BY JOHN SANFORD Reviel Netz, an assistant professor of classics, might not have actually shouted "Eureka!" on a visit last year to the Walters Art Museum in Baltimore, but that's what he was thinking. A scholar of Greek mathematics, Netz was hanging out with one of his colleagues and frequent collaborators, Professor Ken Saito of the Osaka Prefecture University in Japan, when they flew together to Baltimore in January 2001 to look at a recently rediscovered codex of Archimedes treatises. "It was basically just tourism," Netz recalled. On
a lark they examined a theretofore unread section of A section from The Archimedes Palimpsest, which classics Professor Reviel Netz stumbled on during a visit to the Walters Art Museum in Baltimore. Closer examination showed the Greeks understood the concept of infinity. ROCHESTER INSTITUTE OF TECHNOLOGY, WALTERS ART MUSEUM, JOHNS HOPKINS UNIVERSITY
The Archimedes Palimpsest, as the book is called, is in terrible shape. (A palimpsest is a manuscript that has been written on more than once; in this case, a 13th-century Greek prayer book overlays the 10th-century script of the treatises.) The pages have been battered, gouged, scorched by fire and blotched by fungus. Without the use of computer technology, they would be mostly unreadable. But
when the palimpsest caught the attention of the great Danish
philologist Johan Ludvig Heiberg in 1906, the underlying script was
much more legible. At that time, the volume was in a library
collection in Constantinople -- present-day Istanbul -- and, until
Heiberg went to examine it, nobody seems to have realized its
importance; the book contained the ancient Greek mathematician's
previously unknown treatise on Heiberg's find made the front page of the The
buyer permitted the text to be displayed as part of an exhibition
titled
The ancient Greeks developed mathematics into a theoretical discipline. But conventional wisdom has always held that they disliked dealing with infinity because it's a messy concept. "Infinities give rise to all sorts of slippery problems," Netz explained. In the 17th century, however, mathematics underwent a fundamental shift, thanks mostly to the efforts of England's Sir Isaac Newton and Germany's Gottfried Wilhelm Leibniz, who built upon Galileo's work and who are credited with inventing calculus. In the 19th century, mathematicians rebuilt the calculus to create a rigorous and precise "science of infinity," Netz explained. For the past 100 years -- that is, since Heiberg first looked at the palimpsest -- scholars have known that Archimedes toyed heuristically with concepts of infinity. But what Netz and Saito found in the palimpsest was that Archimedes actually had dealt with infinitely large sets in a mathematical proof. "It
has always been thought that modern mathematicians were the first
to be able to handle infinitely large sets, and that this was
something the Greek mathematicians never attempted to do," Netz
writes in an essay on Archimedes published in the Nov. 1 issue of
The proof in question is too complex to explain here, but suffice it to say that its shakes up the historical view of pre-calculus to its very foundations. And this is something Netz particularly enjoys. "As an undergraduate, I was told the given fact that a feature of Greek thinking in general is the abhorrence of infinity," he said. "We tend to think that cultures are monoliths, and I really like this example showing that cultures are not monoliths." In other words, it is impossible to pigeonhole ancient Greek thought or, for that matter, the intellectual culture of any civilization. People like to simplify forces that shape history, and this can lead to conceptually crude, underdeveloped ideas -- such as that the Victorians were sexually repressed. Archimedes, who lived in the third century B.C., is probably
best known as the protagonist in the apocryphal story about water
displacement -- if it were a Hollywood film, it would be titled
"The bath anecdote does not give us the true measure of the
man," Netz writes. "In |
Reviel Netz |

Stanford Report, November 6, 2002