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Gunnar_Zarncke 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.
I don't see a clear verbal vs. non-verbal dichotomy - or at least the non-verbal side has lots of variants. To gain an intuitive non-verbal understanding can involve
visual aids (from precise to vague): graphs, diagrams, patterns (esp. repetitions), pictures, vivid imagination (esp. for memorizing)
acoustic aids: rhythms (works with muscle memory too), patterns in the spoken form, creating sounds for elements
abstract thinking (from precise to vague): logical inference, semantic relationships (is-a, exists, always), vague relationships (discovering that the more of this seems to imply the more of that)
Note: Logical inference seems to be the verbal part you mean, but I don't think symbolic thinking is always verbal. Its conscious derivation may be though.
And I hear that the verbal side despite lending itself to more symbolic thinking can nonetheless work its grammar magic on an intuitive level too (though not for me).
Personally if I really want to solve a mathematical problem I immerse myself in it. I try lots of attack angles from the list above (not systematically but as it seems fit). I'm an abstract thinker and don't rely on verbal, acoustic or motor cues a lot. Even visual aids don't play a large role though I do a lot of sketching, listing/enumerating combinations, drawing relations/trees, tabulating values/items. If I suspect a repeating pattern I may tap to it to sound it out. If there is lengthy logical inference involved that I haven't internalized I speak the rule repeatedly to use the acoustic loop as memory aid. I play around with it during the day visualizing relationships or following steps, sometimes until in the evening everyting blurs and I fall asleep.