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I think we all occasionally come across terminology that we'd like to see supplanted (e.g. by something more systematic). What I'd like to know is, under what circumstances is it reasonable to believe that such a fight is winnable?

Question. What recent programmes to alter highly-entrenched mathematical terminology have succeeded, and under what conditions do they tend to succeed or fail?

Definitions.

  • Recent = In the past fifty years.
  • Highly-entrenched = The literature at the time had the old terminology written all over it.
  • Succeeded = The new terminology is now used almost exclusively in research papers.
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Given that we still use 17th century broken notation in analysis, and people actually think the summation conventions are ok, I wouldn't harbor any hopes that things will get better. –  Andrej Bauer Jul 11 at 8:54
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I am referring to $\frac{dy}{dx}$ and the fact that it's legal to write $\int x^2 dx = x^3/3 + C$ (which exposes the bound variable $x$), and the fact that ${{\Gamma_{ij}}^{kl}}_{mn}$ means something. –  Andrej Bauer Jul 11 at 18:07
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@DonuArapura: Roger Penrose introduced abstract index notation, which combines the expressiveness of index notation with the coordinate independence of coordinate-free notation. –  Ben Crowell Jul 12 at 1:59
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@DonuArapura: I'd argue that anyone who uses the coordinate-free notation has never had to actually compute a tensor. The notation for contractions alone is just nightmarish in them. Abstract indices are absolutely the only way to go, and they are much more superficially similar to Einstein notation. –  Jerry Schirmer Jul 12 at 3:35
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@AndrejBauer: but when you say that dy/dx refers to the standard part, you're dragging in Robinson's non-standard analysis Check out the link in my earlier comment, p. 10. Fermat and Leibniz had a notation, ${}_{\ulcorner\!\urcorner}$, and a term, "adequality," that expressed essentially the same notion as Robinson's standard part (subject to the limitations of the development of mathematics at that time). The idea is much more generic than NSA. –  Ben Crowell Jul 12 at 14:45

5 Answers 5

Roger Penrose's abstract index notation for tensors is a relatively modest example, but I think it fits all the criteria of the question. Around 1952, Penrose invented a personal graphical notation for tensors and tensor operations such as contraction and covariant derivatives. It's been described as "fornicating ostriches;" variations on it are referred to by terms such as "birdtracks." It was manifestly coordinate-independent (in fact it didn't even use letters of the alphabet). Penrose continued to use it for his own purposes, but didn't publish a description of it until [Penrose 1971].

However, he also realized, apparently some time later, that it was possible to make up a notation that looked almost the same as traditional coordinate-dependent index-gymnastics notation (which Einstein invented ca. 1914) but that was in fact isomorphic to his coordinate-independent diagrammatic notation. Very little changed about the way the notation was written; the main changes were to the semantics. In old-style notation, the usual convention was that a Greek index ranged over the values 0 to 3, referring to four coordinates, while a Latin index ranged over only the spacelike indices. The new convention is that Latin indices are abstract, Greek ones concrete. In a notation like $p^c=T^{cd}u_d$, $d$ is a bound symbol and $c$ is not bound. Neither $c$ nor $d$ takes on specific values like 2 that refer to a particular coordinate system. This equation is simply a way of typesetting the following manifestly coordinate-independent diagram:

Penrose birdtracks

Here the square is $p$, the rectangle is $T$, and the circle is $u$. The abstract indices are to be interpreted simply as instructions for how to hook up the plumbing of the diagram, like instructions that say, "insert tab A in slot B."

Abstract index notation was popularized by Penrose, Geroch, and Rindler in the 1970s, and is now nearly universal among relativists. It won because (1) it looked familiar; (2) it was superior to index-gymnastics notation because it could express things in a manifestly coordinate-independent way; and (3) it was capable of conveniently and compactly representing tensor calculations that would have been too cumbersome to express using "mathematician-style" coordinate-independent notation.

Penrose, "Applications of negative dimensional tensors," in Combinatorial Mathematics and its Applications, ed. D.J.A. Welsh, Academic Press, 1971

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Although just beyond your 50-year scope, this may be of interest. Among the series $\mathsf A_n, \mathsf B_n, \mathsf C_n, \mathsf D_n$ in the Cartan-Killing classification of simple Lie groups, everyone (I believe) always agreed to call $\mathsf A_n$ the special linear group, $\mathbf{SL}(n)$, and $\mathsf B_n$ and $\mathsf D_n$ the special orthogonal groups, $\mathbf{SO}(2n+1)$ and $\mathbf{SO}(2n)$.

