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

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

2011-09-05T11:55:19Z

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".

Thanks in advance!

[edit]

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.

Answer by Carlo Beenakker for Notation in Frege's Grundgesetze der Arithmetik: The U with a flourish

2011-09-05T14:22:19Z

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

Answer by axisofeval for Notation in Frege's Grundgesetze der Arithmetik: The U with a flourish

2011-09-05T20:35:48Z

"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.

Answer by Marcus Rossberg for Notation in Frege's Grundgesetze der Arithmetik: The U with a flourish

2011-09-05T21:01:15Z

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.

Answer by Udo Wermuth for Notation in Frege's Grundgesetze der Arithmetik: The U with a flourish

2011-10-17T18:13:12Z

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 156 of http://www.mufi.info/specs/MUFI-Alphabetic-3-0.pdf.