Issues in doing research across areas
Norbert Wiener was an American mathematician who spent almost all his career at MIT. He is famous for being a child prodigy—when he was young; he is famous for being eccentric—his whole life. There are countless “Wiener” stories. And he is famous for deep results in many aspects of mathematics as well as work in applied areas.
Today I would like to talk about the claim that the best research is interdisciplinary research (IR).
For an example of pure theory, Wiener’s proof of the prime number theorem via analysis is one of the shorter proofs—but relies on a powerful Tauberian result of his. For example of IR, he created the area of cybernetics, which fell out of favor, but is back again—at least the word “cyber” is in vogue.
Wiener often worked on problems that were motivated by applied ideas. He would certainly be called one of the greats in this style of research. Edward Block, the Managing Director Emeritus, SIAM, ended a retrospective he wrote in 2005 by quoting from a front-matter statement in a 1964 volume of Wiener’s papers that is attributed to SIAM:
Professor Norbert Wiener (1894—1964) believed that significant research topics are to be found in the ‘crack’ between two fields. Motivated in this way, he spent much of his life in areas bordering on electrical engineering, physics, and biophysics. His exceptional intuition and profound understanding of mathematics exhibited to him a unity where previously only diversity had been in evidence.
Let’s start talking about IR, but first a story.
A not too short personal story. Years ago I was the director of graduate studies for the Department of Computer Science at Princeton. One of my jobs was to help manage the PhD. exam. It consisted of three written parts: one on theory, one on systems, and one on applications. During exam week, each candidate student took all three exams. These exams had been made up by each area group and were graded by them independently.
However, the final pass/fail decision was decided by the whole faculty on the Friday of that week. All the faculty meet for the big decision meeting, which was not just a simple adding up the scores. Faculty had input, and some students who did poorly might still pass if they had a champion. It sounds a bit unfair, but it really was a pretty good system. Since we had the raw scores as well as the additional information from faculty members, the results were pretty reasonable.
I would stand in front with the details on slides of how each student did on each exam, and we would discuss each case. Some years it went fast, most years there was some issues. But one year the faculty were quite upset. Here is what had happened:
- There was a system question, I will call S, that all the system students got right. But all the theory students got zero on it.
- There was a theory question, I will call T, that all the theory students got right. But all the system students got zero on it.
This enraged everyone. There were claims from theory faculty that S was an unfair question, and dually there were the symmetric claims from the system faculty that T was unfair. We began to debate this…
Then I looked at the actual questions S and T, and pointed out a curious property: the questions were really the same. They had the same answer. The only difference was the language that was used. Everyone was taken aback. The system question asked for a method to mark a data structure—garbage collection was the expected answer. The theory question basically asked for a method to search a graph—breadth-first search was the expected answer.
I recall two things. The result was that the students all passed. And we had a lengthy discussion on what was wrong with our teaching if the simple change in language threw the students off so badly. We reached no conclusion, although I made a suggestion that was never implemented.
I suggested that we mix all three days’ questions and divide them into three exams: exam I, II, and III. The students would be given the exams on successive days. Since the questions were mixed together randomly, they would not know which questions were systems or theory or applications. I thought this might be pretty good, and so did almost two thirds of the faculty. But the other two thirds of the faculty hated the idea and so nothing happened. Yes,
sometimes—especially when one counts faculty opinions.
The fundamental issue that I came to realize was this simple insight: Problems do not come with labels. Students should not expect that they will be told in the future that this problem is a theory one, and that is a system one, and so on. They should know that problems in real life come without labels—there may be no simple marker that suggests what type a problem is.
Each field has its own language which acts a barrier for IR. In the exam case the simple switch from asking about graphs to asking about linked data items threw our students. One of the issues that we face in IR is the lack of agreement on what is meant by simple words or phrases. This is obvious, but still a barrier that can be difficult to overcome.
Perhaps more vexing is the views that each area takes. Let’s look at the simple assertion that . What does this mean in different areas?
Mathematics: Here we mean that is exactly equal to . No exceptions.
Physics: Here it may mean that is very close to . In math we might emphasize this by writing
But physics is full of equations that hold approximately and that is just fine.
Biology: Here it might mean that usually is . For example, consider the case of how DNA bases pair up:
This is a famous rule that helped unravel the original structure of DNA and led to the breakthrough of James Watson and Francis Crick. But it is not always true. I worked years ago with biologists and discovered that it was more a guideline than a rule. In nature sometimes wrong pairs did pair up.
Economics: Perhaps here is even more complex. For instance, Ken remembers taking an economics course in 1979 that featured the St. Louis Equation,
Here is the change in national income (reckoned as GNP), is the amount of money extant and the federal spending in time interval , and is a constant while the and are weights.
Not only are there variants on this equation—which is really an approximate model—one can find numerous personifications of these variants. One paper asked, “Does the St. Louis Equation Now Believe in Fiscal Policy?” while another presented “Democratic” and “Republican” versions. The same course began with the equation
where for time period , stands for consumer spending, for investment, for the same thing as , and for exports minus imports. We can combine these equations, but are we intended to? What about the other definitions of on its Wikipedia page?
Cybernetics in Full
Cybernetics can be treated narrowly as meaning control systems for electronic equipment. The original vision and proper meaning is broader and more foundational. Wiener’s 1954 book Cybernetics was subtitled, “the control and communication in the animal and the machine.” In it he stated:
the purpose of Cybernetics [is] to develop a language and techniques that will enable us indeed to attack the problem of control and communication in general, [and] also to find the proper repertory of ideas and techniques to classify their particular manifestations under certain concepts.
Besides putting the emphasis on language, this made the problem one of studying how an animal or machine becomes familiar with its environment and influences it. The advent of the Internet as an organism for research is causing us to re-evaluate the kind of communication and familiarity needed.
Wiener even probed the issue of background and environment in regard to empirical science:
What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.
What assumptions are we making when we formulate problems for a research agenda, and what if they are orthogonal to those of a prospective partner in another discipline? What aspects of communication might fall through the crack and make it hard to find the common research topics inside the crack?
The question is simple: how can we carry out IR with these roadblocks? What is the best way to overcome them?
[fixed formatting of Block quote and crack photo.]