Hi Russell,
> On 5 Jun 2018, at 01:32, Russell Standish <[email protected]> wrote: > > On Mon, Jun 04, 2018 at 03:50:33PM +0200, Bruno Marchal wrote: >> >> You seem to confuse arithmetical realism, used in all branches of science, >> and Platonism (which is part of the consequence). To define mathematically >> what a computation is, we need arithmetical realism. In SANE04, my >> definition is redundant because the Church-Turing thesis makes no sense at >> without arithmetical realism. > > Hi Bruno, I think you need to be aware that your writings do not help > here, and that perhaps you need to clarify the point. For a long time > I understood Arithmetic Realism <=> Platonism of the integers, but now > I understand you make a more subtle distinction. Yes. I have acknowledge that already. It is better to not make “arithmetical” realism into the definition of Computationalism, because people put too much in it, and it is actually the minimal amount of acceptation in math to make sense of the Church-Turing thesis. Since then I define comp by YD + CT, without AR. I have introduced AR originally to emphasise that I do not assume more than arithmetical realism. It was a way to show how few assumption is made. It is not a metaphysical assumption. The only metaphysical, or theological, or psychological assumption uses is in the YD. It was also warning for intuitionists, or ultrafinitists. I > > One way of moving forward is that when you talk about the "Robust" > universe case, you are effectively postulating Platonism of > computations. ? I have used “robust” only for a physical universe. It is a physical universe in which we can run a UD. Obviously, if our expansion continue for ever, we can make a case that our universe is not robust. That would be a problem with step 7, but step 8 (or equivalent) makes this non relevant. I have defined Arithmetical Realism sometimes by saying that an arithmetical realist is someone who does not take his kids back from school when he heard that they have been taught that there is no greatest prime number, like if that was bs). Arithmetical realism implies Realism on computations. By Matiyazevic, we have explicitly that realism on the existence and non existence of solutions of Diophantine equation is enough. > So you can then move on to discussing the non-robust > case, which I take to be some kind of ultrafinitism in fact. At step seven. Yes, a physicalist ultrafinitism can be used to prevent the immaterialist consequence. But only at step seven, and that moves is made into an appeal to magic in step 8, or just with Occam, and the fact that computations exists in arithmetic (and not just as description, which certainly do not exist as some computation can be infinite, and there is no infinite in arithmetic). > > A more detailed discussion of the distinction between arithmetic realism > and platonism would help here. Yes. Note that I have already explained, but maybe you were not there, why I prefer the expression “arithmetical realism” better than arithmetical platonism, used by mathematicians. The reason is that I use Platonism in the sense of Plato, a loose and general sense of skepticism about primary matter or physicalism. It was more a philosophy-calism, or idealism, not excluding forms of mathematicalism, like the starting idea of Pythagorus (only numbers). So arithmetical realism is a recall that in the storyline we will follow you have signed up with the excluded middle principle: either phi_678(890) converges (stops), or it does not. Precisely, either the translation of all facts making phi_678(890) converging are arithmetically true, or they aren’t. I use “Platonism” for a variety of theologies, which have progressed from Plato to Damascus, with two apparent remarkable peak in clarity with Moderatus of Gades (by deduction because we lost his texts) and Plotinus (we have the entire texts!). It is a theology in the greek term of the sense: that is a theory of God, or Truth, guessed as transcendent, with an explanation of where it comes from, and why and how it make your consciousness here and now, and where the illusion comes from, etc. God is just the nickname of “theory of everything”. With Mechanism, it will be, with respect to the sound machine, the sigma_1 truth (but that is a “secret”, a theorem of G*). > For instance, why did you feel the need > to include arithmetic realism as a distinct axiom from the CT thesis > in the first place? To avoid any confusion in case the member of the jury was intuitionist or ultrafinitist. It really does not means much more than believing that (3^3) + (4^3) + (5^3) = (6^3) is certainly true, or certainly false, independently of verifying it or not. To implement a computation, you need a reality, to make sense for example of a proposition like “ at time t, the register was containing the number 5”. The physical reality can implement that. But the arithmetical reality can to, not by just describing that computational fact, but by the fact that some numbers divides some other number. As far as the computational realm is concerned, the arithmetical reality kicks back as much as the physical reality. I use Tarski’s notion of Truth. And Gödel’s arithmetical provability/believability predicate corresponding to the machine x. The theology of x is given by Tarski minus Gödel, when x is self-referentially correct, and believe in enough induction axioms, like PA. The arithmetical reality is more rich than we previously thought before Gödel. The computable is a tiny part of it, and to talk on the computable, the detour to the non computable is necessary, and made constructive (and that is what Emil Post already saw in 1922). The complexity of arithmetic is in large part due to the fact that the universal machine/number, in there, introduces a non controllable mess there. There is no algorithmic way to control the algorithms. To get a universal machine, you have to let her go and explore, even if you are afraid she get lost looking for chimeras, or meet strangers. Free-will exists because it is encrypted in the non computable distribution of halting/non-halting sequence. That is Post number (although I don’t find the reference): it is 0,0100100011011100111000… according to the stopping of phi_i or the belonging of a universal (creative) set in the sense of Post. Unlike Chaitin number, it is highly redundant, and “interesting”, in the sense of Bennett. It Chating numbers maximally decompress, somehow. It is equivalent to a universal dovetailer. Arithmetical Realism is just that the proposition like “it exists a number which has this decidable property” is eaten true of false, or that i belongs to w_i, or not. Platonism is the attitude in metaphysics to remains skeptical with what we see and how we interpret it with the idea that it could result from simpler but invisible principles. Best, Bruno > > > -- > > ---------------------------------------------------------------------------- > Dr Russell Standish Phone 0425 253119 (mobile) > Principal, High Performance Coders > Visiting Senior Research Fellow [email protected] > Economics, Kingston University http://www.hpcoders.com.au > ---------------------------------------------------------------------------- > > -- > 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 post to this group, send email to [email protected]. > Visit this group at https://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/d/optout. -- 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]. 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