The letter ℘: name and origin? (2017) | Hacker News Reader

The letter ℘: name and origin? (2017)

mathoverflow.net

283 pointsIdealeZahlen142 comments

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.

I can relate. Ages ago, before Safe Search and search result tailored to one‘s history and preferences, I was trying to figure out how to write that big union symbol (∪) in LaTeX and googled for Big Cup LaTeX. I got _very_ different and unexpected results.

Googling guitar-related stuff is how I learned there’s such a thing as c-string women’s underwear & bathing suit bottoms, not just g-strings.

That was, briefly, a real WTF moment.

[edit] oh my god, of course that one didn’t come from searching guitar topics, that makes no sense given the standard tuning. I’m pretty sure I was googling strings in the C language when I hit that one, lol. I did probably accidentally land on “g string” after searching without thinking about what would obviously come up, when looking up guitar topics, and must have combined the two incidents in my memory.

This reminds me of the time when searching for “c string” would probably result in “The C Programming Language” at number 1.

That’s what I get right now

In the early days of the internet, searching for things like C++ was really challenging because none of the first generation search engines could search for that particular string.

Back in the mid-1990s I extracted a small part of the GNU C++ String library into a small package, which I called "GString".

I had no idea about the garment.

Haha - my favourite WTF Googling moment was when, as a callow youth first setting out in learning Javascript and HTML, I Googled "How to get head"

Eons ago, I was exploring ways to run some outdoor overhead wire between my house and the shed.

One method I considered involved using those little self-wedging widgets that squeeze down tighter as the thing being suspended is pulled harder. (These widgets were once commonly used with overhead POTS telephone lines.)

So I asked around and the broad consensus in my area was that one of these widgets is called a "horse cock."

And while everyone who knew what I was talking could say it with a very straight face, I did not even bother with trying to Google "horse cock" before deciding to go in a different direction with that project.

Is that the same as a kellums grip or hubbell device?

A Kellems grip is a Chinese finger trap-like thing made with steel wire. Those are useful, but are very different from a horse cock.

A horse cock can also be known as an overhead service entrance wedge clamp, which is a surprising mouthful of nomenclature given the parlancial context.

(I don't know what a Hubbell device is -- searching for that just brings up a million wiring devices (outlets, switches, and such) made by Hubbell.

But maybe that's the point?)

I once noticed a LaTeX installer installing a package called he/she (apparently some sort of pronoun swapper). "Latex he/she" is not great for work search history.

Back in the day, a gay colleague of mine forgot the dash in the website of the then trading venue Chi-X. He got a very threatening page in his browser. I assured him I'd back him up in the disciplinary.

What was particularly funny was his look of complete incomprehension at why he was getting this message.

Yeah I find that I reason about math in my head with names for symbols - the visual shape is not sufficient for the math part of my brain to manipulate it as a symbol.

Same. If I see a symbol and can't "say" the symbol, I can't (easily) process it. I think over time that gets better and eventually it's possible to start to "pattern match" it just off the visual representation, but at least for me when something is new, I need the "name" of it as something I can say to myself or I hit a ParseException. :-(

At least you can send claude or openai images now of the math symbol.

There has been various modernization efforts in the past correct? I wonder why nothing has succeeded. Math has always been somehow very tied to keeping credit and personal achievement close to the "source" it feels. Conjectures always coming with names attached, or this symbol named after a whimsical stoke of a pen.

As a first foot-hold I recommend highly https://detexify.kirelabs.org/classify.html

(I think I saw there was a newer one, but don't remember how)

You draw the symbol and get the TeX symbol name. I tried this one and it does give the right \wp (which in this case is confusing and you'd have to look up more about why it's named that)

But for classic ones, for instance the "upside down A" -> "forall" is very helpful and shakes newcomers to math syntax

Feynman said that his students struggled with a reverse problem: how to know that "harnew", an important part in QM equations that the lecturer talks about, actually stands for hν.

Always thought it was kind of cool how Feynman writes about, when learning calculus and maths as a younger student, would create and use his own symbols for things and how it worked well for him. But kind of realized if he was going to enter the scientific community would need to conform to the standardized notation/symbols for equations etc.

