Sociology of Mathematics

In the Science Wars of the 1990s, 'postmodernist' scholars claimed that the social structure of the scientific community influenced the development of scientific thought and argued that this influence somehow invalidated science. To the working scientist, it is obvious that the course of scientific development is influenced by the humanity of its practitioners and is not merely a function of some predetermined logical path. Most scientists (with the delightful exception of Alan Sokal) ignored their postmodern critics, allowing flying airplanes and nuclear power plants to argue in their stead.

Mathematics, like any science, is a human enterprise and is not driven solely by logical or scientific forces. The social structure of the mathematics community plays an important role in determining what questions get attention and funding. What determines which questions mathematicians find interesting?

For the purpose of discussion, I will divide the universe of mathematical problems into three nebulous subsets. (i) There are problems which arise directly from science and engineering questions. For example, physicists might determine that some aspect of high energy particle physics is modelled by a previously unexplored system of partial differential equations. Do these equations have solutions? Are the solutions physically reasonable? Questions like this fall directly in the laps of mathematicians. (ii) Mathematicians attempting to answer questions arising in science or engineering repeatedly observe similar types of mathematical problems arising in disparate fields. They abstract common elements from these similar problems and try to develop a theory which will provide a general framework for simultaneously answering many problems arising in diverse fields. (iii) Pure mathematics generally refers to questions and investigations whose applications to the external scientific and engineering community are neither obvious nor motivating.

Mathematics has an amazing history of developing tools and ideas decades - or even centuries - before applications of these tools to science, engineering, or business are discovered. For example, in my youth, I often thought of number theory as coffee table mathematics, meaning it was pretty but of no practical value. As so often happens in mathematics, however, techniques developed to answer abstract questions in number theory became the foundation of the modern cryptographic systems employed by the world's financial institutions. This transition from pure mathematics to very applicable mathematics happens frequently and blurs my above division of the mathematical universe.

Mathematicians working in problems of type (i) often rely on the judgments of their nonmathematical collaborators in assessing the relative importance of their work. Hence we will shunt the sociological questions associated with this section of mathematics to other fields of science. Problems of the second type are a hybrid of the first and third; hence we focus on the third species. Problems in pure mathematics gain prominence in several ways, including: longevity, centrality, and social promotion.

Problems which have been well known for many decades or centuries but have resisted the attacks of well known mathematicians become more "valuable" with each passing year, even if their potential applications to science or mathematics have not grown in the interim. Mathematicians simply do not like to be defeated. Solving a well known problem which has survived a half century of attack will usually land you a professorship at a prestigious university.

In the development of a subfield of mathematics, problems often arise whose solution will greatly extend the scope of the subfield or perhaps simply fill an obvious gap in the theory. The greater the extension, or the more important the gap, the higher the solution is valued. Of course, the importance of the extension is still partially a social construct. Mathematicians develop their own sense of the relative importance of various aspects of their field, but it is impossible to ignore the opinions of their teachers and colleagues when forming these judgments.

Some prominent mathematicians are proselytizers. They identify an area of mathematics which they think deserves more attention and push others to devote energy to attacking problems in these areas. The more students, postdocs, invited lectures, and important publications they have, the greater the impact of their proselytizing. The greater your contributions to those problems deemed important by the community, the larger is your role in defining what the community deems important.

Hence, social constructs play an important role in the community's determination of the importance of mathematical work. This determination has coercive power through the grant, referee, and promotion processes. The great strides (as defined by the community...) mathematics has made over the last century argues that mathematics has functioned quite well subject to these social influences. In fact, only a believer in some abstract utopian vision of science should find the social aspect of science surprising.

Local Republican Politics

This summer as the healthcare battle heated up, I decided to explore the local Republican organization. I've voted Republican and contributed to Republican candidates my entire adult life but have never participated in the party itself. (I was also exploring a congressional run this summer and was trying to understand the landscape). So, I began regularly attending county party meetings and exploring local splinter groups too. The exposure has been educational and often amusing.

The county organization is split into two factions. These factions have no discernible ideological difference; they simply hate each other. I have not yet ascertained the origins of their animosity. I am frequently caught in the crossfire and have found myself shunned by one side if I spend too much time speaking to members of the opposing group. The main goal of each monthly meeting appears to be for one faction to gain dominance over the other. Little other business is accomplished.

The local Liberty Caucus is more energetic than the county party apparatus. Unfortunately, the only meeting I have attended thus far was dominated by conspiracy theorists (apparently the government is suppressing the information that our oil reserves are actually as large as Saudi Arabia's) and recommendations for stocking up canned goods for the impending disaster. I plan to return soon to see if this meeting was an aberration.

