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W. James Bradley
Russell W. Howell
Of course, the Pythagorean Theorem is not different for Christians. Even so, we believe there are a number of ways in which a Christian perspective can enrich our understanding of mathematics. Conversely, we think that many ideas in mathematics can enhance our understanding of the Christian faith. But before we elaborate on this thesis, we should clarify what we mean by a Christian perspective. The Christian faith has its roots in Judaism, which has existed from about 2000 B.C. Since that time many Jewish and Christian thinkers have reflected carefully and deeply on the human situation. Guided by their faith, they have developed rich and powerful ideas that are applicable to many different areas. In looking at the possibility of forming a Christian perspective on mathematics, then, we seek to do the same. That is, we make a serious attempt to ask whether any ideas that might spring from a Christian faith commitment can enrich our understanding of mathematics, and whether the ideas of mathematics can contributeto and enlarge our understanding of the Christian faith. Thus, we examine mathematics and the Christian faith from an unfamiliar angle in the hope that such an examination will yield new insights. George Marsden calls such an endeavor Christian-informed scholarship, and we are comfortable with that description of our work here.
What would Christian-informed scholarship mean in our project? Since the time of Galileo (early 1600s), the notion that mathematical and religious thought belong to different realms having little or nothing in common has gained considerable credibility. It is not hard to understand why this is the case. Mathematicians have worked hard to frame their discipline as abstract and formal. Especially in the twentieth century they intentionally separated mathematics from the scientific and cultural contexts in which mathematical questions originated. Many had laudable purposes in doing this. They were pursuing goals such as gaining generality and laying a foundation for mathematics that would be independent of any particular individual or cultural experience. Their belief was that abstraction would free mathematics from the biases and presuppositions that so easily slip into thought from one's personal experience or perspective. Indeed, much of the global credibility that mathematics has enjoyed is due to its success in attaining these goals. From this perspective it is easy to see how the jibe, "Is the Pythagorean Theorem different for Christians?" arises. Religious presuppositions are precisely the kinds of distracters that mathematical abstraction has sought to avoid.
So, when we speak of a Christian perspective, we are not challenging the value of abstraction or attempting to turn the clock back to a different era. To the contrary, we actively support the notions of abstraction and rigor. But we don't want to stop there. Rather, we want to look at the mathematical endeavor, its place, nature, and contribution within the broader context of human thought and culture. We want to ask, "How can a vision of mathematics that incorporates its cultural role, its relationship to other areas of thought, its philosophical foundations, and its technical content enrich our understanding of it?" Conversely, we ask whether mathematics itself might contribute to a Christian worldview. Specifically, we are concerned with issues like the following:
Many claim that Western culture is in a major paradigmatic shift from the "modern" to "postmodern" eras. The intellectual shape of the modern age was, however, highly influenced by mathematical ideals. What exactly were those ideals and how did modernism instantiate them? Does the postmodern spirit affirm or deny them? What are the consequences of this shift for mathematics and for its role in the larger culture?
Mathematics has often been viewed as the epitome of the application of "pure reason." The issue of the proper relationship between faith and reason, however, has occupied the thought of many outstanding thinkers throughout the Christian era. If the view of mathematics as an application of reason is correct, any limitations of reason are also limitations of mathematics; any capabilities mathematics possesses are also capabilities of reason. Thus, consideration of the nature of mathematics may help clarify this relationship.
For the past three hundred years or so "foundationalism" an approach modeled on the axiomatic method of mathematics, has influenced philosophic thinking. Many scholars today, however, regard foundationalism as misguided. Does this change in the status of foundationalism have any implications for the practice of mathematics? What does it portend for the role of mathematics in Western culture?
Mathematicians tend to be Platonists. That is, they view mathematical objects such as groups, vector spaces, or the axioms of plane geometry as having an existence independent of human minds, although it is unclear where this existence can be located. Many Christians would locate mathematical objects "in the mind of God." Some thinkers, however, have been nominalists and deny such an independent existence. Furthermore, postmodern thinkers have tended to move away from Platonism. What, then is the nature of mathematical objects, and of mathematical thinking?
Religious concepts use terms such as 'infinity' or 'paradox' that are also common in mathematics. Is this just a coincidence, or could these mathematical ideas give us insights into aspects of our religious beliefs? What other specific mathematical ideas might say things about the Christian faith? Mathematics has had a profound impact on human cultures, especially contemporary Western culture. Can we articulate what that impact has been? Can we provide a framework of norms and values for assessing that impact? Is it desirable, or even possible, to shape or predict the future form of that impact?
