# When was the function arrow notation $x \mapsto y$ first used?

The notation $x\mapsto \sin x$ and its meaning are well-known to most mathematicians. Less well-known seems to be the fact that $x \mapsto y$ means the same as Church's $\lambda x.y$ and Frege's $\grave x y$ (found in Funktion und Begriff, 1891). At least at an informal level these all seem to denote the same thing.

Both Frege's and Church's work are considered mayor milestones in the development of the notion of functions, but I've never heard of the person who came up with the notation $x\mapsto y$.

By whom and when was $x\mapsto y$ first used?

• Nobody seems to know... – Francois Ziegler Mar 9 '17 at 20:45
• For sure, we can find it into the 2nd French ed of Nicolas Bourbaki, Théorie des ensembles, CCLS (1970), page E.R.6 : "Lorsqu'une relation de la forme $y = \langle x \rangle$ est une relation fonctionnelle in $y$, on désignera parfois la fonction qu'elle determine par la notation $x \mapsto \langle x \rangle$." – Mauro ALLEGRANZA Mar 10 '17 at 7:44
• We cannot find it in the English 1968 translation (I assume of the 1st French edition) : Nicolas Bourbaki, Elements of Mathematics: Theory of sets, Springer (1968), page 352 : "When a relation of the form $y = ⟨x⟩$ is a functional relation in $y$, the function it determines is often denoted by the notation $x \to ⟨x⟩$". – Mauro ALLEGRANZA Mar 10 '17 at 7:56
• I suppose we can find it also in the first (1967) edition of Saunders Mac Lane & Garrett Birkhoff, Algebra, because we have it in the supplementary French 1972 book Algebre: solutions developpees des exercices (translated by Weil & Hocquemiller) companion to the French 1970 edition Algèbre. – Mauro ALLEGRANZA Mar 10 '17 at 8:10

I believe the when can be narrowed down to 1963-64, and by whom to, likely, a Bourbaki member. In the treatise, a lower and upper bound are

Of the volumes published in between, I could only view this relevant snippet (similar to, but different from M. Allegranza's) in Théorie des ensembles, Fascicule de résultats, 4th ed. (1964), pp. 13-14:

Lorsqu'une relation de la forme $y=\langle x\rangle$ (où $\langle x\rangle$ désigne une combinaison de signes dans laquelle peut figurer $x$) est une relation fonctionnelle en $y$, on désignera parfois la fonction qu'elle détermine par la notation $x\to\langle x\rangle$, ou même simplement par $\langle x\rangle$, ce qui est un abus de langage très fréquent (par exemple, on parlera de la fonction $\sin x$ dans $\mathbf R$). On notera que le sens de la flèche $\to$ est alors tout à fait différent de celui qui lui a été attribué ci-dessus; pour cette raison, on remplace souvent la notation $x\to\langle x\rangle$ par la notation légèrement différente $x\mapsto\langle x\rangle$ pour éviter toute ambiguïté. Par exemple, si $\mathrm X$, $\mathrm Y$ sont deux parties d'un ensemble $\mathrm E$, la relation $\mathrm Y=\complement\mathrm X$ est fonctionnelle en $\mathrm Y$, et il y a lieu de désigner par $\mathrm X\mapsto\complement\mathrm X$ l'application de $\mathfrak P(\mathrm E)$ dans lui-même qu'elle détermine, pour éviter de la confondre avec une application de $\mathrm X$ dans $\complement\mathrm X$.

Other early publications of $\mapsto$ by bourbakists are

Finally, the earlier switch from an overloaded $\to$ to the proto-mapsto $\sim\!\rightarrow$ (or $\rightsquigarrow$) seems to have been made by Grothendieck between Bourbaki seminars nº 182 and nº 190 (1959), and in print between EGA Chap. II and Chap. III (1961). (He stuck with $\rightsquigarrow$ much longer, until at least nº 290 (1965, p. 202) and Chap. IV (1967).)

Note added. I asked Michel Demazure, who kindly allows me to share these recollections:

1. In his mind, the innovation (and standardization) came from Bourbaki.

2. He thinks he learned it from them through Grothendieck before joining, not vice versa.

3. He seems to recall Serre using $\rightsquigarrow$ in his Collège de France (blackboard) lectures before 1959.

4. They called $\rightsquigarrow$ “tortillon” and $\mapsto$ “poussoir”.

5. In adopting any symbol, a significant consideration was its availability in the Typit system.

(One can indeed see this system used for $\rightsquigarrow$ and $\mapsto$ in e.g. the 1964 seminars, pp. 368, 403, 418.)