But $\mathsf C_n$? Jordan in his Traité des substitutions (1870) called it (or rather its product with dilations) the abelian group, because of its role in Hermite's "important investigations on the transformation of abelian functions"; p.172:

It is clear that if two [linear] substitutions $S, S'$ multiply $\varphi\ [=x_1\eta_1-\xi_1y_1+\dots+x_n\eta_n-\xi_ny_n]$ respectively by constant integers $m, m'$, $SS'$ will multiply $\varphi$ by the constant integer $mm'$. Hence the sought substitutions form a group. We will call it the abelian group, and its substitutions abelian.

This was well entrenched by the time Dickson wrote his Linear groups (1901); p.89:

A linear homogeneous substitution on $2m$ indices (...) is called Abelian if (...) it leaves formally invariant up to a factor (belonging to the field) the bilinear function $$ \varphi\equiv\sum_{i=1}^m \begin{vmatrix} \xi_{i1}&\eta_{i1}\\ \xi_{i2}&\eta_{i2} \end{vmatrix}. $$ The totality of such substitutions constitutes a group called the general Abelian linear group ${}^2)$ $GA(2m,p^n)$. These of its substitutions which leave $\varphi$ absolutely invariant form the special Abelian linear group $SA(2m,p^n)$.

${}^2)$ To distinguish these groups from the ordinary Abelian , i.e. commutative, groups, we prefix the adjective linear. The Abelian linear group is not commutative in general.

On the other hand, Sophus Lie and his school called $\mathsf C_n$ the linear complex group because it consists of symmetries of Plücker's linear line complex (a degree 1 hypersurface in the 4-dimensional space of affine lines in $\mathbf R^3$; Plücker (1866), p.341: "The latin word complexus, which means an intertwining, an inter-crossing, has seemed to us very appropriate to express the new idea we are presenting here. For lack of a better term, we ask for permission to introduce it in the mathematical language.") Thus for instance one finds in Lie and Engel's Transformationsgruppen, vol. II (1890), p.522:

One can say that the Pfaffian equation (73) represents a linear complex of the space $x'_1\cdots x'_n$, $y'_1\cdots y'_n$, $z'$. The group (72) should therefore be called the projective linear complex group.

Needless to say, both proposed names came into conflict with other spreading usages of these words. Hence Weyl's famous footnote in The classical groups (1938), p.165:

$$\text{CHAPTER VI}$$ $$\textbf{THE SYMPLECTIC GROUP}$$ $$\text{1. Vector Invariants of the Symplectic Group}^*$$ * The name "complex group" formerly advocated by me in allusion to line complexes, as these are defined by the vanishing of antisymmetric bilinear forms, has become more and more embarrassing through collision with the word "complex" in the connotation of complex number. I therefore propose to replace it by the corresponding Greek adjective "symplectic." Dickson calls the group the "Abelian linear group" in homage to Abel who first studied it.

Weyl's change was (obviously) highly successful. Ironically, Plücker's linear line complex term had won over Chasles' proposed focal system (1837). If it hadn't, today's symplectic geometers would probably all be doing "focal geometry"!

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The subject known for decades as recursion theory, studying the class of recursive functions and the recursively enumerable (r.e.) sets and degrees, is now known almost universally, especially amongst the newer generation, as computability theory, studying the computable functions and the computably enumerable (c.e.) sets and degrees. This change, led by Bob Soare, is remarkable in that it was not merely a change in terminology for an isolated mathematical construction or idea, but a change in the name of the entire subject, a wholesale recasting of entire schemes of terminology to focus on what was all along the actual focus, namely, the concept of computability, rather than specifically recursion. The change was strongly resisted at first, and even lampooned, but in about a decade, it has triumphed.

Here are some of the elements that may have been important for success:

  • The arguments for change were very strong. The recursion theory terminology grew out of Gödel's treatment of the primitive recursive functions, which grew into the class of recursive functions with the addition of unbounded search, and the subject became known as recursion theory. But in fact, the theme all along, even for Gödel, was computability, and to cap it off one doesn't even need recursion once one has unbounded search. So computability theory is simply a more accurate, honest name.

  • Bob Soare was a tireless advocate for the change, and spent several years giving talks, writing articles and generally making the argument that the subject should be called computability theory. At first, many people argued against the change, but Soare persisted.

