[jacobolus]

"Bold R" and "Double-Lined R" (i.e. blackboard bold) are semantically equivalent. As your next paragraph hints toward, the purpose of the second one is to be distinguishable from the regular italic or Roman R in handwriting (or on a typewriter).

"Fake-Handwritten R" is an extra fancy calligraphic version which is not hard to distinguish. The Fraktur R is a pain to write, but you can write an upright "Re" as an alternative.

The basic issue is that using single symbols for variables is very convenient (both more concise and less ambiguous than writing out full or abbreviated words when writing complicated mathematical expressions), but there are infinitely many possible variables and only a small set of symbols.

[dhosek]

Yes and no. Generally blackboard bold has come to denote particular number sets while bold usually refers to vectors or matrices. There are a handful of traditionalists¹ who will use *R* for the reals or *Z* or even Z for the integers, but the trend toward blackboard bold is, I think, definitely where things are going.

⸻

1. I would put Donald Knuth in that category, given his choice to not include blackboard bold in his original inventory of characters for Computer Modern, but that might just as much have been a choice based more on limitations of the computing systems he was working with at the time (or his needs for typesetting The Art of Computer Programming which were the primary driver of TeX).

[jacobolus]

Whether you write bold R, Z, Q, C or blackboard bold for these number sets nobody at all is going to be confused – they appear in both ways all over the place in books and research papers – and if you mix ordinary bold R, Z, Q, C next to the blackboard bold versions of the same upper-case letters in a single document then your friends should tell you to knock it off.

As for "where things are going" – this has been changing extremely gradually over the past 60 years. If the trend accelerates maybe you'll stop seeing both variants in wide use in about another century.

[dhosek]

> in about another century.

That sounds about right. Maybe even 50 years, but it is a rather slow process.

[seanhunter]

Springer for example uses capital bold Z, I, Q, R, C, not blackboard versions in most of their books whereas Cambridge University press seems to go for Blackboard bold.

On the other hand "Wolfram" (tspfka "Mathematica") seems to not only use the uppercase blackboard bold for Reals, Integers etc but also use lowercase blackboard bold for i, e, c_x (arbitrary constants) etc. Which is just annoying.

[anigbrowl]

only a small set of symbols

I grind hundreds of flashcards every night to learn Japanese and I can assure you that one thing we are not short of is symbols. Chinese characters use ~218 basic symbols which can be stacked and combined to form tens of thousands of characters. There are 350 symbols just for counting different kinds of things.

https://www.tofugu.com/japanese/japanese-counters-list/

[edanm]

The thing is, these symbols are supposed to represent something, so it's better if they give some intuition. But some words get overused, exactly because of that.

E.g. the two Rs we are talking about here both stand for the same word - Real. Except one is Real as in the Real numbers, and one Real as in the real part of a complex number.

If you go with a random symbol, you're putting a different kind of cognitive overhead because you have to map a random symbol you never saw before to a specific concept. Here you just have to distinguish font, and even that is often not necessary, because you often are dealing with a branch of maths that only uses one of the meanings.

[BalinKing]

Tangentially related: Category theorists sometimes denote the Yoneda lemma by よ.