Further note. Bourbaki ($x\mapsto y$), Church ($\lambda x.y$) and Frege ($\grave xy$) have other antecedents: $\newcommand{SS}{\mathrel{\style{display:inline-block; transform:rotate(.25turn);}{\mathrm S}}}$

• Eduard Study apparently originated $x\to y$, initially typeset $x\rightarrowtail y$ (1905, 1891).
• Peano used $y|x$ and earlier $y\,\overline x$:

(1900, p.38): let $A$ be an expression containing the variable letter $x$; by $A|x$, which one can read « the expression $A$ regarded as a function of $x$ », we indicate the function sign $u$ which, written in front of $x$, produces the given formula $A$. (...) E. g.: $a^n/n!\,|n$ represents the function sign which for value $n$ of the variable has value $a^n/n!$.

(1898, p. VII): The sign $|$ is the inversion sign. In earlier work it had the shape $\overline{\phantom{m}}$.

(1897, pp. 2, 58): let $a$ be an expression containing the variable letter $x$. We indicate by $a\,\overline x$ the function sign $f$, such that one has $fx=a$. One has therefore $$(fx)\,\overline x = f.$$ E. g. $x^2-3x$ is an expression containing $x$; if we put $$f = (x^2-3x)\,\overline x$$ we will have $fx=x^2-3x$. The value of $fx$ for $x=5$, or $f5$, will therefore be indicated by $$(x^2-3x)\,\overline x\,5=10.$$

• Eisenstein used $x \SS y$, which Peano et al. (1901, p. 173) quote as a forerunner of $y|x$ even though its usage isn’t quite the same:

(1847, p. 211): To be able to express myself more briefly, I will represent the concept of Substitution by a special sign, say by the sign $\,\SS$, whose introduction is urgently needed in many investigations, whenever one deals with the values that some expressions assume when other expressions, of which the former are functions, change their form or value. This sign $\,\SS$, which saves a lot of words, reads in a premise (hypothesis): “If the expression on the sign’s left is replaced by the one on the right,” and in the ensuing conclusion (thesis): “Then the left-hand side goes over into the right-hand side.” E. g. the proposition, if $A\SS B$, then $C\SS D$ reads: if one replaces $A$ by $B$, or lets $A$ go over into $B$, then $C$ goes over into $D$, or $D$ replaces $C$.

• Very nice, thank you! In your added Note 2., the not vice versa means that Bourbaki did not learn it from Demazure? – Michael Bächtold Mar 15 '17 at 9:09
• @MichaelBächtold That's right. – Francois Ziegler Mar 15 '17 at 11:37
• Very interesting further findings! I hope to have some time to look into them soon. I suspect they also lead to more insights into $y|_{x=a}$, $y=y(x)$ and the history of notation for substitution – Michael Bächtold Nov 4 '17 at 7:58
• Note added: Porteous’ two editions of Topological geometry (1969, 1981) also use the tortillon $\rightsquigarrow$, commenting (p. 5): “The arrow $\mapsto$ is used by many authors in place of $\rightsquigarrow$.” – Francois Ziegler Jul 10 '18 at 0:46

A good source for these types of questions is Jeff Miller's website on earliest uses of mathematical symbols. This notation is recent (on the other hand, use of arrows for limits goes as far back as Riemann). The original version was introduced by Ore in L'Agebre Abstraite (1936), but without the vertical bar:

"We shall say that two algebraic systems $S$ and $S'$ are homomorphic (with respect to addition and multiplication) if there exists a correspondence $a\to a'$ between the elements of $S$ and $S'$ giving to each element $a$ of $S$ a unique image $a'$ in $S'$ such that each element of $S'$ is the image of at least one element of $S$ and further such that if $a\to a'$, $b\to b'$ we have $a+b\to a'+b'$ and $ab\to a'b'$". [translation mine]

In 1939 Bourbaki started using "la application $x\to f(x)$", and it spread far and wide along with their volumes. The earliest version of the $f: X\to Y$ notation seems to be in a Hurewicz-Steenrod paper of 1940. Algebraic topologists took a liking to it, and it was a blessing for the emerging homological algebra according to MacLane:

"At first the vivid arrow notation $f: X\to Y$ for a map was not available, and homomorphisms of homology groups (or rings) were always expressed in terms of the corresponding quotient group or rings. Thus the familiar long exact sequence of the homotopy groups of a fibration was originally described in terms of subgroups and quotient groups; this is the style used by all three discoveries of the sequence and of the covering homotopy theorem [...] The occurrence of exact sequences of homology groups (though not the name 'exact') was first noted by W. Hurewicz in 1941 [...] The practice of using an arrow to represent a map $f: X\to Y$ arose at the same time. I have not been able to determine who first introduced this convenient notation; it may well have appeared first on the blackboard, perhaps in lectures by Hurewicz and it is used in the Hurewicz-Steenrod paper, submitted November 1940..."

