Applied mathematics: lessons in the moral sciences

Introduction: Mathematics, Mathematicians and War

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Introduction: Mathematics, Mathematicians and War

This work was born on September 11, 2001. Actually, it is probably more accurate to say it was conceived in the wake of the attacks on the World Trade Center Towers and the Pentagon, in the context of an ongoing project of mine to examine the values in and of mathematics. Although in recent years I have published work primarily in science studies, my “family of origin” in academia is mathematics, specifically mathematical physics, a brand of applied, as opposed to pure mathematics. I like to think of my efforts to understand the role of mathematical thinking in shaping and creating our world—to make visible the cultural work done by mathematics—as a new twist on “applied mathematics.” As an intellectual one of my most immediate reactions to this world-changing event was to ask myself: How can what I think about come to matter in the world? What is the relationship between mathematics and war? The first connections that came to mind were examples of individual mathematicians who lent their services qua mathematicians to a war effort. Here we might think of Archimedes (287?-212 B.C.E.) who designed ingenious devices to help defend Syracuse against a siege by the Romans in the third century B.C.E.; or Stanislaw Ulam (1909-1984) and the many other men and women who worked on the Manhattan Project in World War II; or John von Neumann (1903-1957), one of many brilliant mathematicians who worked for the RAND Corporation, a Cold War “think tank” whose original purpose was to conduct research on intercontinental nuclear war.
On the other hand, some mathematicians, like G.H. Hardy (1877-1947),1 have deplored the application of mathematics to warfare. Although a champion of “pure mathematics,” contrary to views often attributed to him, Hardy did not “boast” about or “glory in” the uselessness of his work. Rather, he was comforted by the belief that “real” (abstract, “higher”) mathematics was “harmless and innocent” and he could pursue it with a “clear conscience.” (Hardy, 1967, 140-1)2

If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications. But science works for evil as well as for good (and particularly, of course, in time of war); and both Gauss and lesser mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean. (Hardy, 1967, 120-1)

But as I thought more deeply, I realized that such examples were just one thread in a complex and tangled web of interrelationships. As I pursued my exploration of the connections between mathematics and war, I was drawn into a dense network of interwoven strands that led naturally to my thinking more broadly about the relationship between mathematics and ethics. Several themes emerged from the tangle, and it what follows, I will discuss each of them: 1) the institutionalized rules governing the conduct of mathematical research can be read as lessons on how to live well with each other; 2) values embedded in the methodology of mathematical argument have been idealized to moral values that “must” be followed by any honest intellect inquiring after truth; 3) mathematical models can provide a framework for clarifying ethical issues, and may even lead to moral discoveries; 4) and of course the “knottiest” linkage is that applications of mathematical research often lead to ethical dilemmas, questions, problems and situations. In writing this paper I have put my own spin on the threads I unraveled and used them to weave a new fabric that reflects my personal pattern of questions and meanderings. I hope that this record of my journey is of value to others who share some of my concerns.

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