The aim of this book is to explain, carefully but not technically, the differences between advanced, research-level mathematics and the sort of mathematics we learn at school. The most fundamental differences are philosophical, and listeners of this book will emerge with a clearer understanding of paradoxical-sounding concepts such as infinity, curved space, and imaginary numbers. The first few chapters are about general aspects of mathematical thought.

Logic is often perceived as having little to do with the rest of philosophy, and even less to do with real life. In this lively and accessible introduction, Graham Priest shows how wrong this conception is. He explores the philosophical roots of the subject, explaining how modern formal logic deals with issues ranging from the existence of God and the reality of time to paradoxes of probability and decision theory. Along the way, the basics of formal logic are explained in simple, non-technical terms, showing that logic is a powerful and exciting part of modern philosophy.

Number theory is the branch of mathematics that is primarily concerned with the counting numbers. Of particular importance are the prime numbers, the "building blocks" of our number system. The subject is an old one, dating back over two millennia to the ancient Greeks, and for many years has been studied for its intrinsic beauty and elegance, not least because several of its challenges are so easy to state that everyone can understand them, and yet no one has ever been able to resolve them. But number theory has also recently become of great practical importance.

There are many stories we can tell about the past, and we are not, perhaps, as free as we might imagine in our choice of which stories to tell or where those stories end. John Arnold's addition to Oxford's popular Very Short Introductions series is a stimulating essay about how people study and understand history.

The study of geometry is at least 2500 years old, and it is within this field that the concept of mathematical proof first arose. To this day geometry remains a very active area of research in mathematics. This Very Short Introduction covers the areas of mathematics falling under geometry, starting with topics such as Euclidean and non-Euclidean geometries, and ranging to curved spaces, projective geometry in Renaissance art, and geometry of space-time inside a black hole. Throughout, Maciej Dunajski outlines the role geometry plays in the broader context of science and art.

From finding a spouse to finding a parking spot, from organizing one's inbox to understanding the workings of human memory, Algorithms to Live By transforms the wisdom of computer science into strategies for human living.

In this Very Short Introduction, Simon Horobin investigates how we have arrived at the English we know today and celebrates the way new speakers and new uses mean that it continues to adapt. Engaging with contemporary concerns about correctness, Horobin considers whether such changes are improvements or evidence of slipping standards.

Games are everywhere: Drivers maneuvering in heavy traffic are playing a driving game. Bargain hunters bidding on eBay are playing an auctioning game. The supermarket's price for corn flakes is decided by playing an economic game. This Very Short Introduction offers a succinct tour of the fascinating world of game theory, a groundbreaking field that analyzes how to play games in a rational way.

Bertrand Russell wrote that mathematics can exalt "as surely as poetry". This is especially true of one equation: ei(pi) + 1 = 0, the brainchild of Leonhard Euler, the Mozart of mathematics. More than two centuries after Euler's death, it is still regarded as a conceptual diamond of unsurpassed beauty. Called Euler's identity, or God's equation, it includes just five numbers but represents an astonishing revelation of hidden connections.

For centuries, experts have argued that learning was about memorizing information: You're supposed to study facts, dates, and details, burn them into your memory, and then apply that knowledge at opportune times. But this approach to learning isn't nearly enough for the world that we live in today, and in Learn Better journalist and education researcher Ulrich Boser demonstrates that how we learn can matter just as much as what we learn.

The laws of thermodynamics drive everything that happens in the universe. From the sudden expansion of a cloud of gas to the cooling of hot metal - everything is moved or restrained by four simple laws. This powerful and compact introduction explains what these four laws are and how they work, using accessible language and virtually no mathematics.

From Leonardo Da Vinci to John Dee and Comenius, from George Eliot to Oliver Sacks and Susan Sontag, polymaths have moved the frontiers of knowledge in countless ways. But history can be unkind to scholars with such encyclopaedic interests. All too often these individuals are remembered for just one part of their valuable achievements. In this engaging, erudite account, renowned cultural historian Peter Burke argues for a more rounded view.

Most of us are aware of the hive mind—the power of bees as an amazing collective. But do we know how uniquely intelligent bees are as individuals? In The Mind of a Bee, Lars Chittka draws from decades of research, including his own pioneering work, to argue that bees have remarkable cognitive abilities. He shows that they are profoundly smart, have distinct personalities, can recognize flowers and human faces, exhibit basic emotions, count, use simple tools, solve problems, and learn by observing others. They may even possess consciousness.

This Very Short Introduction presents the field of fluid mechanics by focusing on the underlying physical ideas and using everyday phenomena to demonstrate them, from dripping taps to swimming ducks. Eric Lauga shows how this set of fundamental physical concepts can be applied to a wide range of flow behaviors and highlights the role of fluid motion in both the natural and industrial worlds. This book also considers future applications of fluid mechanics in science.

If you move at high speed, time slows down, space squashes up, and you get heavier. Travel fast enough and you could weigh as much as a jumbo jet, be flattened thinner than a CD without feeling a thing - and live forever! As for the angles of a triangle, they do not always have to add up to 180 degrees. And then, of course, there are black holes. These are but a few of the extraordinary consequences of Einstein's theory of relativity.