Hi guys: It's a general question about mathematics so I figure it best belongs to the general math forum. I would like to know how to be more "AHA" insightful, often time when I look at a math problem/puzzles, there's always these little things (observations, reasoning) that would enable one to solve the problem elegantly. I have came to appreciate them and now often take time to observe and reason before moving on to a brute force route for any problems. I must say that has served me well, but occasionally some problems would still come around and I just can't seem to find an AHA insight no matter how I conjecture, observe, or reason? guess maybe everyone has a limit at thinking out side the box? can our insight be increased? I know one might say: do more problems. but then, I am concerned since my goal is to have some groundbreaking achievement in the science/math/engineering field, but if I am just increasing my problem solving ability by experience then it sort of contradicts my intention. thanks
Well, I can't give you any fool proof procedure, of course; but consider the following idea: If you have solved a particular problem satisfactorily, why not try work at solving it in a DIFFERENT way as well? Mathematicians like new independent proofs of something they know is true, not to bolster their own knowledge (why should they?), but rather, perhaps, to enhance their creativity, and help their brains to be able to think outside the box, even for a very "box-contained" problem?
ah! that way I can practice my problem solving skills without necessarily ruining the surprise(well, since it is known the surprise is already ruined). I guess that's a life aha! for me. but in short, you are still suggesting that it takes practice to enhance one's creativity? any approaches that are not fool proof but would be helpful?
Well, I don't think that there exists solid research on how to make "AHA!" insights more likely in a systematic manner. But, what we call "intuition", or similar sudden flashes of insights are only "sudden" for the conscious mind. How the unconscious mind ACTUALLY works to synthesize, in an extremely rapid fashion, various insights so as to present it to the conscious mind as a sort of AHA! or stroke of intuition, is, of course, still relatively unexplored. BUT, to actively try to think outside the box where you "know" the answer (my essential advice to you) cannot possibly be hampering for your subconscious, but rather, give it some additional experience to draw upon, and thereby become more effective.
Definitely, well I guess there's no harm in thinking more often outside the box,I shall take your advice. To some extend, I think observation is useful for thinking outside the box, since it takes some observations of your attempted approaches to identify the boundary of the box. After all, I know that not everyone can be Euler, but my life goal is to be at least like Wronski, even if my entire life I have been failing an failing again and again, there's one thing that people would remember be for ( Wronskian )
I think physicist Richard Feynman wrote that he always carried around in his mind 5 or 6 problems that he was trying to solve. Sometimes he would run into an unrelated situation that reminded him of one of the unsolved problems and that would give him an unexpected insight into a solution. You could try doing something like that to develop intuition and insight.
As Edison put it, invention is 99% perspiration, 1% inspiration. I think the a-ha moments are somewhat over-rated. Terence Tao talks about how when he was younger, he had this romantic notion that mathematical discoveries come in these sudden flashes, but then when he became a big mathematician, he realized that wasn't really the way it worked. Occasionally, that can happen, but the flash is just one little part of the process. Poincare is said to have had such an occasion when he realized that hyperbolic geometric is deeply related to complex analysis when stepping off a train or something like that (not sure about the train part, but he was just kind of going about his day, not working on math, when it happened). This could only have occurred after a lot of thinking about both hyperbolic geometry and Mobius transformations in complex analysis. When you've actually done research, you start to realize that the romantic picture of the light-bulb going off in your head is a little misleading. Don't think about it as one break-through flash of insight. More often, you need about 1, 000 of those little a-ha moments, as you work through smaller parts of the problem. Usually, each one by itself isn't that important. It usually will just solve one of your tiny little sub-sub-sub-problems, not a big breakthrough. And the insights may or may not come in a flash like that. A lot it is likely to be just consciously working on it. But the flashes sometimes come if you take breaks. Part of your brain is still working on it. Other times, you might get a flash when you start working on it again because you have a fresh start. The romantic idea version of science doesn't quite give you the idea of how much grunt work it takes to write things up, and other painstaking, tedious things. Also, it gives you the wrong sense of scale. Research is more like a marathon, not a sprint. The romantic, light-bulb-going-off story gives you more of an idea that it's a short-distance affair where you just have one little insight, then maybe work out and few details and it's over. That's not how it works, most of the time.
