The digital and the real world is a quite unique text that goes beyond the common clichés of describing digitalization as a global phenomenon. Instead, the book investigates the complex relationships between continuous physical reality and digital representations from a computer science (CS) point of view. As the author shows, the gap between the digital world of logical programming and the complex multidimensional enriched with sensor experience world of sciences can be breached by understanding the synergy (in the form of systematic research procedure) between constructivity, provability, and computability. Indeed, and quite often, the multidimensional problems and large-scale problems do not have explicit descriptions in terms of constructive logic leading to well-defined algorithms. Instead, the mathematical solution provided serves as a “path” to the solution rather than an algorithm to compute it directly. In this case, proof of the solution validity and its computability becomes the real challenge. The book lays the foundation research stemming from mathematics, CS, and philosophy, which aims to build up a framework capable of guaranteeing the security and reliability of the knowledge by proof mining, constructive proofs, and program extraction.

The book’s 16 chapters discuss the various aspects of constructivity, provability, and computability--from the basics provided in chapter 1 to a philosophical outlook in chapter 16. In chapters 2 to 4, computability and provability are discussed in the context of elementary number theory. Here, interested readers will find the classical Turing theory of computability (chapters 1 and 2) and further discussion of the complexity of problem solving (chapter 3) and proof theory (chapter 4).

The rest of the book requires a higher level of mathematics background, so chapter 5 provides a concise introduction to the needed concepts. The most interesting chapter is next: “Intuitionistic Mathematics and Human Creativity” discusses Brouwer’s philosophical understanding of mathematical construction. The remaining chapters discuss proof mining, program extraction, and reverse mathematics.

Chapter 16, “Digital and Real Computing in the Social World,” deals with the problems and challenges related to modeling in the social world.

This excellent book can be very useful for graduate students in various disciplines--from CS, physics, and mathematics to economics and social studies. Overall, the text has a light but precise style, which makes it a very pleasant read.