Sunday, July 25, 2010

The Debate about Church's Thesis and its Converse: Church's Definition/Thesis and its Epistemological Status (Part 4.2 of Effective Versus Algorithmic Computability)

Section 4.2 is devoted to the much discussed epistemological status of Church's thesis. First, (4.2.1) the thesis is introduced in the original form, namely as definition, and it is explained where the term 'Church's thesis' comes from. In 4.2.2 the arguments in favor of the thesis/definition, put forward by various authors, are exhibited and discussed. The present author doesn't want to hold back his opinion that Church's step-by-step argument remains to be the best on the market – in opposition to some recent criticism by Sieg, followed by Soare and Schulz.
The epistemological status of the thesis is further examined at full length in Section 4.2.3. After some preliminary remarks on definitions (real and nominal ones, as propositions and as rules) using Weingartner's theory of definition, the early debate among the initiators of computability theory is recollected and analyzed ( Church, Turing, and Gödel pleaded for 'definition', while Post, Kleene, and Kalmár rejected this view. Already this controversy contained essentially all ingredients of later discussions, up to now. Nothing really new has been contributed since then, as Section shows. In the thesis is compared to other well-known mathematical theses/definitions such as that every circle is the set of all points equidistant to some given point. Also, Quines discussion of the paradigmatic case of the definition of 'ordered pair' (in Word and Object) is examined in order to find some clue what to do with Church's thesis. Further comparisons and analogies are listed in, quite a few involving fundamental physical laws. Finally, (un-)provability and refutability of Church's thesis are discussed in Section In a brief resumé is presented.


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