I very much agreed with Hank’s recent post about this year’s HSS, so I thought I’d add my two cents. In particular, I wanted to say something about the “Making Mathematics: Models, Machines, and Materialities” panel. It was excellent; indeed, one of the best at this year’s HSS!
Although the presentations were quite diverse, the panel had a remarkably tight and coherent theme. Chris Phillips delved into the history of the chalkboard as a ubiquitous tool in American mathematical pedagogy. David Roberts talked about the late 19th century enthusiasm for “linkages,” that is, mechanical instruments that transformed circular motion into a perfectly straight line. Stephanie Dick explored the architecture of a mid 20th century geometry theorem proving machine developed at IBM. And Alma Steingart discussed the role of visualization in topology.
In her talk, Steingart argued that although Stephen Smale had conclusively shown that a sphere can be turned inside out in the 1950s, a large contingent of the mathematical community was not satisfied until a coherent visual account of the transformation had been supplied. This lead to a number of attempts to model the process using everything from chicken wire to advanced computer graphics imaging techniques.
Beyond its coherence, what made this panel so good was the fact that it explicitly engaged in an ongoing conversation about the importance of inscription techniques in mathematical theorizing. In some way or another, all of the talks helped cement the claim that diagrams, three dimensional models, and images are far from mere heuristics. Rather, they are often a constitutive element of theorizing as mathematical practice.
In this regard, I was especially intrigued by Stephenie Dick’s paper, which centered on a geometry theorem proving machine developed at IBM in the 1960s. What made it so exciting is that she added an ontological component to the usual epistemic claims about the role of diagrams in mathematical theorizing.
The program that Herbert Gelernter and his colleagues at IBM developed to generate geometric proofs was modeled on human cognition. In particular, Galertner explicitly tried to make the program mimic the geometric intuitions and proof generating strategies of a high school student. (Indeed, he seems to have had students at the Brooklyn Technical Hight School in Fort Greene in mind.) Among other things this included working backwards from a desired conclusion to the assumptions, rather than the other way around. Most interesting, though, is that it also involved drawing figures, shapes, and diagrams.
For each proof, Gelertner and his colleagues supplied the computer with a visual diagram. This was in addition to a modest set of commonly known results in Euclidean geometry and the usual logical transformation rules that allow you to transform one syntactic expression into another. One way I understood the role of the diagram in Stephanie’s talk is to think of it as a model — as something for a syntactic expressions to be true of. Working backwards from the conclusion, the computer was thus able to eliminate certain fruitless paths to the assumptions by ascertaining whether they violated the supplied diagram.
What’s so interesting about this? To my (admittedly limited) understanding, Stephanie was trying to go beyond just pointing out that here again diagrams and inscriptions played an important role in doing theoretical work. She also raised another, perhaps deeper question: what is it to be a diagram for a computer?
One way to understand the importance of diagrams for humans is that they offer us another way of thinking. Whereas equations help us think analytically, diagrams help us to think synthetically (or perhaps spatially, which may amount to much of the same thing). But computers are only capable of analysis. They are syntactic manipulation engines — transforming one string of digits into another. A provocative way to read Stephanie’s talk is thus to ask whether Gelertner was trying to give the computer a way to do something beyond merely manipulating strings of digits by supplying it with a diagram. The question of course is what this was. If the traditional syntactic symbol shuffling that computers engage in can be said to correspond to analytical thinking, then what does the computer do with a diagram? What is it to be a diagram for a computer?