An award-winning mathematician shows how we prove what's true, and what to do when we can't. How do we establish what we believe? And how can we be certain that what we believe is true? And how do we convince other people that it is true? For thousands of years, from the ancient Greeks to the Arabic golden age to the modern world, science has used different methods-logical, empirical, intuitive, and more-to separate fact from fiction. But it all had the same goal: find perfect evidence and be rewarded with universal truth. As mathematician Adam Kucharski shows, however, there is far more to proof than axioms, theories, and laws: when demonstrating that a new medical treatment works, persuading a jury of someone's guilt, or deciding whether you trust a self-driving car, the weighing up of evidence is far from simple. To discover proof, we must reach into a thicket of errors and biases and embrace uncertainty-and never more so than when existing methods fail. Spanning mathematics, science, politics, philosophy, and economics, this book offers the ultimate exploration of how we can find our way to proof-and, just as importantly, of how to go forward when supposed facts falter.