Welcome to the Journal of the Association of Christians in the Mathematical Sciences! To launch this journal, I have selected twenty-eight articles from the Proceedings of the fourteen biennial ACMS conferences. Within the literature on science and religion, they constitute a unique electronic anthology focusing specifically on the relationship between mathematics and Christian belief.
While preparing this anthology, I was deeply impressed by the importance of the questions the ACMS community is addressing, by how little attention they have received relative to what they deserve, and by the depth of the thought that our members and guest speakers have put into these questions. The papers address a broad range of issues including the ontology of mathematical objects, epistemology, the relationship of mathematics to the physical world, education, values, infinity, modernism and post-modernism, and the relationship of mathematics to the arts.
Reading these articles has convinced me of three things:
- Beliefs that Christians commonly share have profound implications for our understanding of the nature of mathematics and its significance.
- The overall contours of those implications are fairly clear.
- We have not yet adequately addressed some vital questions.
I would like to challenge the ACMS community to address these questions. They are not easy, and many of them will require significant scholarship. But if we can address them well, they have the potential to make a major contribution to the discipline of mathematics, to our students, and to our own thinking. Before I can present the questions, though, I need to expand on the first two convictions.
It seems to me that there are three classes of Christian beliefs that are particularly relevant to our understanding of mathematics — beliefs about God, beliefs about the physical world, and beliefs about humanity. In summarizing these, I am going to address only beliefs that have been widely shared by Orthodox, Roman Catholic, and Protestant Christians throughout the Christian era — I call these beliefs “historic Christianity.”
One critical belief about God is his aseity (that is, God exists and is self-existent — his existence does not depend on anything else). Furthermore, God is the creator of the physical universe and is sovereign over it in that everything that is not God depends on his creative and sustaining activity for its existence. He is eternal, omniscient, and personal. He is loving in that He possesses a consistent posture of good will toward all of his creation and he is eager to reveal himself to humanity.
The most relevant belief about the physical world is that it is entirely, without exception, the product of God’s creative activity and it depends on Him for its ongoing existence. Furthermore, God regards it as a good world and is pleased with what he has done in making it.
Two beliefs about humanity are especially applicable. One is the imago Dei – that human beings are made in the image of God. Theologians are not of one mind about what this means. However, it certainly includes at least the following — that while human bodies are made of the same substance as the physical world and are subject to its physical laws, human beings are distinguished from the physical world by consciousness, moral responsibility, a special role in the physical world as stewards of it, and were created to enjoy a continuous intimate relationship with God. It also probably means that God has imprinted aspects of His nature on human beings, although no consensus has developed as to which aspects of human beings express the image of God. Some classical thinkers emphasized reason; others have emphasized love, moral responsibility, and creativity. The second relevant belief about humanity is that is “fallen.” That is, human beings have broken their relationship with God, seeking an autonomous existence that does not depend on God.
Note that this taxonomy omits one major category of Christian beliefs — those about Jesus Christ. Historic Christian belief has always proclaimed that God seeks to restore his broken relationship with humanity and that his efforts to do so find their fullest realization in the death and resurrection of Jesus Christ. Furthermore, it sees Christ as central both in the act of creation and in its ongoing governance. Nevertheless, important as these beliefs are for Christian thinking, I have not been able to clearly identify what role they play in the relationship between mathematics and Christian belief. Hence, the perspective I’ll articulate below is one that believing Jews, Muslims, and other monotheists could also affirm. I’ll return to this observation in the questions.
