Editing Church–Turing thesis (section)

==Statement in Church's and Turing's words==
{{See also|Effective method}}

{{harvs|txt=yes|author-link=J. Barkley Rosser|first=J. B.|last=Rosser|date=1939}} addresses the notion of "effective computability" as follows: "Clearly the existence of CC and RC (Church's and Rosser's proofs) presupposes a precise definition of 'effective'. 'Effective method' is here used in the rather special sense of a method each step of which is precisely predetermined and which is certain to produce the answer in a finite number of steps".<ref>{{harvnb|Rosser|1939}} in {{harvcolnb|Davis|1965|page=225}}.</ref> Thus the adverb-adjective "effective" is used in a sense of "1a: producing a decided, decisive, or desired effect", and "capable of producing a result".<ref>{{cite dictionary |entry=effective |dictionary=Merriam Webster's New Collegiate Dictionary |edition=9th}}</ref><ref>See also {{cite dictionary |url=http://www.merriam-webster.com/dictionary/effective |entry=effective |dictionary=Merriam-Webster's Online Dictionary |edition=11th |access-date=2014-07-26 |postscript=,}} which also gives these definitions for "effective" – the first ["producing a decided, decisive, or desired effect"] as the definition for sense "1a" of the word "effective", and the second ["capable of producing a result"] as part of the "Synonym Discussion of EFFECTIVE" there, (in the introductory part, where it summarizes the similarities between the meanings of the words "effective", "effectual", "efficient", and "efficacious").</ref>

In the following, the words "effectively calculable" will mean "produced by any intuitively 'effective' means whatsoever" and "effectively computable" will mean "produced by a Turing-machine or equivalent mechanical device". Turing's "definitions" given in a footnote in his 1938 Ph.D. thesis ''[[Systems of Logic Based on Ordinals]]'', supervised by Church, are virtually the same:

{{quote|{{sup|†}} We shall use the expression "computable function" to mean a function calculable by a machine, and let "effectively calculable" refer to the intuitive idea without particular identification with any one of these definitions.<ref name="Turing_1938_thesis_p8">{{cite thesis |author-first=A. M. |author-last=Turing |date=1938 |url=https://webspace.princeton.edu/users/jedwards/Turing%20Centennial%202012/Mudd%20Archive%20files/12285_AC100_Turing_1938.pdf |title=Systems of Logic Based on Ordinals |type=PhD |publisher=Princeton University |page=8 |access-date=2012-06-23 |archive-url=https://web.archive.org/web/20121023103503/https://webspace.princeton.edu/users/jedwards/Turing%20Centennial%202012/Mudd%20Archive%20files/12285_AC100_Turing_1938.pdf |archive-date=2012-10-23 |url-status=dead}}</ref>}}

The thesis can be stated as: ''Every effectively calculable function is a computable function''.<ref>{{harvcoltxt|Gandy|1980|page=123}} states it this way: "What is effectively calculable is computable." He calls this "Church's Thesis".</ref> Church also stated that "No computational procedure will be considered as an [[algorithm]] unless it can be represented as a Turing Machine".{{citation needed|date=November 2019}}
Turing stated it this way:

{{quote|It was stated&nbsp;... that "a function is effectively calculable if its values can be found by some purely mechanical process". We may take this literally, understanding that by a purely mechanical process one which could be carried out by a machine. The development&nbsp;... leads to&nbsp;... an identification of computability{{sup|†}} with effective calculability. [{{sup|†}} is the footnote quoted above.]<ref name="Turing_1938_thesis_p8"/>}}