We solved that though, it's now pronounced “(h)aitch-bar-o-mega.”

Everything is easier with a piece of pi!

See also: <https://shapecatcher.com/>

This is great, thank you! Would be even better if it had a little "click here to hear it said out loud" button.

Detexify appears to be a kNN classifier. https://gist.github.com/kirel/149896/3a13825f826ec91e04d4adb...

Good stuff: worked the first time I tried to draw a ℘.

Very cool, but I tried a plus ("+"), and it didn't show up in the list, even when I clicked "show more" several times.

I have seen math textbooks that have a "table of symbols" or similar, by the table of contents or the index. It's really helpful. Also nice to have - a "map of the book" (I don't know a better name for this) which indicates graphically which sections of the book depend on which other sections. "Dependency graph", maybe?

I wish music books would do this too - I've been self-teaching myself a little classical guitar, and some of the scores I'm reading have various symbols that have taken me quite a while to figure out. I eventually determined that a bold III means to play in third position in some books, but these things aren't consistent between publishers.

Classical music is relatively standardized. Roman numerals to indicate position are standard practice for all string instruments (I remember learning this as a beginning bass player as a kid). You will sometimes see Roman numerals used as a means of identifying chords relative to the tonic of the current key, but that’s uncommon and the notation is a bit different than the position notation in how it’s placed on the staff (if it even appears on the staff and not in a harmonic analysis).¹ I’ve not seen the upper- and lower-case distinction in roman numeral notation I learned somewhere which uses cases to distinguish between major and minor in any classical music harmonization texts, but I may have just paid insufficient attention.

⸻

1. I have a vague notion that it might show up in figured bass once in a great while, but I could be wrong.

This all seems right to me. As for your figured bass comment, you could have something like iii^6_4 for an e minor chord with the B in the bass, when the key is C major. But if you were writing it next to the staff you wouldn’t need to write the iii - it’s implied by the bass note and the figures, and classically figured bass writes the minimum it has to, for example just 6 instead of 63 for a first-inversion triad.

I've played lots of piano and wind music and took lessons for that stuff in school, but I've been self-teaching guitar / ukulele / bass / mandolin in a smattering of different styles, which is probably part of my issue. I go through cycles where I'll focus on one instrument and style for a few weeks - in the last year I've dabbled in bluegrass banjolele, Irish fiddle tunes on mandolin, jazz on bass and keys, rock electric ukulele, funk piano, rock organ... And that's not even an exhaustive list.

But nowadays you can simply ask a vision model

yep, or even just a text model

ChatGPT 01-preview gave me:

User: what is the Math squiggle that looks like a cursive p?

Assistant: The mathematical symbol you’re referring to is likely the Weierstrass \wp function symbol, which resembles a cursive or script “p”:

Uh, symbols and math??? Uh, as I read mathematical physics, I get the impression that certain symbols in certain equations are accepted throughout physics as already defined. But in math, we have, e.g.,

"For the set of real numbers R, some positive integer n, and R^n, with both R and R^n with the usual topologies and sigma algebras, we have function f: R^n --> R, Lebesgue measurable and >= 0."

That is, in writing in math, a popular but implicit standard is, each symbol used, even if as common as R and n, is defined before being used.

Sooooo, right there in the math writing, the symbols are defined. Or, right, an author might assume that the reader knows what "the usual topology" or "Lebesgue measurable" are but does not assume the reader knows what the symbols mean. I.e., the issue is not something about helping readers with their knowledge of math but, instead, just being clear about the symbols. Or, can assume the reader DOES know about the real numbers but does NOT assume that R is the set of real numbers -- and correctly so because R might be the set of rational numbers, some group (from abstract algebra), nearly anything.

Again for physics, E = mc^2 and J = ns^2 are not the same, not within the standards, maybe even if say J is energy in Joules, n is mass, and s is the speed of light!!!

The standard is based on the expectation that you should be able take a definition or a theorem out of its context and it should still make sense. The same symbols are often used for different things in different contexts, while concepts (even advanced ones) tend to remain unambiguous.

