Foreword

WILLIAM P. THURSTON

Mathematics is commonly thought to be the pursuit of universal truths, of patterns that are not anchored to any single fixed context. But on a deeper level the goal of mathematics is to develop enhanced ways for humans to see and think about the world. Mathematics is a transforming journey, and progress in it can be better measured by changes in how we think than by the external truths we discover.

Mathematics is a journey that leads to view after incredible view. Each of its many paths can be obscured by invisible clouds of illusion. Unlucky and unwary travelers can easily stray into a vast swamp of muck. Illusions arise because we cannot directly observe each other think. It is even difficult for us to be self-aware of our thinking. What we cannot see we replace by what we imagine, hence the illusionary trap that can lead to nowhere.

People’s inclination to disregard what they cannot easily see (or say) in order to focus on what can readily be seen (or said) results in predictable and pervasive problems in communicating mathematics. We have an inexorable instinct that prompts us to convey through speech content that is not easily spoken. Because of this tendency, mathematics takes a highly symbolic, algebraic, and technical form. Few people listening to a technical discourse are hearing a story. Most readers of mathematics (if they happen not to be totally baffled) register only technical details—which are essentially different from the original thoughts we put into mathematical discourse. The meaning, the poetry, the music, and the beauty of mathematics are generally lost. It’s as if an audience were to attend a concert where the musicians, unable to perform in a way the audience could appreciate, just handed out copies of the score. In mathematics, it happens frequently that both the performers and the audience are oblivious to what went wrong, even though the failure of communication is obvious to all.

Another source of the cloud of illusions that often obscures meaning in mathematics arises from the contrast between our amazingly rich abilities to absorb geometric information and the weakness of our innate abilities to convey spatial ideas—except for things we can point to or act out. We effortlessly look at a two-dimensional picture and reconstruct a three-dimensional scene, but we can hardly draw them accurately. Similarly, we can easily imagine the choreography of objects moving in two or three dimensions but we can hardly convey it literally, except—once again—when we can point to it or act it out. It’s not that these things are impossible to convey. Since our minds all have much in common, we can indeed describe mental images in words, then surmise and reconstruct them through suggestive powers. This is a process of developing mental reflexes and, like other similar tasks, it is time-consuming. We just need to be aware that this is the task and that it is important, so that we won’t instinctively revert to a symbolic and denatured encoding.

When I grew up I was a voracious reader. But when I started studying serious mathematical textbooks I was surprised how slowly I had to read, at a much lower speed than reading non-mathematical books. I couldn’t just gloss over the text. The phrasing and the symbols had been carefully chosen and every sign was important. Now I read differently. I rarely spend the time and effort to follow carefully every word and every equation in a mathematics article. I have come to appreciate the importance of the explanation that lies beneath the surface. I love to chase signals that direct me toward a theoretical background I don’t know. I have little expectation for the words to be a good representation of the real ideas. I try to tunnel beneath the surface and to find shortcuts, checking in often enough to have a reasonable hope not to miss a major point. I have decided that daydreaming is not a bug but a feature. If I can drift away far enough to gain the perspective that allows me to see the big picture, noticing the details becomes both easier and less important.

I wish I had developed the skill of reading beneath the surface much earlier. As I read, I stop and ask, What’s the author trying to say? What is the author really thinking (if I suppose it is different from what he put in the mathematical text)? What do I think of this? I talk to myself back and forth while reading somebody else’s writing. But the main thing is to give myself time, to close my eyes, to give myself space, to reflect and allow my thoughts to form on their own in order to shape my ideas.

Studying mathematics transforms our minds. Our mental skills develop through slow, step-by-step processes that become ingrained into our neural circuitry and facilitate rapid reflexive responses drawing little conscious effort or attention. Touch typing is a good example of a visible skill that becomes reflexive. It takes a lot of work to learn to touch type, but during that process people can at least see what they need to learn. For me the process of typing is transparent. I think about what I want to say and I put it on the computer screen; I do not ordinarily attend to the typing at all. Although my moment-by-moment mental processing is complex, it seems seamless.

Now think of a skill similar to typing that involves no physical motion and no direct use of words or symbols. How do we observe and learn such a skill—and how do we communicate about it? This kind of skill is sometimes glossed as intuition—that is, a powerful insight, hard to explain. One of the main problems with doing mathematics is that we tend not to be aware that hardly definable skills are important, that they need to be nurtured and developed. Yet practicing mathematics requires them. Mathematical ideas can be transcribed into symbols organized into precise descriptions, equations, and logical deductions—but such a transcription is typically far removed from the mind process that generates the ideas.

These types of problems are pervasive but not insurmountable. Mathematics can be communicated without being completely denatured, if people are attuned to its mental dimensions. One important part of the remedy is reflective reading of thoughtful writing, such as the interesting and varied collection in this book.

For the last two years, Mircea Pitici has invited me to meet with the students in his Freshman Writing Seminar in Mathematics at Cornell University. I was struck by the imaginativeness and the intensity of the participants. It was immediately obvious that Mircea’s writing seminars lead students to think about mathematics on paths different from those available in typical mathematics courses. It reminded me how compelling and interesting the journey of mathematics can be, at all its stages. When mathematics is rendered into technicalities, the story is removed. But discussions of mathematics without technicalities can evidence its poetry and bring it back to life.

We humans have a wide range of abilities that help us perceive and analyze mathematical content. We perceive abstract notions not just through seeing but also by hearing, by feeling, by our sense of body motion and position. Our geometric and spatial skills are highly trainable, just as in other high-performance activities. In mathematics we can use the modules of our minds in flexible ways—even metaphorically. A whole-mind approach to mathematical thinking is vastly more effective than the common approach that only manipulates symbols. And a collection like this, of writings on mathematics, opens up the whole mind toward a more comprehensive understanding of mathematics.

Happy reading!