  • Soare was a major figure an leader in the field, who wrote the standard textbook from which everyone had learned the subject. This text was called, "Recursively enumerable sets and degrees," using the old terminology, but Soare preferred to recast it as "Computably enumerable sets and degrees."

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Any thoughts as to what factors may have contributed to Bob Soare's success in this venture? –  goblin Jul 11 at 17:43
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Persistence, persistence, good arguments, and persistence. –  Andrej Bauer Jul 11 at 18:08
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I agree with @AndrejBauer, except that for me the main thing was that Bob had very strong arguments. The recursion theory terminology was leftover from Goedel's consideration of the primitive recursive functions, where recursion was indeed central, and then that class led to what were called the recursive functions. But in fact, one doesn't need recursion at all to define the class of functions, once one has unbounded search. The main theme of the subject was indeed computability, even for Goedel. I don't think Soare would have succeeded, however, without his untiring advocacy. –  Joel David Hamkins Jul 11 at 18:49
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It must also have been an important element for success that Soare was a major figure and leader in the field, who wrote the standard textbook from which nearly all graduate students learned the subject. –  Joel David Hamkins Jul 11 at 19:47
    
Is it really true that most computer scientists (rather than logicians) use c.e. rather than r.e.? –  Benjamin Steinberg Jul 11 at 20:01

An example of a failure to change notation is the movement by Eilenberg, Jacobson, Herstein and others to replace function notation $f(x)$ with $xf$ and then have the composition $fg$ mean first do $f$ and then $g$. The notation has the advantage that diagrams $X\xrightarrow{f}Y\xrightarrow{g} Z$ don't have to be flipped around. It also has the property that if you want to use left modules, then you don't have to use opposite algebras when proving the Wedderburn theorem.

It seems that this notation caught on for a while in Algebra and is quite popular in permutation group theory, automata theory and semigroup theory. GAP uses this convention.

But I think it has nowadays mostly burned out and perhaps this has contributed to why few people still use Herstein's book.

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We could also draw all the diagrams in the other direction... –  Andrej Bauer Jul 12 at 7:40
    
I do that sometimes. –  Benjamin Steinberg Jul 12 at 10:50
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@AndrejBauer, I actually do write that way in private: my notebooks are filled with sentences like "consider a function $f : Y \leftarrow X$ such that..." My only issue with this otherwise wonderful solution is that lambda-abstraction ends up going "on the wrong side." So we end up saying: let $f : Y \leftarrow X$ denote the function $\lambda x(\tau)$ where $\tau$ is a well-formed term of type $Y$. I still think its the best solution, though. –  goblin Jul 12 at 14:30
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Things could be worse: I once taught an undergraduate discrete math course using a textbook in which composition of binary relations was defined in left-to-right order, except that if the binary relations are functions then the order of composition is right-to-left, except that if the functions are permutations then the order is left-to-right. –  Andreas Blass Jul 12 at 15:32
    
@AndreasBlass Which textbook is this? –  Vidit Nanda Jul 12 at 15:55

In most cases that I am familiar with, successful changes to terminology were accompanied by other more basic innovations. E.g. Grothendieck's language became standard in algebraic geometry in the late 1950's, because he successfully rewrote the foundations of the subject.

An example of a proposed change for its own sake is the word "contrahomology" for "cohomology". I guess because it's contravariant. This occurs in the book by Hilton and Wylie on algebraic topology written in 1960. Happily, it never caught on.

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Another book, which I can't remember, suggested "omology" as an umbrella term for them. –  Jeff Strom Jul 11 at 16:09
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Can anything Grothendieck has done to algebraic geometry really be considered a notational change? I think the things he invented didn't have any notations beforehand... –  darij grinberg Jul 11 at 18:01
    
Speaking of Grothendieck, at some point someone who might very well have been him has modified the terminology "$\lambda$-ring / special $\lambda$-ring" to "pre-$\lambda$-ring / $\lambda$-ring". Neither has fully caught on... :( –  darij grinberg Jul 11 at 18:02
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Although Hilton and Wylie tried to rename cohomology "contrahomology" (presumably because it's contravariant, as you say), they stopped short of noticing that homology is covariant and renaming that in the consistent manner. That would have been really confusing! –  Jeremy Rickard Jul 12 at 15:09
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Since "co-" is so widely used to mean "opposite direction", a more basic change could have been calling covariant functors "variant" and contravariant functors "covariant". –  KConrad Jul 12 at 15:40

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