My county is dominated by University folk, and Democrats outnumber Republicans by about 4 to 1, but I am hoping that disgust over Obamacare will lead to one of those rare elections where Republicans can capture the local Congressional seat (if only for two years). I have been very attentive to the Republican Congressional primary, especially as I had considered entering the race myself. Our two leading candidates are a nativist/racist and a Libertarian. I find the nativist's views repugnant; moreover, he triggers a visceral negative response. I listen to my viscera. (Render unto Math those things that are mathematical and unto the viscera those things that are emotional.) The Libertarian is a pleasant man and is reasonably bright, but he has done a good job of ensuring that I describe myself as a Conservative instead of a Libertarian. While Libertarians share the Conservative desire for smaller, less intrusive government, they strike me on closer inspection as being as otherworldly as the Liberals. They also have a penchant for spherical horses when analyzing foreign policy issues. I find their notions on currencies bizarre. My local candidate advocates allowing the introduction of competing currencies in the U.S. He completely ignores the gross economic inefficiencies introduced when comparisons of relative costs and returns on capital are obscured by multiple units of currency. Who wants to hedge currencies when transporting Texas gas to New York? Our Libertarian candidate also obsesses over the Federal Reserve. While I understand his objections - excessive growth of the money supply - I reveal my conservative inclinations. Rather than throw out the Fed, which has been with us through nearly a century of amazing economic growth, I would first like to remove the mandate for full employment from the list of Fed duties and see what ensues. Placing the stability of our monetary system in the hands of Russian mining interests by returning to the gold standard holds no appeal for me. In response to the Libertarian's attack on Fed generated inflation, our nativist candidate now preaches the virtue of inflation.

Forced to choose between a nativist and a Libertarian, I'll probably support the latter. There is less scope and support for Libertarian mischief in Congress than for nativist mischief. Moreover, I feel a nativist candidate would stain the entire local Republican party and retard its growth in this potentially dramatic election year.

The Economics of Tenure

The conservative blogosphere posts frequent rants against tenure and academic research. In this note I will make some observations about tenure so that future rants are better informed.

I am tenured and therefore have an investment in the current system. I do not assert that tenure is necessarily the best system, but it is the one that has evolved. (As a conservative, I always give extra credit to those mechanisms that the society or marketplace has developed over time.)

As a scientist, I give no weight to the claim that tenure in my field protects me from taking controversial positions. Mathematics is fairly immune from social controversy; hence the only reasonable interpretation of controversy in this case is dispute over the scientific value of my work. Tenure then allows me to do work that others do not value. That sounds like damning with faint praise. Grant seeking and salary protection largely negate the "protection" to pursue work underappreciated by the broader community. As long as a significant portion of my income arises from research grants and as long as my yearly raises (and self esteem) are strongly dependent on the quality of my research (as judged by the community - not me), I have a very strong incentive to produce work that the scientific community highly values.

More positively, tenure allows me to pursue ambitious projects which have high possibility of failure. Andrew Wiles's dogged and ultimately successful pursuit of the Fermat conjecture gives a wonderful - but rare - example of tenure's support of ambitious but risky projects.

If tenure only rarely fulfills its putative role of promoting intellectual risk taking (at least in my field), what role does it serve? First, it lowers the cost of employment. Faculty exchange some salary for employment security. This is especially important in fields like mathematics and physics where there is a mythology asserting that your best work is done early in life. In sports, the expected diminution of talent with age is accompanied with early career salaries that suffice to fund the rest of your life. In academics, your employment is protected, but you are expected to redirect your energies to greater teaching and administrative work if your research productivity declines in later years.

A second role of tenure is to improve the quality of the university faculty, as measured by the impact of their research. (I know this is probably of limited interest outside academe). Why should extremely rigid work rules improve the labor force? Academics are not often gifted managers, and they are very slow to fire poor workers. Offering tenure can be a lifetime commitment, akin to marriage. At elite institutions, faculty are extremely cautious about offering tenure. They vet tenure candidates with great care. Although they make mistakes, the result is a higher quality of researcher than I believe would result from a less rigid system. When departments hire faculty on short term contracts, they still rarely fire them except under extreme conditions. Hence universities receive an economic benefit (reduced salaries) and a quality benefit (better research groups) from tenure without a significant attendant loss in quality due to rigid firing rules. Of course if more academics become better business managers and become more willing to fire incompetents the last claim would be invalid.

The only part of this discussion which could apply to tenure for elementary through high school teachers is the possibility that tenure could be exchanged for lower salaries, but salaries at this level seem to be set more by political considerations than by the marketplace; so, the potential for economic benefit seems negligible.

Exchanging tenure for reduced salaries in the larger workforce would so interfere with employers' ability to respond to changing market conditions that it would surely lead to economic decline. Hmm. Perhaps only my tenure prevents me from seeing the relevance of this last idea to the university setting.

