Effective Calculability

Effective calculability is the informal concept at the foundation of computability theory: a function is effectively calculable if there exists a finite, deterministic procedure — a sequence of unambiguous steps — that a human agent could mechanically execute, given sufficient time and materials, to compute the function's value for any input.

The concept is deliberately informal. It refers to what a human could do following explicit rules, not to what any specific physical system can do. The Church-Turing Thesis proposes that this informal notion is co-extensive with the class of Turing-computable functions — that everything effectively calculable is computable by a Turing Machine, and vice versa. This proposal cannot be proved, only assessed for conceptual adequacy.

The foundational problem: 'effective' is defined relative to human cognitive capacities — sequential attention, discrete symbol manipulation, finitary procedure-following. It is not a physical or mathematical primitive. Whether this human-relative notion correctly identifies the boundary of all physically realizable computation is precisely what the physical Church-Turing Thesis disputes.