R is unambiguous if it's in blackboard bold, but otherwise it can mean almost anything. n is likely interpreted as a non-negative integer if left undefined, but you usually need to establish explicitly if it can be 0.

The "unicode" program (in Ubuntu's package repository) gives the Unicode entry for any character:

  $ unicode ℘
  U+2118 SCRIPT CAPITAL P
  UTF-8: e2 84 98 UTF-16BE: 2118 Decimal: &#8472; Octal: \020430
  ℘
  Category: Sm (Symbol, Math); East Asian width: N (neutral)
  Unicode block: 2100..214F; Letterlike Symbols
  Bidi: ON (Other Neutrals)
  Age: Assigned as of Unicode 1.1.0 (June, 1993)

Or you can ask Wikipedia: https://en.wikipedia.org/wiki/%e2%84%98 (manually URL-escaped for HN)

That assumes you are in a position of being able to paste the character (and as unicode), which is not always the case.

If you already have a copyable version of the character it also works in the original Google search. Or any other place you can put it. The problem is when you don't have the literal character as text (say, an image, video, or non-digital source) and need to reproduce it to do that lookup in the first place.

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

Asking ChatGPT (or the like) is a good solution nowadays though: “The mathematical symbol that resembles a cursive "p" is likely the Weierstrass p, denoted as ℘. […]”

I asked once and it just gave me “p”. I said “no more fancy like” and it produced the correct result. This is the real power of the model for narrowing search, you can interrogate the results.

I heard as many names for the "\partial" symbol as I had math professors in university. At least they all wrote them the same.

"del" is the only way

Except that "del" is also (and I think more commonly) the name for the upside-down Delta used in vector calculus for div, grad and curl.

(I usually say "partial dx by dt" or whatever, which seems OK to me.)

I have this little book that I cannot find right now that is pretty explicitly that. just a little math symbol dictionary. its small enough to fit in a pocket and was invaluable all through college.

As somebody who spends a fair amount of time studying math heavy material that uses math that I never studied formally, this stuff is the bane of my existence. It's one thing to see a random Greek letter, where at least I very likely know what the character "is" (eg, "rho" or "psi" or whatever) and can at least pronounce it to myself and make a mental note "go back and see what rho stands for in this equation". But exactly like you say "squiggle that looks like a cursive P" doesn't easily admit a mental placeholder, AND it's hard to look up later to find out exactly what it is. I've really wanted to tear my last hair out over this a few times. And I am pretty sure one recent such occasion involved this exact character, so this really hits home!

And never mind that cognitive load that comes from managing the use of symbols that are the "same symbol" modulo something the typeface. Trying to read something like

"Little b equals Fraktur Bold Capital B divided by (q times Cursive Capital B) all over Gothic Italic B", blah, blah... then throw in the "weird little squiggle that looks kinda like a 'p' but not quite". It's insane.

The Unicode charts for the mathematical symbol ranges can serve as a visual index: [0]

They open as PDFs and have grids of all the symbols, along with useful metadata such as related and similar symbols, common substitutes, alternate names, etc. You won't find everything in those ranges - some things are elsewhere in Unicode, in their native language or already located elsewhere in an earlier version - but it's a great resource. Even for symbols that are merely letters in some language's alphabet, Unicode sometimes provides a unique codepoint (character) for their use in mathematics.

[0] https://www.unicode.org/charts/

That provides each table as a separate PDF downlod: mathematics covers ~20 PDFs, each named for its contents. It may be faster to download the entire Unicode standard as one PDF (~140 MB):

https://www.unicode.org/Public/16.0.0/charts/

Once there was a lecture at Yale, and Serge Lang, a frequent loud critic of bad notation was in the audience. There was a function Xi, and soon it was joined by its complex conjugation. Then they were divided. Serge Lang walked out.

That's delightful, but I fear many in the audience here may not quite see why. So, at the risk of explaining too much:

The Greek letter xi is one of those where the capital and lowercase versions are very different. Lowercase is a bit like a curly E. Uppercase is (at least if you're writing it in a hurry) basically three horizontal lines on top of each other.

