The nature of truth in mathematics has exercised the minds of thinkers from at least the time of the ancient Greeks. The great advances in mathematics and philosophy in the twentieth century and in particular the work by Gödel and the development of the notion of independence in mathematics have led to new and complex views on this question. Collecting the work of a number of outstanding mathematicians and philosophers, including Yurii Manin, Vaughan Jones, and Per Martin-Löf, this volume provides an overview of the forefront of current thinking and a valuable introduction for researchers in the area.