Odd-looking letters in maths books

Alex asked about the Gothic-script (Fraktur) letters which appear in some maths books, such as:


These are letters from the Fraktur or Gothic typeface, which dates back to the Middle Ages but was used in Germany until the 1940s. One important German daily newspaper still has its masthead in Fraktur:


As I understand it, German mathematicians (unlike other writers in Germany) generally had their work printed in roman typefaces such as we are used to (because they wanted their work read internationally), but if they wanted a special extra symbol beyond the roman and Greek alphabets, the Fraktur typeface was familiar to them.

And Germany was the centre of world mathematics for a century before 1933. (That is why the name for the whole numbers is ℤ – the German word for number is Zahl). So, for example, Fraktur-R


became a common symbol for the set of real numbers.

Most leading mathematicians fled Germany in 1933 and after, and for many uses the Fraktur letters were replaced by “blackboard” or “double-struck” letters. So, for the set of real numbers:


Apparently – I didn’t know this until I looked it up, just now – the common use of double-struck letters in maths books (as distinct from on blackboards) dates only from an influential textbook published in 1965. Before then, if you didn’t use the Fraktur-R, you just used a bold R in print, and double-struck on the blackboard.

Fraktur letters are still occasionally used in maths where a writer is desperate for a special extra symbol. The Hebrew alphabet has also been called on, for the symbols ℵ0, ℵ1, ℵ2, etc., used to represent transfinite numbers.

This use of ℵ comes from a German mathematician, Georg Cantor. Some claim that he had Jewish ancestry, but this seems doubtful; he certainly wasn’t Hebrew-speaking or religiously Jewish.