Eventually, both Cormack and Hounsfield were recognized with a Nobel Prize for their contributions to the development of medical imaging. But it is clear that the ideas and mathematics were independently discovered by these and other scientists mentioned in this example. The key to the application of computed tomography in hospitals was the computer. In 1970, when practical disc operating systems first became available, it was immediately recognized that by using the fast Fourier transform and algorithms based on the work of Radon, Bracewell, and Cormack, a practical medical X-ray CAT scanner could be manufactured.

In the early 1970s, a number of mathematician-scientist pairs commenced a series of discoveries that led to modern medical three-dimensional imaging. Noteworthy contributions circa 1974 came from three teams. First there were the contributions of Lawrence Shepp to the practical implementation of reconstruction. That work was clearly the result of his partnership with physicist Jerome Stein and physician Sadek Hilal in the filtered back projection method of reconstruction now used. Then, the team of Robert Marr, Paul Lauterbur, and Lawrence Shepp showed the power of arithmetic and Fourier techniques in three-dimensional reconstruction from projections in MRI. Grant Gullberg, Ronald Huesman, and Thomas Budinger, another team of mathematician, physicist, and physician, showed solutions to the attenuated Radon transform problem in SPECT. Currently, MRI and SPECT are being applied by teams of mathematicians, computational scientists, and statisticians to problems ranging from earthquake prediction to understanding and treating mental disorders, heart disease, and cancer.

The message from this historical synopsis is that mathematicians and scientists working in separate locations and on seemingly unrelated scientific objectives related to the mathematical inverse problem made slow and generally unrecognized progress. But when both mathematicians and scientists worked together, as did the three teams cited above, progress was rapid and almost immediately significant.

SOURCE: Based on Herman (1979); Natterer (1986); and Deans (1983).

BOX 2.4 Economics and Game Theory

Modern game theory has provided economists with mathematical tools for investigating resource allocation conflicts between groups of adversarial agents (“players”). These agents can be firms competing for market share, governments vying for advantages in trade, or firms and workers bargaining over labor contracts. Mathematician John von Neumann and economist Oskar Morgenstern were founders of modern game theory, and their collaboration offers an intriguing illustration of the linkages between mathematics and sciences. Mathematician John Nash extended their work in ways that made game theory applicable to a rich collection of conflicts of vital interest to the field of economics.

Morgenstern and von Neumann's book, Theory of Games and Economic Behavior, was published in 1944. At the time, Morgenstern was in the Economics Department at Princeton and von Neumann was at the Institute for Advanced Study. Morgenstern was skeptical of the then preeminent role of the Keynesian paradigm because of its naive treatment of incentives and individual decision making. This skepticism was clearly evident in his earlier writings. It was his collaboration with von Neumann, however, that allowed Morgenstern to translate his skepticism into an alternative approach to economic modeling. While the formal results in the book were due to von Neumann, Morgenstern's perspective was vital for attracting the attention of economists. Indeed some economists were quick to recognize the potential importance of game theory to their discipline, although it would take decades before this view was widely held. Von Neumann was a



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