I write this book to convince you of the importance of studying ancient Greek history. I argue that Greek antiquity was exceptional and consequential. It was contingent – it did not have to be the way it was. And it was decisive, helping to shape what followed.

The notion of the “Greek miracle” is problematic for an obvious reason: it implies that some transcendent set of values was present in a parochial section of humanity. While anti-racist arguments serve to historicize this miracle and show how it is explained without reference to the identity of the Greeks, we should be on our guard concerning the potential racist ways in which discussion of the “Greek miracle” may be appropriated. The chapter surveys such racist appropriations and comments that we need, nevertheless, to come up with concrete accounts of the Greek miracle, precisely so as to refute such racism and also, and less obviously, we should recognize the way in which certain processes, begun with the Greeks, have a progressive political valence, the theme of the remainder of the book.

The chapter pursues the consequences of the claim that the Greek canon was made based on the performative qualities of its authors, emphasizing its internal friction. As such, it did not embody any timeless values. Its function could be replicated by other traditions influenced by it: first Roman, then European languages and then globally. There is no function today that is uniquely performed by the Greek literary legacy and, in this sense, there is no need to preserve the particular tradition of classical studies. Greek antiquity is worthy of study simply because of its pivotal role, but it essentially expired. And yet, the attitude of admiration toward this type of liberating past experience is a useful one to maintain, as part of an overall hopeful attitude toward the arc of the moral universe.

Early states converge on a similar, number-based, “algorithmic” theoretical science. In Greek mathematics we see a new science, based not on the anonymous teacher but on the named author, seeking fame. Such authors look for new, surprising results, and therefore couch them in the language of proof. The resulting body of knowledge of many surprising proofs has no precedent in previous societies. The generation of Archimedes adds considerable subtlety and brings this to the cusp of modern science. The new Greek departure intertwines with other traditions, especially in the Near East, giving rise to a number-based but also proof-based astronomy (that of Ptolemy) and to the Arabic algebra. In the Renaissance, efforts to creatively engage with this Greek legacy gave rise to the scientific revolution. The science we know is objectively valid, but also historically contingent; one of the contingencies making it possible was the new departure of Greek mathematics.

Literary canons are neither the embodiment of transcendent values nor the artificial construct of gatekeepers. They are produced by deep social and poetic forces. The Greek emphasis on performativity meant that the new Athenian democracy became hyper-performative with its drama and speeches and outspoken figures such as Socrates. This riveted attention in the Greek world and, as Athenian political power disappeared, Athens’ soft power could not be replaced (since new sites of power were not as publicly performative). This created a Greek model of culture as a sphere autonomous from the state, predicated on the internal friction of the democratic city. This made possible a science based on debate. Replicated by later cultural traditions, it provided the model for Roman otium, for scientific debates from the Islamic world onward and for the rise of modern European civil society.

The chapter ‘Concepts in Greek Mathematics’ by Reviel Netz problematises a set of assumptions commonly encountered in the literature on Greek mathematics, which typically derive from a supposedly objective, a-historical conception of mathematical theory and practice. In sharp opposition to that tradition, Netz raises the possibility that the purpose of engaging with mathematical concepts may have been different in antiquity than what it has been taken to be. He asks central questions afresh, for instance: why do mathematical texts begin with definitions, and what is the purpose of mathematical definitions and of axioms. In connection to these issues, he highlights new aspects of the relationship between Greek mathematics and Greek philosophy, between engaging with mathematical concepts and philosophical thinking. He also advances the thesis that the relations between mathematics and philosophy changed through the various eras of antiquity, as did mathematical concepts and the role of mathematical definitions. We should seriously entertain the idea that even mathematical concepts need to be viewed within a given historical and cultural context.

Greek mathematics may not have been unique, among the ancient mathematical traditions, in its very use of proof (in the last generation, Høyrup and Chemla have shown the role of proof in the Babylonian and Chinese mathematical traditions, respectively1). But in no other ancient mathematical tradition was proof so explicit and foregrounded. Other mathematical civilisations proved: Greeks did so self-consciously.

This chapter moves into a much faster clip. We still look at individual authors, but we take each briefly and consider the authors as tokens for entire civilizations (I also reduce, even further, my use of endnotes: other than direct quotations, I refer in general to the Suggestions for Further Reading). More than this: the chapter is decidedly not “the history of mathematics after the Greeks.” Its subject matter is much narrower: the way in which Mediterranean civilizations, from the Middle Ages onward, responded to the legacy of ancient Greek mathematics. Specifically, our subject matter is the reception of Greek mathematics in three cultures, which we survey in sequence, one section each. “Byzantium and the Making of ‘Greek Mathematics’” looks at the Greek-language tradition of mathematics in the Middle Ages, mostly in the city of Constantinople. Its main theme is manuscript transmission, and so, finally, we get a closer look at some of the primary sources informing the book so far. The following two sections engage with more original developments. “The World Made from Baghdad” concerns the Arab-language tradition of mathematics in the Middle Ages (and glances at its satellites in other languages—which encompass medieval Latin). We mention some essential new contributions—such as the rise of algebra—as we note the continuity of Arab and Greek mathematics. The same is true, to some extent, even of the last major episode—“The Renaissance to End all Renaissances”—the final section, dedicated to the return of Greek mathematics in early modern Europe. The makers of what is now known as the scientific revolution saw themselves not as revolutionaries but as restorers, and it is impossible to understand their project apart from the history, surveyed in this book, of Greek mathematics.