Chapter three analyses the period between 1650 and 1800. Many thinkers see The Enlightenment’ as intoxicated with ideas of reason, control and with building perfect knowledge, organizations and societies. I demonstrate that this view is exaggerated. The ravages of wars produced two opposed intellectual movements: on the one hand, natural law and rationalism whose adherents believed in certain knowledge and abstract schemes; on the other, thinking in terms of probabilities, which recognizes and accommodates uncertainty. Hume, Smith, Voltaire and Montesquieu saw the limits of human reason and foresight and made considerable room for uncertainty. Military thinkers cautioned that war is unpredictable and that systematic knowledge is a pipe dream. Uncertainty and unpredictability occupied the centre stage in European culture; in paintings, the picaresque novel and the popularity of gambling and betting. This era was much contested as three different world views established themselves: The idea that the world could be understood and predicted, the sense that it is entirely uncertain and a pragmatic world view that recognizes and accommodates uncertainty as a part of the world.

This chapter introduces Bayesian probability and the rules of probability theory, emphasizing that Bayesian probability emerges as the uniquely consistent extension of deductive (Boolean) logic to contexts of uncertainty and incomplete information

This chapter starts by reviewing important concepts from probability theory and stochastic processes. Subsequent chapters on probabilistic input and structural uncertainty make heavy use of random vectors and vector-valued stochastic process, so the reader should be familiar with the material included on these concepts. Next, the chapter provides a review of set-theoretic notions. The material on sets in Euclidean space included in this part is key to understanding the set-theoretic approach to input uncertainty modeling. The chapter concludes with a review of several fundamental concepts from the theory of discrete- and continuous-time linear dynamical systems.

Chapter 2 provides a review of probability theory, focusing on the topics that are essential for the remainder of the book. Included are the elements of set theory, the axioms and basic rules of probability theory, the concept of a random variable, discrete and continuous random variables, univariate and multivariate probability distributions, reliability and hazard functions, expectation and statistical moments, distributions and moments of functions of random variables, and extreme-value distributions. Appendix A presents commonly used probability distribution models with their properties, for easy reference. Thorough mastery of the material in this chapter is essential for understanding the remainder of the book.

This is a survey of constructive and computable measure theory with an emphasis on the close connections with algorithmic randomness. We give a brief history of constructive measure theory from Brouwer to the present, emphasizing how Schnorr randomness is the randomness notion implicit in the work of Brouwer, Bishop, Demuth, and others. We survey a number of recent results showing that classical almost everywhere convergence theorems can be used to characterize many of the common randomness notions including Schnorr randomness, computable randomness, and Martin-Löf randomness. Last, we go into more detail about computable measure theory, showing how all the major approaches are basically equivalent (even though the definitions can vary greatly).