# Notation in Frege's Grundgesetze der Arithmetik: The U with a flourish

In the Grundgesetze der Arithmetik, Frege used a number of strange characters for notation. I would be most interested to know anything about the typography (origin, usage and so on) of the strange U with a flourish which occurs in the following.

I am no logician, but I am given to understand that the symbol (U in the following) is used as "a function-name ‘Ux’ in such a way that if y is the extension of a relation, then Uy is the extension of its inverse".

In response to some of the comments as to the relevance of this question to mathematics, I add my motivation for it. I have heard it said (by a rather famous Frege scholar) that Frege chose his notation by taking whatever was available in the [type] box. I have come to the view that this is not the case, and that Frege often chose his notation rather carefully. This rather obscure issue leads me to seek the typographic origins of these symbols. I know the origins of most of those in the Grundgesetze (which are surprisingly diverse: phonetics, commerce, German, Greek, ...) but a few remain unidentified, hence the question.

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Well, if it means "Umkehrung" then a fancy U is a natural choice. – Gerald Edgar Sep 5 '11 at 12:14
I'm not sure, but I see this question as being not just about the sign, but also about what is being signified. If it were a fancy $U$ and Frege meant it to signify 'Umkehrung', that would be one thing. If it were a rotated $A$, and carried other resonances for Frege, that would be another. – Todd Trimble Sep 5 '11 at 15:10
I love typography, too. It's too bad this question is just closed and remain unanswered. @JJGreen, could you re-post your question at graphicdesign.stackexchange.com ? They also talk about typography there. – Yuji Tachikawa Sep 5 '11 at 17:23
Regarding the edit: this seems an interesting undertaking to me; it is not that I dislike the question. Indeed to some extent I am interested in fonts and typography, too. Yet, in particular in view of your motivation, it just seems better/more efficient to me to ask this elsewhere (for example the site mentioned by Yuji Tachikawa). As your question really seems to be 'what is this typographic symbol?' So, I'd ask experts on typographic symbols, as opposed to mathematicians/logicians that are experts on the meaning of this symbol as used Frege. (Hope this makes sense.) – quid Sep 5 '11 at 18:11
SNd, IMO it's not CW. It has (theoret.) a correct answer; in part., it neither tries to create a resource as the totality of small contribs nor does it ask for something subjective (where there thus can be no 'correct' answer) nor is it at all a poll-like question (which typically would also make it subj). AFAIK earlier CW was handled (informally?) slightly differently; yet quite some time it was clarified that 'to avoid' that somebody gets points from not directly math quest. is not the point of CW. And, even if IMO not really on-topic, it is good and 100 is well-derved. +1 to stress this. – quid Sep 6 '11 at 19:47

The symbol stands for the currency Mark.'' It is an old symbol developed in handwritten manuscripts. As far as I know it is a lowercase m with an abbreviation symbol to indicate that letters are dropped. The lowercase m has changed to a simple horizontal bar.

See the OLD FLOURISH MARK SIGN on page 146 of http://folk.uib.no/hnooh/mufi/specs/MUFI-Alphabetic-2-0.pdf

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Thank you so much Udo, you are a star! – J.J. Green Oct 17 '11 at 20:42
The link appears to be expired... – Joel David Hamkins Jul 24 at 2:55
I fixed the link (and added the image, to protect this answer from future broken links) – Carlo Beenakker Jul 24 at 6:32

Frege used an unusual German Fraktur font for the fancy U. This has created many problems for modern typesetters, as one can read in a 1982 edition: "After unrecallable arrangements had been made for composing the book, it proved that Gothic letters (Frege's deutsche Buchstaben) were not available."

The sharp angles and ligatures in the fancy U are characteristic for a Fraktur font, but there are many variations. I have searched the web for precisely this U, and have not found it.

These typographic issues are of course quite unrelated to mathematics, but not entirely; see "Maths = typography?"

http://www.tug.org/TUGboat/tb24-2/tb77lawrence.pdf

Unrelated to the original question, but noteworthy in this context, is the question how to typeset Frege's symbols in a modern document. Fortunately, this is possible with Metafont and LaTeX (the fancy U is \fgeU), see

http://soliton.vm.bytemark.co.uk/pub/jjg/en/code/fge.html

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Thanks for the interesting link, and more generally the informative answer. – quid Sep 5 '11 at 19:05

Frege uses Fraktur ("Gothic letters") for quantified variables, lower case for first-order variables, upper case for second-order variables. The "fancy U" does not belong to these.

The "fancy U" is similar in style to ligatures of standard abbreviations, like old signs for weights and other measures. The sign for "Pre" is pretty close (214C):

http://www.unicode.org/charts/PDF/U2100.pdf

If the crucial 'P'-bit wasn't missing, this could almost be it.

The custom characters Frege uses look much more ham-fisted than his elegant character, compare his character for "Endlos", his sign for the smallest infinite cardinality (right side of the equation):

http://homepages.uconn.edu/~mar08022/pics/1-122-150.pdf

(The character on the left that looks like a script 'N' or 'A' is an overturned 'lb'-ligature, for "libra": pound.)

This speaks for the "Fancy U"'s being a symbol that was present in the type-setters box.

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Compared to Freges symbol, 214C from the unicode table is at least mirror-inverted. – Ralph Sep 5 '11 at 21:41
Ralph -- this is true, but only rotations are possible for metal type. – J.J. Green Sep 5 '11 at 21:59
some more searching on the web turned up this extensive discussion on the origin of the Frege's fancy U: typophile.com/node/31818 – Carlo Beenakker Sep 6 '11 at 8:54
@Calro Beenakker, I did not read the full dicsussion in detail, but the the fancy U linked there at the top is still another symbol, not the one discussed here. (The 'identity picture' of the questioner there is precisely what is talked about here, but what he is asking about seems to be something else.) Indeed chances are the questioner there and here are the same person (compare the names) – quid Sep 6 '11 at 10:07
Carlo, as mentioned by quid, the person asking the question there was me; perhaps the page is confusing as I am using the "fancy U" as a avatar there, but the question is about a different character which also looks like a U but is in fact the archaic German lb (pound) sign mentioned by Marcus Rossberg above. I also asked about the subject of this question on typohile at typophile.com/node/32040 but did not get any leads. – J.J. Green Sep 6 '11 at 10:20

"Umkehren" means to reverse - in the sense of a car turning around, for example. My guess - knowing nothing about it - would be that it is a custom U character, with the two additional lines signifying turning around.

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