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INTERNATIONAL BENCHMARKING OF US MATHEMATICS RESEARCH
2
SCOPE AND NATURE OF THE PANEL'S EVALUATION
Mathematics is the most formal and rigorous of the sciences, but there is no universally accepted definition of mathematics (and the panel has not attempted to formulate one). In this report, to conform with the 1992 report Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States, published by the NRC Board on Mathematical Sciences, mathematics broadly includes pure mathematics, applied mathematics, statistics and probability, operations research, and scientific computing.
Mathematics has several properties that complicate its assessment:
Mathematical research generally has a particularly long “shelf life”: large parts of it do not become obsolete. Not infrequently, a much earlier result or insight—even from a previous century—is suddenly the key to solving a modern problem.
Seemingly diverse subfields of mathematics often form unexpected links.
Throughout science and engineering, mathematics provides a universal language, tools for analysis, abstractions to guide understanding, and methods for solving problems. Consequently, mathematical research has a tightly coupled, two-way connection with other fields: mathematical discoveries influence research in other fields, and developments in other fields provide new problems for mathematicians to study. However, the contributions of mathematical research are often not labeled explicitly as such.
Mathematical training is a central part of the education of all US citizens, from kindergarten through high school and college.
Because of the first two properties, we have not defined and separately assessed subfields of mathematics, but rather have focused on mathematics as a whole. Because of the third and fourth properties, our evaluation reaches beyond mathematics to fields and activities where mathematics research has a direct and visible impact. We also recognize that important mathematical research is conducted by people whose affiliations and titles do not explicitly identify them as mathematicians; some of the difficulties associated with this phenomenon are described in SIAM (1995).
US mathematics research was defined by the panel as research in the mathematical sciences conducted by residents of the United States working in US institutions.
The panel would like to mention 5 key caveats with respect to its analysis. First, following its charge from COSEPUP, this report is based on the qualitative judgments of the panel members informed by both their own knowledge and the sparse quantitative data available. The panel has attempted to be as fair and impartial as possible, balancing the points of view of US academic mathematical researchers with views of leading mathematicians from outside the United States, nonmathematical US researchers, and industrial researchers. In addition, the panel
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OCR for page 19
INTERNATIONAL BENCHMARKING OF US MATHEMATICS RESEARCH
2
SCOPE AND NATURE OF THE PANEL'S EVALUATION
Mathematics is the most formal and rigorous of the sciences, but there is no universally accepted definition of mathematics (and the panel has not attempted to formulate one). In this report, to conform with the 1992 report Educating Mathematical Scientists: Doctoral Study and the Postdoctoral Experience in the United States, published by the NRC Board on Mathematical Sciences, mathematics broadly includes pure mathematics, applied mathematics, statistics and probability, operations research, and scientific computing.
Mathematics has several properties that complicate its assessment:
Mathematical research generally has a particularly long “shelf life”: large parts of it do not become obsolete. Not infrequently, a much earlier result or insight—even from a previous century—is suddenly the key to solving a modern problem.
Seemingly diverse subfields of mathematics often form unexpected links.
Throughout science and engineering, mathematics provides a universal language, tools for analysis, abstractions to guide understanding, and methods for solving problems. Consequently, mathematical research has a tightly coupled, two-way connection with other fields: mathematical discoveries influence research in other fields, and developments in other fields provide new problems for mathematicians to study. However, the contributions of mathematical research are often not labeled explicitly as such.
Mathematical training is a central part of the education of all US citizens, from kindergarten through high school and college.
Because of the first two properties, we have not defined and separately assessed subfields of mathematics, but rather have focused on mathematics as a whole. Because of the third and fourth properties, our evaluation reaches beyond mathematics to fields and activities where mathematics research has a direct and visible impact. We also recognize that important mathematical research is conducted by people whose affiliations and titles do not explicitly identify them as mathematicians; some of the difficulties associated with this phenomenon are described in SIAM (1995).
US mathematics research was defined by the panel as research in the mathematical sciences conducted by residents of the United States working in US institutions.
The panel would like to mention 5 key caveats with respect to its analysis. First, following its charge from COSEPUP, this report is based on the qualitative judgments of the panel members informed by both their own knowledge and the sparse quantitative data available. The panel has attempted to be as fair and impartial as possible, balancing the points of view of US academic mathematical researchers with views of leading mathematicians from outside the United States, nonmathematical US researchers, and industrial researchers. In addition, the panel
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INTERNATIONAL BENCHMARKING OF US MATHEMATICS RESEARCH
was specifically charged not to make recommendations. With more time and effort, additional opinions and data could have been collected, but such efforts would have been expensive relative to the additional guidance obtained.
Second, given the diversity of mathematical sciences, no panel could represent all the subfields of mathematics. The panel has done its best to review all subfields, but some are undoubtedly better analyzed than others. The findings and conclusions here do not apply uniformly to all areas of the mathematical sciences For example, statistics has enjoyed much stronger employment prospects than most other areas. Thus, some variation in interpretation of these results is required when focusing on specific areas.
Third, many statements in this report are not based on numerical data, mainly because of the paucity of statistics that allow meaningful comparisons among countries. For example, undergraduate and PhD degrees in the United States are not directly comparable with all similarly labeled degrees in other industrialized countries. Even when quantitative information is available, sometimes it contains so many ambiguities that we were reluctant to rely on it. To understand the position in other countries when suitable data were unavailable or unclear, the panel relied on the informed judgments of panel members and colleagues from outside the United States.
Fourth, a substantial amount of mathematics is carried out by people bearing other labels: physicists, chemists, electrical engineers, economists, computer scientists, and statisticians. Some Nobel prizes in these fields have been awarded for mathematical work.
Fifth, had this report been written 7 years ago, the Soviet Union would have loomed large as a competitor of the United States. The collapse of the Soviet Union changed that; Russia is in disarray on all fronts, infrastructure is collapsing, and many of the best former-Soviet mathematicians have found employment abroad, particularly in the United States. At present, mathematics in Russia and Ukraine is a shadow of its former self.
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