> On 5 Jun 2018, at 03:34, Bruce Kellett <bhkell...@optusnet.com.au> wrote: > > From: Bruno Marchal <marc...@ulb.ac.be <mailto:marc...@ulb.ac.be>> >> >> 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. > > It is not at all clear what you mean by arithmetical realism -- there seem to > be two distinct concepts that are confused.
I have defined it many time, sometimes with quite lengthy explanations, sometimes much quicker. The shortest definition is that it is the belief that the excluded middle can be applied to arithmetical proposition, even when we cannot test them. In fact I use only “sigma_1 realism”, or pi_1 realism, which can be paraphrased into the the statement that when we run a machine/program, either it stops or it does not stop (with default hypothesis like no asteroid, no lack of memory, etc.). Arithmetical realism is used by almost all sciences, except some subpart of mathematics. > The difference is perhaps most easily captured in the use of the word > "exists". If we say that there "exists" an integer between 2 and 4, then that > could be called mathematical existence. You can do that. > And that is all that is necessary for mathematics to be used in the rest of > science. Yes, but in metaphysics, we need to put all cart on the table, and eventually, with computationalism, we cannot make sense of anything more than elementary arithmetic for an ontological basic existence, the one which has to be assumed. (Or any Turing equivalent machinery). > It is only when you go beyond this concept of mathematical existence and use > the word in the same way as we would say that the moon "exists", that you run > into trouble. On the contrary, there will be indeed a integers in between 2 and 4. But the moon will get only a phenomenal existence. It will definitely not have the same sense as the arithmetical ExP(x), but the moon will only be “observable”, and that will be a modal existence, actually like []<>Ex[]<>P(x), with [] and <> a material modality (I have given three examples of them). > The Church-Turing thesis is nothing magical -- it states only that any > function computable by a human using some algorithm is also computable by a > Turing machine. I am more or less OK. Better to avoid here “using some algorithm” as the Church-Turing thesis is introduced to define just that. > One side of this -- the human computing via an algorithm -- requires physical > existence of the human. I can agree, but the key point is that the physical existence does not be primary, and does not need a physical ontology to be observable and physical for the human. The fact that the relation between some numbers implements (all) computation(s) is enough to doubt (at least) the need to assume an ontological physical reality. There are no evidence at all for such a primary reality. > The other side -- the Turing machine -- does not necessarily require a > physical machine -- the definition of the machine and its operations would > suffice. Nice. > So the Church-=Turing thesis, in itself, contains a confusion of the two > meanings of "exists”. The Church-Turing thesis is agnostic on the physical ontology. But it is based on arithmetical realism, as it is is made conceptually consistent by the closure of an infinite set for Cantor's diagonalisation, and the fact that some program does not halt. For the total computable function, there is no Church-thesis ever possible, and no universal machine, etc. > >> If anyone would believe that arithmetical realism is false, we would have >> heard argument that Rieman hypothesis or the twin conjecture or Goldbach are >> senseless. But that does not exist. > > That is only one meaning of the word "exists" -- arithmetical realism as I > have defined it above. This is not a mode of existence that would allow any > actual computation -- it allows only descriptions of computations. Not at all. The following theory: classical logic + 0 ≠ s(x) s(x) = s(y) -> x = y x = 0 v Ey(x = s(y)) Contains all descriptions of all computations, but implement none of them. Yet, the following theory does: classical logic + 0 ≠ s(x) s(x) = s(y) -> x = y x = 0 v Ey(x = s(y)) x+0 = x x+s(y) = s(x+y) x*0=0 x*s(y)=(x*y)+x It is only through x+0 = x x+s(y) = s(x+y) x*0=0 x*s(y)=(x*y)+x That the theory becomes Turing universal, and essentially incomplete (incomplete-able). You do seem to confuse, like many others, then difference between “having a description in arithmetic”, and being (arithmetically) true. > > But then, I don't expect that this will convince you that platonism is the > confusion of the two meanings of the word "exists", or that the UD in > platonia cannot compute anything. Because that is just false, using the standard definition. Of course, that come true if you define computation by physical computation. But today, no physicists has ever succeeded in providing a definition of a physical computations which do not related on the mathematical computations. They defined physical computation by the physical implementation of a mathematical computation, i.e. the usual computation as defined, by Post, Kleene, Church etc. > >> If you could avoid ad hominem remark, that would be nice. Also. > > I don't think you know the meaning of 'ad hominem'. You seem to think that > any personal remark is 'ad hominem'. But strictly, 'argumentum ad hominem' is > something quite different. Look it up! Bruno > > Bruce > > > -- > 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 everything-list+unsubscr...@googlegroups.com > <mailto:everything-list+unsubscr...@googlegroups.com>. > To post to this group, send email to everything-list@googlegroups.com > <mailto:everything-list@googlegroups.com>. > 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 everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. 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