Who first considered the $f$ in $f(x)$ as an object in itself, and who decided to call it a function?

The question is in the title, but allow me to provide some background.

I’m aware that Leibniz introduced the word “function” into mathematics (around 1673) and that Johann Bernoulli or Euler introduced the notation $f(x)$ (sometimes between 1706-1740). I've also read the Wikipedia article on the History of the function concept and the book Die Analysis im Wandel und im Widerstreit by Spalt, plus a few other sources on the history of functions. But I haven't found the following addressed:

Looking at definitions* and uses of the word function among mathematicians up until the beginning of the 20th century, I see that they consistently call $y$ the function. By $y$ I mean the one in $y=f(x)$. Equivalently they call $f(x)$ the function. Actually they call $y$ a function of $x$, but they eventually drop the “of $x$” when it is clear from the context or irrelevant for the discussion. None of them ever consider $f$ as a mathematical object in itself.

This is in stark contrast to the modern definition, which calls $f$ the function, describes it as a rule or the like, and officially calls $f(x)$ something else (like the output of the function at input $x$ etc.).

The first mathematician I could find to distinguish clearly between $f$ and $f(x)$, with a definition similar to modern ones, was Cantor in 1895. But to my surprise he did not call $f$ a function but a “Belegung” (allocation). Here's my translation of what he wrote in Beiträge zur Begründung der transfiniten Mengenlehre, I, 1895, p.486:

By an ,allocation of the set $N$ with elements of the set $M$' or simply put, by an ,allocation of $N$ with $M$' we understand a law, through which with every element $n$ of $N$ is connected a certain element of $M$, where one and the same element of $M$ may be used repeatedly.

The with $n$ connected element of $M$ is to a certain extent a unique function of $n$ and may be denoted with $f(n)$; it shall be called the ,allocation function of $n$'; the corresponding allocation of $N$ shall be called $f(N)$.

Two allocations $f_{1}(N)$ und $f_{2}(N)$ are called equal if and only if, for all elements $n$ of $N$ the equation $$f_{1}(n) = f_{2}(n),$$ holds, so that if for a single fixed element $n=n_{0}$ the equation does not hold, $f_{1}(N)$ and $f_{2}(N)$ are characterised as different allocations of $N$.

[$\ldots$]

The totality of all different allocations of $N$ with $M$ determines a set with the elements $f(N)$; we call it the ,set of allocations of $N$ with $M$' and denote it with $(N\mid M)$. Hence: $$(N\mid M)=\{ f(N) \}.$$

Notice that he keeps the traditional meaning of the word function, so it makes perfect sense to give $f$ (or as he writes $f(N)$) a new name.

Two questions:

1. Was Cantor the first person to treat $f$ as an object on its own and distinguish it from $f(x)$? In other words, was Cantor the first one to give the modern definition of a function?
2. When and why did it become standard to call $f$ a function, despite the established use of the word for $f(x)$ (and despite Cantor calling $f$ something else)?

* Several such historical definitions can be found at the end of the following MO question: Formalizations of the idea that something is a function of something else?.

• Are you sure that 1895 is the first occurrence in Cantor? – Francois Ziegler Jun 3 '17 at 20:15
• No, I'm not sure. I found that quote of Cantor in Spalts book, but I haven't systematically searched Cantors works. – Michael Bächtold Jun 3 '17 at 21:49
• Cantor created the terminus "Belegungsmenge" in 1895. Besides its appearance in his paper of 1895 Cantor explains the terminus to Klein in a letter of July 17, 1895. – Otto Jun 4 '17 at 9:22
• Not an answer, but in case you're interested in some references to historical papers on the development of the idea of a function, see the list I posted here. – Dave L Renfro Jul 6 '17 at 15:54
• @sand1 based on your assumption, could you explain what became know or relevant about $f$ around 1900, that wasn't known before, and which cause $f$ to be suddenly considered one of the most central objects of mathematics? – Michael Bächtold Sep 5 '17 at 7:44

2 Answers

Cantor 1895 is predated at least by Dedekind in §2 of Was sind und was sollen die Zahlen? (1888) (translation):

21. Erklärung *). Unter einer Abbildung $\varphi$ eines Systems $S$ wird ein Gesetz verstanden, nach welchem zu jedem bestimmten Element $s$ von $S$ ein bestimmtes Ding gehört, welches das Bild von $s$ heißt und mit $\varphi(s)$ bezeichnet wird; wir sagen auch, daß $\varphi(s)$ dem Element $s$ entspricht, daß $\varphi(s)$ durch die Abbildung $\varphi$ aus $s$ entsteht oder erzeugt wird.

