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I'm wondering if there are any mathematical objects that were given a name when first discovered (and wildly used at their time), but then got renamed to match their characteristics later?

Counter examples:

  • Imaginary number was coined by Descartes in the 17th century describing roots of polynomial involving $\sqrt{-1}$.
    • I've seen some rename suggestions: lateral, orthogonal, perpendicular, vertical number. None of them replaced the original word, yet.
  • Irrational number was coined by the school of Euclid around two thousand years ago to demonstrate the number that cannot be written as a ratio.
    • Luckily(?), the word ratio ($a/b$) comes after rational (make sense). So irrational can have both meanings and, thus, does not need to be renamed(?).

Related examples:

  • In terms of notation change, originally the equals sign was two vertical bars, then became two horizontal bars.
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    $\begingroup$ I don't think the ancient Greeks would have talked about irrational numbers; such things weren't numbers for them. They'd have talked about two quantities being incommensurable. $\endgroup$ Mar 16 at 17:17
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    $\begingroup$ Essentially, B* algebras have been renamed to C* algebras (both terms used to exist, but later it was found that they are essentially the same thing). For details: en.wikipedia.org/wiki/C*-algebra#Some_history:_B*-algebras_and_C*-algebras $\endgroup$
    – jacques
    Mar 17 at 10:09
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    $\begingroup$ Are there any examples of renamings motivated by extra-mathematical concern? For example, has anything gotten renamed because it was originated by a Nazi? $\endgroup$ Mar 18 at 14:53
  • $\begingroup$ @MichaelLugo that's an interesting point! Potentially related: Has there ever been a case where someone wished a theorem or important result wasn't named after them? Has it happened more than once? $\endgroup$
    – uhoh
    Mar 18 at 21:16

9 Answers 9

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Another example is the notion of orbifolds. Originally, it was introduced by Satake in 1950s under the (nondescriptive) name V-manifolds. Then, beginning in 1980s, under the influence of William Thurston, it was replaced by the name orbifolds to indicate that they are related to manifolds via the (local) "orbit space" construction. Thurston in his Princeton Lecture Notes describes the process of introduction of the new terminology thusly:

This terminology should not be blamed on me. It was obtained by a democratic process in my course of 1976–77. An orbifold is something with many folds; unfortunately, the word "manifold" already has a different definition. I tried "foldamani", which was quickly displaced by the suggestion of "manifolded". After two months of patiently saying "no, not a manifold, a manifoldead," we held a vote, and "orbifold" won.

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Newton referred to his concept of a derivative as a "fluxion". He called time-varying functions "fluents".

Generally speaking, it is common for important mathematical concepts and entities to be given one name by their discoverer, but later renamed after that person. I think that is true for virtually all entities that are known as "Somebody's number/theorem/group/etc".

Just off the top of my head: I don't know what term Diophantus used, but he likely didn't call them "Diophantine equations".

Or similarly, Lie called Lie algebras "infinitesimal groups".

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    $\begingroup$ And as far as I remember the name "derivative" only took off with Lagrange, while before they were called "differential coefficients" by people like Leibniz, the Bernoullis and Euler $\endgroup$ Mar 16 at 5:47
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Gregoire de Saint-Vincent, along with his student Alphonse Antonio de Sarasa, developed hyperbolic logarithm, now known as natural logarithm, by relating logarithms to the quadrature of the hyperbola $xy=1$.

His findings led to the natural logarithm function, once called the hyperbolic logarithm since it is obtained by integrating, or finding the area, under the hyperbola.

Later, when Euler treated a logarithm as an exponent of a certain number, called the base of the logarithm, he identified hyperbolic logarithm as the natural logarithm.

Leonhard Euler changed that when he introduced transcendental functions such as $10^x$. Euler identified $e$ as the value of $b$ producing a unit of area (under the hyperbola or in a hyperbolic sector in standard position). Then the natural logarithm could be recognized as the inverse function to the transcendental function $e^x$.

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Monads were introduced as "standard constructions" (an awful, non-descriptive name), then were called "triples" (even worse), until the term "monad" was popularized by Categories for the working mathematician.

The history of the term is explained in more detail here.

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  • $\begingroup$ "Triple" is at least descriptive, given enough context :) $\endgroup$
    – chepner
    Mar 16 at 13:42
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    $\begingroup$ Famously, it is the only concept in math that has three things in it. :P $\endgroup$ Mar 16 at 21:56
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It seems that the constant formerly called $\zeta(3)$ is now called "Apéry's number".

