On Disciplinary Boundaries

W. James Bradley

Let me begin with a story. I spent the spring semester of this year on sabbatical leave at Wilfrid Laurier University in Waterloo, Ontario. On my last day, I was invited to join a visiting speaker from Israel for lunch at the Waterloo University faculty club. Four us sat around the table – two Canadians, an Israeli, and myself, an American. We introduced ourselves and exchanged business cards. One of the Canadians looked up from mine and said, “Calvin College. Is that Presbyterian?” I explained that it was affiliated with the Christian Reformed Church, but that its roots were the same as Presbyterianism. He then commented, “But of course, the church affiliation is pretty much nominal today, isn’t it?” I replied that it was not and that Calvin College seriously tried to integrate Christian ideas with every academic discipline. Eyebrows raised all around the table. So I continued. “Let’s be honest,” I said. “None of us is ever really neutral about anything. We all have presuppositions and basic beliefs that serve as filters for every idea that comes our way. For instance, in the academy we are very familiar with feminist scholarship and (a few years back) Marxist scholarship. Christian scholarship asks what these various filters are, what role they play in people’s thinking, what presuppositions Christians bring to scholarship, and how they might lead to conclusions that are sometimes the same, sometimes different than one would draw from other presuppositions.” My Canadian colleague replied, “Oh, that’s very interesting. I can see how that would be very applicable to disciplines like history or psychology. But it certainly doesn’t seem to apply to mathematics.” “Well,” I said, “I think it does.” Now I really raised some eyebrows. I went on. “Let’s talk about the Enlightenment.” And I sketched briefly the Enlightenment perspective on mathematics and science, as sure sources of truth, as sufficient to solve whatever problems humanity might face, and as a sound basis on which to build societies that would provide for peace and the well being of their citizens. I asked whether the academic community still believes that and why. We went on from there and discussed David Hilbert’s concept that mathematics was a source of “presuppositionless knowledge.” My Israeli colleague had an extensive knowledge of Maimonides, a Jewish thinker that was a contemporary of Thomas Aquinas. He explained how difficult it had been for Maimonides (and other Jewish thinkers) to reconcile the abstractness of Greek thought with the Hebraic tradition. After about an hour, I sat back for a moment and looked around the table. Everyone was leaning forward, energetically engaged in the conversation. And I thought, “I am sitting in the faculty club of one of the leading technical universities of North America and we are discussing some deeply Christian ideas.” And I was so excited, I could hardly sit still.

Two principles stand out for me from this experience. First, there exists a strong felt need in the mathematics community for perspective on our discipline. Secondly, God has given us some tools that we can use to engage that community in discussion of issues that are important to Christians.

Let’s talk about perspective. Every discipline has to draw its own boundaries. Its representatives, through professional organizations and through dialogue in journals identify the discipline’s areas of concern and responsibility – its turf, if you like. Thomas Kuhn speaks of the “disciplinary matrix’ – a shared understanding of the kinds of questions members of the discipline ask, the kinds of methods acceptable for addressing these questions, and a value system as to what constitutes good work. Let’s consider computer science as an example. I’ve been around long enough now that I have been able to watch the evolution of its disciplinary matrix first hand. Computer science originated with the work of mathematicians and electrical engineers. Business people got involved very early and the language COBOL was written around 1960. A field of applied computing sometimes called Information Systems began to emerge. Many computer scientists were uncomfortable with this being regarded as part of computer science. In the 1980s, information systems had advanced to the point where it changed names – to software engineering. Computer scientists now included it within their boundaries but the engineers objected that anything with the word “engineering” in it was their turf. The result was many high level meetings. All of this reminds me of the adage about sausages – even though they taste good, you wouldn’t want to watch one being made. Nevertheless, this “nitty gritty” is how the boundaries of disciplines come to be defined.

Now let’s turn to mathematics. I’m going to jump directly to my bottom line. My thesis is that the disciplinary boundaries of mathematics have been drawn too narrowly. In fact, I see this conclusion as a consequence of attempting to think through a Christian perspective on mathematics. That’s my thesis. Here’s what I want to do with it:

1) Sketch how the boundaries of mathematics have come to be drawn where they are.

2) Illustrate what it would mean to broaden them.

3) Suggest some implications of these ideas for us.

In the Greek era, mathematics was closely tied to philosophy. Both Plato and Aristotle were inspired by mathematics and wrote a great deal about it. For Aristotle, each discipline had its own subject matter and its own laws. Mathematics was the science of number and space. He would not have thought in terms of mathematical models underlying subjects like physics, biology, or political science. Nevertheless, writing around 1600, Galileo expressed a different perspective. In a famous statement, he wrote:

Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.

A generation later, Rene Descartes wrote, “The long concatenations of simple and easy reasoning which geometricians use in achieving their most difficult demonstrations gave me occasion to imagine that all matters which enter the human mind were interrelated in the same fashion.” So mathematics had moved from being the science of number and space to the alphabet of the universe to the basis of all human thought, at least in the thought of a few scholars. But the work of Isaac Newton near the end of the seventeenth century opened the door for much wider adoption of such ideas. Using mathematics, Newton solved the problem of humanity’s place in the physical universe, a problem that had engaged the best minds for all of recorded history. Newton’s tomb in Westminster Abbey is one indicator of the extent of Newton’s impact on Western thought – it is much more prominent than that of many of the English kings! Within one or two generations after Newton, scholars put forth the basic ideas of the Enlightenment – that all problems of importance to human beings were solvable by reason and that such use of reason primarily meant the activities of mathematics and science.

