Gödel had his doctorate at 23, completing all his university studies in about 5 years. In the U.S., for example, if one goes through the usual path of four years of undergraduate studies (after being admitted to a university at 18), a 23-year-old student would likely be a first-year graduate student (assuming he or she does not take any break after the undergraduate years) in the midst of taking several core classes, which might demand enough time of study that even the most dedicated student might not be able to think of other mathematical questions not related to those classes. After taking those core classes, which might take two years, the prepared student is told to take more classes from different areas for a broad mathematical knowledge, which might take an additional year or more, before he or she is allowed to focus on some research. Hence, getting a doctorate at twenty-three is rather an exception. But, are all those classes really needed to become a mathematician?
I suppose the reason why graduate students are required to take core classes is to make sure admitted students have preparation to handle more advanced work in mathematics. Those classes might be of particular importance to students who have their degree in a non-mathematical area or to students who unfortunately have not had an adequate mathematical training in their undergraduate years. But, what about students who were lucky to have such a strong training? Should they be subject to the same treatment? Has there been any evidence that shows to introduce such prepared students to research leading to their dissertation at an early stage would impede their training? Of course, I would think there needs to be some kind of evaluation to ascertain of the students’ capability, but does it need to be done by taking several classes?
Beyond the core classes, I suspect the justification for obliging students to take additional classes unrelated to their area of interest is to expose them to different sides of mathematics, which ultimately could help their research by allowing them to frame a problem in different perspectives. If accurate, while such endeavor can be seen as very valuable and can again be of great importance to students without prior mathematical training or with a training with lacunae, the number of classes and their content seem to be arbitrary: would it make any difference to take 2 topology, 2 analysis, 3 algebra classes instead of 2 topology, 3 analysis, 2 algebra, and 1 logic classes? Would it not ultimately depend on how those classes are taught, how much work the students are willing to put in, and what their final choice of specialization is? Instead of trying to squeeze many classes in the limited time one has to complete the dissertation, would it be seen as risky to relegate the study of different topics to a more widespread timeline, which could be during and after the years of studies? Anyway, isn’t it true that more mathematics can be learned while someone is doing research? All of this seems to suggest that there might be many opportunities during one’s mathematical career to learn a good amount of mathematics, so maybe more emphasis needs to be put on writing a high-quality dissertation by investing more energy learning what is needed for its completion rather than trying to uncritically absorbing knowledge, which may end up being irrelevant to the doctoral studies. Again, I do not mean that learning a new subject for its own sake is a waste of time; au contraire, I think it’s a very good way to learn something, however idealistic this might seem. Nevertheless, one also might need to be realistic enough to realize that one has a limited resource (mostly financial) to complete a dissertation, so to be selective about what one learns at this time might be wise.
So, what do you think about all of this? Do you think graduate programs are retarding students by making them take all those classes that may be irrelevant to their research? Do mathematics students really need to take all those classes to become successful mathematicians?