students if some of the very small programs were eliminated, leaving only those with a critical mass in faculty and students. Although practices in some sciences tend to encourage such an outcome, few mathematicians are prepared to argue in favor of this approach, believing that such a system would be elitist; and, in any case, almost no one is willing to support steps that would bring it about.

One important reason departments cling to doctoral programs even though the programs might be marginal may be related to the different working conditions that usually prevail for departments with and without such programs. Although cogent arguments are made linking the presence of doctoral students to the mathematical vitality of a department, the small number of students and the vanishingly small number of successful students in some departments can make one wonder about the relevance of the arguments in some cases. A second reason often concerns the role graduate students have in teaching the lion's share of lower-division mathematics at universities. However, should we decide that we need young, energetic, nonpermanent instructors and then use graduate programs to provide them?

Usually when the suggestion is made that the graduate enterprise may be too big, one has in mind the great difficulty that young mathematicians have today in finding jobs. In making this connection, two assumptions are implicit: the first is that the purpose of doctoral training is to prepare students for a university, or at least a college, position and the second is that the system has a responsibility to new doctorates to guarantee them such a job. Neither assumption has always been held in mathematics, and they are certainly not held currently in all other fields. That is why, for example, most colleagues in the humanities have little sympathy with the current plight of young scientists, since few new doctorates in fields such as history or English can expect to get an academic position. However, it is depressing to see eager, talented, young mathematicians leave the field for lack of a suitable position. Let me make one other observation. Most mathematicians strongly agree that a doctorate should be required for teaching mathematics at 4-year colleges and that it is highly desirable for teaching mathematics at 2-year colleges. However, many faculty view doctoral graduates who have taken such teaching positions as having left the community of mathematicians.

An issue related to the view of the doctorate as preparation for a job concerns the completeness of the training provided. That is, if candidates are being prepared for teaching positions, then shouldn't their graduate education reflect that? Although some recognition of this possibility is now made in most departments, I wonder what message is being given at most places on the relative importance of teaching.

Let us return to the question of size and the purpose of graduate education. Since there seems little likelihood that the number of university and college positions in mathematics will increase, at least in the foreseeable future, the profession must make some choices. Either the community must somehow reduce the size of the doctoral system, or it must broaden its views on appropriate jobs for graduates and both stimulate demand and prepare students for these alternative positions, or students must be told that one studies mathematics because one likes it. It is then the students' responsibility to find employment, and if they are fortunate, they will make use of what they have learned. Most groups that have thought about this issue have opted for the middle course, but this has consequences for graduate training. It would have to be broader and more responsive to the needs and desires of business, industry, and other disciplines. In particular, students would need to learn how to communicate what they have learned to the educated public. The realization that graduate education would have to adapt is particularly hard



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