War II tend to conclude: (A) The enormous achievements in air plane construction, air raid operation and strategic bombing were, by large, according to the postwar Allied Bombing Survey, not very effective for diminishing the adversary's capacity to continue the war or shortening the war.

(B) The two atomic bombs on Hiroshima and Nagasaki have clearly triggered the final Japanese capitulation. However, negotiations about the surrender were already on the way and the immediate military relevance was negligible compared to the Japanese break-down on all fronts.

(C) The decisive importance of the code breaking by the "Colossi" is attested for the naval war, mainly positioning of and defense against German submarines. It also provided some information for the land battle, but apparently not of real operational relevance. Signal intelligence rapidly expanded to an "industry" of some 30.000 people engaged on all sides, though without any further certified true war relevance. In recent wars and violence (for instance in Iraq, Near East, Kosovo, Afghanistan) not many coups were attributed to passive electronic surveillance and not one single coup to its active counterpart, cyberwar, though enormous means have been invested and though it has attracted enormous public interest.

Perhaps Lev Semyonovich Pontryagin's (1908-1988) mathematical work has had the greatest influence on peace and war in the 20th century. His mathematical career began with quite outstanding achievements in pure mathematics: The Pontryagin numbers for instance laid ground to modern global analysis and differential geometry and came to inspire research in algebraic geometry and number theory up till our days. In the late 40s he changed his field to the control theory of dynamical systems and became, especially with his "Maximum Principle", the mathematical father of trajectory control of satellites and intercontinental ballistic missiles. Providing the Soviet Union with means of intercontinental nuclear retaliation he contributed to putting an end to a protracted period of British and American nuclear war preparation. Until 1949 (mainly) London wanted to exploit the nuclear monopoly and from 1950 until 1955 (mainly) Washington wanted to use the advantage of B-36 and B-47 encirclement for nuclear black mailing and possible physical extinction of the Soviet Union.

A variety of mathematical algorithms are implemented in modern terminal guidance systems for missiles and bombs. The most important techniques are terrain contour matching by Fast Fourier Transform, Laser guidance by Fourier optics, satellite based Global Positioning System, precision timing for free fall bomb dropping. They have reduced the Circular Error Probable (CEP) from 1100 m in World War II to 13 m by the turn of the century. Here CEP denotes the radius of a disc around the goal point such that (on average) 50% of the bombs hit inside the disc.

C4ISR is the acronym for modern military signal processing. It stands for Control, Command, Communications, Computers, Intelligence, Surveillance, and Reconnaissance, that is to gather, process, hide, and disturb a huge variety of information about own and adversary forces.

From the point of mathematics also stealth is a kind of signal processing = reducing radar reflection by shaping the surface of a flying body in a particular way, namely moving eigenvalues from the resonance side to the non-resonance side.

The military use of virtual reality visualization techniques can be characterized as "auto-stealth" since the point is to hide confusing or unwanted information for personnel in combat, commanders and the public.

Some military relevance may or may not be attributed to various types of war games, combat simulation and dynamical models of arms race. Critique will point to the self-deception aspects of this kind of modelling. However, the founder of numerical weather prediction and of numerical analysis at large, Lewis F. Richardson (1881-1953), himself an ardent pacifist, holds it that rigorous mathematical thinking about conflict never can do harm.

What changes? Roman warfare and Modern warfare since Machiavelli, Maurice of Nassau, Gustavus II Adolphus, and, foremost, the Napoleonic Wars and the World Wars, have been characterized by the rationality of goals and means; quantization of troops, inventory, distances, order of battle; and discipline and shared goals of fighting troops, commanders and the hinterland.

Today, most military historians agree that the rise of modern war in the period 1500-1945 was accompanied by a rise in mathematical technology and other technical innovation, but not driven by it. Rather, the new ways of warfare came from the mathematical idea: "to liken war to business competition" (Clausewitz). Discarding differences and emotions between adversaries, the basic assumption of modern warfare is that both sides are guided by the same kind of logic, rationality, reason.