The pedantic distinction between $\to$ and $\mapsto$ is the work of later textbook authors. Cabillon comments on the NCTM Math Forum (credit to Francois Ziegler for linking the thread):

"The notation "a barred arrow f(a)" [i.e. $a\mapsto f(a)$] is fairly modern, I would dare to say. I have not been able to determine who first introduced this notation, but my guess is that it came with the "New Math" vogue of the early 60's. The first time I encountered this notation was in French books."

• I question both Miller’s attribution to Ore (1936) and MacLane’s attribution to Hurewicz-Steenrod (1940). – Francois Ziegler Jun 25 '17 at 19:47

Thanks for the useful answers so far. I find the answer of William Waterhouse in the discussion linked to by Francois Ziegler very interesting.

He gives evidence for the following

1) The "barred arrow" $\mapsto$ almost certainly was introduced between 1960 and 1965.

2) It arose after a period of at least a decade (probably more) in which the plain arrow $\to$ was used ambiguously [to denote both the type of a function (domain/codomain), as well its element-to-element description].

3) I do not know who invented it, but it definitely did not come from the "New Math" school reform movement in the United States.

4) My guess (though I cannot fully support this) is that need for a distinct symbol became felt around this time just because there were more situations arising where the ambiguity was a possible source of confusion.

and he also writes the following

There is one additional piece of evidence that supports both 1) and 4) and is probably unfamiliar:

In the very early 1960's, one sometimes found the notation

/\/\/\-->

(i.e., a wiggle at the start of the arrow) in the meaning that was soon taken over by |-->. (I must say this looks very much like a suitable place to recall Victor Borge's "phonetic punctuation" routine.)

I don't know whether this appeared in "formal" publications. (Checking back, I see that I was using |--> already as a graduate student in 1966.) I have it in my notes from an algebra course taught by John Tate in Spring 1962, and it occurs in a semi-formal publication that should be in some libraries:

Dieudonne, Jean Alexandre Algebraic geometry [College Park], 1962 Maryland. University. Dept. of Mathematics / Lecture notes ; no.1 (Library of Congress number) QA564.D5

Then there is also an interesting answer by Emili Bifet about Riemann using the plain arrow $x\to y$ to denote the element-to-element action of a function, even before Ore (1936). (And not to denote the passage to a limit as in $\lim_{x\to 0}$.) Here is my translation of what Riemann wrote:

II. The integrals of a linear differential equation of second order at a branching point (From a lecture winter semester 1856/57.)

If $a$ is a branching point of the solution to a linear differential equation of second order and, as $x$ moves clockwise around $a$, $z_1$ goes over to $z_3$ while $z_2$ goes over to $z_4$, what shall be denoted with $z_1 \to z_3$ and $z_2 \to z_4$ for short, then $$z_3 = t z_1 + u z_2$$

$$z_4 = r z_1 + s z_2.$$

If $\epsilon$ is any constant, then $$z_1 + \epsilon z_2 \to z_3 + \epsilon z_4.$$

...

Since now $$z_1 (x - a)^{-\alpha} \to z_3 (x - a)^{-\alpha} e^{-2 \alpha \pi i}$$

it must hold

$$z_1 (x - a)^{-\alpha} \to z_1 (x - a)^{-\alpha} + (z_1 + \epsilon z_2)k(x - a)^{-\alpha}.$$

Since further

$$\frac{k}{2 \pi i} (x - a)^{-\alpha}(z_1 + \epsilon z_2)l(x - a) \to \frac{k}{2 \pi i} (x - a)^{-\alpha}(z_1 + \epsilon z_2)l(x - a) + k (x - a)^{-\alpha}(z_1 + \epsilon z_2),$$ ...

In modern terminology he is apparently talking about the monodromy map along a path in a complex domain. Bifet also writes that the arrow notation in analysis (limits) ist due to Leathem (around 1905) which agrees whit what Jeff Millers website says on this.