> On 3 Jun 2019, at 11:47, 'Cosmin Visan' via Everything List > <[email protected]> wrote: > > You just said in another post that mechanism = computationalism.
Yes, I use computationalism as a synonym of “Indexical Digital Mechanism”, or simply Mechanism to be short. > Now you say that mechanism = partially computationalism. Can you make up your > mind ? Important remark. It is just an (amazing, and amazingly simple) theorem (formalisable in elementary arithmetic!) that there is no effective theory of the entire range of the total computable (see below). So, to define mathematically/precisely what is (total) computability, we need to accept partial computability. Then, typically, the universal machine behaviour will NOT be entirely computable. A universal machine is necessarily a code for a partial computable function; it is undefined on some natural number. A total computable function from N to N is defined on all natural numbers. It is a function from N to N. A partial computable function from N to N can be undefined on some natural numbers. It is a function from a subset of N in N. Note that a total function is a particular partial function, when the subset of N on which it defined is the st N itself. A total function is a particular case of partial function. We use "strictly partial” to refer to a partial function which is not total. We can effectively, mechanically, enumerate all partial computable function, through their codes in some universal system (like a universal programming language). But as I have proved more than ten times in this forum, the price of having a universal machine is that we cannot mechanical decide in advance if a code is program computing a total or a partial function. If we were able to distinguish mechanically the codes for the total computable functions from the (strictly) partial one, we would be able to enumerate all total computable function from N to N: F_0, F_1, F_2, F_3, …. F_I, …. Then the function G(n) = F_n(n) + 1 would be a total computable functions, but then there would be a F_k equal to G, and we get: G_k(k) = G_k(k) + 1, and each G_k(k) are numbers, so by subtracting G_k(k) at both sides, we get 0 = 1 (contradiction). OK? (Ask for any possible further clarification) This means that the only way to get all total functions is to get all partial computable function in the list, and the total functions will be mixed in an NON algorithmetical way, among the partial function. The problem above will no more appear as the reasoning will only show that G_k(k) is not defined, and so cannot be subtracted on both sides. You can program G_k, and verify, on all programming language of computer that it crash the machine, making it running "forever”. I have defined mechanism by the idea that we can survive with a digital (universal) machine at the place of the brain (in a generalised sense which can include the body possibly with any finitely describable part of the environment). This needs the notion of universal machine, which are necessary *strictly* partial computable, so mechanism cannot avoid that partialness of the machine’s behaviours. So if I am a universal machine (and the universal part of this is provable, it is “being a machine” which is not provable), I can only be (strictly) partial computable. Bruno > > On Monday, 3 June 2019 10:35:27 UTC+3, Bruno Marchal wrote: > If we assume mechanism, we assume to be at least Turing Universal, and the > Turing universal being are only *partially computable”. > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected] > <mailto:[email protected]>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/d7fdcb4a-f453-4bc5-b089-3142e76dc12c%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/d7fdcb4a-f453-4bc5-b089-3142e76dc12c%40googlegroups.com?utm_medium=email&utm_source=footer>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/2DD3276D-9D47-4406-8E5B-816335C77DFC%40ulb.ac.be.