The operation called complex conjugation can be notated in two ways, but the more common one among mathematicians is to put a horizontal bar above the thing being conjugated.

So the conjugate of Xi is ... four parallel horizontal lines.

And now we divide Xi (three horizontal lines) by Xi-bar (four horizontal lines), getting: eight horizontal lines.

You were explaining this for someone like me. Thanks for that!

In fact, here is the dreaded letter: Ξ And its lowercase version: ξ

The letter in actual use seems more likely to be printed as a two-tiered Z.

    -----
       /
      /
     /
     ---
       /
      /
     /
    -----

For Ξ or ξ?

Back in my grad school days ξ was almost always handwritten as just some sort of squiggle you'd struggle to associate with the character "ξ" unless you already knew what it was supposed to be. My classmates were very surprised that I could (can!) somehow write it as something that actually is recognizable as ξ, quickly and consistently.

(But they missed what my hand muscles do to ζ, so it all balances out.)

The form I gave you is the capital. It's noted in the wikipedia article, but not discussed: https://en.wikipedia.org/wiki/Xi_(letter)

I spent a while browsing a Flickr group of "Greek signs", which unfortunately tended toward formality. I noticed zero examples of the joined form. There were a handful of standard Ξ, and two oddities - three examples of the letter being printed 王, and one example in which Ξ was used as the lowercase form. ( https://www.flickr.com/photos/robwallace/11888342614/in/pool... ) I guess it's easier to write than ξ is?

The letter doesn't appear to be all that common in general, unless you live in an Alexandropolis.

---

Having done some more looking, I still haven't found a joined Ξ, but I have found some examples of the standard form in graffiti and handwriting (e.g. https://www.flickr.com/photos/telemax/4257624412/in/pool-gre... ). The 王 form also appears in graffiti. It's probably displacing the double-Z form.

Thank you. That actually was helpful. In print it might have been legible, but on a blackboard that would been difficult to read.

A lovely story, but sadly this recollection from Paul Vojta disagrees on his reaction: https://www.ams.org/notices/200605/fea-lang.pdf (see p547)

It was a joke. The lecturer (Barry Mazur, at Harvard) had made a T-shirt with Lang's catchphrase "This notation sucks" and was trying to get Lang to say it with the most over-the-top example of bad notation he could come up with so he could bring the shirt out, but Lang didn't say anything so the whole thing was a bust.

I also find it frustrating, but I’ve come to appreciate that it’s a way to at least partially sidestep the hard problem of naming things. There are still idioms and choices to make, but using abstract symbols makes it easier to play with the abstract concepts being presented.

My most-used programming language is Go, but I’ve been writing mainly Swift for the past year or so. While there’s a lot I like about Swift, its verbosity leads me to waste an inordinate amount of time pondering what the correct verbiage ought to be, and I often miss Go’s more terse, often single-character naming convention.

> My most-used programming language is Go, but I’ve been writing mainly Swift for the past year or so. While there’s a lot I like about Swift, its verbosity leads me to waste an inordinate amount of time pondering what the correct verbiage ought to be, and I often miss Go’s more terse, often single-character naming convention.

Huh. I was expecting that comparison to go the other way given Go's notorious verbosity in terms of error handling, generics etc.. Maybe people compensate for verbosity in one area by being more concise in others (though that doesn't explain e.g. APL).

I would say that Go is extremely explicit, but I wouldn’t say it’s verbose.

Or, I suppose you could say that Go is semantically verbose (explicit error handling, no/low use of generics, no operator overloading), but syntactically concise (short variable names). Swift is the opposite, being semantically concise (extremely heavy use of generics, default arguments) but syntactically verbose (labeled arguments, English-like clauses, result builders).

I distinctly remember the first time a lecturer used the "dx/dt" "symbol" in normal algebraic operations (that is, multiply both equation sides by dt and so on). I was so shocked it's actually not a elaborate differentation symbol, but something with actual division. Next time it was similar with integration, where the dx was substituted by some other function of du.