Acknowledging an adversary's logic and reason has also been a decisive step in seeing him as an equal and finally in developing international law (for instance Grotius' maxim to think about war "sicut mathematici" – like mathematicians) and international humanitarian law (for instance the Red-Cross concept of protection of equally valuable lives and civil objects).

In all that period, roughly speaking, warfare became gradually more calculable, whereas war itself, this grandiose piece of psychology, chaos and destruction, remained incalculable with a principally unpredictable outcome.

Accompanied - or driven - by the aforementioned new mathematical technology (this must be decided separately in each instant), the character of war, preparing for war and executing war have changed after World War II.

Irrational concepts like nuclear genocide threat or collective nuclear suicide are considered political and military options. There is no rational human argument about nuclear war or a nuclear first strike capacity.

However, on the positive side, the outcome of nuclear war became so predictable that it could support the principal of "Mutually Assured Destruction". In a period of international tension (in the 1960s and 70s) this paved the way for at least psychologically important talks and agreements between the two nuclear superpowers of that era. Practically, nuclear war has been averted hitherto.

Perceived predictability is also behind the newly emerging concept of Hit'nd Run Wars, based on a perceived full control of risks and accepted casualties and attributed to mathematical advances and revolutionary changes in mathematical technology (first of all ISR and precision munition). Examples are the recent HighTech-(Own)LowCasualties wars against Iraq, Yugoslavia and Afghanistan which resume the asymmetric warfare of the past colonial wars and wars against the American Indians.

The promise of a "limited" character of these wars is dangerous because it lowers the barriers of waging a war which may not stay as limited and clean as planned.

However, the limited punishment by Hit'nd Run actions is no longer the privilege of mathematical technology supported former colonial masters. It may become a habit also among segments of the world population which do not identify with the present day masters and are willing to sacrifice their lives in LowTech-ManyCasualities terrorist attacks against centres of the dominant world part (case WTC).

Due to the real limitations of nature and natural laws it is very hard to invent and design new ways to kill, hurt, and disable humans or to destroy buildings and other human artifacts. The chemistry of energy storage has already been optimized during WWI. There is nothing "better" than TNT. The physics of energy release has since then been optimized during and shortly after WWII: there is nothing "better" than the atomic bomb and the hydrogen bomb. Engineering has provided better and better materials for the hardening of projectile tips and for strengthening tank armours. But this, too, is a principally limited process of metallurgic advance.

Now the special capacity of mathematics in modern armament and warfare is its role in enhancing the efficiency of delivery (traditionally the domain of engineering, for instance the construction of canons, tanks, and warhead enforcement). Here there are no natural limitations because we progress mainly through new ways of symbolic manipulation. In such a way mathematics promises an infinite curve of innovations in precision; timing and coordination; miniaturization; new physical ways of delivery (case sound beam); and, worst of all, new dreams of invincibility.

1) Frontispiece from N. Tartaglia, Nova Scientia... Venice, 1537

2) AWACS air plane, interior

3) Circular Error Probable (bombing simulation)

4) "In America we like our bombs smart and our youth stupid",

Newsmakers photograph, reproduced in The Bulletin of the Atomic

Scientists, March/April 2000, p. 52

5) John von Neumann at Los Alamos

6) Order of battle in the Kosovo War - sketch

7) Joint Strike Fighter - advertisement model

Note: This is only a draft and not for publishing in this form. Editing will be necessary for co-ordination with some of the other historical and application oriented essays of the Hutchinson Companion Encyclopedia of Mathematics.

Annotated Bibliography

General

Booss, Bernhelm, and Jens Høyrup.

Von Mathematik und Krieg.

May 1984. 74 p.

Ser. Title: Schriftenreihe Wissenschaft und Frieden. no. 1.

[ISBN 3-924684-00-6]

English: On mathematics and war: an essay on the implications, past and present, of the military involvement of the mathematical sciences for their development and potentials.

In: Measure, Number and Weight: Studies in Mathematics and Culture, Høyrup, J. SUNY Series in Science, Technology, and Society. 1994. p. 225-278 + notes, bibliography. [ISBN xxx]

Godement, Roger.

