De Rerum Natura

The Archimedes Codex is turning out to be a great read on the importance of diagrams in Greek mathematics, the transmission of ancient knowledge to present times and modern document forensic techniques. The book is written by two principals of the Archimedes Palimpsest Project. I just finished reading Reviel Netz’s explanation of why visual thinking has become so reviled in modern mathematics and it was so simple to understand that I want to share it with readers

As Netz explains it, the problem with diagrammatic proofs hinges on the fact that diagrams do not have the generality of language. For example, if one wants to discuss triangles in general, a diagram of a triangle thwarts that generality since, by construction, it represents a specific triangle. The ambiguity of language is turned to good use by turning it into an encompassing generality.

Netz argues that Greek diagrams are schematic not illustrative. Evidence from the Palimpsest and later medieval documents strongly suggests that Archimedes drew a polygon inscribed inside a circle with circular arcs rather than straight lines in his “Spheres and Cylinders”. He also drew straight lines for sections of a spiral in “On Spirals”. All of this is the reverse of what modern diagrams in editions of Archimedes would do. Our diagrams are illustrative, theirs were schematic.

This schematic versus illustrative distinction is Netz’s explanation for why the Greeks never made a logical mistake in their mathematical works even though the diagrams are central to their exposition.

This bias in 20th-century mathematics to visual proofs (Coxeter mentions that Hilbert thought geometry should be taught in a darkened room!) is now creating a backlash that could bring diagrams back into the heart of proofs. Take a look at Euclid and His Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry for how that most non-geometric machine — the computer — is making visual proofs rigorous.