Mathematics in a Postmodern Age

The discipline of mathematics has not been spared the sweeping critique of postmodernism. Is mathematical theory true for all time, or are mathematical constructs in fact fallible? This fascinating book examines the tensions that have arisen between modern and postmodern views of mathematics, explores alternative theories of mathematical truth, explains why the issues are important, and shows how a Christian perspective makes a difference.

Table of Contents
Acknowledgments
Introduction 1
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
Conclusion 360
Bibliography 385
Index 394
Read a Sample Chapter

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'.
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