Science et défense

Gazette des Mathématiciens 61 (1994), 2-60.

Old History

Onians, John.

War, mathematics, and art in ancient Greece

History of the Human Sciences 1989, 2: 39-62.

Notes: Argues that 'the interest in mathematics and all the products that interest in Greek art, Greek science, and Greek philosophy derive primarily from the dominance of military values’.

Source of data: Isis Current Bibliography of History of Science.

Subjects: Mathematics--in war--Greek and Roman antiquity.

Genre Or Form: Journal article.

ISIS ISIS Record ID: XISI43275-H

Sayili, Aydin.

Ibn Sina and Buridan on the motion of the projectile.

In: From deferent to equant: A volume of studies in the history of science in the ancient and medieval Near East in honor of E.S. Kennedy / [edited by] David A. King, George Saliba.

Published: New York : New York Academy of Sciences, 1987.

Physical Details: xxix, 569 p. : notes.

Series: Annals of the New York Academy of Sciences ; 500.

Subjects: Science--Islam.

Genre Or Form: Monograph.

ISIS ISIS Record ID: XISI68997-H

Voss, Mary J.

Title: Between the cannon and the book: Mathematicians and military culture in 16th-century Italy.

In: Dissertation Abstracts International 1995, 56: 2373-A.

Notes: Dissertation at Johns Hopkins Univ., 1995. Univ. Microfilms order no. 95-33336. 525 pp.

Examines how new uses of mathematics in military technologies led, in part, to the reconceptualization of physical nature on a new mechanical model. Focuses on the work of Niccolò Tartaglia and Guidobaldo del Monte.

Source of data: Isis Current Bibliography of History of Science.

Subjects: Military technology--Renaissance (15th and 16^th centuries)--Italy.

Mathematics--Renaissance (15th and 16^th centuries)--Italy.

Tartaglia, Niccolò, c. 1500-1557.

Del Monte, Guidobaldo, 1545-1607.

Genre Or Form: Journal article.

ISIS ISIS Record ID: XISI71135-H

R Bennett, Jim.

The Geometry of War, 1500-1750: Catalogue of the Exhibition.

Published: Oxford : Museum of the History of Science, 1996.

Physical Details: xiii, 85 p. : ill.

Notes: Includes bibliography.

Source of data: HTE Current Bibliography in the History of

Technology.

Summary: Reviewed by W. R. Laird in Isis 88 (June 1997): 331-32.

Other Authors: Johnston, Stephen.

Subjects:

Ballistics.

Mathematics.

Artillery.

Engineering, Military.

Surveying.

Instruments.

Military technology--Renaissance through 17th century.

Genre Or Form: Monograph.

ISIS ISIS Record ID: XHTE19972730-H

Willmoth, Frances / Mathematical Sciences and Military Technology: The Ordnance Office in the Reign of Charles

II.

In: Field, J. V.

Title: Renaissance and Revolution: Humanists, Scholars, Craftsmen and Natural Philosophers in Early Modern Europe.

Published: Cambridge and New York, Cambridge Univ. Press, 1993.

Physical Details: 291 p, 117-131.

Notes: Includes bibliography.

Subjects:

Science and Technology.

Mathematics.

Military Technology.

Crafts and Craftsmen.

Technology and Art.

Astronomy.

Physics.

Painters and Painting.

Medicine.

General and Collected Works--Renaissance through 17^th century.

Genre Or Form: Monograph.

ISIS ISIS Record ID: XHTE15953-H

Ausejo, Elena.

Mathematicians and politicians: The case of Spanish military men (1789-1848).

Bollettino di Storia delle Scienze Matematiche 1995, 15(1): 15-26.

Notes: Source of data: Isis Current Bibliography of History of Science.

Subjects: Mathematics--political aspects--19th century--Spain; Portugal.

Genre Or Form: Journal article.

ISIS ISIS Record ID: XISI70149-H

World War I

Russell, I. / Purely by Coincidence: The Rangefinders of Barr and Stroud at the Battle of Jutland, 1916.

