Heh.
The "tabs vs spaces" debate of mathematical typography is whether these symbols should be used at all in printing, or reserved for actual blackboards. In LaTeX you can have actual boldface letters, so you should write a boldface letter R to represent the real numbers and so on. Using "blackboard bold" in print looks terribly off to some (many?) people.
But there is certainly some dependency on the font. For example, the Baskerville used by the Publications de l'IHÉS is a much thicker font, and the bold letters flow very naturally. Look for example here
http://www.numdam.org/item/PMIHES/
The publications in the sixties and seventies, which are scanned, are very beautiful and the bold letters mix smoothly in the text. You cannot see them "from far away" as happens with Computer Modern.
Curiously, the modern pdfs look slightly different, with apparently thinner fonts. This may be an effect of printing, that smears the ink a bit and produces slightly thicker type?
https://en.wikipedia.org/wiki/Blackboard_bold
Summary: boldface came first, in print, and was widely used. Much later, some mathematicians started to write blackboard bold on blackboards (because actual bold would be very untoward to write on a blackboard). Then, the usage was backported to print.
I find it ugly, a sort of breaking "suspension of disbelief", I don't know how to explain why. Some sort of "anachronistic" feeling, like watching a movie about the roman empire and some soldiers wear watches.
EDIT: as for having "different symbols", this is not the case. They are exactly the same symbol, with different faces. Like when you write the letters "a" or "g" very different on a blackboard as they appear on print.
Italic typefaces were created by Italian type designers to mimic the cursive handwriting of the time. They weren't meant to be mixed with roman style typefaces. However, typesetters who had access to both roman and italic fonts started to use italic for emphasis when typesetting texts in roman type.
And of course blackletter is also just a style that originally tried to mimic earlier handwriting.
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.
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.
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.
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.
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 :)
[0] https://expo-patrimoine.ihes.fr/?page=9&lang=en
[1] http://www.numdam.org/item/CG_1999___32_5_0/
You also have this semi-anonymous blog post [2], that laments the decline in quality when the journal in question was acquired by Springer. Quote:
> How good was the typeface? It was so good, some people submitted their papers to IHES precisely for the beauty of the typesetting
[2] https://www.galoisrepresentations.com/2014/08/15/the-decline...
https://en.wikipedia.org/wiki/The_Fifteen_Decisive_Battles_o...
See: the entire Latin alphabet.
Sans-serif symbols for screens, serif symbols for print.
You can find many old and new math texts using either convention. Most often readers won't even notice. There are authors who are adamant against blackboard bold, most famously Donald Knuth and Jean-Pierre Serre. Others simply use regular boldface because of tradition, or because it's the default style of the journal. Most people don't care too much. But I have also heard people expressing strong opinions on each side, that's why I made the analogy with tabs-vs-spaces. Nothing too serious.
Elegance is entirely subjective. I have the opposite opinion: if we can do real boldface in print, which is the original usage, what's the point of using a black-board version? We have the real thing! But I also understand the opposite opinion, voiced elsewhere in this thread, that open bold lettering is a new case, like italics or fraktur, and we can use it freely in print.
Almost [1] every single student absolutely hates that notation in scripts. Bold is used to draw attention, it should not be abused as being part of the variable/type. Just use the proper symbol.
[1] And I'm only saying "almost" to account for the possibility of there being like 5 super weird people on this planet that think otherwise. I don't know a single person who thinks this is a good idea, even the professors writing their scripts like that think it's stupid and are only doing it due to some nonsense fear of it otherwise not printing correctly due to one single bad experience with a shitty printer in 1950. Or maybe there's some other historical reason for this, but in 2023 I'd classify not using the better notation as malicious.
Why use the wrong symbol, when you can just use the correct (ie, blackboard bold) symbol?