Why We Graph Functions: A Story of Cultural Exchange
The ideas of functions and their graphs are so central to modern mathematics that many mathematicians, or at least mathematics students, are surprised to learn how slowly and recently the ideas were developed. The objects we now study with as graphs of equations, from vectors and parabolas to polynomials and linear systems, were imagined by their original discoverers or inventors in forms that we would scarcely recognize today. My intent in this article is to give a short history of how we got where we are today, what is special about the functions we practice graphing in high school algebra and geometry classes, and how these ideas fit into modern mathematics.
I will begin with the development of geometry by the Greeks and the development of algebra by Mesopotamian and later Persian and Arab mathematicians. I will then discuss the development of ideas leading to analytic geometry in the first half of the second millennium A.D., from Nishapur to the Netherlands. I will then give a sketch of the history of the function concept, and try to tie it together with the role of graphs since Newton and Leibniz’s work on calculus. Throughout this article, I will try to emphasize real-world motivations. However, this topic provides a good opportunity to illustrate how abstract, theoretical ideas are developed in mathematics, and it’s an opportunity that I intend to seize.
Mathematics as we know it began in Babylon, roughly a thousand years BC. The agricultural, engineering, and administrative projects involved in building societies between the volatile Tigris and Euphrates rivers required systematic methods for dividing up quantities of land and resources, for tracking the seasons according to the floods and the stars, and for designing buildings. It is not known exactly how these first mathematicians obtained their tables of numerical results, but it is likely that used some proto-algebraic procedures and some geometric techniques based on measurement of constructed figures.
Several Ancient Greek mathematicians later developed a highly sophisticated body of geometrical knowledge known as synthetic geometry, beginning with Pythagoras and culminating in the work of Archimedes, Euclid, and Apollonius. Theirs was a geometry focused on the intrinsic qualities of two- and three-dimensional figures: proportion, angle, area, volume, and circumference. They did not make reference to any kind of external coordinate system of the kind that we would now recognize, constructing their shapes not with coordinate axes but with a compass and an unmarked ruler. They did not have any real concept of coordinate geometry: they would discuss the length of a line, but would not find it meaningful to label points on that line with numbers akin to milestones or to markings on a coordinate axis. It is important to recognize as well that the Greeks did not think of these pictures as graphs of anything resembling a function. They were interested in cubes, spheres, and conic sections because these shapes appeared in engineering and astronomy and because they could be easily constructed with a compass and straightedge, not because they represented relationships between abstract quantities. That conceptual development would have to wait.
The rapid expansion of Muslim empires across North Africa, Spain, and the Middle East in the 7th to 16th centuries has often been called the Islamic Golden Age. It featured stunning accomplishments in medicine, architecture, governance, military strategy and technology, natural science, finance, law, poetry, and mathematics. Medieval Islamic mathematics borrowed heavily from the work of Indian mathematicians like Aryabhata and Bhaskara, who focused much more on arithmetic and what we would call word problems than on the geometry that so obsessed the Greek intellectual community. In order to simplify decisions in Islamic inheritance law, a mathematician named al-Khowarizmi wrote a book explaining the basics of what we now call algebra, after the Arabic word “al-jabr”, which means something like “completion” or “balancing”. Al-Khowarizmi (whose name also gave us the word “algorithm”) did not use any special notation: he wrote his textbook in plain Arabic and had no concept of a function or a graph. He was among the first, however, to solve numerical problems as their own entities, separate from geometry, and with a general procedure, as distinct from the ad hoc, case-by-case methods of the Babylonians. These techniques were expanded by other Indian, Persian, and Arabic mathematicians, who developed ways of solving algebraic equations that cemented the independence of the field from geometry.
The gap between algebra and geometry began to close early in the second millennium AD with the work of Omar Khayyam, a Persian mathematician and poet who used Greek techniques to solve certain special cases of the cubic equation, though not with anything resembling what we would call a graphical method in the sense of a graph of a function: his solution involved the intersection of a circle and a hyperbola. His key insight was that algebraic problems could be treated as geometric problems and solved with geometric techniques. Later, in Europe, the 14th-century philosopher Nicholas Oresme, who was probably not familiar with the work of Khayyam, conceived of a way of representing the intensity of a quality of an object, such as whiteness or heat, as heights above the surface of the object or along its length. This was a primarily philosophical consideration, relatively unconcerned with practical problems, but it contained the germ of the idea of a function as a quantity varying over an interval or a set of points.
