On Sun, May 19, 2019 at 04:12:00PM +1000, Bruce Kellett wrote: > On Sun, May 19, 2019 at 10:27 AM Russell Standish <[email protected]> > wrote: > > On Fri, May 17, 2019 at 10:47:36PM +1000, Bruce Kellett wrote: > > 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 > > > > > > 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. > > > Axioms 1-3 define the successor operator s(x). It is enough to > generate the set of whole numbers by repeated application on the > element 0. As a shorthand, we can use traditional decimal notation (eg > 5) to refer to the element s(s(s(s(s(0))))). 4&5 define addition, and > 6&7 define multiplication on these objects. > > > That is where the problem lies. If these axioms generate the set of whole > numbers, then that is a constructvist or nominalist account of arithmetic. If, > however, the integers exist independently and are thus just a model for these > axioms (a domain in which the axioms are true), then you have arithmetic > realism. You can't have it both ways.
It is clear that application of the successor function is sufficient to generate all whole numbers (given sufficient resources, of course). The definitions of addition and multiplication give a contructive way of computing these operations. I can't see why one can't also suppose that those entities exist independently of whether I bother to run a program that generates them or not - so one can have it both ways AFAICS. Realism vs nominalism is a choice. > > > Goedel's incompleteness theorem demonstrates there are true statements > of these objects that cannot be proven from those axioms alone. > > In that sense, the whole numbers are a consequence of those axioms, > whilst also being separately existing entities (having a life of their > own). > > > That is an independent assumption, not implied by the axioms above, as I have > pointed out. > No I was parroting another argument that Goedelian incompleteness entails an independent existence - that some things are true (exist) even if you cannot generate that thing algorithmically. I'm a little ambivalent on this argument - it forms the core of the argument a friend of mine is writing a book about, but he's only shown me the first chapter (which I've critiqued), so I haven't got to the meat of it. > > There are also nonstandard airthmetics, that involve adding additional > elements (infinite ones) that cannot be created by successive > application of s. > > Given these 7 axioms can also be viewed as an algorithm for generating > the whole numbers, acceptance of the Church-Turing thesis (ie the > existence of a universal Turing machine) is sufficient to reify the > whole numbers. > > > That remains to be proved. Church-Turing is about calculable numbers, not > about > reification. It also works in a purely nominalist account. > Hmm - possibly I went too fast here. The existence of a running universal dovetailer is sufficient for the whole numbers to be reified, as the abovementioned constructive program will eventually be run for all such whole numbers. In order for the dovetailer to fail to generate all whole numbers, it must be starved of some resources, which is an ultrafinitist move. I think that the CT thesis requires that all possible programs can be run in order for a machine to be considered truly universal. Ultrafinitism makes a nonsense of that, of course. Hopefully, we're still on for that beer in Carlton on Friday afternoon! -- ---------------------------------------------------------------------------- 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 view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/20190519064509.GL5592%40zen.
