The question in the title is actually what puzzles me a lot, and since I’m living on campus, I have heard how PhD students and young researchers are suffering. Some of these complaints, especially those about programmers, are of course jokes, but I also know people who are so depressed by the hardships of research that they eventually give up. For fear that I may experience the same kind of depression in the future, it is quite pressing to ask and contemplate the difference between being a student and a researcher. Is it just a difference of skills, knowledge and experience, or are there different mindsets, methods and philosophies involved?

Today I finished reading a classic textbook on abstract algebra, written by John Fraleigh. In the meantime I read quite a lot about the history of algebra, including some original works by the founding people. It has been a one-year self-study journey, so it is quite different from our usual learning methods of going for lectures, doing homework and taking tests. It was more like a free individual study like what researchers can experience, and this can be a great case study on the transition from a student to a researcher.

And yes, I think there ARE fundamental changes, not just in the method of learning but in what we focus and how we think.

Probably the most prominent change is that we focus more on how to raise a question or set a theoretical background than how to find the answer. When I read through the textbook, I found that it is not the proofs that are hard: most them are standard and can be done by brutal force. (I don’t even really read the proofs given in the book. I usually write the proof ideas myself instead.) It is how people even started to think of concepts like isomorphisms and groups that intrigued me. Actually these concepts come from good questions, like: How do we describe the feeling that two systems have the same structure? What is the minimum requirement for a system, such that we can solve linear equations in it? Why can we reorder the numbers in multiplication without changing the result, but not in exponentiation? After we have the right question, the answer is not that hard and can lead to significant findings. My opinion is that our education focuses very little on how to find a good question and how to develop a theory – even the best lecture is about understanding the theory itself, not on the underlying rationale, i.e. what problems triggered people to establish the theory and why people choose to shape the theory like that. Simply introducing modules on questioning doesn’t help. True change should be seen in every lecture for every faculty.

And secondly, it is about problem-solving. The problems in research is probably one that we have never seen before (otherwise we don’t need the research), and unlike in exams, there is no syllubus to cram overnight; the problems won’t care whether we have learned the relevant stuff or not. That’s why we need to be extremely diverse. I am amazed at how abstract algebra takes its roots in traditional algebra (i.e. polynomials), number theory, vector and deteminant theory (today called linear algebra), as well as set theory. In very general terms, abstract algebra came into being because people found strikingly similar patterns in several branches of mathematics, and decided to “abstractify” them and start a new branch. (Of course, it was not a one-person decision, but ran across about a century, 1830-1930.) And, walao, this new branch is later found to be so powerful that it can solve century-old problems in its ancestors! Therefore, it is a very bad idea to only stay in one area. We really need the new perspectives of other seemingly irrelavant fields.

Thirdly, it is extremely important to respect the conditions, constraints and special cases. In algebra, thinking small is almost always the forerunner of achieving big. For example, there was this very uncomfortable discovery (due to Kummer) that in some number systems we can’t factor a number uniquely into “primes” as we can for integers. It triggered the serious discussion of what kind of systems has the unique factorization property, and the epic theory of rings resulted from it, which still finds its use in today’s data processing and information security. Also, it is always those limitations that yield give beautiful structures and new findings. For example, isomorphism (a one-to-one function that preserves structure) in itself is just a name change, similar to translating “one, two, three” to “eins, zwei, drei”. But when we add the constraint that the isomorphism must be a function from a set to itself, i.e. an automorphism, a flower-like pattern can appear. This pattern turns out to be the cornerstone of the explanation that quintic polynomial equations are not all solvable by radicals, an extremely important result!

So asking good questions, being diverse and respecting limitations: these are what I can say for now. They can be useful not only in research, but in any career. Also, I think the key above all is to never give up! Cracking something hard really helps to build personality, boost confidence, change perspective, make friends, and so on!!!