How do recursive functions describe computations that can be carried out by a machine?

In the context of the foundations of mathematics, recursive functions describe computations that can be carried out by a machine because they represent the formalization of effective calculability.12

A function is considered recursive if its values can be determined through a finite, mechanical procedure—essentially a program—that applies routine operations to known values. The Church-Turing thesis asserts that this formal definition of recursive functions perfectly captures the intuitive, pretheoretical notion of what is "effectively computable" by a machine.321

The connection to machine computation is established through the Turing machine, a theoretical model that uses a finite set of instructions to manipulate symbols on a tape. It has been proven that the class of functions computable by a Turing machine is identical to the class of partial recursive functions. Consequently, any process that can be executed by a computer or a human following a finite, step-by-step algorithmic method is, in theory, a recursive function. This equivalence allows mathematicians to study the limits of computation, such as the decision problem, by analyzing the properties of these functions.421