> I’ve literally never heard anyone advocate your position until you did, just now
Many mathematicians concerned by typography use real boldface. For example: Terence Tao's blog, Donald Knuth, Paul Halmos (author of "how to write mathematics"), and the famous journal "Publications Mathématiques de l'IHÉS" which is the undisputed gold standard in mathematical typography. They use real boldface for the number sets N, Z, Q, T, R, C.
I've never seen a boldface R to mean a set different than the real numbers.
People use both, for reasons of tradition, ergonomics, practicality, available toolset. Boldface is obviously common in places where BB is unavailable, more restrictive, more difficult. Web publishing is a great example.
BB is common, in my experience, in hand-written text where bold isn't really an option. It is, to my tradition, the most common and recognizable way to indicate the most common field sets and also generally is a good stand-in for any "large category" of interest.
Bold is used intermittently in my experience, probably due to its inability to be hand-written. To me, it tends to mean "vector" or "matrix" much more than set. In hand-written forms, I sometimes see "arrow hats" instead, especially for vectors.
> "Publications Mathématiques de l'IHÉS" which is the undisputed gold standard in mathematical typography.
It is? Do you have any supporting evidence for this claim?
I just had a look at a bunch of recent articles, and I would very much dispute it. I saw nothing extraordinary, and found the fonts they used rather ugly (though of course that is highly subjective). The use of bold face to highlight theorem/definition/etc. numbers is IMHO very questionable. The boldface letters you praise stick out like a sore thumb, feeling as if they were being emphasized and highlighted when they clearly are not meant to be.
> It is? Do you have any supporting evidence for this claim?
I don't have any evidence to support this claim. I always thought about it as self-evident, because it was in that journal that Grothendieck published his work, and the same style is used in by the legendary Hermann editor from Paris and by Bourbaki. But I cannot find any non-partisan source of my claim. As for non-neutral sources, you have for example the congratulations on the typesetting by Dieudonnée [0] (who was a member of the IHES), or a more recent article by Haralambous about the Baskerville variant used by the institute [1]. I will retire my claim of "undisputed" if you find a source that says that the pinnacle of mathematical typesetting is something else :)
The point is to be understood.
Use blackboard bold when it’s what people are most familiar with and will engender the least confusion.
You call this the “tabs vs spaces” debate, but I’ve literally never heard anyone advocate your position until you did, just now.