Chapter 2 — Functions and Dynamic Relations¶
2.9 — From Quantity to Change: the Function as an Object of Dynamic Thought¶
A concept is not born defined: it is defined, centuries later, in order to keep talking about what was already being used without a name.
Long before there was a definition of function, there was the need to predict. In the last centuries before our era, Babylonian astronomers computed tables of positions of the Moon and the planets in columns of numbers that followed regular patterns: to each date, a position; from each position, the next, obtained by adding differences that grew and shrank according to fixed arithmetical rules. There was neither variable nor formula in the modern sense, which would take many more centuries to appear, and yet there was a correspondence: to each moment, a single state of the sky. The function, in its oldest form, was not a definition but a practice: that of tabulating, with astronomical patience, what follows what. One may ask whether that tabulated correspondence already deserves to be called a function, or whether something more, a rule and not just a list, is needed for the name to be justified.
That something more came first from physics, not from algebra. Around 1350, Nicole Oresme proposed representing how a quality, such as the speed of a moving body, varies over time. He called it the latitude of forms: a horizontal segment represented time, and over each point of that segment rose a vertical segment whose height was the intensity of the quality at that instant. The upper ends of those segments traced a figure; for a uniformly accelerated motion, the figure was a triangle. And information could be extracted from the figure: Oresme identified its area with the distance traveled, without the algebraic vocabulary to say so precisely. It is, in essence, the idea behind the graph of a function as we defined it in this chapter, a set of pairs, one for each instant, almost three centuries before the Cartesian plane gave it numerical coordinates. Oresme had no equations; he had something perhaps more fundamental: the conviction that change itself, and not only quantity, could be represented and measured. Is the graph prior to the function it represents, or is the function, as we maintained when defining it, the one that was always logically first, the graph merely making it visible?
The algebraic synthesis came with Descartes and Fermat in the seventeenth century, whose analytic geometry, studied in §1.8, dissolved the distinction between curve and equation: a circle and the equation \((x-h)^2+(y-k)^2=r^2\) became the same thing seen from two languages. At the end of that century, Leibniz introduced the word itself, functio, for quantities that depend on a point moving along a curve. And in the eighteenth century, Euler gave it its first lasting definition: a function was an analytic expression, a formula built from variables and constants by means of the operations of the algebra and calculus of his time. From this point of view, \(x^2\) was a function because there was a formula producing it, and whatever admitted no formula fell outside the category. It was a fruitful definition, on which much of the calculus of the eighteenth and nineteenth centuries was developed, and at the same time restrictive in a way its own practitioners were slow to notice.
The crack appeared with trigonometric series, in particular with Fourier's work on the propagation of heat at the beginning of the nineteenth century. Fourier maintained that functions of very irregular appearance, even defined by completely different pieces, could be represented as infinite sums of sines and cosines, and he convinced many of his contemporaries of it. The question was unavoidable: what is a function, if an object so unlike an analytic expression in Euler's sense could behave like one? Dirichlet's answer was not to defend Euler's definition but to abandon it. In 1837 he proposed that a function is any correspondence that assigns to each value of one variable a single value of another, regardless of whether it admits a formula. Already in 1829 he had given an example showing how far that generality reached: the function that equals \(1\) at each rational number and \(0\) at each irrational. It cannot be drawn, it resembles no reasonable algebraic formula, and yet it satisfies, point by point, the existence and uniqueness this chapter took as its definition. Dirichlet's function was not a whim: it was a thought experiment designed to separate the essence of the concept, the univocal correspondence, from everything that until then had seemed inseparable from it, the formula, the absence of jumps, the possibility of tracing it with a continuous movement of the hand.
One last step remained, taken between the end of the nineteenth century and the middle of the twentieth, with Cantor's set theory and the work of Peano, of Hausdorff and, later, of the group of mathematicians who signed with the collective pseudonym Bourbaki. If a function is any univocal correspondence, and a correspondence is nothing but a set of pairs, then the function could, and for the formalist school should, be defined entirely in the language of sets: as a subset of the Cartesian product of two sets, subject to existence and uniqueness, and accompanied by the sets of departure and arrival. It is the definition with which this chapter opened, including the convention of §2.1 that the codomain is part of the function. What is remarkable about this step is not only its rigor but its economy: no new object had to be invented. It was enough to use what already existed, sets, ordered pairs and membership, together with a precise condition on how to combine them. The function, which had begun as a nameless astronomical table, ended up, more than two thousand years later, as a particular case of the most elementary notion of modern mathematics: the set.
This journey, from the table to the curve, from the curve to the formula, from the formula to the arbitrary correspondence and from the correspondence to the set, is not a succession of technical refinements in which each stage corrects an oversight of the previous one. It is the story of an idea that resisted, at each stage, being captured by the language available: the Babylonians had the correspondence without the rule; Euler had the rule without the generality; Dirichlet had the generality without the language of sets to ground it. One may ask whether the set-theoretic definition with which this chapter began is really the end point of that story, or whether it is, like each of the previous ones in its time, the most precise one available until a new class of objects, as Fourier series did in their day, forces the concept to be stretched once more.
There remains, to close, a question that already appeared at the end of §2.1 and that this journey only deepens. The requirement of uniqueness, that each cause correspond to a single effect, is the condition that separates a function from an arbitrary relation. But does that requirement describe something nature already possessed, which Dirichlet and his successors managed to name precisely after more than twenty centuries of partial attempts? Or is it a decision, as artificial and as fertile as any other axiom of this book, that mathematics imposes on the world in order to begin calculating about it?