American Academy of Arts and Sciences
October 6, 2007
Robert J. Zimmer
President, University of Chicago
One of the questions I am frequently asked concerns the relationship between being a mathematician and being a university president. And as a mathematician, I am also frequently asked about the relationship between mathematics and music. These questions are generally asked with rather different tones. The question about the roles of mathematician and president often has an inadequately masked undertone of incredulity. On the other hand, the question about mathematics and music is generally asked with an optimistic hope of insight into some deep level of cognitive function.
An analogy I like to use about mathematics and music, and indeed about university presidencies as well, is that of a conductor of an orchestra. If you were a naïve person who knew nothing about an orchestra and you saw one play, you might comment that all the music is actually being made by the persons with the instruments. You might wonder why that person is standing there with a stick, waving his or her hands. Is the conductor actually contributing anything? One can ask an analogous question about university presidents. Isn’t all the real work of the university being done by the faculty and students, with the president doing something analogous to just waving his or her arms about? Some in this room may even harbor such suspicions.
One of the functions of a conductor is to illuminate the structure of the music. By structure, I mean how the components fit together and relate to each other to form a greater whole. The whole is not merely the union of the parts; it incorporates, in addition, the relationship of the constituents to each other. The orchestra is no more a collection of independent musicians playing than a city is simply the collection of its inhabitants or a person the union of cells. Similarly, a university is much more than simply a collection of talented faculty and students. Universities have a structure whose purpose should be to create a research and educational environment that enhances the work of individuals through a sometimes complex set of relationships, thereby making the whole greater. In fact, this structure makes possible what we understand as a university, and it is the health of this structure that is ultimately the president’s responsibility to foster and oversee.
Now let me turn to mathematics for a moment. A great deal of mathematics is in fact concerned precisely with structure. To take a familiar example, let us consider the humble triangle, which we all remember from plane geometry. At its simplest level, a triangle is just a geometric shape with three straight line segments as its sides. A naïve person, in looking at a triangle, might think there is not much more to say. If this were the case, much of plane geometry would amount to drawing straight lines and counting. But with a little thought, we realize that sides have lengths, and with a little more thought, we discover angles, which is really a subtler notion about the relationship of two lines. Now one has three sides, three lengths, three angles, and one can ask about the relationship of all these. In fact, the geometry of triangles that we all learned about many years ago is about the relationship of these constituents and how they relate to the whole, where the “whole” includes the question of what it means for two triangles to really be the same. If one simply observes the parts, namely three sides, and that they are there, without focusing on the relationships of the parts, the loss in understanding is dramatic.
This focus on structure and relationships pervades a great deal of mathematics. So as a mathematician, much as with an orchestra conductor, one’s job is to illuminate structure through the understanding of the relationship of the constituents, and how the various forces and constituents at play become incorporated into the whole. Writing a sophisticated mathematics proof is akin to orchestrating a collection of relationships between ideas into something more meaningful and illuminating than these ideas are by themselves.
Some of you are surely sitting there thinking that these remarks about structure and relationships could apply to almost any subject or activity that has any complexity and depth. This is largely true, but as a society we give inadequate attention to this perspective. Albert Einstein made an oft-quoted remark about trying to make everything as simple as possible but no simpler. The public discourse on a wide array of important topics most often focuses on only the first part of this admonition–making everything as simple as possible–but often ignores the latter caution–but no simpler.
Public discourse and public policy often lack a structural perspective, approaching problems by isolating one or two components. The multiple components of the problem, and, importantly, their relationships to each other, are often unacknowledged, unanalyzed, or unappreciated. Universities have a key role to play in these matters not only because they provide analytic understanding of these components, which itself is often not easy. They also can focus attention on and analyze the total structure and set of relationships, particularly (and this is an important caveat) if their own internal structures foster this activity. In other words, universities, at their best, can and should be a venue for the second part of Einstein’s admonition–but no simpler.
Interestingly, for certain problems, mathematics makes a return entry here due in part to evolving technology. Although the components of a triangle and their relationships entail a relatively small amount of information, many modern problems, while still about the relationship of components to each other and to the whole, entail managing massive amounts of data. The power of the digital computer has led to a new capacity for computationally oriented mathematics to contribute to reconceptualizing and analyzing complex structural problems, particularly as a tool for integrating the properties of components and their relationships into properties of a whole complex system. The increasing sophistication of modeling global climate change or the relationship of the human genome to organism-level properties such as health and disease, and the increasing sophistication of spatial or geographic methods in the social sciences, are but a few salient examples of this newfound power. The computational mathematics approach to structural complexity is promising in many areas; however, when applied to many others, it is still in its infancy, with its ultimate utility yet to be explored. Ensuring that universities are structured and equipped to deal with these evolving intellectual opportunities is itself an example of a challenge of university leadership.
My comments today have focused on a conceptual relationship between two sectors of my professional life. The one further comment about my professional life that I would add is how much I appreciate joining this distinguished collection of individuals who, taken together, form such an extraordinary whole.
The address appears in the Winter 2008 Bulletin of the American Academy of Arts and Sciences.