Archimedes, particle accelerators and being visual

I feel like I was pulled into a little whirlpool of interesting bits of info this morning.  I was attracted to the title of David Castelvecchi’s blog: Archimedes and Euclid?  Like String Theory versus Freshman Calculus.   The blog reports the opening of an exhibition at the Walters Art Museum in Baltimore, showcasing one of three medieval copies of Archimedes’ works, the Archimedes Palimpsest.   The first interesting bit is that the text, which had been erased and recycled into a prayer book, was brought back to life with the use of a particle accelerator!

As the exhibition will display on panels and videos, imaging experts were able to map much of the hidden text using high-tech tools—including x-rays from a particle accelerator—and to make it available to scholars.

As Castelvecchi describes this treasure, he takes note of some of its significance as interpreted by Reviel Netz at Stanford University:

Reviel Netz, a historian of mathematics at Stanford University​, discovered by reading the “Method of Mechanical Theorems” that Archimedes treated infinity as a number, which constituted something of a philosophical leap. Netz was also the first scholar to do a thorough study of the diagrams, which he says are likely to be faithful reproductions of the author’s original drawings and give crucial insights into his thinking.

An LA Times article (linked to the Palimpsest website itself) contains a similar observation from Netz:

The X-ray image also revealed a section of “The Method” that had been hidden from Heiberg in the fold between pages. It contained part of a discussion on how to calculate the area inside a parabola using a new way of thinking about infinity, Netz said. It appeared to be an early attempt at calculus — nearly 2,000 years before Isaac Newton and Gottfried Wilhelm Leibniz invented the field.

One passage he studied several years ago involved the innumerable slices and lines that could be made from a triangular prism similar to a wedge of cheese. Netz said the passage, which was unreadable to Heiberg, showed that Archimedes was grappling with the concept of infinity long before other mathematicians.

Archimedes’ use of infinite processes, in addressing the value of pi for example, is well known.  But there are two details that get my attention in Netz’s evaluation of these ancient works.  One is that Archimedes imagined the numerical nature of infinity and the other is the perhaps newly appreciated visual aspect to his work.

Reviel Netz is a professor of classics and specialist in ancient mathematics and cognitive history at Stanford University.  In the book he wrote, The Archmedes Codex, Netz makes the argument that ancient Greek mathematics was visual in a way the modern reader of science and mathematics may fail to appreciate.  For the modern reader, mathematics ideas may be illustrated visually, but they are not established visually.  But Netz explains that the drawings of Archimedes are not illustrations, they are arguments.

Ancient diagrams are schematic, and in this way they represent the broader, topological features of a geometric object.  Those features are indeed general and reliable; a diagram represents them just as well as language represents them.  And so, ancient diagrams can form part of the logic of an argument which is perfectly valid.

We have learned, therefore, something crucial and surprising about Archimedes’ thought process, about his interfaces.  He essentially relied on the visual; he used schematic diagrams that can be used in perfect logical rigor without danger of error based on visual evidence. When Archimedes gazed at his diagrams along the Syracusan seashore…..I know that what he saw there was a crucial part of his thought process—one of the most basic tools that made Greek science so successful.

The entire discussion of Archimedes, inspired by this unexpected find, brings mathematics back, again, to the intuitive, even the sensory.  Archimedes was an extraordinary thinker, but his ancient world magnifies, for me, the very internal nature of his investigation.

There is a NOVA piece on the discovery of the hidden manuscript and on that page a link to a piece on infinity.

This piece contains a number of paintings by Quita Brodhead and contains the following note:

No fear of the infinite: Quita Brodhead was an abstract painter who investigated infinity in her “Endless Circle” series and other paintings, several of which illustrate this article. Brodhead died in September 2002 at the age of 101.

You can listen to Quita Brodhead herself here.  I recommend it.