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    $\begingroup$ Conversely, the constant formerly called "Legendre's constant" is now called $1$. $\endgroup$
    – usul
    Mar 16 at 11:26
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    $\begingroup$ That seems more like giving a name to an otherwise anonymous function value. $\endgroup$
    – chepner
    Mar 16 at 13:41
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See here for a few examples, such as recursion theory becoming computability theory, (linear) complex groups becoming symplectic groups (change due to Weyl), bicompact becoming compact, and pre-schemes becoming schemes.

Other examples: Gaussian integers used to be called complex integers, the term "group" initially referred only to what we'd call finite groups (existence of an inverse was not initially part of the definition, since it followed from the definition using finiteness), Lie algebras were previously called infinitesimal groups (the change to Lie algebras is due to Weyl), cyclic algebras were initially called "algebras of type D" by Dickson, tensors were called "systems" by Ricci and Levi-Civita and "direct products" by Murray and von Neumann, topology was called analysis situs (change due to Lefschetz), and commutative algebra was initially called ideal theory.

The meaning of homomorphism and isomorphism did not settle down for a while: in Jordan's 1870 textbook on group theory (a subject that for a long time was called the "theory of substitutions" using the old-fashioned term substitution for what we'd call a permutation), he used isomorphism for what we'd call a group homomorphism and holoedric isomorphism for what we'd call a group isomorphism (the term simple isomorphism was also used for modern-day isomorphisms). A discussion of this is here. Normal subgroups used to be called invariant subgroups.

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  • $\begingroup$ 'existence of an inverse was not initially part of the definition, since it followed from the definition using finiteness' I think an extra condition is needed here, since there are finite monoids. Were people assuming that groups had a faithful action by bijections? $\endgroup$ Mar 18 at 14:18
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    $\begingroup$ @OscarCunningham Galois (1830s) defined a group to be a set of permutations closed under composition, where the permutations were of a finite set. Much later, Weber (1882) defined a group to be a finite set with a binary operation that is associative and has cancellation on both the left and right. The first time inverses were explicitly included in the definition of a group was by von Dyck (also 1882). He defined groups in terms of generators and relations. See jstor.org/stable/2690312?seq=1 and mathshistory.st-andrews.ac.uk/HistTopics/Abstract_groups for further details. $\endgroup$
    – KCd
    Mar 18 at 15:37
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In 19th-century (term) logic, it used to be common to talk about "classes". With the recognition of the idempotence law $b\cup b=b$ by William Stanley Jevons, Charles Sanders Peirce, and Robert Grassmann, those classes correspond to what we nowadays call "sets" (thus making the algebra of sets an interpretation of Boolean algebra, a branch of abstract algebra which in retrospect was founded by Jevons after reforming George Boole's algebra of logic to include that idempotence law, which Boole never accepted). And now, in ZFC set theory, "classes" mean something different: they are more general collections than "sets" (e.g. the [proper] class of all sets).

Moreover, in the 19th-century German mathematical literature (e.g. in Richard Dedekind's writings), it was common to use the terms [de] "System" ([en] "system") and [de] "Mannigfaltigkeit" ([en] "manifold"/"multiplicity") to refer to what we know as [de] "Menge" ([en] "set") nowadays. So the term wasn't still standardized at that time.

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  • $\begingroup$ Zermelo's "definite Eigenschaft" $\endgroup$
    – Spencer
    Mar 18 at 17:01
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"Polynomial" is a relative late denotation, from the end of the 19th, start of the 20th century. The word itself was coined by Viete around 1600 as extension of "binomial" and "trinomial", for instance to describe the expression under a root. At about the same time "multinomial" was introduced for the same role.

Then there is some short use of "polynomial coefficients" for the combinatorial numbers better known as "multinomial coefficients".

The functions themselves become separate objects of interest around 1800 and are denoted as "numerical functions" by Galois, some variation of "entire rational algebraic functions" with Gauß. (Afaik the german school system is the only institution to keep this denotation alive, "ganzrationale Funktion".)

And as said, in the literature on extensions of function fields suddenly around 1910 the word "polynomial" is used. Around that time there was in general a shift in algebraic notation, for instance Kronecker around 1880 used "module" for ideals.

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Kohonen maps are now called self-organizing maps.

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