The letter ℘: name and origin? (2017)

One thing I've always struggled with Math is keeping track of symbols I don't know the name of yet.

Googling for "Math squiggle that looks like a cursive P" is not a very elegant or convenient way of learning new symbol names.

I wish every proof or equation came with a little table that gave the English pronunciation and some context for each symbol used.

It would make it a lot easier to look up tutorials & ask questions.

This. Related to that, I'll also never get used to mathematicians' habit to assign semantic meaning to the font that a letter is drawn in. Thanks to that, we now have R, Bold R, Weirdly Double-Lined R, Fake-Handwritten R, Fraktur R and probably another few more.

All of those you're of course expected to properly distinguish in handwriting.

I'm sure most of them have some sort of canonical name, but I'm usually tempted to read them with different intonations.

(Oh and of course each of those needs a separate Unicode character to preserve the "semantics". Which I imagine is thrilling edgy teenagers in YouTube comments and hackers looking for the next homograph attack)

"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.

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.

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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).

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.