> On 17 May 2019, at 14:47, Bruce Kellett <[email protected]> wrote: > > On Fri, May 17, 2019 at 10:14 PM Bruno Marchal <[email protected] > <mailto:[email protected]>> wrote: > On 16 May 2019, at 03:27, Bruce Kellett <[email protected] > <mailto:[email protected]>> wrote: >> On Thu, May 16, 2019 at 12:59 AM Bruno Marchal <[email protected] >> <mailto:[email protected]>> wrote: >> The first order theory of the real numbers does not require arithmetical >> realism, but the same theory + the trigonometrical functions reintroduce the >> need of being realist on the integers. Sin(2Pix) = 0 defines the integers >> in that theory. >> >> If you reject arithmetical realism, you need to tell us which axioms you >> reject among, >> >> 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 >> >> Some people add some metaphysical baggage in “realism” which is not there., >> “Arithmetical realism” is just the doctrine according to which the axioms >> above make sense. Usually, they are implicitly taught in primary school. >> It is used only for the Church-Turing thesis and the (mathematical) >> definition of “digital machine”. >> >> Bruno >> >> You are just using your personal Humpty-Dumpty dictionary to define >> "realism". Arithmetical realism is a bit more than just the axioms above -- >> it is a metaphysical notion. > > “Metaphysical notion is fuzzy”, but I have given a precise definition of > realism in arithmetic, the one used in the work. Realism is just the belief > in the truth of the axioms above (and a bit of logic). > > You say that "realism" is just acceptance of the axioms of arithmetic above. > But then you say that arithmetical statements are true in the model of > arithmetic given by the natural integers. There is a problem here: are the > integers the model of your axioms above, or is it only the axioms that are > "real". If the integers are the model, then they must exist independently of > the axioms -- they are separately existing entities that satisfy the axioms, > and their existence cannot then be a consequence of the axioms, on pain of > vicious circularity.
If you agree with the axiom above, you can follow the derivation of physics from them. But to understand why we have to do this, and why there are no other ways, we need Mechanism, which, as you remark correctly, ask for much more than those axioms, the YD leap of faith. The integers are not the model, but the element of the model. The model is the set theoretical structure (N, 0, s, +, x). The symbol “0” is interpreted by the number 0, s by the function x + 1, etc. No theories can prove or even talk about its model, by results by Gödel and Tarski. We use always stronger theories to stay the semantics of smaller theories. That is obligatory by incompleteness. A bit like Riemann who use complex number to study the distribution of the primes. There are simply no effective theories at all of the natural numbers. Arithmetic is essentially undecidable: there are no complete or total theories of the natural numbers. That is why, in part, we don’t need to assume more than the numbers. Any universal machinery would be OK. Physics and theology are independent of the choice of the ontology, which is only on the terms of any Turing universal machinery. You assumption that there is a primitive reality simply does not work, or you need to be realist on much larger part of mathematics, so to make sense of an infinite body, to get back the identity Brian-mind, but then you just propose a non mechanist theory (more exactly, you point on a theory which do not yet exist), and this just to avoid a simpler solution based on a simpler theory. That is speculation with the goal to avoid testing a simpler theory. That is not the scientific attitude. Bruno > > The alternative is to say that the integers are defined by the axioms, and > cannot, therefore, be a model (in your sense, viz, independent entities that > satisfy the axioms). > > 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 [email protected] > <mailto:[email protected]>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/CAFxXSLQkbFdDHToiXjOGeVdWAmJ3x5o1GwaSnTBiB5E18QPFHQ%40mail.gmail.com > > <https://groups.google.com/d/msgid/everything-list/CAFxXSLQkbFdDHToiXjOGeVdWAmJ3x5o1GwaSnTBiB5E18QPFHQ%40mail.gmail.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/25D14E9F-50A4-4D78-925B-A9D19941194D%40ulb.ac.be.