Notice the pattern. The perspective on the range of mathematics concerns widened in the seventeenth and eighteenth centuries from Aristotle's relatively narrow view to the very broad Enlightenment perspective. But then in the 19^th century it narrowed to where for much of the 20^th century mathematicians saw the scope of their discipline as only being concerned with formal, abstract mathematical systems. How did this happen? In a sense, it was well intentioned. In much of the nineteenth century, mathematics faced some serious crises, first with the foundations of geometry, then the foundations of analysis, and then the foundations of logic. So separating mathematics from other disciplines was a way of stripping away all distracters and isolating the core issues. But this effort had the unpleasant side effect of isolating mathematics from the contexts in which it historically arose and mathematicians from their colleagues in other disciplines. For example, about a decade ago my own alma mater, the University of Rochester, faced a financial crisis. It had expanded its graduate programs too rapidly and found the cost of its graduate school exceeded its financial resources. So the administration decided to cut some Ph.D. programs. It carefully surveyed the faculty and found only one faculty member outside of the mathematics department who had had significant professional interaction with faculty in the mathematics department. So the administration concluded that mathematics was an isolated department, not making a contribution to the broader university and its graduate program could be eliminated. The mathematics department rallied its professional colleagues at other universities who protested loudly. The American Mathematical Society recruited several Nobel winning scientists who met with the Rochester administrators and argued that no major university could be without a graduate program in mathematics and the program was saved. However, major changes were made – the size of the program was reduced, some mathematical physicists were added to the department, and today the Rochester graduate program is probably stronger than it was in the early 90s. But it seems to me to be a very clear example of the unfortunate side effect of turning inward – isolation.

In fact, I see a number of hopeful signs that the mathematics community is becoming less isolated. For example, the National Science Foundation has been very intentional about balancing an emphasis on "applicable" mathematics and "core" mathematics. Mathematical biology has become increasingly important in the past 5-10 years, is introducing some new questions into mathematics, and is pushing many mathematicians to learn some things outside their field. The MAA now has special interest groups in the history of mathematics, philosophy of mathematics, and several other non-technical areas. Nevertheless, the legacy of a mathematics isolated from a broader context lives on.

So what would characterize a broader vision of mathematics? I want to switch gears here from the historical approach I've been following to a conceptual approach. I see two major problems. First, the discipline lacks a vision for God's purposes for mathematics. This is not surprising – the mathematics community is a secular community and its membership includes atheists, agnostics, and people from a wide variety of religious traditions. Furthermore, its historical roots precede the Christian era by centuries. Nevertheless, as Christians, we believe that God is purposeful and makes his purposes known to us. He has a purpose for mathematics whether people know it or not. And the lack of a vision for that purpose has unhappy consequences. Secondly, most of the historical development of mathematics has occurred within the context of western culture. One quality that has characterized most intellectuals in the West has been a belief in the power of autonomous human reason. Christians affirm that reason is indeed a gift of God, but also affirm that it cannot be separated from a broader context that includes emotions, beliefs, and personal commitments. That is, when used in a context in which individuals and communities sincerely desire to do God's will, reason is a valuable tool in helping us fulfill that desire. However, when separated from God's will, reason becomes a servant of selfishness and, when elevated as high as Western culture has elevated it, becomes an idol in which people worship themselves, or perhaps I should say, one aspect of themselves. Mathematics, because of its emphasis on precise definition, careful reasoning, and abstraction is very attractive to people with a belief in the autonomy of human reason. Thus mathematics became a principal expression of this belief, especially in the Enlightenment era.

So what then is God's purpose for mathematics? There's no direct answer to this in Scripture. One can't look in a concordance under "mathematics, purposes of." Rather one has to look at broader revelations of God's purposes and reason deductively to mathematics. The principal formulation that 20^th and 21^st century evangelicals have used of God's purpose in the world is the "Great Commission" – "…go and make disciples of all nations." Reformed people have tended to emphasize the "cultural mandate" – beginning from the "fill the earth and subdue it" verse in Genesis, Reformed thinkers have focused on building the Kingdom of God, including building communities and cultures and developing technology and the arts. But Joel Carpenter, Calvin's provost argued in a talk in September 2002, that these are two perspectives on the same mission. That is, making disciples means more than just making converts, but means people who are able to do God's will in all areas, that is, are able to build the Kingdom of God in all the ways the cultural mandate emphasizes.

