Bruno Marchal wrote: > If you prefer I should > have said "associate" instead of "identifying".
Hi Bruno, welcome back. The terminological distinction you now make above is important - maybe it's another case of Franco-English faux amis (false cognates), but when you say 'identify' I think it steers Peter (and others) towards notions of 'reification' of number. To 'associate' (in your sense of point-for-point commensurability) digital machines with number is a different matter (literally) than to 'identify' (i.e. posit an absolute identity between) them. Perhaps for the future, vis-a-vis reification, we should simply conclude, with Father Jack: "that would be an ecumenical matter"! > Computability is an absolute notion (with CT), but provability is a > relative (with respect to a machine) notion. Put in another way: > computations admits a universal dovetailer which generates and run all > computations, but there is no universal dovetailer for proofs. Point taken. The EC 'axioms' may be better conceived as primitive computations (like the UD), not theorems. In terms of comp, is there any necessary distinction between a UD and a parallel distributed 'architecture'? It seems to me that this must depend on your treatment of time within comp, which I must confess I'm not clear on. It seems critical that the UD 'rotates' stepwise between programs like a multi-tasking OS so that all programs do in fact emerge regardless of their stopping characteristics. As the execution 'proceeds', the differential distribution of specific program segments determines how much 'time' in total each is in effect executed by the UD. But can't this distribution just as well be over a 'block' structure? I look forward (I think) to your more detailed comments! David > Hi, > > I have come back from Bergen (it was very nice) and I have read the > last posts and I will make some comments in order. > > Peter D. Jones said some time ago, after I said that I will identify > "(digital) machines" with number; he said: > > "You can't". > > Of course I can. This is a key point, and it is not obvious. But I can, > and the main reason is Church Thesis (CT). Fix any universal machine, > then, by CT, all partial computable function can be arranged in a > recursively enumerable list F1, F2, F3, F4, F5, etc. It is the list of > the Fi, which has this fundamental and amazing property that it is > close for the diagonalization operation. I have explain this at length > in some posts to George and Tom. The identification between number and > machine is similar to the geometric identification of real numbers and > points once a coordinate system has been fixed. If you prefer I should > have said "associate" instead of "identifying". In computer science, a > fixed universal machine plays the role of a coordinate system in > geometry. That's all. With Church Thesis, we don't even have to name > the particular universal machine, it could be a universal cellular > automaton (like the game of life), or Python, Robinson Aritmetic, > Matiyasevich Diophantine universal polynomial, Java, ... rational > complex unitary matrices, universal recursive group or ring, billiard > ball, whatever. Then just list the programs describable in the language > of that machine to get the Fi. The domain of the Fi gives the Wi which > can be shown to be the mechanically generable sets (of numbers, or > entities nameable, associable or identifiable with numbers in some > context like the partial recursive (computable) functions). > > > David Nyman wrote: > > > [Scene: Night-time. Fathers Ted and Dougal are in bed. > > > > Ted: "Dougal, that's a great idea! Can you tell me more?" > > Dougal: "Whoa, Ted - I want out! I can't take the pressure."] > > > I love them :) > > > Bruno > > PS For a while I will let Colin and David continue their discussion > before interfering. I have other comments but will regroup them for > making minimal the number of posts. Just try not to confuse > computability and provability (in a formal system or by a machine). > Computability is an absolute notion (with CT), but provability is a > relative (with respect to a machine) notion. Put in another way: > computations admits a universal dovetailer which generates and run all > computations, but there is no universal dovetailer for proofs. By > Godel's theorem for any proof system (with checkable proofs) you can > build a richer proof system. Without this all 'hypostases' would > collapse, for example, and the interview with a universal machine would > be ... infinitely boring, and probably unrelated to both quanta and > qualia. > Not also that the relative aspect of provability does not prevent the > finding of universal feature of provability (like obeying G and G* for > example). Note also that most provability systems (like RA and PA or > ZF) are universal computers, but still only relative (and different) > theorem provers. Seen as a universal machine, RA and PA and ZF can > simulate each others. As provability systems, ZF extends properly PA > which extends properly RA. > ZF and PA are lobian machines. RA isn't. > > > http://iridia.ulb.ac.be/~marchal/ --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. 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