Does more knowledge enhance your chance of doing better and 'original' research (in mathematics)?

Question

NOTE: I had asked this question in here, in my edit, but since it is irrelevant to that question, I moved it here as it is a separate question...

I chose to study bulky books instead of attending lectures or studying lecture notes, because I thought that studying every line of the books would enable me to have a rich access to the largest possible knowledge in my mind (not in the library) to have more creativity in time of doing research. By that, I mean I thought that the more techniques I would know, the more able I could be to test many 'keys' to open impenetrable problems. Unfortunately, due to never being in a university environment (as I am a high school student) I couldn't ask it whether I am wrong or not. I would truly appreciate it if someone would please guide me regarding that issue: Does more knowledge enhance chance of solving challenging problem in mathematics?

(Supposing that someone who has learned a lot of stuff will not forget it with the passage of time)

Added to express my question more clear - My personal opinion is that the more problems (few from each of many different types not many from each of limited diversity) you solve the more expert you become in maths. This can include 'slow'-extension (i.e. not a breakthrough); e.g. after learning definitions and some theorems regarding different types of spaces in General Topology finding their relationships with each other as exercise. This approach can be (IMHO) helpful for many different situations from IMO participants to graduate research students. But, Does it helpful or harmful for being sharp to do a breakthrough research in Mathematics (like works of Gödel or Perelman)?

Answer 1

Intuitively, of course the answer should be yes. Surprisingly, however, in reality the answer is sometimes no.

A common saying, which to the best of my knowledge has no known attribution, goes something along the lines of: "The fools didn't know it was impossible, so they did it!"

To achieve a significant original scientific contribution, you need to know enough to have a sound basis to build upon. You also need to avoid immersing yourself too deeply in conventional wisdom and groupthink, or else you are likely to have your originality bleed away from you.

Moreover, every hour spent reading the works of another is an hour not spent producing works of your own. But of course, you can waste a huge amount of time if you end up reinventing another's idea or miss an important clue in somebody else's work. So across a scientific life, you will end up wobbling back and forth on a balance between spending more time acquiring knowledge and spending more time generating it yourself. Early in your career, you will spend most of your time acquiring knowledge, because you need to get to a point where you can understand what is unknown well enough to begin grappling with it meaningfully. Later you may spend more time working on creating new knowledge, but must also keep up with your reading to keep from going stale.

Even in old and well established fields, however, it's surprisingly easy to get out to the edge of human knowledge and find yourself tangling with some of the interesting things we do not know. Like, how do cats drink water? We only learned that a couple of years ago. Heck, we're still figuring out how bicycles work!

If you don't have enough knowledge, you may not notice that there's a problem worth solving there, and you certainly won't have the tools to solve it. If you're too saturated in the knowledge of others, you might not notice the problem either, because you might end up accepting their explanations instead.

In sum: you must know your fundamentals, know your field, and know how to go looking for specific pieces of additional knowledge. But knowledge is not originality, and although knowledge often supports originality, it can also interfere with it.

Answer 2

My own experience and observation of how some fellow students became very successful researchers suggests that doing what you describe is actually detrimental.

As someone else mentioned, every hour you spend studying these thick books is an hour away from research. But the actual effect is more insidious than that. In mathematics, there are two issues. One is that often to write a mathematically accurate text, you have to build up a lot of formalities, like definitions of how you use symbols, how you will handle edge cases, and proofs of little things that can actually obscure the main ideas. The whole point of attending a seminar or lecture is to see how an expert, possibly even the discoverer of the ideas in question, sees the subject. For research, sometimes this is all you need to know if it turns out you don't need to delve further into a topic, but even if you do delve deeper, knowing what are the main ideas is crucial and often not at all clear from a text, which is usually designed more to be an authoritative reference or complete in its exposition.

The other issue has to do with there being many ways to view one thing in mathematics. Experts build up very personal ways of viewing mathematics. I think this might be truer than in more experimental sciences. Even a well-studied area can be viewed in a way that's still correct but unusual enough to lead to new insights. A lot of famous mathematicians have arrived at new theories by just trying to understand old theories their own way, without being unduly influenced by the standard ways. I personally find it difficult to understand a proof that is coming from a different approach than what I'm comfortable with. To put it simply, sometimes your brain might be more hard-wired to understand something geometrically than algebraically or vice-versa. By learning to fill in the gaps in lectures your own way, you learn what works for you and you start to develop your own style of doing mathematics. If you're reading a very detailed or complete text, it can take you longer to understand simply because the author(s) might be more comfortable with using a particular tool than you are. One could argue that learning that tool is important, but I think there's something to be said for learning tools that come to you naturally. At the least, it's probably a more efficient approach.

Answer 3

It is a delicate compromise really.

The benefits of having an encyclopedic knowledge of all areas of math are pretty clear. However, life is short, and you will be judged on the research you do from an early age.

Learning can be a form of procrastination. You also risk spreading yourself too thinly (learning the easy parts of many subjects but mastering none). Empirically, I observe that many mathematicians (ignoring Tao etc) are experts in a narrow field, and only have a superficial understanding of other areas.

With practice, it becomes easier to isolate the relevant parts of a proof, and extract the minimal technical information to apply it. You don't really need book "knowledge" of it.

It's also a matter of taste. I favour the "just enough education to perform" model. I am increasingly bored by learning, and prefer to work on new ideas all the time ("new" to me, anyway. I rely on chatting to people regularly as a screen against reproducing known results). I find it more efficient and fun to talk to people, than learning. Perhaps it comes down to your philosophy of life https://www.youtube.com/watch?v=WrhzX3dRRiI ;)

On the other hand, many ideas and techniques can be considered "core" mathematics. I think you basically must know these, if only so that you can talk to other mathematicians efficiently.

Answer 4

I think that, under all other conditions being equal, being exposed to larger knowledge base is (might be) beneficial for quality of research. The "other conditions being equal" condition (no pun intended) is important, as there might be other, more important in a particular context, factors that have greater positive effect on research quality and/or originality than just volume of knowledge.

Several such famous factors come to my mind. The first is the Eureka effect. The second is the deliberate practice concept, which includes the famous 10000 hours heuristic, based on research by Anders Ericsson and popularized by Malcolm Gladwell in his book "Outliers". However, note that the role of deliberate practice in mastering a subject domain (to an expert level) recently has been actively challenged by research studies (i.e., see this one). Finally, the third factor that I believe is important (and in many cases might be more important than other factors, including the volume of knowledge) is the ability to see and/or analyze a topic through different mental/conceptual "lenses" (for details and references, see Wikipedia articles on perspectivism and cognitive perspective).

Therefore, considering the points above, I think that the answer to your question is "Maybe".