Most contemporary societies invest an enormous amount of their wealth and their children's time in mathematics education. Substantial investments are also made in mathematical research. That is, contemporary cultures have placed an enormous amount of trust in their mathematical communities; these communities in turn have an enormous responsibility to their respective cultures. How can mathematical communities best meet this responsibility?
David Hilbert, one of the greatest mathematicians of the twentieth century, has claimed that mathematics is a "presuppositionless science." Is this the case, or does mathematics have its own values and presuppositions? If so, can we clearly articulate those? Can we constructively critique them? It seems to us that consideration of issues like these is important for a broader understanding of mathematics. As Christian thinkers have thought deeply about many of these matters, we believe that Christian concepts have a great deal to contribute to the discussion of such issues. We hope that this book will stimulate further conversation about them.
We also want to be straightforward concerning our own presuppositions. While we will try throughout the book to acknowledge our assumptions wherever they are used, they provide a general overarching framework that shapes even the way we ask the questions we seek to address, so it is im- possible to point out every instance of their application. Here, then, is a brief summary of the main features of our presuppositions. First, we believe in the God of the Old and New Testaments, and believe that he is the creator of the universe. In particular, he is our creator, and for reasons we don't fully understand he has created us with the capacity to engage in mathematical inquiry. Thus, we reject the notion that this capacity is neutral, a tool to be used for good or ill. Rather we affirm that such capacity is inherently good and hence has intrinsic value. Second, we believe that humans are made in God's image, but they are "fallen" - that is, they have rejected God's authority over them and have replaced it with a claim of autonomy. This attempt at autonomy is a source of human evil. Hence, although mathematical capacity is good, humans can engage in mathematical exploration for evil ends. Third, we believe it is the responsibility of human beings to work for "redemption" - that is, to work to overcome the consequences of the fall. In a fundamental sense, of course, true redemption is possible only by the work that Christ did on the cross for us. But out of gratitude for this gift of God, we as believers desire to help fulfill God's purposes for creation, and that is the sense in which we work for redemption. For Christians interested in mathematics, this entails endeavoring to discern God's purposes in giving human beings the capacity to engage in mathematical activity, and seeking to help the mathematics community fulfill those intentions.
What exactly were God's intentions in giving us the capacity to engage in mathematical inquiry? Scripture does not provide a direct answer to this question, so the answer must be inferred from broader purposes that have been revealed. If we go back to Genesis 1 and 2, we see that God's original purpose was that we be co-creators with him in two ways: that we be stewards of this world, walking closely with him in using his creation to build cultures and to care for this world, and that we ourselves would be built into "sons of God." But carefully studying anything in this world often involves forming precise definitions, measuring, and thinking deductively about the way things are and the way they might be. Thus mathematics is an essential component of co-creating. While such a vision does not give precise answers to every value question we might ask, it gives us a framework from which to start. If, as we affirm, the capacity to do mathematics is a good gift of God, it reveals something about his nature, for example, his subtlety, order, beauty, and variety. When people respond to these qualities of mathematics with awe and joy and turn to God with reverence and thankfulness, they are fulfilling this purpose. Thus, we immediately see one consequence of a Christian perspective - mathematics does not need to be "applicable" to be of value. However, applicability is also important, as a second purpose of mathematics is its use in helping to build human cultures, to serve people, and to care for the earth. Thus a second consequence of a Christian perspective is that the mathematics community cannot stop at considering the abstract, formal aspects of mathematics, but must consider the consequences of these abstractions when they are reintroduced into the human community.
This concludes our brief sketch of our underlying assumptions. We do not expect every reader to agree with all of these - some will reject them in toto. Some will disagree with certain points or with emphases. Furthermore, we do not expect that those readers who do agree with our assumptions will necessarily agree with every position put forth in this book. Indeed, the individual authors of various chapters do not uniformly agree with each other about what they have said. We believe, however, that the ideas we present here are credible, and we hope that this book succeeds in demonstrating that the kind of thinking we are seeking to do can enrich our understanding of both mathematics and the Christian faith.
