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B. Jack Copeland, Carl J. Posy, Oron Shagrir.
Computability: Turing, Godel, Church, and Beyond.
MIT Press. 2013

In the 1930s a series of seminal works published by Alan Turing, Kurt Gödel, Alonzo Church, and others established the theoretical basis for computability. This work, advancing precise characterizations of effective, algorithmic computability, was the culmination of intensive investigations into the foundations of mathematics. In the decades since, the theory of computability has moved to the center of discussions in philosophy, computer science, and cognitive science. In this volume, distinguished computer scientists, mathematicians, logicians, and philosophers consider the conceptual foundations of computability in light of our modern understanding.

Some chapters focus on the pioneering work by Turing, Gödel, and Church, including the Church–Turing thesis and Gödel’s response to Church’s and Turing’s proposals. Other chapters cover more recent technical developments, including computability over the reals, Godel’s influence on mathematical logic and on recursion theory and the impact of work by Turing and Emil Post on our theoretical understanding of online and interactive computing; and others relate computability and complexity to issues in the philosophy of mind, the philosophy of science, and the philosophy of mathematics.

Oron Shagrir.
The Nature of Physical Computation.
OUP. 2022

Computing systems are ubiquitous in contemporary life. Even the brain is thought to be a computing system of sorts. But what does it mean to say that an organ or a system computes? What is it about laptops, smartphones, and nervous systems that they are deemed to compute—and why does it seldom occur to us to describe stomachs, hurricanes, rocks, or chairs that way? These questions are key to the conceptual foundations of computational sciences, including computer science and engineering, and the cognitive and neural sciences.

Oron Shagrir here provides an extended argument for the semantic view of computation, which states that semantic properties are involved in the nature of computing systems. The first part of the book provides general background. Although different in scope, these chapters have a common theme—that the linkage between the mathematical theory of computability and the notion of physical computation is weak. The second part of the book reviews existing non-semantic accounts of physical computation. Shagrir offers an in-depth analysis of three influential accounts, and argues that none of these accounts is satisfactory, but each of them highlights certain key features of physical computation that he eventually entwines into his own account of computation. The last part of the book presents and defends an original semantic account of physical computation, with a phenomenon known as “simultaneous implementation” (or “indeterminacy of computation”) at its core.