The Church-Turing thesis is a foundational claim in computability theory, proposed independently by Alonzo Church and Alan Turing in the 1930s. A standard formulation holds that every effective computation can be carried out by a Turing machine.
By an effective method the thesis means a procedure with finitely many exact instructions, that produces a result in a finite number of steps, that could be performed by a human working with paper and pencil, and that demands no intuition or ingenuity. The thesis asserts that any problem solvable by such a systematic procedure can be solved by a Turing machine.
What gives the thesis its force is convergence. Church identified effective computability with the recursive functions and with lambda-definability, while Turing defined it in terms of what his abstract machines could do. Although these approaches looked very different, they proved to pick out exactly the same class of computable functions.
This equivalence across several independently developed formal definitions is taken as strong evidence that they all capture the correct, intuitive notion of mechanical computation. The thesis is not a theorem that can be proved, because it links an informal idea, what can be calculated, to a precise mathematical definition; instead it is supported by this accumulated evidence and is widely accepted.