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wadavis comments on Open thread, Oct. 27 - Nov. 2, 2014 - Less Wrong
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I have a question for anyone who spends a fair amount of their time thinking about math: how exactly do you do it, and why?
To specify, I've tried thinking about math in two rather distinct ways. One is verbal and involves stating terms, definitions, and the logical steps of inference I'm making in my head or out loud, as I frequently talk to myself during this process. This type of thinking is slow, but it tends to work better for actually writing proofs and when I don't yet have an intuitive understanding of the concepts involved.
The other is nonverbal and based on understanding terms, definitions, theorems, and the ways they connect to each other on an intuitive level (note: this takes a while to achieve, and I haven't always managed it) and letting my mind think it out, making logical steps of inference in my head, somewhat less consciously. This type of thinking is much faster, though it has a tendency to get derailed or stuck and produces good results less reliably.
Which of those, if any, sounds closer to the way you think about math? (Note: most of the people I've talked to about this don't polarize it quite so much and tend to do a bit of both, i.e. thinking through a proof consciously but solving potential problems that come up while writing it more intuitively. Do you also divide different types of thinking into separate processes, or use them together?)
The reason I'm asking is that I'm trying to transition to spending more of my time thinking about math not in a classroom setting and I need to figure out how I should go about it. The fast kind of thinking would be much more convenient, but it appears to have downsides that I haven't been able to study properly due to insufficient data.
As someone employed doing mid-level math (structural design), I'm much like most others you've talked to. The entirely non-verbal intuitive method is fast, and it tends to be highly correct if not accurate. The verbal method is a lot slower, but it lends itself nicely to being put to paper and great for getting highly accurate if not correct answers. So everything that matters gets done twice, for accurate correct results. Of course, because it is fast the intuitive method is prefered for brainstorming, then the verbal method verifies any promising brainstorms.
Could you please explain what you mean by "correct" and "accurate" in this case? I have a general idea, but I'm not quite sure I get it.
Correct and Precise may have been better terms. By correct I mean a result that I have very high confidence in, but that is not precise enough to be useable. By accurate I mean a result that is very precise but with far less confidence that it is correct.
As an example, consider a damped oscillation word problem from first year. You are very confident that as time approaches infinity that the displacement will approach a value just by looking at it, but you don't know that value. Now when you crunch the numbers (the verbal process in the extreme) you get a very specific value that the function approaches, but have less confidence that that value is correct, you could have made any of a number of mistakes. In this example the classic wrong result is the displacement is in the opposite direction as the applied force.
This is a very simple example so it may be hard to separate the non-verbal process from the verbal, but there are many cases where you know what the result should look like but deriving the equations and relations can turn into a black box.
Right, that makes much more sense now, thanks.
One of my current problems is that I don't understand my brain well enough for nonverbal thinking not to turn into a black box. I think this might be a matter of inexperience, as I only recently managed intuitive, nonverbal understanding of math concepts, so I'm not always entirely sure what my brain is doing. (Anecdotally, my intuitive understanding of a problem produces good results more often than not, but any time my evidence is anecdotal there's this voice in my head that yells "don't update on that, it's not statistically relevant!")
Does experience in nonverbal reasoning on math lend actually itself to better understanding of said reasoning, or is that just a cached thought of mine?
Doing everything both ways, nonverbal and verbal, has lent itself to better understanding of the reasoning. Which touches on the anecdote problem, if you test every nonverbal result; you get something statistically relevant. If your odds are more often than not with nonverbal, testing every result and digging for the mistakes will increase your understanding (disclaimer: this is hard work).
So, essentially, there isn't actually any way of getting around the hard work. (I think I already knew that and just decided to go on not acting on it for a while longer.) Oh well, the hard work part is also fun.