> On 5 Jun 2020, at 00:39, Lawrence Crowell <[email protected]> > wrote: > > On Thursday, June 4, 2020 at 6:07:45 AM UTC-5, Bruno Marchal wrote: > > > On 2 Jun 2020, at 19:34, 'Brent Meeker' via Everything List > > <[email protected] <javascript:>> wrote: > > > > > > > > On 6/2/2020 2:49 AM, Bruno Marchal wrote: > >>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List > >>> <[email protected] <javascript:>> wrote: > >>> > >>> > >>> > >>> On 6/1/2020 2:08 AM, Bruno Marchal wrote: > >>>> Brent suggest that we might recover completeness by restricting N to a > >>>> finite domain. That is correct, because all finite function are > >>>> computable, but then, we have incompleteness directly with respect to > >>>> the computable functions, even limited on finite but arbitrary domain. > >>>> In fact, that moves makes the computer simply vanishing, and it makes > >>>> Mechanism not even definable or expressible. > >>> That's going to come as a big shock to IBM stockholders. > >> > >> Why? On the contrary. IBM bets on universal machine > > > > No, they bet only on finite machines, and they will be very surprised to > > hear that they have vanished. > > They bet on finite machines … including the universal machine, which I insist > is a finite machine. That is even the reason why I called it from times to > times universal number. > > I recall that once we get the phi_i, which can be defined in elementary > arithmetic, we get all the universal numbers, that is all u such that there > phi_u(x, y) = phi_x(y), and such u can be used to define all the recursive > enumeration of all digital machines. > > The implementation of this fine but universal machines are called (physical) > computer, and is the domain of expertise of IBM. > > Bruno > > > Of course any computation is going to be finite or involve a finite number of > bits.
Any halting computation. Some non halting computation requires infinite time and space, virtual, arithmetical or physical. > This happens as well with quantum computers, but there is one difference. Two > states can be prepared and entangled so they have a continuum of > probabilities depending upon measurement angle. This is what separates QM > from classical mechanics. This separates entanglement of spins from the > Bergman's socks, where knowing the left sock is in one box the right must be > in the other. So while there is a finitude to the entanglement entropy or the > quantity of quantum information, the possible ways an entanglement can > register outcomes is infinite. This is what gives a violation of Bell's > theorem in QM. With the measurement of a quantum system the pair of a state > and measurement forms a type of Godel numbering. This connects QM foundations > with the phi_u(x, y) = phi_x(y),you state above. OK. But you assume some quantum universe, where the UDA explains why you have to derive the quantum from arithmetic or (Turing) equivalent. > > A classical computer will always be finite, and you can't have an infinite > Cantor diagonalization. The Kleene diagonalisation is constructive. It shows the inexistence of some finite machine having some “arithmetical omniscience”. It requires potential infinities, not the cantorian infinities. > The computers that are manufactured are done so to solve certain problems, > RSA encyrption, user interfaces for service personnel from travel agents to > sales, word processors, games, cell phone signal shifters, data processors of > medical measurements and on it goes. All computers exists in arithmetic, and all computations exist in an internal limit of arithmetic (by step 2, actually!). With mechanism, the physical reality is not the fundamental reality. The physical reality emerges from the computation executed in virtue of the number relations, like the prime number distribution, for example. > Even with quantum computers this will take off, and in fact I have thought > quantum computing would be a way of managing a dynamics network defined by > millions of drones over a city. Even if as I think the Godel-Turing result > underlies obstructions between entanglement types quantum computers will in > time become the province of engineering and business applications. No doubt on this. It is just that with mechanism, the physical universe is not ontological, but more like a collective hallucination made by the relative universal number relations which are as true, and independent of the physical laws, than 117 is composite and 317 is prime. Bruno << Of this reality, as I explained […], I take a 'realistic" view. At any rate (and this is my main point) this realistic view is much more plausible of mathematical than of physical reality, because mathematical objects are so much more what they seem. A chair or a star is not in the least like what it seems to be ; the more we think of it, the fuzzier its outlines become in the haze of sensations which surrounds it; but '2' and '317' has nothing to do with sensations, and its properties stand out the more clearly the more closely we scrutinize it. It may be that modern physics fits best in the framework of idealistic philosophy---I do not believe it, but there are eminent physicist who say so. Pure Mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical is built that way. >> G. H. Hardy, "A Mathematician's Apology", Cambridge University Press, 1940 (1998). > > LC > > > > > > > Brent > > > >> and know well what is a computer: a finite arithmetical being in touch > >> with the infinite, and indeed, always asking for more memory, which is the > >> typical symptom of liberty/universality. IBM might be finitist, like > >> Mechanism, but is not ultrafinist at all. Anyway, mathematically, > >> Mechanism is consistent with ulrafinitsim, even if to prove this, you need > >> to go beyond finitism, (but then that’s the case for all consistent > >> theory: none can prove its own consistency once “rich enough” (= just > >> Turing universal, not “Löbian”). > >> > >> Bruno > > > > > > -- > > 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] <javascript:>. > > To view this discussion on the web visit > > https://groups.google.com/d/msgid/everything-list/3068f558-7f61-56cb-61fe-44832ec28a91%40verizon.net > > > > <https://groups.google.com/d/msgid/everything-list/3068f558-7f61-56cb-61fe-44832ec28a91%40verizon.net>. > > > > > -- > 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/1cde9ecd-fc90-4031-a117-3e67a43af888o%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/1cde9ecd-fc90-4031-a117-3e67a43af888o%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/47281CA0-10D0-496D-B65C-EACA27D11BBE%40ulb.ac.be.
