The proposition that the set of functions on the natural numbers that can be defined by algorithms is precisely the set of functions definable in one of a number of equivalent models of computation. These models include Post production systems, Church's lambda calculus, Turing machines, Kleene's mu-recursion schemes, Herbrand-Gödel equational definability, Shepherdson-Sturgis register machines, the while programming language, and flow charts. The proposition is a scientific hypothesis, subject to empirical and theoretical confirmation rather than mathematical proof. The evidence that it is true is roughly the following.
First, a large number of disparate methods (e.g. those listed above) for computing functions have been shown to be equivalent in power when computing on the natural numbers. Second, there has been a failure to find a function and a convincing method of computing it that has not been computable by one of the known models of computation. Third, philosophically distinct notions — mechanical computability, digital and analog computability, definability in a formal calculus, definability in an algorithmic language — have been investigated and interrelated. Fourth, a generalization of the theory of computable functions to an abstract computability theory for algebras has revealed new connections and distinctions between models, but confirmed the primary nature of the features of the computation theory on the natural numbers.
The Church–Turing thesis leads to a mathematical theory of digital computation that classifies what data can be represented, what processes simulated, and what functions computed (see computable algebra). It provides a scientific foundation for a discussion of the scope and limits of computable processes in the physical and biological sciences, and hence attracts the attention of philosophers, scientists, and engineers.