I have sometimes heard speakers at seminars refer to named theorems or definitions by prefacing them with the phrase "so-called". For example,

  • "the so-called Theorem of Highest Weight. . ."
  • "the so-called Hilbert Theorem 90. . ."
  • "the so-called Galois group of an extension. . ."

Some of my teachers at my institute also use this phrase in the above manner quite regularly. This is not done repeatedly with the same object, of course. I think that the speaker uses the phrase "so-called" to emphasize that a theorem is being named, or that a definition is being made. I have certainly never seen the phrase being used in textbooks.

My question is whether this is an acceptable convention in the mathematical community. I am not a native speaker of the English language, but as far as my understanding goes the words "so-called" are typically used negatively, to put down the object being referred to. This is the colloquial usage of "so-called" that I am aware of. It is quite possible that conventions differ in technical fields, and even among different technical fields.

So, does the phrase "so-called" have any negative connotations when used in mathematics as in the examples above? And if it does not, is the usage common enough that the chances of being misunderstood when using it are small?

I checked the English Language and Usage SE for a discussion on the use of this phrase and came across this question. It seems the consensus there is that it is context dependent, and it is usually used neutrally in technical writing. But I still felt it would be better to ask here to get a more mathematics-specific viewpoint.

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    $\begingroup$ I personally dislike the phrase. To me it sounds a bit like you're showing off. I would not say rude though. Not sure you'll get a definite answer on this. $\endgroup$ – Karl Aug 25 '18 at 6:36
  • $\begingroup$ I think that "so called" does not always have a negative connotation. To my ear it sounds fine. I agree with the conclusion of the English language stack exchange. $\endgroup$ – littleO Aug 25 '18 at 6:42
  • $\begingroup$ @littleO just to be clear, could you specify which is the conclusion that you agree with? That "so-called" is used neutrally, or that it is context-dependent? $\endgroup$ – Brahadeesh Aug 25 '18 at 6:45
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    $\begingroup$ I agree that it's context dependent, and also that "so called" is usually used neutrally in technical writing. Usually in math at least it's just a way to indicate that we are now providing the name of something. $\endgroup$ – littleO Aug 25 '18 at 7:01
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    $\begingroup$ In Dutch we have two: "zogenaamd" (often used in negative sense) and "zogeheten" (always neutral information). $\endgroup$ – drhab Aug 25 '18 at 7:26

It is common use, even in written mathematics, even in text books. Examples from this site include

So-called Artin-Schreier Extension

Is “:” so-called Frobenius inner product?

Inversion of so-called probability-generating functional

Show that f solves the so called wave equation

EDIT. See "this is the so-called representation theory" in Rotman J.J. Introduction to the theory of groups.

  • $\begingroup$ Yes, I had checked for examples on the site and found the above and plenty more. This does not surprise me because the writing on this site is very conversational. But, do you have examples from textbooks, too? $\endgroup$ – Brahadeesh Aug 25 '18 at 6:28
  • $\begingroup$ I just added such an example. $\endgroup$ – J.-E. Pin Aug 25 '18 at 6:33
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    $\begingroup$ It seems that the equivalent German term "so genannten" is used more often. Specifically by Felix Klein. $\endgroup$ – Somos Aug 25 '18 at 11:32
  • $\begingroup$ @Somos does the German equivalent generally carry any negative connotation? $\endgroup$ – Brahadeesh Aug 27 '18 at 4:36

I'm a native English speaker and mathematician.

You're right that it does have a negative connotation. The phrase "so called" suggests that the name you are referring to is misleading. For example I might say "the so-called irrational numbers" if I wanted to remind the audience that there's no connection between the numbers and the common English meaning of "irrational" (irrational numbers aren't illogical). I don't think the phrase "so-called" can be used without this negative connotation, even if some people try. Despite this, the audience will almost certainly work out the intended meaning after a moment's thought.

The phrase I'd use instead is "known as". For example "the result known as The Theorem of Highest Weight". It conveys that the name you've given is the one that people commonly use, which I think is what you want to say.

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    $\begingroup$ Alternative point of view: talk about "so-called" irrational beliefs; after all, there is nothing about illogicality that has anything to do with not being expressible as a ratio. $\endgroup$ – John Gowers Aug 26 '18 at 22:03
  • $\begingroup$ Thank you for the answer! The alternative that you mentioned is quite useful; I think one can replace most (if not all) instances of the phrase "so-called" with "known as" or something similar that is unambiguously neutral. $\endgroup$ – Brahadeesh Aug 27 '18 at 4:35

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