*) Vergl. Dirichlet's Vorlesungen über Zahlentheorie, dritte Auflage, 1879, §. 163.

(The footnote refers to a supplement in Dedekind's edition of Dirichlet.) Also, Dedekind in §11 clearly distinguishes between a map $\psi$ and what he still calls the function $\psi(n)$ it determines:

135. Erklärung. Es liegt nahe, die im Satze 126 dargestellte Definition einer Abbildung $\psi$ der Zahlenreihe $N$ oder der durch dieselbe bestimmten Funktion $\psi(n)$ auf den Fall anzuwenden, wo ($\ldots$)

Edit: To your two questions,

1. Attribution to Dedekind 1888 seems confirmed by W. Sieg and D. Schlimm, Dedekind's Abstract Concepts: Models and Mappings, Philosophia Mathematica nku021 (2014) 1-26. They also quote from an earlier draft of 1872-78, where Dedekind wrote $x\,|\,f$ instead of $f(x)$.

2. I don't know who (rather unfortunately, I'd say) suppressed his careful distinction between Abbildung (mapping, application, "Maler", $f$) and Funktion (function, fonction, "Bild", $f(x)$). Perhaps von Neumann (1925, pp. 222, 223):

($\ldots$) aus einer Funktion $f$ (die von ihren Werten $f(x)$ wohl zu unterscheiden ist!) und einem Argumente $x$ den Wert $f(x)$ der Funktion $f$ für das Argument $x$ zu bilden. ($\ldots$) $\mathfrak M$ sei eine Menge, $f(x)$ eine (in $\mathfrak M$ definierte) Funktion.

or more influentially van der Waerden in Moderne Algebra (1930, §2, §6, §9):

Wenn durch irgend eine Vorschrift jedem Element $a$ einer Menge $\mathfrak M$ ein einziges neues Objekt $\varphi(a)$ zugeordnet wird, so nennen wir diese Zuordnung eine Funktion ($\ldots$) Man nennt eine solche Zuordnung ($\ldots$) auch eine (eindeutige) Abbildung der Menge $\mathfrak M$ auf die Menge ($\ldots$) Ist $\varphi$ eine eineindeutige Abbildung ($\ldots$) Eine solche Funktion $\varphi$ gibt es ($\ldots$)

• Is my understanding correct, that the supplement of 1879 was written by Dedekind, that he defines Abbildungen already there, although he does not write $\phi(\omega)$ but only $\omega'$? Moreover in the supplement, he did not make explicit the relation between Abbildungen and Funktionen? So he does not call $n'$ a function of $n$, nor does he say that the $f$ from a function $f(x)$ is a map? – Michael Bächtold Jun 5 '17 at 0:20
• Yes, I think that's it exactly. After 1888 Dedekind names many maps ($f$) and many functions ($y=f(x)$), but the $\psi(n)$ quote above is the only place I've seen him relate both in the same breath. Such quotes by anyone seem hard to come by – one would be by C. S. Peirce, is a 1911 dictionary article where he proposes to translate Abbildung by imaging: "Any mathematical function of one variable may be regarded as an image of its variable according to some mode of imaging." – Francois Ziegler Jun 5 '17 at 4:15
• I was wondering if the terminology "Maler" for map is something you have seen in the literature of that time. – Michael Bächtold Jul 4 '17 at 12:13
• @MichaelBächtold Not seen by me in anything published at the time. Sieg-Schlimm quote abbildende Maler vs. Bild from a letter of Dedekind (1890) printed in Sinaceur (1974, p. 276). – Francois Ziegler Jul 4 '17 at 17:51
• @MikhailKatz Per wiki quote linked in the question, Dirichlet called $y$, not $f$, the function. – Francois Ziegler Sep 5 '17 at 17:10

Answering 2: Maybe the idea of calling $f$ the function goes back to Frege? In Was ist eine Funktion? Festschrift Ludwig Boltzmann (1904, pp. 656-666) we find on p. 663:

Und haben wir hier nicht das, was wir suchen, die Funktion? Also wird auch »$f$« genaugenommen eine Funktion andeuten.

