Incomputability after Alan Turing

TL;DR

This paper reviews the historical and contemporary significance of incomputability, originating from Turing's foundational work, and explores its implications across various scientific disciplines and understanding of the natural world.

Contribution

It provides a comprehensive Turing centenary perspective on incomputability's role in science, technology, and understanding complex phenomena beyond mathematics.

Findings

Incomputability is central to understanding natural and artificial systems.

Turing's work presaged modern interdisciplinary applications of incomputability.

The legacy of incomputability influences current research in AI, physics, and biology.

Abstract

Incomputability as a mathematical notion arose from work of Alan Turing and Alonzo Church in the 1930s. Like Turing himself, it attracted less attention than it deserved beyond the confines of mathematics. Today our experiences in computer science, physics, biology, artificial intelligence, economics and the humanities point to the importance of the notion for understanding the world around us. This article takes a Turing centenary look at how the interface between the computable world and the incomputable formed a central theme in Turing's work - from the early establishment of the standard model of the stored program computer, to the late work on emergence of form in nature, and the first approaches to understanding and simulating human intelligence. Turing's thinking was remarkably prescient, and his legacy still impacts on much of our work. The incomputable may turn out to be a specially important part of the legacy, with consequences that Alan Turing himself could not have envisaged.