In: Blondel, Christine (ed.)

Title: Studies in History of Scientific Instruments.

Published: London, R. Turner, 1989.

Physical Details: 290 p. ill., tables.

Notes: Papers from the 7th symposium of the Scientific Instruments

Commission of the Union Internationale d'Histoire et de Philosophie des Sciences,

Paris, Sept. 15-19, 1987.

Source of data: HTE Current Bibliography in the History of Technology.

Subjects:

Instruments.

Telescopes.

France.

Publishers and Publishing.

Electricity and Electrical Power.

Medical Technology.

Military Technology.

Electronics, mechanical and electro-mechanical technology--20th century.

Genre Or Form: Monograph.

ISIS ISIS Record ID: XHTE7250-H

Dauben, Joseph W.

Mathematicians and World War I: The international diplomacy of G. H. Hardy and Gosta Mittag-Leffler as reflected in their personal correspondence.

Historia Mathematica 1980, 7: 261-288.

Notes: Source of data: Isis Current Bibliography of History of Science.

Subjects:

Mathematics--in war--20th century.

Hardy, Godfrey Harold, 1877-1947.

Mittag-Leffler, Gosta, 1846-1927.

Genre Or Form: Journal article.

ISIS Record ID: XISI6743-H

Volker R. Remmert:

"Offizier - Pazifist - Offizier: der Mathematiker Gustav Doetsch (1892 bis 1977)"

Militärgeschichtliche Zeitschrift, 59 (2000) Heft 1, pp. 139-160.

Moritz Epple and Volker R. Remmert:

"Eine ungeahnte Synthese zwischen reiner und angewandter Mathematik: Kriegsrelevante mathematische Forschung in Deutschland während des II. Weltkrieges"

In: Geschichte der Kaiser-Wilhelm-Gesellschaft, Wallstein Verlag 2000.

Rees, Mina.

The mathematical sciences and World War II.

American Mathematical Monthly 1980, 67: 607-621.

Notes: Source of data: Isis Current Bibliography of History of Science.

Subjects: Mathematics--in war--20th century.

Genre Or Form: Journal article.

ISIS Record ID: XISI4192-H

Dahan Dalmedico, Amy.

L'essor des mathématiques appliquées aux États-Unis: L'impact de la Seconde Guerre Mondiale.

Revue d'Histoire des Mathématiques 1996, 2: 149-213.

Notes: Source of data: Isis Current Bibliography of History of Science.

Subjects: Applied mathematics; Mercantile mathematics; Industrial mathematics--in war--20^th century--North America: United States; Canada.

Genre Or Form: Journal article.

ISIS Record ID: XISI74615-H

Hodges, Andrew.

Alan Turing: the Enigma

Burnett Books / Hutchinson, London, 1983.

McArthur, Charles W.

Operations analysis in the U.S. Army Eighth Air Force in World War II

Providence, R.I. : American Mathematical Society, 1990.

Physical Details: xxiv, 349 p. : ill., notes.

Series: History of mathematics ; 4.

Notes: Includes index.

Source of data: Isis Current Bibliography of History of Science.

Subjects: Mathematical analysis--in war--20th century—North America: United States; Canada.

Genre Or Form: Monograph.

ISIS Record ID: XISI67509-H

Owens, Larry.

Mathematicians at war: Warren Weaver and the Applied Mathematics Panel, 1942-1945.

Published: Boston : Academic Press, 1989.

In: history of modern mathematics. Vol. 2 / David E. Rowe, John McCleary (eds.) p.287-305.

Notes: Source of data: Isis Current Bibliography of History of Science.

Subjects: Mathematics--in war--20th century--North America: United States; Canada.

Weaver, Warren, 1894-1978.

Genre Or Form: Monograph.

ISIS Record ID: XISI60890-H

Pyenson, Lewis.

On the military and the exact sciences in France

In:

P. Forman and J. M. Sánchez-Ron (eds.)

National military establishments and the advancement of Science and technology studies in the 20^th century, Kluwer, Dordrecht, 1996,

135-152.

