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In general, when a paper references an object discovered/defined in another paper by author X, it goes something along the lines of:

"Let $\tau$ be the constant defined by X in 1999 [1]$\ldots$",

or

"Let $f_{\mu}$ denote the function that generalizes the case $\ldots$ (X, [1])".

At what point does the literature start talking about "X's constant" or "The X function"?

Who/what determines that an object discovered by somebody deserves to take his name?

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    $\begingroup$ It starts this way when an author decides to do so... some authors make efforts to find an intuitive and convenient terminology when introducing something, other don't and sometimes paradoxically get rewarded by having their name attached $\endgroup$
    – YCor
    Commented Nov 9, 2017 at 13:19
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    $\begingroup$ A ludicrous example I have in mind is "Grothendieck group". Grothendieck introduced a number of incredibly subtle concepts, notably in algebraic geometry, generally with an adequate terminology which is generally duly used. But also at some point he used (without claiming credit!) a somewhat much older concept, basically the same as the one to pass from natural to relative numbers... this is a simple construction to pass from an abelian semigroup to an abelian group, which is now often referred as "Grothendieck group". $\endgroup$
    – YCor
    Commented Nov 9, 2017 at 13:22
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    $\begingroup$ @JochenWengenroth Isn't that a bit pretentious? I thought such naming was "awarded" by other authors in related papers. Or was it adequate to call it an $x$-pair in the given context? $\endgroup$
    – Klangen
    Commented Nov 9, 2017 at 13:29
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    $\begingroup$ @Pickle It looks pretentious, but it might not actually be. It's at least possible that the terminology "$x$-pair", though first published in the $xyz$ paper, had been used informally earlier by others (in discussions, in lectures, etc.) and had become reasonably well established in the relevant community. $\endgroup$
    – Andreas Blass
    Commented Nov 9, 2017 at 15:00
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    $\begingroup$ I would like to take this opportunity to remind people about Stigler's Law. $\endgroup$
    – J. M. isn't a mathematician
    Commented Nov 9, 2017 at 19:49

1 Answer 1

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This is determined by informal consensus of researchers in an area. Anyone can propose a name for a mathematical object, just by using this name in a paper. Then the proposed name either sticks to the object or not. This depends on the opinions of other people working in the area. Eventually the name of the object becomes established. Sometimes several names for the same object become established, and all of them popular.

For example Fatou set is the same as the set of normality. The name Fatou set was proposed in two influential surveys in 1980s, and many people in the area use it.

The name does not have to be the name of the discoverer of the object. For example, Drinfeld introduced an object which he named Yangian, in honor of Chen-Ning Yang. M. Lyubich and I gave the name "Baker's domain" to an object that we defined. Unlike some other "personal names" we proposed in the same paper, this one is used by everyone who writes on the subject. Arnold once stated a principle that "If the thing is named after someone, this indicates that the person had nothing to do with the thing". M. Berry remarked that "Arnold Principle applies to itself".

Sometimes a multitude of names reflects a priority dispute or nationalistic feelings (Young diagram vs Ferrers diagram, Schwarz inequality vs Cauchy or Bouniakowski etc.) The name Casorati–Sokhotski–Weierstrass theorem reflects some priority research, but the theorem in question was earlier stated by Briot and Bouquet. Same happened with "Gauss-Manin connection" which was in fact discovered by Legendre.

Sometimes the accepted name is changed ($\pi$ used to be "Archimedes number" and $e$ used to be "Euler's number"). A funny story happened with "Koebe constant". It turned out that it is equal to $1/4$, so one mathematician wrote "Now it cannot be called Koebe constant anymore, because it turns out that it already has a name, namely $1/4$").

Sometimes a name is based on a mistake. For example Abel's equation is the accepted name for $f(x+1)=g(f(x))$. It is based on a manuscript of Abel, but the manuscript itself turns out to be just Abel's personal notes on the work of Napier, where the equation was introduced.

People usually do not give their own names to mathematical objects. However, in his influential book on functional analysis, Banach introduces "spaces of type H", "spaces of type F" and "spaces of type "B". As his "spaces of type H and F" are nothing but Hilbert and Frechet spaces, people probably understood the hint:-)

G. Julia intensively lobbied (not publicly, in private correspondence, which is now published) that the "irregular set" be called Julia set. He succeeded.

Remark. I just found the paper, arXix:1204.4716v1 (Bernard Ycart, A case of mathematical eponimy: the Vandermonde determinant), which shows that there is a whole research area related to this question: it is called "mathematical eponimy". The paper cites several other publications on this subject.

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    $\begingroup$ $\pi$ is (or was) sometimes called the "Ludolphine number" ("Ludolphsche Zahl" in German), after Ludolph van Ceulen, who computed 35 digits of $\pi$. $\endgroup$
    – Goldstern
    Commented Nov 9, 2017 at 21:39
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    $\begingroup$ No I want to use the term Koebe constant for 1/4 for now on. $\endgroup$
    – Christopher King
    Commented Nov 9, 2017 at 23:07
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    $\begingroup$ Great answer Alexandre. Just want to point out that Young diagrams and Ferrers diagrams ARE slightly different in that they have different orientations. On a personal note, the one time I witnessed a mathematician referring to himself in the sense of this question, was in a talk by Anosov, where he mentioned Anosov maps. It was clear to everyone that he could have not done differently :) Oh! and do you have a reference for Julia's effort? $\endgroup$
    – Rodrigo A. Pérez
    Commented Nov 10, 2017 at 0:45
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    $\begingroup$ "e used to be "Euler's number"". Is it not still called Eulers number? $\endgroup$
    – Dale C
    Commented Nov 10, 2017 at 3:55
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    $\begingroup$ @Tilefish, calling it "Euler's number" does run the risk of confusion with the Euler numbers, or the Eulerian numbers, or the Euler-Macheroni constant... he was that prolific after all. $\endgroup$
    – J. M. isn't a mathematician
    Commented Nov 10, 2017 at 4:58

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