In Latex, you get these symbols with \mathbb{}. They are most commonly used to represent to represent number sets, like the set of all integers Z (from the German), set of all natural numbers N, set of complex numbers C, set of all real numbers R, rationals Q (for “quotient”), set of quaternions H (named after Hamilton), or an unknown set F (for “field”). This explains why many of the letters in this series exist in the basic multilingual plane—because R is very commonly used, but A is not. You can find ℝ in the basic multilingual plane at U+211D.
I don’t know why people are interested in the X symbol. It’s just there to complete the alphabet. There are many other ranges like this used for writing mathematical formulas, like the range of bold letters, fraktur, script, etc.
Can you please not break the site guidelines by fulminating, name-calling, snarking, and so on? It's not what this site is for, and destroys what it is for.
You may not owe $CEO better, but you owe this community better if you're participating in it.
I'm a PhD student in algebraic geometry and have seldom seen this. IME "X" is a generic name for all kinds of spaces (metric, topological, complex analytic, algebraic...), but I don't remember ever having seen "𝕏".
That said, in principal everyone is free to call their spaces how they like. It just makes it easier to read if one sticks to familiar notation.
Now that I think about it, most double stroked characters really represent "concrete" objects, not generic ones. Like ℙ for projective space, or 𝔸 for affine space. Also 𝔽 is often used together with an index 𝔽_q for the field with q elements.
On the other hand, Wikipedia[1] agrees with you, that 𝕏 is "occasionally" used for metric spaces.
As a counterpoint, 𝔽 when not fixed to a particular size is arguably still generic over "any field", and 𝔾 is most definitely generic for "any group", even if fixed to a particular size.
There might be small cultural differences at hand here. As a German, I prefer "k" or "K" for fields (as in "Körper"), and I think I've seen 𝔾 mostly as 𝔾_a or 𝔾_m for the additive and multiplicative group. shrug
My experience with 𝔽 is “vector space over a field 𝔽”, which is not a concrete object, but I have probably seen F written in a normal italic style more often.
I’ve also seen ⅆ as the differential operator and 𝔼(X) as the expected value of X. I think I’ve also seen 𝟙_x, and ⅈ, and ⅉ but I don’t remember for what.
I know 𝟙_X as the indicator function (also called characteristic function) of X in measure theory and probability theory, but I’ve also seen 1_X or χ_X.
ⅈ and ⅉ can be basis vectors or imaginary units although I’ve never seen them in blackboard bold in my lectures.
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.
I strongly disagree that the bold symbols are better, even though they’re what blackboard bold was originally conceived to approximate. The reason: Bold font draws attention (ask a typographer about type color), but usually, the bolder symbols are not the ones requiring that attention. (If I read a text about real analysis, I know that the domains and codomains will be the real numbers, for example.) The blackboard bold versions stick out much less when looking at the greater composition of the page.
I respect your opinion, and agree that with some particularly "thin" fonts (most notably, Computer Modern and its variants), the associated bold symbols are too exaggerated and produce and ugly effect.
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
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?
I hated when textbooks/papers did this. Half the time you can't tell if they meant to use the bold letter or if the printer was just being generous on that character. Made legibility quite a bit more difficult. BB letters are unambiguous.
Funny. Coming from a non-math background, having only one symbol to deal with feels less confusing than switching around symbols based on the medium it happens to be written on. Did one of the symbols come "first"?
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.
That's pretty typical for how typography develops. Majuscule and miniscule letters were just two different styles of writing the same letter. But eventually, people started using majuscule (i.e uppercase) letters to give emphasis to the beginning of sentences or certain words, like names.
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.
> 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 :)
Do you have a source for this? I have never come across this debate and personally find the blackboard bold convention to be useful, as well as very visually elegant.
It seems to me that it is part of the folklore. Note: not the folklore of math! The folklore of mathematical typesetting, which is a tiny community.
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.
> In LaTeX you can have actual boldface letters, so you should write a boldface letter R to represent the real numbers and so on.
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.
The "bb" in "mathbb" isn't for "math, baby" (although accurate), but short for "blackboard bold". These letterforms don't originate with blackboards though, using outlines for thick verticals goes back centuries. They're sometimes called open face.
I think you missed the point: the domain has not been in Musk's control that whole time, as I had thought. His company X.com had it, then PayPal got it when it was created in the merger of X.com and Confinity, and PayPal kept it after Musk left, but then Musk got it back.
This doesn't contradict what you're saying, but note that the company was only re-branded to PayPal after Musk left it. I think it was just named Confinity after the merger.
This whole Twitter saga is like what happens when a cat catches a bird and doesn't know what to do with it so just slowly tortures it to death for the hell of it
It sounds like Elon had the domain and it was burning a hole in his pocket. That's the only explanation, honestly.