Ulam, Stanislaw M.

Adventures of a Mathematician

Charles Scribner's Sons, Nwe York.

Autobiography

Present

Mackenzie, Donald, and Garrel Pottinger.

Mathematics, technology, and trust: Formal verification, computer security, and the U.S. military, In:

Annals of the History of Computing 1997, 19(3): 41-59.

Notes: Source of data: Isis Current Bibliography of History of Science.

Subjects: Computers--freedom and secrecy; censorship; espionage--20th century—North America: United States; Canada.

Genre Or Form: Journal article.

ISIS Record ID: XISI76890-H

Elizabeth Urey / The Physical Sciences and Mathematics

In: Author:

Wilson, David A., ed.

Title:

Universities and the Military.

Published:

Newbury Park, Calif., Sage, 1989.

Physical Details:

202 p.

Series:

Annals of the American Academy of Political and Social

Science;

502.

Subjects:

Universities.

Research and Development.

Computers and Computing.

Education, Military.

Science and State.

Technology and State.

General and Collected Works--20th century.

Genre Or Form:

Monograph.

ISIS Record ID:

XHTE3979-H

Booss, Bernhelm and Wolfgang Coy.

Computer fuer den Krieg.

In: von Randow, G. (Hrsg), Das andere Computerbuch. p. 173-198.

Weltkreis Verlag. Dortmund. 1985.

[ISBN 3-88142-327-3]

Booss-Bavnbek, Bernhelm, and Glen Pate.

50 Jahre militaerische Verschmutzung der Mathematik. Undurchdringliche Komplexitaet, ruecksichtslose Kreativitaet und taeuschende Vertrautheit.

In: M. Tschirner, H.-W. Goebel (Hrsg.), Wissenschaft im Krieg - Krieg in der Wissenschaft, Tagungsband eines Wissenschaftlichen Symposiums an der Universität Marburg (1989). p. 157-172.

Schriftenreihe für Friedens- und Abruestungsforschung, Bd. 15., Marburg

Neufeld, Michael J.

The Guided Missile and the Third Reich: Peenemünde and the Forging of a Technological Revolution.

In: Renneberg, Monika.

Title: Science, Technology and National Socialism.

Published: Cambridge and New York, Cambridge Univ. Press, 1994.

Physical Details: xix, 422 p. ill.

with a Spatial Order

other authors:

Mehrtens, Herbert

The Social System of Mathematics and National Socialism: A Survey.

Mehrtens, Herbert

Irresponsible Purity: The Political and Moral Structure of Mathematical Sciences

in the National Socialist State.

Notes: Reviewed by J. D. Hunley in Technology and Culture 37 (Jan. 1996): 196-97.

Source of data: HTE Current Bibliography in the History of Technology.

Subjects:

National Socialism.

Germany.

Technology and State.

Technology and Politics.

Engineering.

Armaments Industry.

Science.

Rockets and Rocketry.

Military Technology.

Aeronautics, Military.

Biology.

Physics.

Kaiser-Wilhelm-Institut (Germany).

General and Collected Works--20th century.

Genre Or Form: Monograph.

ISIS Record ID: XHTE16612-H

Machiavelli, Niccolò:

The Art of War

A Revised Edition of the Ellis Farneworth Translation

With an Introduction by Neal Wood

Da Capo Press, New York, 1990 (Bobbs-Merrill, Indianapolis, 1965).

"The Art of War" was originally punlished in 1521.

von Clausewitz, Carl

On War

Translated by Col. J. J. Graham

Edited with an Introduction by Anatol Rapoport,

Abridged Version

Penguin Books, Harmondsworth, 1968 (Routledge & Kegan Paul Ltd, 1908)

"On War" was originally published in 1832.

Hartcup, Guy.

The effect of science on the second world war

Macmillan Press Ltd, Houndmills, 2000.

Aldrich, Richard J.

The Hidden Hand. Britain, America and Cold War Secret Intelligence

John Murray, London, 2001.