The Digital and the Real World

In the 21st century, digitalization is a global challenge of mankind. Even for the public, it is obvious that our world is increasingly dominated by powerful algorithms and big data. But, how computable is our world? Some people believe that successful problem solving in science, technology, and economies only depends on fast algorithms and data mining. Chances and risks are often not understood, because the foundations of algorithms and information systems are not studied rigorously. Actually, they are deeply rooted in logics, mathematics, computer science and philosophy.

Therefore, this book studies the foundations of mathematics, computer science, and philosophy, in order to guarantee security and reliability of the knowledge by constructive proofs, proof mining and program extraction. We start with the basics of computability theory, proof theory, and information theory. In a second step, we introduce new concepts of information and computing systems, in order to overcome the gap between the digital world of logical programming and the analog world of real computing in mathematics and science. The book also considers consequences for digital and analog physics, computational neuroscience, financial mathematics, and the Internet of Things (IoT).

Errata(s)
Errata (183 KB)
Contents: IntroductionBasics of ComputabilityHierarchies of ComputabilityConstructive Proof TheoryComputational Mathematics and Digital Information SystemsIntuitionistic Mathematics and Human CreativityProof Mining bridging Logic, Mathematics, and Computer ScienceReverse Mathematics Bridging Logic, Mathematics, and Computer ScienceFrom Intuitionistic to Homotopy Type Theory — Bridging Logic, Mathematics, and Computer ScienceReal Computability and Real AnalysisComplexity Theory of Real ComputingReal Computing and Neural NetworksComplexity of Algorithmic InformationComplexity of Information DynamicsDigital and Real PhysicsDigital and Real Computing in the Social WorldPhilosophical Outlook
Readership: Undergraduate and graduate students, scientists and readers who are interested in foundational, interdisciplinary, and philosophical questions of mathematics, computer science, and science in general.
Keywords:Computability;Complexity;Constructive Mathematics;Proof Mining;Real Computing;Analog Networks;Information System;Digital PhysicsReview:Key Features:Compact introduction into the foundations of modern mathematics and computer scienceBridging the gap between digital, real and analog computing by new concepts of information systemsConsequences in natural and social sciences with respect to scientific computing