Brian Clegg’s new book Are Numbers Real? The Uncanny Relationship of Mathematics and the Physical World caught my eye somewhere and I ordered a copy for the library where I work. When it came in I borrowed it and began it with interest, but it turns out it has nothing to do with whether numbers are real, and is an idiosyncratic, cursory and ultimately unfulfilling overview of the history of the relationship of mathematics and physics.
The book begins promisingly:
Our journey in this book will explore a question that is fundamentally important to scientists—and for that matter the rest of us. Yet it’s a question that most people, including scientists, rarely give a moment’s thought to. Are numbers real, and is the wider concept of mathematics, real?
You can point to two oranges or two chairs, but can you point to two? Does two exist? That seems to be the question the book is going to be about … except it isn’t. And this question is not “fundamentally important” to scientists or anyone else except mathematicians. People interested in the philosophy of mathematics care about this, but no one else really does. They use numbers—numbers that work very well to describe the world—but don’t wonder about where numbers come from or if they would exist if there were no intelligent beings in the universe. Did one or five or π exist just after the big bang? Do people discover mathematical truths, or invent them?
Who cares? Not many people, but for those who do, don’t read this book, because it’s not about that. There’s nothing serious about mathematical platonism or any other philosophy of mathematics. You get a cursory history of math and physics, with attention to the ancient Greeks, Roger Bacon (about whom Clegg also wrote a book—there are many mentions of his other books), Newton, Georg Cantor, and quantum mechanics and string theory, with some discussion of how in physics advances have been made by just dealing with the math and not caring about what it represents, and there’s mention of Wigner’s The Unreasonable Effectiveness of Mathematics in the Natural Sciences. That’s all well enough, but it’s a bit ramshackle.
To come back to the question in the title of the book—numbers, I would suggest, are real at their most basic, but most of mathematics isn’t. It’s a fantasy world that sometimes mirrors and parallels our own, and as such can help provide us with tools to help understand reality. But it needs to be kept in its place. And as long as we (and scientists) remember this, we can’t go far wrong.
There is no discussion, much less justification, in the book for the idea that numbers are real. If they are, how do we apprehend them? Can we touch them? This is a serious issue. This book was no help. Time to go back to Benacerraf and Putnam’s classic Philosophy of Mathematics: Selected Readings.