Two more developments, both involved notation, were required before anything we would recognize as analytic geometry could emerge in Europe. The first was the widespread acceptance in Europe of Hindu-Arabic numerals. This process began with a seminal book by the well-traveled Italian Leonardo Bonacci, also known as Fibonacci in the early 13th century. However, the adoption of the decimal place-value system of the medieval Muslim world was not really complete in Europe until the 16th century, in time for the second great development: that of algebraic notation. Until this time, algebraic problems were stated and solved in plain language or in abbreviations within a text. There was no systematic symbolic notation, and as a result (or perhaps as a cause) no concept of operations acting on variables. This changed with the work of Francois Viete in the late 16th century, who solved several classes of polynomial equation with a notation that comes very close to our own. His notation was an obvious improvement, much like the Hindu-Arabic numerals, and it spread quickly. Crucially, both of these developments relied on the technology of printing, which transformed intellectual culture in a way that we can scarcely imagine.
The story, as far as we are interested from the perspective of high school mathematics, continues in the work of Rene Descartes and Pierre de Fermat, two French scholars of the 17th century, both interested in the behaviour of polynomial equations and curves in the plane. Descartes took what we might call an basically Greek approach, taking the revolutionary step of describing geometric figures in the plane, primarily conic sections and cubic curves, in terms of algebraic equations involving the lengths of line segments in the figures. Fermat went in essentially the opposite direction, examining algebraic equations in their own right and then constructing the curves that satisfied them. Descartes’ method did not become popular until it was translated into Latin, the European language of scholarship at the time, from the original French, but it was soon adopted universally and was central to the development of calculus by Newton and Leibniz.
Fermat and Descartes both conceived of rectangular coordinates, following a tradition that was begun by Apollonius of Perga 1800 years earlier. However, this is not the only way to describe curves in the plane: another method is known as polar coordinates, which specifies the position of a point in terms of its radial distance from a fixed origin and the angle between that radius and a specified polar angle. Moreover, when we consider more than two variables, we must go beyond two dimensions and imagine graphs in three dimensions, which are surfaces, or in four dimensions, which can be considered as changing volumes graphed against a time axis.
The graphical and notational advances of the Renaissance led to casual, intuitive discussion of dependent and independent variables, but there was no real attempt to formalize this idea until the 19th century. Most mathematicians for several hundred years were content to understand a function as a quantity whose value changed depending on another value, or perhaps as the algebraic expression that governed that change. These ideas were refined through the study of differential equations, which have functions as their solutions, and infinite series, in terms of which great classes of functions are defined. The formalism of the late 19th century and the increasingly general ideas in algebra and logic, however, required a reformulation of the idea of a function, which is now understand as a particular kind of set: that is, a set of pairs of elements of sets known as the domain and codomain, defining a mapping from the domain to the codomain.
The preceding discussion is long and abstruse, but now I come to the real crux of the issue: why do we study the functions of high school algebra and geometry classes? I gave something of an answer in the articles on trigonometry on polynomials: trigonometric and polynomial functions have simple, regular behaviour and they are easily evaluated, which makes them ideal for theoretical and numerical work. Moreover, we can now see that they have great historical significance: it was in terms of polynomials that the graphical techniques of algebraic geometry were formulated. Finally, I would argue that these functions are ideal pedagogically: exponential functions look like the kind of curve we draw at random when asked to illustrate a quantity growing at an accelerating rate. Polynomials, with their intercepts, peaks, and troughs, give an intuitive visual insight into the analytic problems of zeroes and extrema of functions. These functions are the building blocks of more complicated objects, not because of the nature of those objects but because of the nature of our system of algebra.
I hope that this has helped to clarify what we talk about when we talk about functions, and what we are doing when we draw their graphs. We are now ready for the next article, in which I will discuss linear algebra, a topic that provides undergraduate students with significant conceptual difficulties because of the formal and abstract style of its typical presentation.