> On 2 Oct 2018, at 17:11, Philip Thrift <[email protected]> wrote: > > > > On Tuesday, October 2, 2018 at 9:25:17 AM UTC-5, Bruno Marchal wrote: > >> On 2 Oct 2018, at 09:53, Philip Thrift <[email protected] <javascript:>> >> wrote: >> >> >> >> On Tuesday, October 2, 2018 at 2:20:10 AM UTC-5, Bruno Marchal wrote: >> >>> On 1 Oct 2018, at 14:20, [email protected] <> wrote: >>> >>> >>> >>> On Monday, October 1, 2018 at 11:47:47 AM UTC, Bruno Marchal wrote: >>> >>>> On 30 Sep 2018, at 16:30, Philip Thrift <[email protected] <>> wrote: >>>> >>>> >>>> >>>> On Sunday, September 30, 2018 at 4:50:01 AM UTC-5, Bruno Marchal wrote: >>>> [Re:] forcing theory in set theories with classes. >>>> >>>> >>>> Bruno >>>> >>>> >>>> >>>> Do you follow the work of Joel David Hamkins (forcing applied to >>>> set-theoretic "multiverse", etc.) >>>> >>>> (I have a basic idea of a type-theoretic parallel to this.) >>>> >>>> The set-theoretic multiverse >>>> >>>> https://arxiv.org/abs/1108.4223 <https://arxiv.org/abs/1108.4223> >>>> >>>> Joel David Hamkins >>>> @JDHamkins >>>> Professor of Logic, University of Oxford, and Sir Peter Strawson Fellow in >>>> Philosophy, University College Oxford. Formerly of New York. >>>> http://jdh.hamkins.org <http://jdh.hamkins.org/> >>>> >>> >>> The math is interesting, and could be of some use, but it is a priori far >>> too much Aristotelian to be coherent with the mechanist hypothesis. That >>> should follow “easily” from the result described in most of my papers on >>> this subject. The author does not seem aware of the mind-body problem, >>> which put extreme constraints on what the physical reality can come from. >>> Even Peano arithmetic, although integral part of the notion of observer, is >>> too much rich for the ontology, where not only the axiom of infinity is too >>> strong, >>> >>> Since you want to banish the concept of infinity from mathematics, how >>> would you define, say, the limit of an "infinite" series? How would you >>> even discuss this series in the context of finite mathematics? AG >> >> >> Good question. >> >> The answer is not simple technically. The point is that using only the >> theory Q (Robinson Arithmetic) or SK (the combinators), I can define the >> universal (Turing, Church) machine, and the concept of infinity will be a >> tool used by them in their mathematics. >> >> I do not ban anything from mathematics, nor from physics. I ban only >> infinity from the ontological terms. I ban only infinity in the >> metaphysics/theology. (Even God is not ontological, like in Proclus or >> Plotinus theology). >> >> Have you understand the post on Church’s thesis. You might tell me as this >> will help me to see how to proceed to make you grasp all this. >> >> Bruno >> >> >> >> >> What do you think of bounded arithmetic and other "finitist" approaches? >> >> https://en.wikipedia.org/wiki/Bounded_arithmetic >> <https://en.wikipedia.org/wiki/Bounded_arithmetic> >> see bibliography: http://jeanpaulvanbendegem.be/home/papers/strict-finitism/ >> <http://jeanpaulvanbendegem.be/home/papers/strict-finitism/> > > I wrote a paper on this, in a book in honour to Jean-paul Vanbendegem. But > its approach is more than finitist, and a bit less than ultra-finitism. It > does not fit the study of the “theology” of the machine, and is thus useless > for deriving physics. That does not mean it is not interesting pragmatically, > on the contrary, it is well fitted with the goal to make usable programs. I > do think that mathematically, it is also a restriction of Post creativity > (Turing universality in set theoretical terms) to sub creativity. There is no > possible universal machine there. > > > > >> >> Computable real analysis (one can teach computable calculus instead of >> "conventional" calculus) is essentially finitist: >> https://en.wikipedia.org/wiki/Computable_analysis >> <https://en.wikipedia.org/wiki/Computable_analysis> >> >> One can formulate the Axiom of Infinity [ >> https://en.wikipedia.org/wiki/Axiom_of_infinity >> <https://en.wikipedia.org/wiki/Axiom_of_infinity> ] in a type of bounded set >> theory (Jan Mycielski [ https://en.wikipedia.org/wiki/Jan_Mycielski >> <https://en.wikipedia.org/wiki/Jan_Mycielski> ], described in >> https://books.google.com/books/about/Understanding_the_Infinite.html?id=GvGqRYifGpMC >> >> <https://books.google.com/books/about/Understanding_the_Infinite.html?id=GvGqRYifGpMC> >> ]. What results is an "ontology" of bigger and bigger finite sets of >> numbers with gaps in them. > > > Yes, and that is interesting. But not so much for the mind-body problem, > where we cannot bound anything, except by omega. > > The weaker theory known from which my approach can work, is the > delta_0-induction based on Q + the axioms for the exponential, known as > Delta_0Exp. That is Q: > > 1) 0 ≠ s(x) > 2) x ≠ y -> s(x) ≠ s(y) > 3) x ≠ 0 -> Ey(x = s(y)) > 4) x+0 = x > 5) x+s(y) = s(x+y) > 6) x*0=0 > 7) x*s(y)=(x*y)+x > > + > > 8) x^0 = 1 > 9) x^s(y) = x * (x^y) > > + the scheme of induction axioms: > > P(0) & [For all n (P(n) -> P(s(n)))] ->. For all n P(n), > > with P restricted to the delta_0 (= sigma_0 = pi_0 = recursive, decidable, …) > formula. > > > > That is the weaker Löbian machine known today. > > In between Q and Delta_0Exp, you have all the bounded arithmetics. > > An excellent book on this is (without the many accent for the names): > > Hajek, P. & Pudlak P., 1993, Metamathematics of First-Order Arithmetic, > Springer-Verlag. > > But no need of this for the mind body problem, which needs at least > Delta_0Exp (Löbianity), for the observer. Of course I use the fact that Q can > mimic Delta_0Exp. But Q does not believe what Delta_0Exp is saying, and the > theology is for Delta_0Exp and all its consistent extensions, like PA, ZF, > and you, and me … > > I need the sigma_1 completeness. It is not for practical computational > application, but only for guessing what is fundamental to assume, to > understand where the appearances come from. It might have application in the > foundations of physics, though, and is the best way to figure out the > structure of the afterlife or parallel life, etc. > > Bruno > > > > > When if comes to just physics, what is there in the application of any theory > of physics (QM, GR, The Standard Model, ...) to experiments can't be done in > replacing the theory with a Python program, of a Go program or whatever.
Reality, even just the arithmetical reality is beyond what can accomplish a program. The partial computable is the tiny sigma_1 reality (the true proposition having the shape ExP(x) with P decidable). The arithmetical reality is the union of all sigma_i reality (i = 0, 1, 2, …). It contains the truth of proposition like (x)(Ey)(z)(Eu)P(x,y,z,u), which might be decidable or not. To apply a theory for a prediction in the physical reality, you need also an identity brain/mind, which cannot been afforded in the arithmetical reality, a priori. > > Physicists take a theory T and replace it with a program P that then is used > to match with data D. How? You first person state of mind is realised by an infinity of computations in the arithmetical reality, so the identity used by the physicalist does not work. A vague consciousness of this is reflected in the Boltzman brain problem, which is a very particular case in the universal dovetailing that is isomorphic (for computability) with the sigma_1 arithmetical reality. > The theory T is completely dispensable. Only P matters, because it is only P > that us used to say whether a theory T matches D in the results sections of > papers. The theory will corresponds to the observer. To say that the theory is dispensable, is like to say that both a brain and a telescope is dispensable for the existence of the far away galaxy. But brain, telescope are also natural process that we have to explain. Proving, knowing, observing, … are different from computing, even if they are definable in term of computations and their relation with truth. Eventually, the physical reality is a non computable things emerging from all computation. I assume Digital Mechanism all along, to be sure. Bruno > > - pt > > -- > 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 post to this group, send email to [email protected] > <mailto:[email protected]>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <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]. 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.
