Hi John,

In computer science there is something interesting which can be seen as a critics or as a vindication of what you are saying. That thing is the Church thesis, also called Church-Turing thesis, (CT) and which has been proposed independently by Babbage (I have evidence for that), Emil Post (the first if we forget Babbage), Kleene, Turing, Markov, but not by Church (actually).


The thesis has many versions. One version is that ALL computable functions can be defined in term of lambda expressions, or in term of Turing machines, or in term of Markov algorithm, or in term of Post production system, etc. All those versions are provably equivalent.

Such a thesis *seems* to be in opposition with your idea that complete knowledge is impossible. But it is not.

The contrary happens. Indeed the thesis concerns only completeness with respect to computability, and then, as I have already explain on this list, it entails the incompleteness of any effective knowability concerning just the world of what machines can do. Church thesis makes it impossible to find *any* complete theory about the behavior of machines. I explain this in the first footnote of the Plotinus' paper. I can explain if someone ask more. It is proved by a typical use of the (Cantor) diagonalization procedure.
It vindicates what you say, really. We can sum up this by

Completeness with respect of computability provably entails a strong form of incompleteness for our means of knowability and provability about machines' possible behavior.

This can be proved rigorously in few lines. It is stronger and easier than Gödel's incompleteness, and it entails Gödel's incompleteness once we can show that the propositions on the computable function can be translated into arithmetical propositions (the lengthy tedious part of Gödel's proof).

Not only Church thesis makes it possible to think about 'everything', but it makes us able to prove our (machine's) limitation about the knowledge about that everything. In any case, this makes us modest, because either CT is wrong and we are incomplete for computability, or CT is true and we are incomplete about our knowledge about computability, machines, and numbers.

Best,

Bruno



On 16 Mar 2011, at 17:10, John Mikes wrote:

In my opinion an oxymoron.
We cannot even 'think' of it without a complete knowledge of everything, the entire wholeness, call it 'totality' underlying such 'theory'. "All possible" anything, (algorithms, descriptions, assumptions, whatever) - encompass only those 'possibilities' we can think of in the volume of our acquired knowledge (of yesterday). Even (our?) 'impossibilities' are impossible within such framework. We cannot step out from our circle of knowledge into the unlimited unknown world. Any comp we can identify (or even just 'speak' about) is within our world of known items and their relations. Includable into our ongoing mindset. Compare such framework of yesterday with a similar assumption of 1000, or 3000 years ago and the inductive development will be
obvious.
There is no way we could include the presently (still?) unknown (but maybe tomorrow learnable) details of the world (including maybe new logical ways, math, phenomenological domains, etc.) into our today's worldview of "all possible". [Forget about sci-fi]

Maybe even the ways of composing 'our' items (topics, factors, relations and even 'numbers') is a restricted limitational view in the 'model' representing the present level of our development - of which conventional sciences form a part. Comparing e.g. the caveman-views with Greek mythology and with modern 'scientific' futurism (like some on this list) supports this opinion. So I would be cautious to use the qualifier 'COMPLETE'.

John Mikes

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