The most important thing to know about ancient Greek mathematics is how little anyone knows about it. The scant evidence available today is tremendously indirect: reconstructions from unrepresentative survivals of fragments of translations of transcriptions of commentaries on compilations of summaries of allusions to refutations of excerpts of documents produced as part of an oral culture of learning in which the original writing may never have been expected to encapsulate what really mattered. Many centuries separate the people we want to know about and even the oldest materials we have to know them with, and most of what they did and thought is simply and definitively lost.
But that is only the start of the difficulty. Ancient thinkers lived in a very different world, mathematically and otherwise. Much of what we take for granted in mathematics today pushes that world farther away, from such rudiments as numbers and arithmetic to grand conceptions of the universe and the reasoning of mathematical demonstrations. Every generation rewrites the knowledge of its forebears, and Greek mathematics has been rewritten by more generations than it’s even possible to identify.
Reviel Netz, a professor of classics at Stanford University, California, caught the attention of more than just fellow classicists when he confronted this hard problem by making it vastly harder. His breakout book, The Shaping of Deduction in Greek Mathematics (1999), proposed that the key to getting into the heads of ancient mathematicians was to focus on the facet of their work that has survived least well across the eras: their diagrams. Marshalling an erudite array of highly indirect inferences about how diagrams must have functioned – conceptually, argumentatively, socially and so on – Netz demonstrated an exciting method to draw new conclusions from a vexingly small and problematic set of sources.
Netz’s elucidations of ancient scholarship have continued in that vein from provocation to provocation, stirring the interpretive pot with inventive ways to reimagine what old and obscure documents tell us about far older and obscurer worlds. Many of those provocations show up in his New History of Greek Mathematics, a lively introduction to ancient Greek mathematics and mathematical astronomy that strives, by the author’s signature methods, to help readers glimpse the minds of the iconic thinkers of Greek antiquity.
Rather than replace encyclopedic expositions of the available evidence, this new history attempts a more selective world-building toward a higher-level account of what mattered and why. Netz joins continuous attention to how scholars know what they know with an expansive willingness to speculate when evidence is indecisive or unavailable. Well-informed guesses and rough estimates, in his hands, prove to be powerful guides.
The book tacks back and forth between mathematical, biographical, sociological, philological and other ways of characterising the past. Its extended technical discussions won’t teach you much mathematics, but they succeed in conveying something more elusive: a sense of the weirdness and excitement of a long-gone way of reasoning. Netz tells us what an ancient reader ought to have found obvious or difficult or surprising. (Though he is authoritative, his views are not always the consensus in a field with much room for disagreement.) This, combined with explanations of the material, political, philosophical and other contexts of these texts, provides the key to their social and intellectual significance.
In Netz’s view, the association between ancient mathematics and a small set of iconic authors is not just an accident of how few ancient texts survive. Rather, the distinctive mathematical accomplishments of Greek antiquity derived from the emerging importance of named authors and the social motivations to stand out by claiming exciting and provocative results. (As someone who built his own name and career on erudite provocations, Netz is not shy in empathising with like-minded ancient colleagues.) These authors formed generational communities, and engaged in intergenerational projects to comprehend, reimagine and outdo their contemporaries and forebears.
This model, driven by ambitious and creative named individuals, thrived only briefly in the grand scheme of world history. It set ancient Greek mathematics apart in style and content from the mathematics of bureaucracy and statecraft that appears to have dominated much of the rest of the ancient world, and that would later transform the Greek mathematical heritage.
The hero of the story is Archimedes, a generation-defining genius who epitomises for Netz the provocative spirit of the Hellenistic heyday of Greek mathematics. As for the other famous names of classical mathematics, readers who have mainly encountered these as isolated eponyms in maths class will find them refreshingly posed as participants in historical conversations that put the concepts and works associated with them in new light. With some famous figures, notably Pythagoras, the illuminating point is rather their absence from these conversations.
Halfway through, Netz’s treatment of (briefly) worldly and (at length) heavenly mathematics forms a shorter book within the book, starting over chronologically and revisiting the story with different emphases. Where the ancient mathematics that most interests Netz was largely indifferent to the concerns of these middle chapters, they are necessary to explain the legacy of that mathematics from Late Antiquity into the Renaissance and even the steam-powered industrial revolution. These chapters also allow Netz to explore the Antikythera Mechanism, a remarkable device whose interpretation has challenged generations of historians of ancient science.
The hero of the story is Archimedes, who epitomised the provocative spirit of Greek mathematics in its heyday
His provocations are effective, even thrilling, regarding the times and places he knows best. His deep familiarity with the world of Archimedes in particular allows him to venture hypotheses that convey rewarding insights even when the claims are arguable.
His virtuosity with limited and indirect evidence of the ancient Greeks is based on a profound expertise that regrettably does not extend to other ancient worlds or to modern history, with its very different sources and methods of interpretation. It is deeply questionable, for example, to use modern ‘simple societies’ and ‘underprivileged students’ as proxies for ancient cultures that did not leave written mathematics, especially in the haphazard way Netz invokes them.
He is exactly right to emphasise that efforts to understand ancient Greek mathematics were vital to early modern science, a fact that left a lasting mark on the interpretive challenges he lucidly presents regarding ancient Greek sources. But his conclusion extends this essential point to an almost flippant hypothesis about the Greek origins of the scientific and industrial revolutions, premised on a view of monolithic achievements of distinct ‘civilisations’ that Netz’s own analysis of the interconnections and heterogeneity of the ancient world so clearly undermines.
Rather, the unequivocal legacy of ancient Greek mathematics that shines brightly in the book is the perennial importance of trying to empathise with those from radically different worlds. This was as true for the generations of ancient Mediterranean authors who struggled ingeniously to make sense of each other as it has been for the later scholars who helped make the modern world in part by trying to reimagine a very peculiar ancient one. New histories of Greek mathematics are vital beyond such specialist enclaves as Stanford’s classics department because the challenge of putting oneself in another’s mind is always being renewed, from one generation to the next.
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