Concept of a function and Idea of a formula as a function

Question

Enderton Elements of Set Theory, p. 43 (1977, Academic Press), writes:

There was a reluctance to separate the concept of a function itself from the idea of a written formula defining the function.

What is the basis for the above historical claim? And at around what point did the concept of a function itself from the idea of a formula become firmly separated?

It seems interesting that what is today regarded as an elementary mistake had a strong historical basis.

Fuller quote from Enderton:

This question originally posted over at Math.SE.

Answer 1

You can see History of the function concept.

For Euler (1748) :

a function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities

i.e. a function was a “symbolic expression” that, received a value as “input” allows us to calculate a corresponding “output” value.

It seems that it is in Dirichelet (1837, page 135), that we can find the first explicit definition of the concept of function as “arbitrary correspondence”:

If now a unique finite $y$ corresponding to each $x$, and moreover in such a way that when $x$ ranges continuously over the interval from $a$ to $b$, $y=f(x)$ also varies continuously, then $y$ is called a continuous function of x for this interval.

It is not at all necessary here that $y$ be given in terms of $x$ by one and the same law throughout the entire interval, and it is not necessary that it be regarded as a dependence expressed using mathematical operations [emphasis added].

Answer 2

I recommend an excellent account by Luzin in the Monthly: MR1615544, MR1613935 (American math Monthly 105 (1998), 1 59-67 and 3, 263-270.

It is usually overlooked that there are actually several different notions of function in modern mathematics. One is the Dirichlet's definition which is usually cited (where two sets are given X and Y, and a rule which to each element of X puts into correspondence an element of Y). Notice that X is a part of the definition!

So the problem of the type "find the domain of $\log((x-1)(x-2))$ makes no sense from the point of this definition.

In 18 century, Euler understood a function as some analytic expression whose domain is not given in advance. This different notion (from Dirichlet's definition) is not "out of date". It evolved into a modern definition of an "analytic function". Roughly speaking, an "analytic expression" has a "natural domain of definition", which is not given in advance. And problems of the type "find the domain of definition" of an analytic function make perfect sense in modern mathematics.

There are also other notions of functions in modern mathematics (generalized functions, or distributions), which also do not fit in the Dirichlet definition. Moreover, these generalized functions are in some sense closer to what physicists and engineers mean by a function than the Dirichlet definition.

Answer 3

Just to add. After giving a standard, modern definition of the function, Stephen Abbott (in p.7 of Understanding Analysis) notes:

This definition of function is more or less the one proposed by Peter Lejeune Dirichlet (1805–1859) in the 1830s. Dirichlet was a German mathematician who was one of the leaders in the development of the rigorous approach to functions that we are about to undertake. His main motivation was to unravel the issues surrounding the convergence of Fourier series. Dirichlet’s contributions figure prominently in Section 8.3, where an introduction to Fourier series is presented, but we will also encounter his name in several earlier chapters along the way. What is important at the moment is that we see how Dirichlet’s definition of function liberates the term from its interpretation as a type of “formula.” In the years leading up to Dirichlet’s time, the term “function” was generally understood to refer to algebraic entities such as $f(x) = x^2+1$ or $g(x) = \sqrt{x^4 + 4}$. [The above definition] allows for a much broader range of possibilities.

Answer 4

I don't think it is quite so clear cut as the popular opinion ("Euler thought only of symbolic expressions, while Dirichlet gave the modern definition") makes us believe. Consider for example this definition of functions from Eulers later work, Institutiones calculi differentialis, 1755, Preface p.VI::

Thus when some quantities so depend on other quantities, that if the latter are changed the former undergo change, then the former quantities are called functions of the latter ; this definition applies rather widely, and all ways, in which one quantity could be determined by others, are contained in it. If therefore $x$ denotes a variable quantity, then all quantities, which depend upon $x$ in any way, or are determined by it, are called functions of it.

Examples are $x^{2}$, the square of $x$, or any other powers of $x$, and indeed, even quantities that are composed with these powers in any way, even transcendentals, in general, whatever depends on $x$ in such a way that when $x$ increases or decreases, the function changes. From this fact there arises a question; namely, if the quantity $x$ is increased or decreased, by how much is the function changed, whether it increases or decreases?

To my mind this is not substantially different from what Dirichlet said.

Moreover, Dirichlet never talked of sets or the domain or codomain of a map, nor did he call the "rule" the function, as the modern definition you find in all books does. See also Who first considered the $f$ in $f(x)$ as an object in itself, and who decided to call it a function?