I don't really draw that distinction. I'd say that my thinking about mathematics is just as verbal as any other thinking. In fact, a good indication that I'm picking up a field is when I start thinking in the language of the field (i.e. I will actually think "homology group" and that will be a term that means something, rather than "the group formed by these actions...")

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.

I usually think about math nonverbally. I am not usually doing such thinking to come up with proofs. My background is in engineering, so I got a different sort of approach to math in my education about math than the people who were in the math faculty at the university I attended.

Sometimes I do go through a problem step by step, but usually not verbally. I sometimes make notes to help me remember things as I go along. Constraints, assumptions, design goals, etc. Explicitly stating these, which I usually do by writing them on paper, not speaking them aloud, if I'm working by myself on a problem, can help. But sometimes I am not working by myself and would say them out loud to discuss them with other people.

Also, there is often more than one way to visualize or approach a problem, and I will do all of them that come to mind.

I would suggest, to spend more time thinking about math, find something that you find really beautiful about math and start there, and learn more about it. Appreciate it, and be playful with it. Also, find a community where you can bounce ideas around and get other people's thoughts and ideas about the math you are thinking about. Some of this stuff can be tough to learn alone. I'm not sure how well this advice might work, your mileage may vary.

When I am really understanding the math, it seems like it goes directly from equations on the paper right into my brain as images and feelings and relations between concepts. No verbal part of it. I dream about math that way too.

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.

Be careful about keeping descriptive and normative separate.

The correlation that we are talking about is descriptive and has to do with observable reality. What you think should be done and how is normative and has to do with your value judgments.

I feel that the cohesiveness of a community and its effectiveness at maintaining its norms is directly and strongly correlated to the disincentives that it provides for deviating from these norms. Just presence of symbols is not enough.

Of course things like self-selection and evaporative cooling are major factors as well.

Well, yes, but I think that in practice living within a religious community imposes a lot of pressure to conform to the religious norms. Some of that pressure is social (from not being invited to the right cocktail parties to outright shunning) and some can be direct and violent. I recall that the haredim are not above throwing stones at cars on a Saturday...

The mark of totalitarianism is not force, but rather complete control over all aspects of life.

"He loved Big Brother".

Done, though sadly without the digit ratio due to lack of equipment. I'm a newbie and I just thought that was really cool.

One of the main strengths of religious institutions is their sheer pervasiveness; by inserting itself into every facet of life, religion ensures that its disciples can't stray too far from the path without being reminded of it.

That's called totalitarianism, by the way. Not many people consider it to be a good thing.