# How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics

William Byers
Pages: 424
https://www.jstor.org/stable/j.ctt7s98c

1. Front Matter
(pp. i-iv)
(pp. v-vi)
3. Acknowledgments
(pp. vii-viii)
4. INTRODUCTION Turning on the Light
(pp. 1-20)

A few years ago the PBS programNovafeatured an interview with Andrew Wiles. Wiles is the Princeton mathematician who gave the final resolution to what was perhaps the most famous mathematical problem of all time—the Fermat conjecture. The solution to Fermat was Wilesʹs life ambition. ʺWhen he revealed a proof in that summer of 1993, it came at the end of seven years of dedicated work on the problem, a degree of focus and determination that is hard to imagine.ʺ¹ He said of this period in his life, ʺI carried this thought in my head basically the whole...

5. SECTION I THE LIGHT OF AMBIGUITY
• CHAPTER 1 Ambiguity in Mathematics
(pp. 25-79)

This chapter begins the process of developing a new way of describing what mathematics is and what mathematicians do. One might think that this is an easy task—just ask a mathematician what it is that he or she does. Unfortunately this will not work, for the business of mathematicians is the doing of mathematics and not reflecting on the subject of what it is that they do. Davis and Hersh note that there is a ʺdiscrepancy between the actual work and activity of the mathematician and his own perception of his work and activity.ʺ¹ The only thing I can...

• CHAPTER 2 The Contradictory in Mathematics
(pp. 80-109)

This chapter is about the role of the contradictory in mathematics. One of the themes of this book is that mathematics does not inhabit a world that is disjoint from the world of human experience but is in continual interaction with that larger world. The contradictory is an irreducible element of human life as we all experience it. It is not only that we often disagree with others; it is also that we human beings seem capable of simultaneously sustaining two contradictory points of view. The obscure Buddhist dictum ʺLife is sufferingʺ is best understood as the claim that there...

• CHAPTER 3 Paradoxes and Mathematics: Infinity and the Real Numbers
(pp. 110-145)

In this chapter we pursue our investigation of ambiguity in mathematics by beginning a discussion of the paradoxes of infinity. What is a paradox? According to theEncarta World English Dictionarya paradox ʺis a situation or proposition that seems to be absurd or contradictory but is or may be true.ʺ It is interesting that the word ʺparadoxʺ has the contrary meaning to ʺorthodox.ʺ Orthodox means ʺfollowing established or traditional rules.ʺ Orthodoxy operates within acceptable limits. Paradox, on the other hand, breaks the limits of the orthodox. It appears ʺabsurd or contradictoryʺ from the perspective of the orthodox, but it...

• CHAPTER 4 More Paradoxes of Infinity: Geometry, Cardinality, and Beyond
(pp. 146-188)

In the last chapter I discussed the manner in which the Greeks achieved a stable notion of infinity, how that equilibrium broke down as a result of developments in mathematics such as the invention of the calculus, and the way a new equilibrium was forged. This new way of thinking —the system of real numbers—remains definitive for most mathematicians to this day. Nevertheless, as we shall see, there are other legitimate intuitions about infinity that can provide the basis for further mathematical development.

This chapter continues the discussion of infinity by considering a series of paradoxes that are associated...

6. SECTION II THE LIGHT AS IDEA
• CHAPTER 5 The Idea as an Organizing Principle
(pp. 193-252)

What is the core ingredient of mathematics? Is it logic or precision? Is it ʺnumberʺ or ʺfunctionʺ? Is it ʺstructure,ʺ or ʺpattern,ʺ or the subtlety of mathematical concepts? Perhaps it is abstraction? In our search for the inner nature of mathematics we might do well to listen to the words of mathematicians. Not just the words they use when we ask them to explain the nature of their subject. The language they use in such an artificial situation is alien to the language that they use when they are discussing mathematics among themselves.

When discussing a particular piece of mathematics,...

• CHAPTER 6 Ideas, Logic, and Paradox
(pp. 253-283)

The previous chapter initiated a discussion of mathematical ideas. I considered a number of fairly straightforward examples of mathematical ideas from the areas of mathematics that I had developed in previous chapters. In this chapter I wish to extend this discussion in directions that are unexpected and perhaps even a bit shocking. This will demonstrate what differentiates the perspective I am taking from the usual one. Ideas arise from a context that may include ambiguity, contradiction, and paradox.

The proper domain of mathematics is this expanded region. This domain from which ideas may arise fits nicely with the expanded domain...

• CHAPTER 7 Great Ideas
(pp. 284-322)

Here we come to a key element in the view of mathematics that is being built up in this book—a point of view that will be made more explicit through a discussion of a certain variety of mathematical idea that will be called a ʺgreat idea.ʺ The discussion of ʺgreat ideasʺ was anticipated in the introduction and, in passing, in Chapters 1 and 5. How could anyone have the effrontery to attempt to define what is great in mathematics? This is clearly an impossible task from the point of view of the content of the mathematical idea. What will...

7. SECTION III THE LIGHT AND THE EYE OF THE BEHOLDER
• CHAPTER 8 The Truth of Mathematics
(pp. 327-367)

Mathematics is about truth: discovering the truth, knowing the truth, and communicating the truth to others. It would be a great mistake to discuss mathematics without talking about its relation to the truth, for truth is the essence of mathematics. In its search for the purity of truth, mathematics has developed its own language and methodologies—its own way of paring down reality to an inner essence and capturing that essence in subtle patterns of thought. Mathematics is a way of using the mind with the goal of knowing the truth, that is, of obtaining certainty.

Unfortunately the very notion...

• CHAPTER 9 Conclusion: Is Mathematics Algorithmic or Creative?
(pp. 368-388)

The objective of this concluding chapter is to summarize and draw out the implications of the view of mathematics I have been developing in the preceding chapters. I have been describing mathematics as an activity that is dynamic and creative, pulsating with the life of the mind. It is a ʺway of knowingʺ that is quite unique.

But mathematics is not off in some obscure corner of human activity; it is central to human experience and human culture. Thus it is not surprising that many of the great questions of the day have a certain reflection within mathematics. The question...

8. Notes
(pp. 389-398)
9. Bibliography
(pp. 399-406)
10. Index
(pp. 407-415)