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RyanCarey comments on The value of learning mathematical proof - Less Wrong
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Depending on the post, there are other reasons that people who share your intellectual background may be better at understanding a post, other than the study making them smarter. It could be that they were smarter already, or that similar effects are apparent for people with shared non quantitative backgrounds as well.
A bit more intuition:
In pure math, it's not uncommon for the statement of a theorem, when unpackaged (with all of the definitions spelled out, relative to what the reader already knows), to span several pages. It's often the case that if a mere word or two in the statement of a theorem were changed, the statement would be false. So you need to read every word carefully – it's a sink or swim situation.
Programming has the same character too, but it contrasts with pure math in that it's easy to place it in a different category from verbal communication in general. Some mathematical proof consists of symbolic manipulation, but the more theoretical areas involve a huge amount of verbal-type reasoning. So you would expect people with background in pure math to have this skill, even if they didn't before they started learning.
I agree that mathematical thinking and communication has a special and interesting character. Communicating in a mathlike way is a skill, to be sure. But if meticulously careful communication was superior in a wide range of scenarios, I would have at least a weak reason to expect more people to use it, at least when talking to other math people. But it's not clear to me that math people talk in mathlike ways to math people anymore than sociologists talk in sociologist-like ways to sociologists.
For success in writing, I imagine it's more important to think in a writer-like way. For success in business, I imagine it's again not mathlike thinking that is necessarily most crucial...
Yes, there are multiple possible causes. I'm just expressing my intuition (based on a huge amount of evidence that I can't easily share) and I'm saying that there's reason to make a Bayesian update in the direction of what I'm saying. Naturally, the size of the update depends on how compelling my perspective seems.