Minimax and sophomoric politics

A fundamental problem with our modern congress, troubling to both liberals and conservatives, is the corruption of the legislative process by monied interests and the attendant evils of crony capitalism. The conservative perspective is that this corruption is a consequence of congress's excess of power and its strong incentives to legislate favors for business interests in return for campaign contributions. The purpose of this post is to explore, in a naive manner, the root of this difficulty, and to formulate a framework for analyzing the problem. Our conclusion is that the fundamental conservative principles that (i) government legislation should restrict individual freedom as little as possible and (ii) should adhere to constitutional strictures, needs to be supplemented by a third principle, which unfortunately, will be as difficult to realize as the first two. We call this third principle the minimax principle.

Legislation which is narrowly focused and legislation which picks winners and losers is particularly likely to be motivated by or to stimulate campaign donations. The conservative perspective is that the government should generally not be in the business of picking winners and losers, but it is difficult to imagine any legislation which avoids doing so. Hence, at best our goal should be to minimize this problem.

The mathematical framework for considering such a problem cannot truly apply to any messy real world human endeavor, but it suggests tests for determining the reasonableness of legislation from a fiscally conservative viewpoint. The procedure that comes to mind is the minimax process, familiar from linear algebra.

Consider a piece of proposed legislation. Presumably it is possible to articulate the goal of the legislation. Given the goal, the fiscal conservative will want to minimize implicit restrictions on liberty and free enterprise caused by the legislation and also, of course, he wishes to minimize the cost of the legislation. Next he will wish to maximize the category of entities benefitted by the legislation. For example, if the purpose of the legislation is to boost employment, then why focus on Detroit auto workers versus Southern auto workers? Why focus on auto workers versus airplane manufacturers, etc. There should be a compelling reason for restricting to one of these subcategories. Of course there may not be enough money to fund all programs that fit into the desired category. So, perhaps the goal is modified and some element of the legislation is removed to economize. After removing part of the legislation, the natural category of affected entities may increase again, and we iterate the process. Either the process terminates with acceptable legislation or its scope blows up and becomes unworkable. To summarize, we view legislation as flawed if the beneficiaries of the legislation constitute a smaller group than the broadest group that naturally falls within the goals of the legislation.

The current solution to unaffordable natural scope is to fund those entities which lead to the most campaign donations. Perhaps the correct answer is that if such a minimax process never leads to an affordable answer, the legislation should be abandoned. Any such legislation is too particularistic to be worth pursuing. Important national goals such as the construction of interstate highways that one may wish to develop over many years as funds become available are not excluded by this scheme. One simply specifies the criteria used to prioritize the order in which the roads are built.

Perhaps the minimax principle is not readily amenable to the sausage factory world of legislation, but it provides one more axis for evaluating the quality of legislation.

Definitions, Tea Partiers, and Accusations of Socialism

Every year or two I teach a Fall multivariable calculus class. The enrollment in these classes is typically at least 40% freshmen. These freshmen must have earned a 5 on their high school AP BC calculus exam to be eligible to enroll in this class. Despite this credential, I often give a diagnostic quiz the first day of class. Question 1: define a derivative. Question 2: define an integral. Most of the students fail this quiz, even though a high school AP BC calculus class spends a semester studying derivatives and their applications and a semester studying integrals and their applications. I always find it disturbing that the students spend so much time computing derivatives and integrals but don't know what they are.

I suspect that one aspect of mathematics is not well known to the general public: mathematics is a language - the most precise language known to man. Mathematicians are obsessed with definitions and the precise use of words. Of course when we venture outside mathematics to the real world, all definitions become fuzzy, but we can think more clearly when we use and understand precise language.
In the war of words between tea partiers and liberals, we hear the tea partiers calling Obama a socialist and liberals deriding them for this rhetoric. Let's consider the substance of this exchange. As a mathematician, I begin any such discussion with a definition.
Definition: Socialism is an economic system in which the government owns or controls the means of production.

Fundamental elements of production which the government might wish to control are choice of what to produce, how much to produce, and at what price to sell. In a capitalist economy these elements are decided by the profit seeking owners responding to individual choices made by suppliers, workers, and thousands of consumers. In the socialist ideal, these decisions are made by a central planning body, motivated not by profit, but by the government's perception of the greater good of the society.
Because centralized planning need not reflect the desires of the suppliers, workers, and consumers, the classic liberal (now called conservative) contends that the government must coerce suppliers, workers, and consumers to supply, labor, and purchase at the rate the government desires. The outcome is a loss of individual freedom in order to achieve the "greater good."

Does this description fit Obama's signature achievement, Obamacare? The legislation mandates purchases of health plans, provides for government control of the content of health plans, and dictates limits on what medical providers receive for their services. Clearly this meets the criteria for a socialist enterprise. Obama has in fact moved a sixth of our economy to the centrally planned column. The left only disputes this because socialism has negative connotations for the majority of Americans. They have no argument on the merits. The much derided tea partiers are on the winning side of this exchange.