Observing Whiteboards in Mathematics

What I saw at Marco Panza and collaborators’ Euclid session at Chapman University this Friday October 13th:

Looking, right now at mathematicians workshop looking for a non-standard model to a formal system of axioms, it is very interesting to see how the whiteboard is used. For example, by explicitly writing out the formulas, it raises the salience of the operations involved. Thus, once it was shown, by explicitly writing the relevant formula, that a certain parameter was being calculated by a quadratic equation, this immediately suggested the use of irrationals to find the desired non-standard model. 

Similarly, using ad-hoc formulas i.e. displays that belong to no actual standardized formal language but share a superficial grammar with them, for example, at some time Marco Panza wrote “SAS : Ang → Sea” to represent the fact that, in Euclid, the so-called Segment-Angle-Segment axiom (which they never called that way, always using the acronym “SAS”) states that, given certain background assumptions, the measure of an angle in a triangle determines the length of the opposite segment in that triangle. Someone even had to explicitly ask what the arrow meant. Under it, he wrote the analog case: “SSS : Seg → Ang” to represent the dual fact that, in Euclid, the so-called Segment-Segment-Segment axiom (which, again, they never called that way, always using the acronym “SSS”) states that, given certain background assumptions, the length of the opposite segment of an angle in a triangle determines the measure of that very angle. The choice of representation is to make salient only what is relevant. As you can see, a lot of information is elided, and very little is left: the initials of the name of the axiom – but not the whole name –, the direction of the relation of ontological dependance – but not the nature of the relation –, etc. Also, notice the “formal” analogies among both “formulas” means that they both, together, must be taken as a unit. Some may see something diagrammatic about this display, which again bears witness to the blurriness of the diagram/formula dichotomy in mathematics.

Also, this is a seminar on the history of mathematics, so some representational practices were observed that most likely would not be appropriate in other sub-fields of mathematics. In particular, I was surprised when José Gil wrote one of Euclid’s propositions in full in English prose, before proceeding to analyze the logical structure of its proof. After the seminar I had to chance to ask him why he had done so and he explained that since their goal was to reconstruct Euclid’s work, as presented in the Elements, it was important to have the Euclidean text, by itself, present. This, of course, raises a lot of historiographical questions about the appropriate representations of historical texts, for example, why was it appropriate to have the proposition in English instead of the original greek?

Thus, even though it is true that we can classify most of the representations on their whiteboards (there were five all over the room, but they only used two of them, which they erased more than once; most likely, because they were in front of the large monitor displaying the remote participants in the session) as either geometric diagrams or algebraic formulas, not all mathematical representations belong to one category of the other. These two examples are clear counter-examples.