I remember a night in graduate school when a friend in the humanities asked a few of the math graduate students whether we were Platonists or Empiricists, and this led to a spirited discussion of differing philosophies of mathematics.

And then I remember a decade passing where I just went about my business proving theorems and computing examples without thinking about the issue again.

I expect that the experience of most mathematicians has been similar to mine, and that most of you don’t spend much energy thinking about the meta questions of mathematics, such as whether it is more useful to think of numbers from the cardinal perspective or the structural perspective^{1}. Or maybe you have lively conversations with friends and students at coffeehouses, but it doesn’t seriously impact the work that you do on a daily basis.

And this is fine. At the same time, the philosophy of mathematics is a large area, and there have been lots of interesting papers written about these ideas by mathematicians and non-mathematicians alike. Ernest Davis and Philip J. Davis have compiled a number of recent essays on the meaning and ontology of mathematics in the collection *Mathematics, Substance and Surmise*, published by Springer.

There are seventeen essays in the collection, and the list of authors includes mathematicians, philosophers, cognitive psychologists, historians, and others. The essays are all generally concerned with questions about the nature of mathematical objects: what do we mean by them, in what sense can they be said to exist, and where do they come from? More importantly, how has our answer to these questions evolved over time, and what implications do our answers have for the world of mathematics or the world more generally?^{2} While considering different aspects of these questions, the authors bring a wide range of mathematics into their essays, and the book features appearances from ideas number theory, category theory, analysis, algebra, and geometries both Euclidean and Noneuclidean.

I won’t go through a blow-by-blow description of the essays in the collection, but instead I will point out a few of the essays that I found particularly interesting:

- The opening essay, by Ursula Martin and Alison Peace, looks at the evolution of mathematical collaboration over the years, and in particular it describes and then compares the collaborations between Hardy and Littlewood in the early twentieth century to the polymath project started a few years ago by Timothy Gowers (and others) and continuing to this day.
- In addition to co-editing the volume with his father, Ernest Davis contributes a chapter entitled “How should robots think about space?” that argues that people working in artificial intelligence need to program their creations with some understanding of time and space, and while it is often useful to model time with the real numbers and space with three-dimensional Euclidean space, there are times that it is actually better to use other models.
- Fans of his many textbooks (and I consider myself one), know that John Stillwell’s writings are typically very clear expositions that combine mathematics and history in beautiful ways, and his contribution to this volume is no exception. In particular, he writes about the nature of continuity and addresses the question “what does the continuum do for mathematics?” as well as looking at how our evolving ideas about infinity have led to changes in mathematics as a whole. Along the way, Stillwell describes the difference between “large cardinals,” “informal infinities,” and the continuum itself in a way that is both rigorous and very readable.
- It is often said that mathematics is a universal language. Micah Ross looks at this notion by considering what the nouns, verbs, and adjectives of the language of mathematics would be, and how the linguistic strategies that different cultures have adopted to describe numbers and quantities have varied over time.

As a reviewer, I would be hard-pressed to say that reading this book will dramatically change the way that most readers do mathematics. At the same time, many of the essays changed the way I thought about mathematics, at least momentarily. They are well-written and engaging, and have many interesting ideas while generally being short enough to not overwhelm the reader. In summary, while it won’t be for everyone, I think that people who would enjoy reading this book will enjoy reading this book. And if that is the kind of statement that intrigues you rather than annoys you, I suspect you belong to the set that I am referring to.

**Notes**

**1**. As Lance Rips writes in an essay in this collection, a Cardinalist would define the number five to be the last number you say when counting a collection of five apples, while a Structuralist would define the number five in terms of its relations to other numbers (5=1+4, 5=2+3, etc). If you are like me, you are currently going back and forth between finding this distinction incredibly fascinating and annoyingly pedantic.

**2**. One interesting feature of the book worth commenting on is that the authors chose to put the chapters in the order that mazimizes the mean similarity metween consecutive chapters. This feature alone is intriguing, and is discussed in detail on the book’s website

Darren Glass is an Associate Professor of Mathematics at Gettysburg College, whose mathematical interests include graph theory, algebraic geometry, and cryptography. He can be reached at dglass@gettysburg.edu.