Even though having aha moments doesn't always help with research But wouldn't being more receptive to aha moments be helpful ? If I am not very receptive to aha, does that mean I'm not going to be good at research work at mathematics? I mean physics you can do experiments and such, but for math it doesn't seem like there's a place for mediocres like me.
This is exactly right. The big A-HA moments are fun, but for any substantial problem, you need to have a bunch of little a-ha moments before you are allowed to have one of the big ones -- and the little a-ha moments don't necessarily feel like a-ha moments while you're having them, because they're very small and incomplete. And yes, you have to work to earn the a-ha moments at all. The more time you spend working on a problem, the more likely you are to have a sudden insight regarding it, even if your work seems to be going nowhere during that time. Sometimes you have to get 95% of the way through a brute-force attempt at the problem before you have an insight that makes most of that work obsolete. Sometimes you even get 100% of the way through without having the insight -- but that's okay, because then the problem is solved. Someone else may have the insight later; this is why the first proof of a result is often technical and difficult, but eventually a simple proof is published and everyone wonders "Why didn't we see that in the first place?" The reason is that you had to earn the ability to come up with the simple proof, and you do that by putting all that effort into coming up with or understanding the complicated one. Note also that for every amazing insight you have that solves a problem, you may have several insights that don't solve the problem, or even contribute to the solution, because they lie on dead-end paths.
that's actually a good news to me if it is true that you can solve a problem through hardwork in the mathematical research zone, since that would mean that even little guys like me can make contributions.
Yes, it can help. As I said, it's mainly a matter of working hard on something and then taking a break to let your brain work on it subconsciously, let things sink in more, or come back with a fresh perspective. I don't know. I don't really remember having that many a-ha moments in my thesis work. Mostly conscious effort, I would say. And the effort prepares you for the a-ha. It's mainly a matter of putting things on the back-burner. Like those few important problems Feynman kept in his mind, so that if a solution presented itself, he was ready for it. I don't think there's necessarily anything special about the light-bulb going off. Anything the light-bulb can do, conscious thought can probably do, too, under the right conditions, although I think the light-bulb is often helpful in getting past bad conscious ways of thinking that are holding you back. Sometimes, it comes that way, sometimes, it's more conscious. The light-bulb effect can be useful, but I don't think it's really magic, either. Here's another trick you can use. Whenever you do have the light-bulb go off, write down the idea as soon as you can. That will reinforce the process, and you should have more a-ha moments if you keep doing it. You can even carry around a little notepad and pen. This can also make sure you don't forget your ideas. There's some truth to that. Someone I know had an adviser who met Andrew Wiles, the guy who helped prove Fermat's last theorem and was not very impressed with him, as a mathematician. Apparently, Wiles just throws tons and tons of ideas at the problems until they crumble. So, persistence can sometimes be a deciding factor. However, mathematical ideas can be pretty deep and hard for a lot of people to grasp, so it can help to be intelligent, and not just that, but to have the right approach to learning it. If you want to do something that's a little on the easier side, from what I hear, something like numerical analysis might be a good fit. You can succeed sometimes, even if you are not the greatest mathematician, with the highest ability to understand deep concepts, but you are great at programming.
Sort of non "a-ha" type thought. "A-ha" seems like it comes from nowhere, but with conscious reasoning, it comes from a process, like this implies that, therefore that, or maybe you visualize things, and so on. Stuff that you are actually aware of as you are doing it. Not the behind-the-scenes stuff your brain is doing that you are not aware of. Of course, conscious thought is also backed up by skills and experience, which are kind of automatic processes that you don't have to be conscious of, but you get that by practicing consciously first.
Ah, so reasoning, well at least it is something that you can actually get better by practicing right?
Yes. Or by reading a book like Visual Complex Analysis that gives you a good example to follow or talking to professors and grad students to see how they think about things.