On Fri, May 17, 2019 at 10:14 PM Bruno Marchal <[email protected]> wrote:
> On 16 May 2019, at 03:27, Bruce Kellett <[email protected]> wrote: > > On Thu, May 16, 2019 at 12:59 AM Bruno Marchal <[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. 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]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/CAFxXSLQkbFdDHToiXjOGeVdWAmJ3x5o1GwaSnTBiB5E18QPFHQ%40mail.gmail.com.
