ThinkArt Lab Animation: A.T. Kelemen© November 12, 1998 Dr. Rudolf Kaehr

After having produced a picture of the intuition of proemiality and polycontexturality of natural number systems, the obvious questions arises, what can we do with all that? and especially, what can we do with all that what we cannot do with the classical approach?

It is more than crystal clear, that everything would be changed if we would have been able to introduce, in a convincing way, the slightest change in the very concept of formal systems, say, logic and arithmetics. Logic and arithmetics have not to be confused with the big business of all sorts of logical and arithmetical systems or the immense multitude of formal systems based on the very concept of logic and arithmetics. (Whatever this exactly means.)

## 2.4 Relativization of Inductive Definitions (David Isles)

First of all we should remember that the concept of a Turing Machine is a paper-and-pencil concept. More a program, than a physical machine. Its purpose was purely mathematical, that is to give a formal explanation of the intuition of the notion of algorithm in mathematics, especially in number theory.

This opens up the possibility of questioning Turing´s explication in the context of new mathematical intuitions and their own explications. First of all it is also about

and not about computerGedankenexperimente