Historical Definitions of Functions -- Jan 13, 2010

QUESTION: What is a function?

ANSWER: This answer has varied over the course of time. Most of you probably think of a function as an expression involving variables; if this is the case, then you are happily living in the 1700's from a mathematical point of view. Over time, the idea of a function has developed significantly...

• 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.


• Example: f(x) = 3x² + 2x + 1


• Lacroix, 1810: Every quantity whose value depends on one or more other quantities is called a function of these latter, whether one knows or is ignorant of what operations it is necessary to use to arrive from the latter to the first.


• Example: g(n) = Γ(n) + n! + ζ(eⁿ)


• Fourier, 1822: In general, the function f(x) represents a succession of values or ordinates each of which is arbitrary. An infinity of values being given to the abscissa x, there is an equal number of ordinates f(x). All have actual numerical values, either positive or negative or null. We do not suppose these ordinates to be subject to a common law; they succeed each other in any manner whatever, and each of them is given as if it were a single quantity.


• Dedekind, 1888: A function f on a set S is a law according to which every determinant element s of S there belongs a determinant thing which is called the transform of s and denoted by f(s).


• Stewart's Calculus book, page 11, 2010: A function f is a rule that assigns to each element x in a set D exactly one element, called f(x), in a set E.

OBSERVATION: As we see in the quotes on the website for MA 113 sections 7,8,9, Calculus originally was a calculational tool, and it was not fully understood why it worked from a rigorous logical point of view. However, since we live in the 21st century, a lot of the precise language that was developed in the 1800's and 1900's to understand the logic behind Calculus has come into our textbooks and teaching methods.

The quotes above are from "A History of Mathematics," by V. Katz.