The notion that mathematical statements should be 'well-defined' had been argued by mathematicians since at least the 1600s,[3] but agreement on a suitable definition proved elusive.
[5] The best-known variant was formalised by the mathematician Alan Turing, who defined a well-defined statement or calculation as any statement that could be expressed in terms of the initialisation parameters of a Turing machine.
[6] Other (mathematically equivalent) definitions include Alonzo Church's lambda-definability, Herbrand-Gödel-Kleene's general recursiveness and Emil Post's 1-definability.
It remains an open question as to whether there exists a more powerful definition of 'well-defined' that is able to capture both computable and 'non-computable' statements.
An alternative account of computation is found throughout the works of Hilary Putnam and others.
This notion attempts to prevent the logical abstraction of the mapping account of pancomputationalism, the idea that everything can be said to be computing everything.
Gualtiero Piccinini proposes an account of computation based on mechanical philosophy.
"Medium-independence" requires that the property can be instantiated[clarification needed] by multiple realizers[clarification needed] and multiple mechanisms, and that the inputs and outputs of the mechanism also be multiply realizable.
A rule, in this sense, provides a mapping among inputs, outputs, and internal states of the physical computing system.
[13]: ch.1 He maintains that a computational system is a complex object which consists of three parts.