Induction, pure and simple

Toward a theory of intelligence

“…Gold [10] and Putnam [19] showed how this idea could be formalized. The use of limiting computable procedures to do induction has been studied by many, including Burgin [2] and myself [14].…”

“…Burgin [4] has called abstract machines that use them "inductive Turing machines of the second order" and I have called them "hyper trial and error" machines [14]. A more general, and more abstract, discussion of such procedures in logical, rather than machine, terms can be found in Kleene's classic paper on the arithmetic hierarchy [12].…”

Thinking may be more than computing

“…Some of the things that philosophers, such as Dreyfus (1979), have suggested that "com- puters cannot do", might find formal models in Recursion Theory. Several philosophical problems associated with induction might be clarified in terms of trial and error models (Kugel, 1977). Uncomputable models of philosophical proposals might increase their precision, a merit that should not be taken lightly.…”

“…But there is a sense in which this is not true when it comes to time. This is easiest to see if we reformulate the Hierarchy Theorem in terms of the 2 and n quantifiers (Kugel, 1977) where '(3x)' . .…”

“…Each 2 quantifier makes us go off in different dimensions of time. (For further discussion, see Kugel, 1977;Clote, 1986.) It is probably easier to think of ways that this kind of thing might go on, than ways that infinite oracles could be implemented in our finite brains.…”