A Christian perspective on a discipline usually does not address technical matters within the discipline. For example, the Pythagorean theorem is the same for Christian believers as for others who do not share that belief. Rather, it provides a framework within which questions of meaning, value, and purpose can be addressed. Such questions necessarily involve concepts not accessible to mathematics (or science). I believe that the main characteristics of a Christian approach to mathematics follow from the framework of Christian belief outlined above. The characteristics are these:
- A discomfort with both Enlightenment foundationalism and post-modern relativism. Enlightenment thinkers sought a non-controversial, literal description of the physical world that would be verifiable (or at least non-falsifiable) in terms of sensations and logic. Its methodology was empirical and rational, and statements that could not be reduced to sensations such as religious beliefs were excluded from such a description. Furthermore, it regarded empirical methods and reason as sufficient to solve any problems that human beings needed to solve. From a Christian perspective, the Enlightenment project is commendable for its commitment to truth and its desire to build prosperous, secure, and peaceful societies. However, from a position of Christian belief, the Enlightenment’s optimism about human capabilities is excessive and its exclusion of divine revelation and religious experience is a fatal flaw. Post-modernism is a less coherent project. It values experience, intimacy, emotion, and diversity over abstraction and empirical observation. It is very skeptical of truth claims, often viewing them as expressions of a desire for power; it tends to regard ontological questions as meaningless, preferring epistemological questions. From a position of Christian belief, post-modernism’s rejection of the Enlightenment’s presumption of the human capacity to find foundational truth solely by reason and careful observation is laudable as is its emphasis on alternative ways of knowing, on diversity, and social justice. Nevertheless, Christian beliefs do not comport well with post-modernism’s denial of transcendent truth, its denial of the possibility of certitude, and its turn away from ontology. While modernism and post-modernism are very different, they tend to focus solely on human capabilities while excluding God. Thus Christians typically find neither approach satisfactory.
- A tendency toward Augustinian positions on fundamental philosophical questions. Because God is and always has been omniscient, he has always known all mathematical truth. Thus on ontological questions, Christian belief tends toward realism — that mathematical objects such as numbers have an objective existence in the mind of God, totally apart from human knowledge or experience. Furthermore, mathematical truth is seen as more than mere consistency; it expresses transcendently true statements about objects that objectively exist. A Christian approach to epistemology tends to focus on human beings as created by God with the capability of obtaining certitude about mathematical objects because of capacities that God has embedded in human beings, because of the nature of the physical world, and because of how God has configured the interaction between them. The “unreasonable effectiveness” of mathematics is seen as the result of God using mathematical ideas in creating the world and putting the capacity for understanding those ideas in the human mind in order that human beings might exercise the stewardship over the creation entrusted to them.
- Concern with context. From a Christian perspective, God’s truth is a seamless garment — it possesses a fundamental unity. Christian belief affirms a fundamental tension between mathematics’ distinctive identity and its embeddedness within a much broader context. For example, formalist approaches to axiomatics are valuable explorations of what can be accomplished by syntax alone. But Christians tend to be uncomfortable with excluding meaning (which is derived from context) from mathematics. Thus Christian belief suggests that all of mathematics’ interactions with the physical and social worlds, with history, the arts, and with ethical issues are essential parts of the discipline.
- Resistance to reductionism. Many reductionistic statements have been made about mathematics. For example, mathematics is just a formal game played with symbols, it is all part of logic, it is just the expression of various neurological activities in human brains, and it is nothing but a social convention. Christians have no difficulty affirming that mathematics is each of these things but feel decidedly uncomfortable with the statement that it is “only” one or more of these things. For the Christian mathematician, mathematics contains a strong element of mystery that always enables it to elude precise definition and is a source of delight and awe. Mathematics points beyond itself to God.
- Resistance to arbitrary limitations on mathematical activity. Intuitionists have objected to the law of the excluded middle and have insisted that all proofs be constructive. Other mathematicians have expressed discomfort with the notion of infinity, sometimes speaking of potential infinities as contrasted with actual infinities. Christian belief affirms the goodness of all of God’s creation including the human imagination. Thus as long as use of the excluded middle and non-constructive proofs do not lead to inconsistencies, Christians typically have no problem using them. Furthermore, they have no problems with the notion of an actual infinity or even with the notion that the entire Cantorian hierarchy may image concepts fully realized in the mind of God.
- Concern with social responsibility, ethics, and values. Christian belief sees God as purposeful and as having given humanity responsibility to care for the physical and social worlds in which we live. He has equipped human beings with a variety of gifts to enable them to fulfill this responsibility and mathematics is one of them. Thus the mathematics community has been entrusted with a valuable gift from God and is accountable to him for its stewardship of this gift. Careful thought about social responsibility, ethics, and values is necessary to carry out this work of stewardship.
