| |
|
|
|
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 Gedankenexperimente and not about computer science or technology.
On the other side, the today reality of computation is far beyond of what is conceived by Turing Machines. Instead of algorithms, one of the new metaphors and challenges seems to be interactivity, in all its forms.
Therefore, it is possible to start a more deconstructing reading of the concept of Turing machines and to introduce step by step a new type of machines, the polylogic machines, without being forced to enter a debate engulfed by the orthodoxy/heterodoxy dramaturgy of academic referees. Nothing is wrong with the classical concepts. Neither with the known extensions, like o-machines, etc., of Turing himself and others. And nevertheless there is no reason to not to try another approach, surely not to the exactly same challenges, but strongly related to each other and interwoven in some family resemblance(similarity, likeness).
"As a final example, let us consider the changes effected when one uses different NNN´s (Natural Number Notation Systems) in place of the intuitive natural numbers in a standard argument from recursion theory. In what follows a <--- n means that Turing machine a is given input n. Recall the standard
Theorem (unsolvability of the halting problem
Let T be the class of Turing machine programs and /a/ be the Gödel number of a T.
There is no test" Turing machine b T such that b <--- /a/ halts in state
Y if a <-- /a/ halts
N if <-- /a/ halts doesn't halt.
Proof
If there were, define the contradictory machine
b* = b v {<YSRY>/S any tape symbol of b} #
In this argument the intuitive natural numbers are used in at least three distinct constructions.
1) in the inductive definition of the class of Turing machine programs. Here a given inductive definition will have a length and we may speak of l(a), the shortest length of the Turing machine program a;
2) the class of inputs to the Turing machines; and
3) to measure the length of Turing machine computations (this is implicit in the words "halts" and "doesen´t halt").
Now whatever may be our preconceptions, there is nothing in this argument that requires the use of the "same" natural numbers in all three constructions.
Indeed all that is required is that if a T, then /a/ should be defined, that is, should be available as an inout. Hence it is consistent with the structure of the argument to suppose that we have three different NNN´s, N1, N2 and N3 and that for a prticular stage k we consider the class of Turing machine programs T(Nk1) (where a T(Nk1) means l(a) (Nk1 ), the class of inputs Nk2 and relativize the notion of "halting" to "halting as measured in Nk3.
Theorem
The point of this example is to suggest that the peculiarly "absolute" character of a result such as the unsolvability of the halting problem may be chimerical and have its origin in certain unrecognized assumptions (the uniqueness of the natural numbers)."
in: F. Richman (Ed), Proc. New Mexico 1980, LNM873, 1981, p 133
It is of great importance, to keep exactly the traditional wording in the process of deconstruction. Therefore it is pointed to "in this argument" and not in another agument. Often people change the wording and proof the classical result wrong. Deconstruction has nothing to do with this attitude of "Besserwisser". The classical argument is in no sense wrong; contrarily, it is correct in all steps.
The difference is in the decision of the preconditions. If we accept them, then everything is correct. If we don't accept them, e.g. the uniqueness of the natural numbers, then trivially our results will differ correspondingly.
On the other side we have the enormous problem to introduce the new and different concept, here the anti-traditional concept of NNNS. If it fails, the whole argument was only a Gedankenexperiment in the sense, suppose we have a well founded theory of NNNS then look what happens with our heavy weight theorems. And this (hypothetical) argumentation and its constructions is of enormous importance at least to learn how to over come classical limits of thinking.
Unfortunately, Yessenin Volpins introduction of his NNNS has not convinced many of his colleagues. This is, by the way, one of the reasons of my own research, based on Gunthers concept of kenogrammatics, in this field. May be my own attempt brings the whole idea and intuition some steps further to realization.
|
ThinkArt Lab http://www.ThinkArtLab.com sales@ThinkArtLab.com |
| |
|
|
|