Musings on Turing's Thesis

Summary

I take Turing's thesis (equivalently, Church's thesis) to assert that those functions on the integers which can be computed by a human being following any fixed algorithm on pencil and paper can also be computed by a Turing machine algorithm (or alternately by a lambda calculus algorithm). This thesis can be formulated using any of the many definitions of algorithm developed in the past eighty years which compute the same functions of integers. This has often been implicitly replaced by what I would call the physical version of Turing's thesis. This asserts that those functions on the integers which can be computed on any physical machine can be computed by a Turing algorithm. If the brain is regarded as a physical machine, this version subsumes the first version. But not everyone regards the brain as entirely physical (“Mathematics is a free creation of the human mind”—Brouwer). So we separate these formulations.

The meaning of Turing's thesis depends on determining what algorithms are possible, deciding whether algorithms should be defined to allow unbounded search using potentially infinite time and space, and what algorithms the brain can execute. The meaning of the physical Turing thesis depends in addition on determining what can be manufactured in the physical world. Neither the capabilities of the brain nor the capabilities of physical materials have been or are likely to be characterized by science. These questions have an intuitive, informal, and inexhaustibly open character.