Twitter as a brand becomes less valuable the more he alters it, since he's such a polarizing figure, so changing the brand to break continuity seems pretty counterproductive. It just reminds people that Elon is still fiddling with it.
> It sounds like Elon had the domain and it was burning a hole in his pocket. That's the only explanation, honestly.
I think he just thinks he’s getting his revenge and that everyone will see how his terrible idea of x.com was actually genius back in the day. Like a particularly expensive midlife crisis. A bit like some dudes who start playing guitar because they could not back in the day, when it was a useful tool to make friends and look cool.
Also people less likely to assume its the brand for a pornography site? Feel like people have developed a habit of avoiding (or seeking out) anything with a prominent "X" in the name on the Internets.
They are an accessibility nightmare. I read a comment here recently from a blind person that had to give up on math education entirely due to all of the symbols not being supported by screen readers. It was pretty sad.
That's on screen readers developers. And I don't know when that comment you seem to remember dates back to, but I have had several blind students in my math lectures over the past decade. They seemed did fine.
I avoid using oddball Unicode characters for this reason,[1] but I also feel like at this point the can of worms has been unleashed, and the screen reader authors are going to have to adapt to a more hostile environment.
It seems like one approach might be something like this, assuming none of them are doing it already. If this is TLDR, the key feature relating to the topic at hand is the point about handling mathematical symbols, so skip everything except that one.
* Pre-parse the entire page and count the number of characters that would be read using a multi-word description. While doing this, create sub-counts of characters that fall into each Unicode script (Latin, Cyrillic, mathematical notation, etc.).
* If the number of characters that would be read using a multi-word description exceeds a user-specified threshold, prompt the user for how to proceed.
* First, tell the user how many different Unicode scripts (and blocks if necessary[2]) are contained on the page. For each set, prompt using the following logic if the user hasn't already set a default:
** If a script/block/whatever consists of abstract symbols, like the mathematical set, offer to read it using the closest approximation in the character set it's derived from. e.g. for 𝕏, read it as "X" instead of "mathematical double-struck capital X". Read both 𝜸 ℾ as "gamma" instead of their lengthy "mathematical..." descriptions.
** If a script/block/whatever represents a spoken language, offer to do one of the following:
** Read it as if spoken by someone fluent in the language.
** Precede it with "the following [n] [script name] characters" and read each one individually, without any leading per-letter indicators, e.g. for "ΑΒΓΔ", "the following four Greek characters: alpha, beta, gamma, delta". Optionally include per-letter capitalization indication, but only if the user has previous enabled that option.
** Replace it with a count and the script name, e.g. "five hundred characters of Chinese script". Provide a way for the user to interrupt the narration and expand that section instead of omitting it.
* Allow the user to store all of the prompt answers as defaults and not prompt them again unless the user resets the option. e.g. "always read mathematical symbols as the closest Latin letter approximation".
* Maybe remind the user that it is an option in edge cases like "this entire page would be read as 'a series of 53,198 symbols you've suppressed reading'".
[1] One of my first full-time jobs, decades ago, was doing UI development with a focus on accessibility.
[2] Pretty unbelievable that the Unicode consortium has taken the position that an apparently unlimited number of stylistic variations on Latin letters should get their own Unicode renditions, but that basically every other culture's character sets gets merged into one style of character even if there are extensive historical reasons to have different renderings, and also has the time to keep adding corner-case emojis, but can't be bothered to provide a single way to easily differentiate "Cyrillic" from "Latin" as well as differentiate "mathematical notation" from "Latin".
The central theme of this paper is the variational analysis of
homeomorphisms $h \colon \mathbb X \xrightarrow []{{}_{\!\!\mathrm
{onto}\!\!}}\mathbb Y$ between two given domains $\mathbb X ,
\mathbb Y \subset \mathbb R^n$.
Elmo (and specifically the value-extraction-by-merchandise craze CTW created around him) pretty much single-handedly destroyed the original ensemble cast spirit of Sesame Street, so it's actually pretty apt.
In Latex, you get these symbols with \mathbb{}. They are most commonly used to represent to represent number sets, like the set of all integers Z (from the German), set of all natural numbers N, set of complex numbers C, set of all real numbers R, rationals Q (for “quotient”), set of quaternions H (named after Hamilton), or an unknown set F (for “field”). This explains why many of the letters in this series exist in the basic multilingual plane—because R is very commonly used, but A is not. You can find ℝ in the basic multilingual plane at U+211D.
I don’t know why people are interested in the X symbol. It’s just there to complete the alphabet. There are many other ranges like this used for writing mathematical formulas, like the range of bold letters, fraktur, script, etc.