I swear I treated those as some grammar token, which doesn't hold any real meaning. I've been using those as such for years before.

Technically, dx/dt is not a fraction, but, but, ...

https://mathoverflow.net/questions/73492/how-misleading-is-i...

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.

⸻

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.

> in about another century.

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

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.

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/

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

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.

> I'll also never get used to mathematicians' habit to assign semantic meaning to the font that a letter is drawn in

You never learned to use capital and lowercase letters differently? Why did you capitalize the 'i' in "I'll"?

Case and font are different

How?

Well because I can have upper and lowercase letters in the same font. Aa. Upper and lower cases have different significance, ie. starting letters for honourifics and proper nouns. Unless you want to take a very unconventional view that uppercase is simply a different font from lowercase, which is not how anyone else in the world uses the word.

> Upper and lower cases have different significance

Yes, that's why they're a good example of assigning semantic significance to the font that something is written in.

> Well because I can have upper and lowercase letters in the same font.

What do you think that means?

Try this one: I can have Arial and Calibri letters in the same font.

Where a 'font' is a data file used by word processors to render ASCII or Unicode, it's just as true.

If you think a 'font' means something other than that, what is it, and how does your definition preserve the idea that a capital A and a lowercase a are distinct in some dimension that isn't their 'font'?

This is what you get if you insist on using single letters for every variable. Why do that? Well, because otherwise a variable name might be confused with a bunch of variables multiplied together, because we don't use multiplication signs. Why not? Well you see, the signs might be confused with the variable x.

We had a guy in class at high school who was a math prodigy,but hated the greek alphabet. He always said things like "If we multiply the symbol that I don't know with the square root of the other symbol that I don't know..." He was proof for me that you can be really good at (high school) math without knowing all the symbols' names.

Well rarely do you see a symbol with zero context.

You're reading something. "Oh this has something to do with Weierstrass...."

If you can get it on your clipboard, you can usually paste it into Google or append to `https://en.wikipedia.org/wiki/` and get the answer.

This is something I dislike intensely about pro mathematicians. If you need that many symbols, document them properly and work to ensure they're on the flyleaf of every mathematics textbook. You can easily picture a math paper that reads like

  [    ], but [    ]; however [   ] and [   ], so [   ]....

with the spaces being filled in by signatures of famous mathematicians.

Claude worked it out from that description.

I keep on saying that math is not hard.

It's just so disorganised that it's hard for people who are not passionate about it to study.

It’s what drives me nuts about people making custom operators on Haskell.

A word is easy to search, but something like “~~>>=>” doesn’t really give anything and it’s not nearly as cute as the writers of the libraries seem to think it is.

I know about Hoogle but that’s not a solution, as that only searches documentation, not stuff like Stackoverflow.

I randomly chose *> to search for on Google. It does pretty well, and yes, the top result is StackOverflow.

https://www.google.com/search?q=*%3E&iflsig=AL9hbdgAAAAAZzeB...*

Huh, fair enough, I feel like that must be recent because I was having trouble with that before, but maybe it was never as bad as I thought it was.

I'll admit I was wrong!

I was asking ChatGPT about a "squiggled s" yesterday. It thought I meant ß, but the character I was actually interested in was §. Context was obscure keyboard layouts.

Local LLMs have helped me a lot in this regard. Even something as simple as "how do I say this formula out loud in words?" helps tremendously.

I was going to say "Use Google Lens" but then I tried it on the character above and it utterly failed :D

In any case, there are lots of camera based apps for either OCR (So you could maybe search for the character) or image search or LLM search (I didn't try ChatGPT).

I did try zooming 300% and grabbing that character as well as the whole title as an image at 100% zoom with Google Lens. It still failed in both cases.

I must confess that I have an irrational fondness for the use of weird symbols in math and technical documents, whether it's for a homework assignment in school or a white-paper for work.

My unit tests are literally full of hieroglyphics. My favorite design doc to this day is one where I sprinkled Sumerian cuneiform throughout the text, e.g. 𒀭𒄑𒉋𒂵𒎌 and 𒂗𒆠𒄭 (Gilgamesh and Enkidu) instead of Alice and Bob.