My translation: "And don't we have here what we are looking for, the function? Hence also »$f$« will strictly speaking signify a function."

The whole article is a very interesting read, since Frege also tries to answer the questions "What is a variable quantity?" (he concludes they don't belong in mathematics), "What is an indeterminate number?" (he concludes there are none) and addresses the distinction between $f(x)$ as a syntactic expression (a term), versus $f(x)$ as a semantic entity (the variable quantity). I don't agree with all of his arguments, but it's fascinating to see how he single-handedly demolishes concepts that had been an integral part of mathematics for 200 years.

Added later: In the comments Francois makes the relevant observation, that Lagrange called $f$ the caractéristique of the function $f(x)$. One also finds this in Johann Bernoulli (in Remarques sur ce qu'on a donné jusqu'ici de solutions des Problemes sur les isoperimetres, 1718, p. 243) and Euler (in Sur la vibration des cordes, 1748, p. 78).

I interpret this similarly to Francois: $f$ is being treated merely a sign, that might helps us distinguish two functions, say $f(x)$ and $g(x)$. But $f$ was not yet considered as a mathematical object on its own. On the other hand in Peano's Arithmetices Principia (1889, p. XIII) we find:

Let $\phi$ be a sign, or aggregate of signs, such that if $x$ is an entity of class $s$, the sign combination $\phi x$ determines a new entity; we also suppose that equality is defined between the entities $\phi x$; and if $x$ and $y$ are entities of the class $s$, and $x=y$, we suppose that it can be deduced that $\phi x = \phi y$. Then the sign $\phi$ is said to be a prefix sign for a function in the class $s$, and we write $\phi \, \backepsilon \, F`s$.

(Translation adapted from Johan Granström's PhD thesis.)

My interpretation here is that Peano does treat $f$ as a mathematical object on its own, distinct from $f(x)$, just like his contemporaries, Frege, Dedekind and Cantor, yet he calls $f$ a sign for a function, just like Bernoulli, Euler and Lagrange.

(So I'm left with the feeling, that I first need to define more precisely what it means to treat something as a mathematical object, in order for my interpretations or even my question to make sense. But currently I'm not able to provide such a definition.)

• ...so Frege tries to argue that it's wrong to talk of variable quantities in mathematics and that $y$ should not be called a function, as it had been done for 200 years. – Michael Bächtold Jun 5 '17 at 22:24
• Interesting! However, I feel that whatever he’s calling function does not fit our “modern definition” as in Q1 (e.g. as graph). In fact, contrast even the title question with what he literally writes in Function und Begriff (1891, p. 19): “Graphs of functions are objects, whereas functions themselves are not.” – Francois Ziegler Jul 5 '17 at 6:28
• He details that even more forcefully in Grundgesetze der Arithmetik II (1903, §147, footnote 2): for him, $f(\xi)=g(\xi)$ for all $\xi$ does not imply $f=g$. – Francois Ziegler Jul 5 '17 at 6:28
• A somewhat relevant discussion on intensional/extensional functions: mathoverflow.net/questions/156238 – Michael Bächtold Jul 5 '17 at 7:22
• I should add that before he started using caractéristique (apparently in 1773, p. 191), Lagrange was much looser: e.g. in 1761 one finds both la fonction φ(x) or similar (pp. 13, 70, 91, 99, 116, 119,...), and la fonction φ (pp. 44, 49, 63, 69-73, 86, 91-92, 95-96, 119, 154, 160,...). So that's a possible answer to your Question 2. – Francois Ziegler Jul 7 '17 at 16:15