Returning to the title of this book, what exactly do we explore? Painting with broad strokes, we ask if our Christian faith has any bearing on how we look at the nature of mathematics, from our views of its truth claims to the status we ascribe to mathematical objects. Some of our thinking will be based on a comparative study of modern and postmodern thinkers, as well as a historical cultural study relating to how mathematics progressed in various settings. We also ask if our Christian faith has any bearing on how we might view the place of mathematics in culture today. We propose to review, from the stance of the Christian faith operating within the mathematics profession, some of the key historical epochs in mathematical thinking, including the philosophical assumptions on which that thinking is based. We look at how mathematical ideas can directly interact with our Christian faith, and finally give an appreciative critique of mathematics, and suggest to the contemporary mathematics community that it needs to cast its performance net more broadly, and its self-critical gaze more intently, if mathematics is to do all it ought to do.
Specifically, we begin in Chapter 1 with a discussion of what we mean by modernism and postmodernism and then look at representatives of mathematical thinking from those traditions. In particular, we explore the ideas of Gottlob Frege, a prominent logician, and Paul Ernest, a contemporary philosopher of mathematics. Frege (1848-1925) predates Ernest (1944-), and Frege's desire to put all of mathematics on an indisputably firm logical foundation did not succeed. It would be a mistake to infer, however, that because of this we somehow think Frege's views are no longer credible. Today, Frege's work is still recognized as very perceptive and influential. We chose him because he is a good example of how a contemporary thinker in the modern school of thought might view mathematics. What, then, were Frege's views, what caused part of his program to fail, and what benefits can we glean for ourselves today by studying his ideas? Questions such as these, of course, must be answered in context, and Paul Ernest provides an interesting contrast with Frege. Whereas Frege views mathematical propositions as true independent of experience, Ernest emphasizes the importance of social agreement in the acceptance of mathematical theories. On the one hand, Frege would hold that a mathematical theory is true for all time. On the other hand, Ernest views mathematical constructs as fallible, reminding us that even published proofs are unreliable guides to any sort of universal truth as they are very commonly flawed.
How should we adjudicate these claims? We do not intend to present Frege and Ernest as polar opposites, forcing our readers to choose between them. Honest thinkers may side completely with one or the other, of course, but they may also favor some middle ground. Or, they may opt for an entirely different approach. But the position a person finally adopts must come from an informed historical perspective. With this in mind, Chapter 2 traces the development of mathematical ideas in three cultures, ancient Greece, medieval Islam, and pre-modern China. Favoring Ernest, we present differences between Greek and Islamic mathematics that appear to depend on world and religious views, even though these two cultures commingled. Favoring Frege, we show that, perhaps surprisingly, the isolated Chinese mathematicians developed many of the same classic theorems (such as the Pythagorean Theorem) common to other cultures. Their methods of proof were quite different from those of Western mathematicians, but this cross-cultural commonality makes it hard to hold a position that mathematical knowledge is merely a social or linguistic convention. Such knowledge may not be universal in a Platonic sense, as the commonality we noted may speak more about the structure of our brains than anything beyond them. Nevertheless, this observation seems to counter the complete relativism that often accompanies an extreme form of postmodern thinking.
But how would we account for this observed universality of mathematical ideas? In particular, what do we think about the status of mathematical objects such as propositions, relations, and properties? Are they simply a construct of human thought (however universal that might be) that would cease to exist if humanity perished? If not, are they located within the mind of God, or do they exist independently of him? We take up these questions in Chapter 3, where we discuss the ontology of mathematical objects. We tentatively argue for a certain form of mathematical realism. This position creates problems, which we discuss. The position also raises numerous interesting paradoxes relating to the nature of God, specifically to the meaning of terms such as 'omniscient' and 'omnipotent'.
Excerpted from Mathematics in a Postmodern Age Copyright © 2001 by Wm. B. Eerdmans Publishing Company . Excerpted by permission.
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|I||The Nature of Mathematics|
|1||Mathematical Truth: Static or Changing?||15|
|2||Mathematical Truth: A Cultural Study||45|
|3||God and Mathematical Objects||65|
|4||The Pragmatic Nature of Mathematical Inquiry||98|
|II||The Influence of Mathematics|
|5||Mathematization in the Pre-modern Period||133|
|6||Mathematization and Modern Science||162|
|7||The Mathematization of Culture||193|
|III||Faith Perspectives in Mathematics|
|8||Mathematics and Values||223|
|9||Creativity and Computer Reasoning||250|
|10||The Possibility of Detecting Intelligent Design||278|
|11||A Psychological Perspective on Mathematical Learning and Thinking||309|
|12||Teaching and Learning Mathematics: The Influence of Constructivism||338|