- A distinctive motivation. Christians tend to feel uncomfortable with statements such as Hilbert’s “Wir mussen wissen wir werden wissen” (We ought to know we will know). While they applaud the desire for knowledge that Hilbert’s statement expresses, they don’t see knowledge as an end in itself. Furthermore, the funding of mathematical research and the support of mathematics education are often justified by mathematics’ role in producing wealth. Christian belief certainly affirms the goodness of material things. However, the goal of life is not the acquisition of knowledge or material things but the development of humanity’s relationship with God. Thus, from a Christian perspective, mathematics is a source of joy and delight leading to useful service and a deepened love for and appreciation of God, not merely a means to acquire knowledge and wealth.
Nevertheless, I find this broad framework to be relatively clear, there are a number of questions that we need to address. Here are some that I have been able to identify:
- What are the implications of the person and work of Jesus Christ for mathematics? If there are none, or if they are not central, what does this tell us about mathematics?
- One can hypothesize that a central direction of nineteenth and twentieth century mathematics has been an effort to establish an autonomous mathematics, independent of belief in God. For example, the movements away from mathematical realism, toward relative rather than objective truth, and toward a more formal mathematics separated from meaning all suggest this. Does historical analysis support this hypothesis? If so, what are the implications for our understanding of mathematics today?
- On what basis can a sound understanding of the social and ethical responsibility of the mathematics community be built? Once that basis has been laid, what are its implications?
- Christian thought emphasizes the fundamental unity of God’s truth and hence the importance of placing mathematics in a broad context. But the details of this notion have not been adequately explored. For example, in the psychological realm, mathematics employs both deductive and inductive reasoning. But human beings employ many other modes of thinking, including case studies, metaphors, and valuing. A more complete taxonomy of such modes of thinking is needed, along with a careful exploration of how mathematical thinking interacts with them. Analogous studies are needed in the social realm — that is, how has mathematical thought influenced various cultures, and how does it shape both culture and public policy today?
- Mathematical realism has been widely attacked recently, notably by mathematical writers sympathetic to post-modern perspectives. Some of the papers presented here reply to these writers. Nevertheless, a careful, systematic response to the full range of critiques is needed.
- Some Christian approaches to realism views mathematical objects as part of the nature of God; others view them as created. What are the issues at stake here? What difference does a choice between these positions make in how we function as mathematicians?
- While a Christian perspective affirms the notion that “truth” in mathematics means more than consistency with a set of axioms, considerably more needs to be done to clarify the precise sense in which it exceeds consistency. Furthermore, what is the status of mathematical truth vis-a-vis the empirical truths of science and the revealed truths of Scripture?
- Educational theorists have been strongly influenced by the “genetic epistemology” of Jean Piaget. This concept appears to have the potential to help us clarify the relationship between realism and constructivism and to provide a more nuanced answer to the unreasonable effectiveness question than given in the general contours above. Someone needs to explore this notion.
- Constructivists’ turn away from ontology leave constructivism with a serious gap as an ontological theory. What phenomena is it unable to account for as a result? What are the educational consequences of this turn?
- Mathematicians frequently state that one of their principal motivations for their work is that they find mathematics of great beauty. What is the concept of aesthetics being used here? How does it compare and contrast with aesthetic concepts in the visual arts and other fields? Christian thinkers have often emphasized the beauty of God. Is there a relationship between these concepts of beauty? If so, what is it?
- Some thinkers (perhaps influenced by process theology) have asserted the idea that God’s creation is not a finished work but that he creates new mathematical objects through mathematicians. Is this idea theologically sound? Is it helpful for our understanding of mathematics?
- Relatively little has been done thinking through a Christian perspective on probability and statistics.
- While the general framework of a Christian perspective has become clear, it has largely emerged from a conversation among scholars. We need educational materials at both the college and pre-college levels that can appropriately articulate this perspective to young people.
- Given that a Christian perspective has significant implications for our understanding of mathematics, what is the responsibility of the ACMS to the larger mathematical community? How can it best carry out this responsibility?
We have done a great deal of good work this past twenty-seven years, but we have a lot more to do! The task before us is far more extensive than any one individual among us can carry out-many members of our community will need to contribute and much dialogue is needed. Some of us may want to incorporate these questions into long-term research projects. May God be with us as we work together to know him better and to proclaim his wonders!
James Bradley, Calvin College, Founding Editor