I'm not sure if this is a good idea, especially in code; apart from adding unnecessary confusion for the reader, this will also confuse some monospaced fonts.

Glad I’m not GPs coworker. Time to refactor Gilgamesh.

I just see boxes, not the cuneiform.

`pacman -S noto-fonts noto-fonts-cjk noto-fonts-emoji`

I've seen emoji used for Alice/Bob/Carol/etc which is a bit more widely supported than old dead scripts.

I await the inevitable mathematical constant unicorn emoji.

I was following you up until that first comma.

Reminds me of Lamport's original paper on Paxos, in which he eschewed Alice, Bob, et al. in favor of ∆˘ικστρα, Γωυδα, Πνυeλ˘ι, and a whole host more.

I left college with a math degree and a profound antipathy for weird cursive symbols. The one that nearly killed me was the Greek "xi". I couldn't pronounce it, and I couldn't write it with any fluency, and in some of the classes I took it was everywhere.

Despite only having a CS degree, I was always especially fond of ξ due to its distinctiveness (and also didn’t have trouble writing or pronouncing it), moreso than letters like ν or ι, which are too close to v or i/j visually for my taste.

I think iota is fine because it's missing the dot that an i has, but nu is terrible, yeah. In fact, in some fonts nu is exactly v: https://en.wikipedia.org/wiki/Nu_(letter)

But then a lot of capital greek letters at least are identical to latin letters in those fonts, so I guess you have to choose carefully anyway... and pick the proper font/handwriting if you absolutely have to use nu. (Hopefully you don't.)

I encounter ξ (xi), and also ζ (zeta) a lot. Honestly, when I write them out by hand, I just make a "wild squiggly line" for ξ and a "simplified squiggly line" for ζ.

If I write it out by hand, it's most likely just for my eyes anyway, and I'd type it out on a computer if I'd want others to have a look at it. But even if I gave someone else my handwritten note, I think from context it would be pretty clear what the "squiggly lines" are supposed to be.

ζ is essentially a cursive z. ξ is near enough to a backwards 3.

ξ is literally three horizontal bars underneath each other, in cursive.

Indeed. Try to write Ξ sloppily using connected strokes and you'll end up with something vaguely like ξ.

> ξ is literally three horizontal bars underneath each other, in cursive.

And? So is 3.

The unstated point is to explain the connection between lowercase ξ and uppercase Ξ.

When I was an undergraduate typing it out on a computer wasn’t an option, not with the hardware available at my school. It was handwritten, or nothing.

You guys do know that Greek speakers write those by hand all the time, right? It’s really not that difficult…

"All the time", yeah. Relatively to that, I have to write individual greek characters very rarely, as they are not part of my usual script. Almost never, compared to a greek person.

The same applies to the scripts used by e.g. Chinese, Arabian, and Korean speakers.

So naturally I am not a "fluent" writer of ξ and ζ, and since I virtually always write those characters in isolation instead of as part of words, it's a different mode of practice even when I do use them.

It's the same on the keyboard, by the way. My keyboard does not have ξ or ζ keys, it's all special.

I actually find xi easy to write, whereas zeta is really hard for me. I think the middle loop of the xi provides an anchor for what I'm aiming for, but zeta ends up as a nondescript squiggle. Sometimes I can't even properly picture what zeta looks like in my head. Is it like a 2, a 5, an S, or a Z? Or a cursive C or an italic G? It's all undifferentiated in my head.

I do still remember the day our math professor taught us both symbols. He did it very purposefully, like he knew it was all riding on him, and we'd all be lost if he didn't pass the arcane knowledge down.

There is a famous anecdote [0] about Barry Mazur coming up with the worst notation possible at a seminar talk in order to annoy Serge Lang. Mazur defined Ξ to be a complex number and considered the quotient of the conjugate of Ξ and Ξ. (Click link to view 8 lines on top of each other)

[0] https://mathoverflow.net/questions/18593/what-are-the-worst-...

