Fdtxjr

How do you express the common phrases "by an abuse of notation", "abusing the notation", etc. in German without invoking negative connotations? (Without this addendum, I'd have asked at german.se.) "Durch/Unter Missbrauch der Schreibweise/Notation" looks horrible to me, in particular due to sexual allegations of the word "Missbrauch". How do they (e.g., folks such as Harro Heuser, or, from the further past, Edmund Landau, David Hilbert, Emmy Noether, von Neumann, etc.) typically write it in German books on mathematical subjects?

An example of the original sentence:

Abusing the notation, we write ℚ⊂ℝ, viewing the rational numbers as particular real numbers.

Related: "by an abuse of terminology" in German

The English "abuse" is adequately translated by the German "Missbrauch". In both languages it may have a sexual connotation, but if you read it in a mathematical text you would never interpret it like that. It often occurs as a compound with other nouns, for example "Alkoholmissbrauch" or "Machtmissbrauch".

An example of a mathematical text containing the phrase is https://www.math.uni-hamburg.de/home/riemenschneider/anvorl3.pdf p.568 (the spelling is old orthography). It seems to me that the following variants are not bad:

• Unter Missbrauch der Notation schreiben wir $mathbbQ subset mathbbR$




• Wir schreiben missbräuchlich $mathbbQ subset mathbbR$




• Wir verwenden die missbräuchliche Schreibweise $mathbbQ subset mathbbR$

Note that the following does not sound nice:

• Unter Missbrauch der Schreibweise schreiben wir $mathbbQ subset mathbbR$
  

However, perhaps you should also ask a question in https://german.stackexchange.com/

Edited:

The phrase does not occur that frequently in German. I cannot remember that I have ever seen it in older literature, but I admittedly I am not sure. This indicates that it might be an Anglicism. Another hint is this:

A Google search with "Missbrauch der Schreibweise" produces only a few results, but if you do it with "Missbrauch der Notation" you will get a lot more. And the latter is the most literal translation of "abuse of notation".

I would say "Durch Missbrauch der Notation" but "Notationsmissbrauch" is also quite common as far as I am aware so "Durch einen Notationsmissbrauch wird es als ℚ⊂ℝ geschrieben, damit die rationale Zahlen als bestimmte reelle Zahlen darzustellen." Or something like that.

The German language features the wonderful word zweckentfremden (literally: to purpose-estrange), which means to use something for another purpose than intended. For me as a native speaker, it has no connotation of sexual or drug abuse. Rather, in everyday language it is often used for lifehacks and similar.

Abusing the notation, we write ℚ⊂ℝ, viewing the rational numbers as particular real numbers.

could be translated to:

• 
  

  Unter Zweckentfremdung der Notation schreiben wir ℚ⊂ℝ, …

  
  




• 
  

  Die Notation zweckentfremdend schreiben wir ℚ⊂ℝ, …

  
  

It is naturally hard to find examples of this usage, but here is one from Jänich’s textbook Mathematik 2:

Das Symbol $langle , , rangle$ haben fir oft für die Bezeichnung eines Skalarprodukts benutzt, es wird hier also ein wenig zweckentfremdet, aber in einer praktischen nud berechtigten Weise.

You could also use "missbräuchliche Notation". This does not sound as clunky and does not have any connotations related to sexual/ drug-related abuse.