The Mathematical Canon
If one of our students progressed through a classical education without reading any Homer, or any Chaucer, or any Austen, or any O’Connor, we would raise an eyebrow. As Austen might say, it is a truth universally acknowledged that there is a canon of great literature, a collection of authors and books pointing us towards truth which every person ought to read. The opening lines of Pride and Prejudice are, commensurate with Austen’s wit, satirical. That every man in possession of a good fortune is in want of a wife is neither true nor universally acknowledged, so too is the universality of respect for the canon. As a result, the canon has waned in the last century as literature programs turn their gaze to the shiny baubles of fashionable texts.
Classical education, however, has rightly resisted that impulsive gaze. In fact, we build our literature programs on the back of that canon. If we instead built a literature program on the back of SparkNotes (and argued amongst ourselves whether Cliffs Notes was not the superior notes), we would serve our students ill. But that’s exactly what we do in mathematics.
The mathematics textbooks we offer to our students are not the canon. No one would mistake them for great writing. If they contain great and lasting truths, they work very hard to conceal them in the camouflage of “relevance.” They are, indeed, something like SparkNotes—a product of committee work and continuous reduction. Math textbooks, though, are worse even than SparkNotes, whose organization at least acknowledges that there is a canon while mediating student access to it.
Our math textbooks, on the other hand, are written as if a canon of great texts and ideas simply does not exist. Any reference to a mathematical canon they acknowledge is parenthetical – perhaps a callout box noting that Euclid was a Greek geometer or that the triangle theorem was named after a certain Pythagoras.
“Yes”, some might argue, “but there is no other way. There is a canon of great literature, and music, and history. Those are, after all, arts. But mathematics? How could there even be a canon of great mathematical writing?
“Mathematics is (surely, of all the fields) a cumulative and progressive field: the newer work improves upon the older. We prove more theorems, adding to our total knowledge of the world. This picture of mathematics undercuts the very idea of a canon, which hinges on some old works being great. But old mathematics is a curiosity, supplanted by newer, more correct, work. Old mathematics is the domain of the antiquarian, not the mathematician. After all, Euclid didn’t even know about zero!”
If we wanted to respond to that line of argument, the historian and philosopher of science, Thomas S. Kuhn, offers us a framework for thinking about literature and mathematics. In his The Structure of Scientific Revolutions and The Copernican Revolution, he suggests that the rosy enlightenment picture of natural science as a cumulative and progressive endeavor is not quite right. Marshalling examples from chemistry and astronomy, he argues that science rather proceeds through periods of “normal” work and “revolutionary” work—what the Greek mathematicians might call “continuous periods” and “periods of discontinuity”.
Normal work is puzzle solving and exploration within a particular framework. The work of an author in a normal period is broadly continuous with the work of other similar authors, answering the same sorts of questions with the same sorts of tools. We might think in history of the paradigm of ancient biographers. What Tacitus does in his Agricola is broadly similar to what Plutarch does in his life of, say, Flamininus. In mathematics we might see something like Calkin and Wilf’s delightful Recounting the Rationals as an example of work in a normal period. In contrast, revolutionary change from one framework to another is what Kuhn calls a “paradigm shift.” During periods of revolutionary work, we see authors breaking out of the frameworks and tools that have been predominant and doing something distinctively new.
How does Kuhn’s framework apply to the idea of a canon? Well, the idea of a literary canon makes sense because literature is a paradigmatic pursuit. Authors explore the space of a particular paradigm. Austen explores the space of the novel as comedy of manners. From time to time, an author chafes at the constraints of a paradigm and writes something revolutionary, creating a new paradigm for other authors to explore. We might think of the movement from chorus-based Greek drama to the Greek and Roman comedy of stock characters in this way. Literature, then, is not wholly cumulative and progressive. Whatever Catullus is about, it is not the result of continuous progress from Homer. Authors in different paradigms are doing different things.
Can we think of mathematics, then, not as only cumulative and progressive, but also as a paradigmatic enterprise? If so, all of a sudden, the idea of a mathematical canon makes sense. Just as in one age the Romantic novel dominates, and in another, people are quite taken by realism, so too in one age, the Greek conception of arithmetic dominates, while in another the geometrical algebra of Descartes rules the day.
Indeed, one of the things that our unjustified notion of mathematics as wholly cumulative and progressive obscures is that there have been several revolutions in mathematical thought of similar magnitude to that which Copernicus ushered in.
Let me offer two instances of revolutionary change by way of example.
First, is the revolution in the idea of number that Diophantus brought to arithmetic. Euclid offers us a clear (if perplexing to modern ears) definition of number: a number is a multitude composed of units. For Euclid, things like 2/5 were not numbers. Numbers were the result of counting, like two and three and so on. The root of number is the unit—one—a principle of unity. So when we count, we count by some unit, say feet or cats or pure monads.
But Diophantus, in his Arithmetic, works arithmetical puzzles where the result is something like 9/2 (II.6 is one such puzzle), and while he is careful to impose conditions on his puzzles that exclude negative results, he seems fine with fractional results. How should we understand this? Well, Euclid is keen that all number should have a unit, a principle of unity. 9/2 is the result of counting a principle of unity: halves. We count nine of them. So Diophantus, working in the Euclidean tradition of Arithmetic, brings a paradigmatic change to the conception of number.
Another example: the humble parabola, which we introduce in seventh or eighth grade, has undergone revolutionary change. Apollonius thought of the parabola as a conic section—a continuous curve produced by cutting a cone. But after Descartes, we think of the parabola as a collection of discrete points that obey a certain algebraic rule. The two approaches are as different as Ptolemaic and Copernican astronomy, and both are valuable.
Even if all of this is true, what does it matter? Well, if the goal of our education is to prepare students for standardized tests, not a whit. But if our goal is to prepare students to live good lives, we ought to engage with the mathematical canon as surely as we engage with the literary canon. Imagine an education in English that never involved reading any great works of literature. Who would love that? Who would emerge from that with a desire to read as an adult? Many people, young and old alike, proudly declare, “I am not a math person.” Perhaps some of them might find in the mathematical canon something to love.