Paul Vojta p546 http://www.ams.org/notices/200605/fea-lang.pdf

    >>> import unicodedata
    >>> unicodedata.name('℘')
    'SCRIPT CAPITAL P'
    >>> ord('℘')
    8472

https://en.wikipedia.org/wiki/Letterlike_Symbols

Good enough for me.

Notably, this is distinct from ("MATHEMATICAL SCRIPT CAPITAL P").

> Books were printed in Fraktur, where the p looks quite normal, i.e., quite different from a handwritten Sütterlin p which could explain, why it hasn't been replaced in the publication of Amandus Schwarz.

Indeed. ("MATHEMATICAL FRAKTUR CAPITAL P") is also separate (but also, Unicode considers these mathematical symbols to exist separately from "text written in Fraktur script". So you get separate characters allocated for these symbols, but they're not intended to be suitable for printing in Fraktur - which is supposedly a presentation (i.e. typeface selection) issue.

Personally I'm not convinced that mathematical symbols derived from Latin or Greek (or other) scripts really have any claim to being separate "characters". Surely that's what variation selectors are for?

I think the answer by teika kazura on the linked page explains pretty thoroughly why this is not “good enough”. Most importantly:

> In Unicode the letter ℘ is given the codepoint U+2118 in the block "letterlike symbols", named "script capital p". But in fact it's lowercase.

Unicode technical note https://www.unicode.org/notes/tn27/ clarifies:

> Should have been called calligraphic small p or Weierstrass elliptic function symbol, which is what it is used for. It is not a capital "P" at all. A formal name alias correcting this to WEIERSTRASS ELLIPTIC FUNCTION has been defined.

I think one’s getting a lot of upvotes from people who meant to click on the link.

Yes, this happened to me

You can always click 'unvote' in the detail line.

Strangely, the most comfortable I've felt with symbols was when learning quantum computing. At the time, there was no established standard (perhaps it has a standard now), but the symbols were used more intuitively than any other math class I've taken.

In the same Unicode block is "2129 ℩ TURNED GREEK SMALL LETTER IOTA" with explanation "unique element fulfilling a description (logic)".

That seems a ridiculous choice for a symbol — turning one of the most symmetrical letters upside down!

Background: https://philosophy.stackexchange.com/questions/51563/what-is...

It could be motivated by the fact that Russell and Whitehead needed a symbol for printing that the printer had in his type case but could not be confused with anything else. Taking a iota and simply turning it upside down would then be a rather ingenious idea. But that is just my speculation ...

https://math.stackexchange.com/questions/1885938/whats-meani... says 'in the article Frege, Peano and Russell on Descriptions: a Comparison, Francisco A. Rodríguez-Consuegra tracks down the source to Peano's Studii di logica matematica (1897) as where the operator first appears'

That page in 'Studii di logica matematica' appears to be https://archive.org/details/peano-studii-di-logica-matematic... .

It also uses an upside-down C and upside-down E.

And the Lambda looks like an upside-down V. The bases of all these upside-down letters do not match the baseline of the text. Obviously there were no special upside-down moveable types available of freshly cast for this book. Peano had to creatively repurpose what types were available.

There are loads of examples of this, e.g. ∀ and ∃, but iota is a really poor choice.

In the font on my phone it looks confusingly reminiscent of Hebrew vav.

My first thought on seeing this title was, "this should totally be the name of a programming language descended from Go"

There is an old convention in physics - from the time when Germany was world-leading in physics - to write vector-valued variables in Fraktur. Using cursive (old German cursive is weird) seems related, though AFAIU the "vectorness" of the ℘ function is just the two components of a complex number.

And algebraists often use lowercase fractur for ideals.

To me \wp looks like a plain cursive p. Had I never seen it referred to as a special character I would have thought it was a lower-case p. There are many "styles" of cursive writing. But it's nice to have specific styles of these letters for use in mathematics.

One thing I like about programming languages is that they usually constrain themselves to strings of ASCII characters, instead of using lots of more or less inscrutable symbols like mathematics does. For example, where a mathematician writes "Σ", a programmer simply writes "sum".

You are holding up code as an example of clarity and scrutability, and because it is mostly restricted to ASCII? Hex code is even simpler - only 16 characters.

