Editor’s note: Welcome to another installment of our weekly War Books series! The premise is simple and straightforward. We ask an expert on a particular topic to recommend five books on that topic and tell us what sets each one apart. War Books is a resource for MWI readers who want to learn more about important subjects related to modern war and are looking for books to add to their reading lists.

This week, we are pleased to share a War Books installment from our director, Colonel Patrick Sullivan, a former math assistant professor at West Point. We gave him the following prompt: What topic and reading would you recommend for military leaders at every level to better understand math and its role in military operations?

As the language of the universe, mathematics is a foundation of every warfighting domain that the military leader will engage with in the emerging global security environment. Moreover, every significant technological development that promises to shape said environment—automation presently, and artificial intelligence in the near future—has mathematics at its core. Unlike other fields relevant to security studies, however, mathematics can seem especially intimidating with a high barrier to entry. Regardless of the intimidation factor, the military leader nevertheless has a responsibility to be at least conversant in key mathematical concepts and uses. In that context, here is a list of popular math books that every military leader should read, developed with an eye toward accessibility.

Journey Through Genius: The Great Theorems of Mathematics, by William Dunham

This book presents mathematics as art, with Dunham intimating that the solutions to great mathematical problems throughout history are just as sublime expressions of creativity and ingenuity as the compositions of any Renaissance master. Any reader with a high school education (and an open mind) can appreciate the simplicity, elegance, and beauty of these solutions, as well as be intrigued by what modern mathematicians still seek to solve and the implications of those potential discoveries.

What is Mathematics? An Elementary Approach to Ideas and Methods, by Richard Courant and Herbert Robbins

I was introduced to this book seventeen years ago as a junior faculty member in the Department of Mathematical Sciences at West Point. First written in the 1940s, this book eschews the mechanistic problem-solving approach typical in mathematics pedagogy in favor of deeply and intuitively connecting the reader with concepts. As British mathematician Ian Stewart said in the 1996 revision, “It doesn’t matter what mathematical things are; it’s what they do that counts.” What is Mathematics? really is unlike any mathematics text that you have probably ever used and can help anyone reconnect with the mathematics of their past toward being a better thinker in the present.

Chaos: Making a New Science, by James Gleick

Chaos is a common but misused term in military leadership and security studies. We presume chaos to be highly contextual—something in the eye of the beholder—but there is surprising mathematical precision to anything in the physical world that appears chaotic. The difference, as Gleick so eloquently argues, is that human ordering is based on Euclidean geometry—the lines and shapes we all know from childhood—whereas the universe’s order reflects fractal geometry. The nature of the patterns in fractal geometry explains everything about our physical world, to include why the seemingly simple, (e.g., predicting weather) is actually complex to the point of being impossible. Understanding what chaos actually is, from a mathematical standpoint, is vitally important to real-world problem framing and solving.

The Evolution of Cooperation, by Robert Axelrod

Game theory is the subfield of mathematics that is arguably the most relevant to military decision-making, and these decisions often involve variations of the famous prisoner’s dilemma. Axelrod’s analysis of the prisoner’s dilemma in a computer tournament more than forty years ago showed empirically that cooperative strategies prevail even in environments of unrestrained competition. His insights proved hugely influential in military strategy formulation and remain significant to the understanding of human agency, behavioral incentive structures, and peer interaction dynamics.

When Genius Failed: The Rise and Fall of Long-Term Capital Management, by Roger Lowenstein

I recommend this book less for understanding of financial markets (although there is significant benefit here) and more for the military leader’s conception of risk. Long-Term Capital Management was a hedge fund that almost collapsed the global financial system during the 1998 Russian ruble crisis. The circumstances of the near collapse all related to low-probability events in pricing models for the derivatives market that turned out to be not so improbable. We often talk about risk calculus in the military, designing campaigns with performance measures to quantifiably minimize risk. Lowenstein’s account provides a real-world cautionary tale about what can go wrong when the calculus is wrong. The bottom line, and reinforcing the ideas of complexity and chaos from other books on this list, is that you cannot manage risk entirely, despite the apparent power of mathematical modeling or, in the military context, of prudent planning.

War Without Fronts: The American Experience in Vietnam, by Thomas Thayer

Related to the above, mathematical modeling intersects with military planning through operations research and systems analysis (ORSA). This book is perhaps the first contribution to the ORSA canon. Thomas Thayer was an analyst in the Vietnam-era Office of the Assistant Secretary of Defense (Systems Analysis), which collected data on military, diplomatic, and development programs in Vietnam, and then assessed the emergent patterns in near real time to inform strategic and political decision-making. Thayer’s book, written ten years after the fall of Saigon, explains the war in the data and its patterns, therein providing the military student with a novel historical frame as well as showing the evergreen potential of systems analysis to aid military decision-making. Systems analysis is fundamentally a leader’s tool, and so its potential utility cannot be realized without the leader prescribing what data should be collected and how to assign meaning to it. Are there patterns? If so, what do the patterns tell us? This sounds simple (and this is really basic math that we are talking about), but easy to screw up in practice as Thayer’s autopsy of America’s Vietnam experience clearly shows.

Colonel Patrick Sullivan is the director of the Modern War Institute at West Point.

The views expressed are those of the author and do not reflect the official position of the United States Military Academy, Department of the Army, or Department of Defense.

Image credit: Spec. Charles Leitner, US Army