I want to suggest that mathematics is an essential tool for carrying out this culture-building mandate. The role of mathematics seems to take two forms. First, recall the statement from Galileo I quoted earlier. Today we probably would not see the building blocks of our physical universe as simple geometric figures but rather as equations. Nevertheless, they are mathematical objects. We can't build a culture in this world without understanding something of the world in which we are building – its physical and social dimensions and its abstract structures. Second, if we want to construct a building, we need a design first. So someone prepares a blueprint – a mathematical model of the building we plan to construct. In fact, it's hard to see how we could build anything well without defining precisely what we want to build, trying out alternative designs, and seeing which works best. That is, one must engage in precisely the kind of thinking that characterizes mathematics. Hence I see mathematics as one of the fundamental tools that God has given us to carry out his purpose that we build the kingdom of God in this world.

This is a simple idea. But I think it provides a perspective that dramatically changes the way we draw the boundaries of the discipline of mathematics. That is, many issues that have been pushed to the periphery or even beyond it take on a new importance. For instance, the first questions I find myself asking are these: "If mathematics is one essential intellectual tool that God has given us to build his kingdom, what are the others? How does mathematics interface with them? What unique characteristics does it have that other modes of thought do not? What can it not do?" I’ve been able to find remarkably little written on these questions. Secondly, the history of mathematics acquires a new importance. That is, how has mathematics been used successfully to build culture? How has it been used inappropriately? What can we learn from such instances of the past? A few years ago, I examined the catalogues of the twenty top mathematics graduate programs in the United States. Not one required a single course in the history of mathematics of is graduate students. Their objective is to bring students to the frontiers of research, which is certainly a laudable goal. But this goal has been divorced from a broader context and an understanding of God's broader purposes for the discipline. Some knowledge of history could help give that context. Thirdly, the philosophy of mathematics becomes more important. For example, Christians have at times looked at reality as consisting of material entities and spiritual entities. So which is mathematics? It doesn't seem to be either. We could say, "Mathematics consists of ideas – abstractions." So then do they exist merely in people's heads? If so, do we each have our own concept of (say) a circle? Or does that concept transcend our individual minds? If it's transcendent, where does it reside? Is it merely a social construct? We could go on. The psychology of mathematics, its sociology, and its relation to language more broadly all become important. It's not the case that every mathematician needs to think about every one of these questions. Rather, it seems to me that such questions ought not to be marginalized. The boundaries of the discipline should be broad enough that such questions are seen as every bit as much a part of the discipline as formal axiomatics. Scholars who study such questions should be every bit as tenurable in mathematics departments as scholars who study more technical mathematical questions.

Furthermore, the perspective has many practical implications for us as mathematicians. Let's start from ourselves and work outwards.

Christian students often say, "I want to serve God, but I don't see how being a mathematics major would help me do that." Young mathematics faculty say, "I got into mathematics because I liked it and could do it well. I never thought about how it connected to my faith." They then add ruefully, "And now I've got to write this statement on the integration of faith and learning." One implication of the perspective on mathematics I am expressing here is that we can approach our discipline with the confidence that we are exercising a wonderful gift of God, that we can feel free to delight in it and thoroughly enjoy it. Understanding and applying mathematics are part of God's will for humanity. We can be grateful that God has given us the ability to do mathematics and has put us in life situations where we have the resources to pursue it. When we do mathematics in an attitude of putting God's will first, we can know that the joy and satisfaction we experience comes from the fact that we are doing something he has created us to do and that he is pleased with our doing it.

Secondly, we can use these ideas to help us select an area of mathematical research. We can say, "Where particularly are my gifts? What dimensions of building God's kingdom especially interest me? How can I use my gifts to serve in that area?" For example, a few years ago I heard a young engineer speak of her fascination with mathematical models of fluid flow, especially turbulence. She delighted both in the abstract concept of these models and in their applications. She said with confidence, "This is what I was created to do. And it is very good."

Thirdly, I think this approach has implications for mathematics curricula in Christian colleges and for course selection by students. We need to provide students a broad perspective on mathematics that takes into account its history, philosophy, and relationship to other disciplines and asks how we can use the gift of mathematics to serve God. I think this is especially important for any of our students who will go on for graduate work since they most certainly will not get such a perspective elsewhere.

And fourthly, I believe we have a message to the broader mathematics community. Recall my story at the beginning of this talk – people in the mathematics community are hungry for perspective. Here are some questions we can talk with people or groups about: Is mathematics (as Hilbert claimed) really a presuppositionless science? Can there ever be a mathematical social science as rigorous as mathematical physics? How do we account for the so-called "unreasonable effectiveness" of mathematics? The Enlightenment vision of a rationally ordered society has fallen out of favor in recent years. Should it have? Why or why not? Mathematical models are being used today for all kinds of decision-making. Does the mathematical community have a responsibility to society as to how these are used?

In conclusion, when I first began to think about the question of how to integrate faith and learning in mathematics, I found the question very hard. I recall praying "Lord, why is this question so difficult?" I subsequently decided that the very difficulty was the key that would unlock the door to seeing how integration might go. That is, I concluded that the way the boundaries of the discipline of mathematics had been drawn, it excluded most of the questions of concern to Christians. By examining the possibility of redrawing the boundaries, I was able to see God in every aspect of mathematics. Thus, I don't think that Christians need to see themselves and their concerns as being on the fringe of the discipline. We have a message that can help the discipline and can connect with even the most secular members of the mathematical community. It can even open the door to spiritual conversations with some.