> where a mathematician writes "Σ", a programmer simply writes "sum".

Communities develop shorthand and terms of art for things they write a lot. Mathematicians need to write lots of sums; programmers have their own shorthand and terminology.

Hex code doesn't allow you to write words. And "sum" is simply better than "Σ". There is no way to know in advance what the latter means, while for the former understanding of verbal English is enough. Mathematicians basically use an iconographic writing system like Chinese.

I wish they did, because then it would be more consistent and properly documented.

We can think of many other strings used by programmers that are not common English, and many strings used by mathematicians that are.

I think the difference is that you are a programmer and not a mathematician (I'm guessing) and are saying, effectively, that what you are subjectively familiar with is objectively more universally understood.

This argument works both ways, apart from it's the mathematicians who are wrong.

Yes it does work both ways. Any mathematician or programmer who uses it is, afaict, just imagining their subjective perspective is some objective universal truth.

> We can think of many other strings used by programmers that are not common English, and many strings used by mathematicians that are.

Are you saying special symbols aren't more common in mathematics than in programming? I simply disagree. Mathematicians hardly use strings at all, e.g. for function names or variables, while they are very common in programming. Mathematicians mostly use single letters in Roman or Greek alphabet, and sometimes with various strange styles like fraktur, double strokes etc.

> Are you saying special symbols aren't more common in mathematics than in programming?

No, I agree that programming uses more ASCII. I'm saying that using a smaller alphabet (e.g., hex), doesn't make it easier to understand. Programming is just as arcane and difficult to understand - even programmers have trouble understanding each other's code, and generally it's believed that the understanding requires documentation in English.

Programmers also write something like *[3]int instead of "pointer to array of 3 integers" in most PLs.

(Modula-2 tried the latter, but it didn't stick.)

Yeah, though such special symbols are less common. I think they are still too common though. E.g. using the unnecessarily obscure "if (A) B" instead of "if A then B".

So this letter ℘ is distinct from another unicode symbol (that I can't copy-paste here?), which we often use for "power set" in math; it's given by U+1D4AB.

"The letter formerly known as p"

If you click the link to the wikipedia page on Sütterlin[0] that is mentioned in one of the answers, there's a link to another wiki page about the Antiqua-Fraktur dispute[1]. Apparently 19th and early 20th century Germany had a whole nationalistic debate about which handwriting script should be used, with the nazis ending it by preferring (somewhat surprisingly, to me) the international choice of the Latin alphabet.

Combine that with Göttingen being the capital of the maths world at the time [2], and I wouldn't be surprised if that dispute had some (now mostly forgotten) influences on funny maths squiggles in general.

Tangentially, the original question feels somewhat asked in bad faith imo, calling many names "bad" with unearned authority, and implicitly seeking popular votes to support their position. Also sentences like:

> BTW Abramowitz & Stegun uses P. Wow. See p 629.

It's great if you're passionate about maths, but clutching pearls over the use of "P" instead of "℘" is a bit much (reminds me of the "π vs τ" debate and how upset that seems to make some mathematicians. Meanwhile Euler, who came up with using "π" as a circle constant, wasn't consistent about what value he gave it at all[3] - he'd just pick whatever circumference-to-radius ratio worked best for his proof at hand).

It's pretty clear that "℘" essentially originated as a Fraktur-based glyph that most Germans of the time would intuitively read as the equivalent of "P" in Antiqua. The letter "P" is pronounced "Pe" in German. No mathematician would have been confused by Abramowitz & Stegun's notation, just like writing "R" instead of "ℝ" won't confuse anyone either.

Also Milton Abramowitz was a Jewish man. He might have felt a certain way about using letters associated with German nationalism, but that's just me speculating.

[0] https://en.wikipedia.org/wiki/S%C3%BCtterlin

[1] https://en.wikipedia.org/wiki/Antiqua%E2%80%93Fraktur_disput...

[2] https://theconversation.com/how-one-german-city-developed-an...

[3] https://www.youtube.com/watch?v=bcPTiiiYDs8

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