This probably applies to other disciplines, but currently for math, students essentially study lots of math in undergrad, learn enough to have a solid foundation for research, then go into grad school, specialize in something, and spend a bunch of time tackling a research problem. Most undergrads, however, cannot do significant research because they don't have enough background. Is it possible that as the field expands even grad students won't have enough background to do significant research? Will the system for getting a Ph.D. eventually have to be modified?
Answer by Alon Amit, Ph.D. in mathematics, math circler:
The way I read the question, and the part that I find interesting, is this: It is becoming increasingly hard and time-consuming to master the knowledge needed for conducting research at the forefront of mathematics. Could it be that it will become so hard that progress in mathematical research will grind to a halt?
I think this is theoretically possible but unlikely. It's certainly harder today for a 19-year-old to produce stunning original results than it had been for Gauss or Galois, but there's room for optimism.
Our lifespan continues to improve, as does the general quality of life. A hundred or 500 years from now, humans may actually live and be able to learn and think original thoughts for far longer than we can today.
We will find ways to augment our intellectual capacity. Computers are already very helpful in some areas of mathematical research. Consider what might happen when we start assimilating computational devices into our own living brains.
Humans will continue to evolve, and our brains will improve. It's unclear if this is still really happening (evolutionary pressure isn't what it used to be), but over long periods of time, it probably will.
Mathematics doesn't just get deeper; it also gets wider. Mathematical research today includes parts that are a direct continuation of research conducted in the 19th and 20th centuries, but it also includes parts that are brand new and seeks answers to questions that weren't even asked a few decades ago.
Judging by the growth in mathematical content and research in the 20th century, I think the current trend is quite positive.
Anyway, I don't think this has much to do with the academic system, which I actually think is quite optimized for getting as many people to the frontiers of knowledge as efficiently as possible.
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