**On â€œoverspecificationâ€ in medieval mathematical diagrams**

*Gregg De Young*

Abstract:

In a recent paper, ChristiÃ¡n Carman advanced a tentative

explanation for â€œoverspecificationâ€ in medieval mathematical diagrams.

Carman argues that the original (â€œcorrectâ€) diagrams were corrupted,

presumably through incompetent copyists, while preparing the initial

copiesâ€”often before the tenth consecutive copy. The diagrams then stabilized in an overspecified form and resisted further changes, sometimes

for centuries of copies thereafter. I feel hesitant about this hypothesis

for several reasons: (1) it assumes that the first Greek diagrams were essentially identical to modern diagrams; (2) pre-modern overspecification

is ubiquitous and is rarely reversed; (3) the hypothesis ignores differing traditions of perspective; (4) the informal tests used to support the

hypothesis do not precisely mirror the medieval copyistâ€™s activity.

Please could you explain the concept of ‘overspecification’ in more detail?

Basically, it means that diagrams are drawn with greater restriction than needed or required by the mathematics. So , for example, Euclid’s famous “Pythagorean Theorem” proposition says that the squares on the two sides of a given right angle are together equal to the square on the hypotenuse. But a majority of medieval diagrams that I have examined draw the right triangle as an isosceles right triangle. The use of the isosceles triangle is more restrictive than is required by the basic mathematical principle. So when J. L. Heiberg edited the Greek text of Euclid in the 1880s he drew the diagram of proposition I, 47 as a scalene right triangle, ignoring the evidence of the manuscript evidence where almost every copyist has produced an isosceles right triangle.

Professor Carman wants to claim that the overspecification in the diagrams is an accidental result of the copying process. I am unconvinced because I see overspecification in medieval diagrams as far to frequent to be explained as an accidental effect. It does not seem likely to me that all this overspecification arose simply as an accident in copying the early manuscripts. And even if overspecification did arise in the copying process, why should it then be preserved and recopied and repeated in so many manuscripts?

Incidentally, Professor Carman has recently published a follow-up article on his hypothesis in the newest issue of Centaurus — in case you would like to follow up on his idea.

Thank you for your interesting paper and presentation! Iâ€™d like to echo Leonieâ€™s question. As a cognitive scientist I am unfamiliar with this use of the term â€˜overspecificationâ€˜. Would it be accurate to say that the diagram at the left of your first slide is â€˜overspecifiedâ€™ because employs a geometric case (isoceles triangle) that is not strictly required by-is more specific than-the corresponding proposition? What are the criteria for a component of a diagram being overspecified? I find this concept to be very interesting with respect to how we might evaluate information equivalence between propositional and diagrammatic representations. Thank you!

The idea of over-specification with respect to diagrams vs sentential representation is well-studied. A canonical example is the sentence “Kyoto is west of Tokyo”. Any distance-preserving map-like representation is going to be unable to represent just this sentence because it must represent a specific distance between the two cities.

I apologize that I have not yet read this paper, so perhaps this is noted, but the same problem occurs with geometric diagrams. To illustrate the Pythagorean theorem you have to draw _some_ right triangle. If you draw a 3-4-5 right triangle, then you are being over-specific, if you draw an isosceles right triangle you are similarly over-specifying.

The mystery, it seems to me, is not that the diagrams are over-specific (that is necessary) but the stability of the particular over-specific example that was chosen (perhaps for aesthetic reasons?)