2 The reductionist/holist debate
"...thirty years of costly experience with this strategy has produced no evidence of this kind of stability... despite a great deal of work...the problem is still pretty much where it was in 1960...this is worse than being unsuccessful; it is simply contrary to experience...". [14]This is a perfectly valid pragmatic observation, justifying the search for alternative approaches to the subject. It does not, of course, disprove the CTT and in itself supplies only weak support for extrapolations to broader classes (e.g. all living organisms).
An example of the latter is Fishler and Firschein [5], where they give the spaggetti, the string, the bubble and the rubber-band computer as examples of machines that "go beyond" the Turing machine. These examples follow a section on the Busy-Beaver problem, which is interpreted as being a function that grows too fast for any mechanistic computation. The examples themselves do not compute anything a Turing Machine can not, but merely exploit some parallelism in the mechanism to do it faster*1. The implication is that since these "compute" these specific problems faster than a single Turing machine, this is sufficient to break the bounders of the Busy-Beaver problem. Of course, the speed up in these example (which are of a finite polynomial nature) is not sufficient to overcome the busy-beaver limitation, which would require a qualitatively bigger speed-up.
Another example is that used by Kampis [8], that humans can transcend the Goedelian limitations on suitably expressive formal systems. He argues that because any such formal system will include statements that are unprovable by that system but which an exterior system can see are true, and humans can transcend this system and see this, that they thus escape this limitation. He then site's Church's example of the conjuction of all unprovable statements as one we can see is true but that is beyond any formal system.
The trouble with this is the assumption that humans can transcend any formal system to see that the respectively constructed Goedelian statement is true. Although us humans are quite good at this, the assumption that we can amounts to a denial of the CTT already, so this can not possibly used as a convincing counter-example! If you state that the truth of the above is evident to us from viewing the general outline of Goedel's proof, i.e. from a meta-logical perspective, then there will be other unprovable statements from within this meta-perspective. Here we are in no better position than the appropriate meta-logic for deciding this (without again assuming we are better and begging the question again).
Church's conjunction of unprovable statements gets us no further. We can only be certain of its truth as an reified entity in a very abstract logic (which itself would then have further unincluded unprovable statements at this level) - otherwise we are merely inducing that it would be true based on each finite example, despite that fact that such a trans-infinite*2 conjunction is qualitatively different from these (and undefinable in any of the logics that were summed over).
There are many such examples. To deal with each one here would take too long and distract from the purpose of this paper. Suffice to say that all of these (that I have seen) seem flawed if intended as an absolute counter-example to the Church-Turing Thesis.
The basic trouble that the holist faces in arguing against reductionism, is that any argument is necessarily an abstraction. This abstraction is to different degrees formal or otherwise. To the degree that it is informal it allows equivocation and will not convince a skeptic. To the degree that it is absolute/formal it comes into the domain of mathematics and logic where the Church-Turing thesis is very strong (by being almost tautologous). While informal arguments can be used with other holists, in order to argue with a reductionist a more formal argument seems to be required*3.
It appears that it is a necessity limitation regarding the nature of expression itself that makes any such complete demonstration impossible.
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