Use and Misuse of Godel's Theorem
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概要
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Wang (1974) reports that in Godel's opinion the two most interesting results proved with rigor about minds and machines are these: (1) The human mind is incapable of formulating (or mechanizing) all its mathematical intuitions. I.e.: if it has succeeded in formulating some of them, this very fact yields new intuitive knowledge, e.g. the consistency of this formalism. This fact may be called the 'incompletability' of mathematics. On the other hand, on the basis of what has been proved so far, it remains possible that there may exist (and even be empirically discoverable) a theorem-proving machine which in fact is equivalent to mathematical intuition, but cannot be proved to be so, nor even be proved to yield only correct theorems of finitary number theory. (2) Either the human mind surpasses all machines (to be more precise : it can decide more number-theoretical questions than any machine) or else there exist number-theoretical questions undecidable for the human mind. In the following I will call these two results "G(1)" and "G(2)" respectively. Some of the questions which will be answered in this paper are these: Question 1: Why is it that there are two results, G(1) and G(2), which were proved with rigour? What is the difference between G(1) and G(2) ? Questoin 2: Which is it that Godel's theorem are to be applied to, minds or brains ? In other words, which do we intend a Turing machine to be a model of ? Questuon 3: Each of the results G(1) and G(2) have the form of disjunction. Which alternative should we choose ?
- 科学基礎論学会の論文
- 2003-11-25