How does the Church-Turing thesis connect algorithms with recursive functions?

The Church-Turing thesis serves as a foundational bridge between the informal, intuitive notion of an algorithm—or effective procedure—and the formal, mathematical definition of recursive functions.12

In the early 20th century, logicians sought to define "effective computability" precisely. Several independent characterizations emerged, including Alonzo Church’s lambda-definable functions and Alan Turing’s abstract Turing machine. The thesis asserts that these distinct formalisms—along with the class of partial recursive functions—are equivalent and fully capture the pretheoretical concept of what can be computed by a mechanical process.31452

By demonstrating that these formal systems describe the same set of computable functions, the thesis provides a rigorous basis for logic and computer science. It allows logicians to conclude that if a function can be computed by an intuitive algorithm, it is necessarily a recursive function and thus computable by a Turing machine. This identification is essential for understanding the limits of computation, as it establishes that any process replicable by a machine is bounded by the capabilities of these recursive definitions.15324