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Bundle_Gerbe 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 with a Ph.D. in math, I tend to think verbally in as much as I have words attached to the concepts I'm thinking about, but I never go so far as to internally vocalize the steps of the logic I'm following until I'm at the point of actually writing something down.
I think there is another much stronger distinction in mathematical thinking, which is formal vs. informal. This isn't the same distinction as verbal vs. nonverbal, for instance, formal thinking can involve manipulation of symbols and equations in addition to definitions and theorems, and I often do informal thinking by coming up with pretty explicitly verbal stories for what a theorem or definition means (though pictures are helpful too).
I personally lean heavily towards informal thinking, and I'd say that trying to come up with a story or picture for what each theorem or definition means as you are reading will help you a lot. This can be very hard sometimes. If you open a book or paper and aren't able to get anywhere when you try do this to the first chapter, it's a good sign that you are reading something too difficult for your current understanding of that particular field. At a high level of mastery of a particular subject, you can turn informal thinking into proofs and theorems, but the first step is to be able to create stories and pictures out of the theorems, proofs, and definitions you are reading.