> On 31 Dec 2018, at 02:37, Russell Standish <li...@hpcoders.com.au> wrote: > > On Sat, Dec 29, 2018 at 09:02:26AM -0800, Brent Meeker wrote: >> >> >> On 12/29/2018 1:58 AM, Bruno Marchal wrote: >>> I use number because people are more familiar with them. Most people can >>> easily conceived that “17 is odd” is true independently of them, but >>> would have an harder time to conceive that KKK=K independent of them, >>> even if this does not mean much more than the first (left) projection of >>> (K, K) is K. >> >> That's a good example. People would immediately recognize that KKK=K is >> just a convention. They are deceived that "17 is odd" is an eternal truth >> independent of human thought because they generalize from their experience >> with physically countable things. >> >> Brent > > This would be a dry argument if instead we simply insisted that all > that is fundamental is a universal machine and all its > computations. That immediately bootstraps the Robinson integers > IIUC. So in a way - integer arithmetic is a convention, our convention > if you like, for looking at computations.
We can see it that way. I have illustrated this recently in showing how the combinators laws Kxy = x Sxyz = xz(yz) Entails the existence of numbers and their addition and multiplication laws. And of course, that was just a step when showing that S and K are Turing universal. All what a (digital) mechanist needs to assume is one universal machinery. The rest is either redundant, or contradictory. I use numbers because it is the “universal system” better known by most, even if many ignore that (N,0,+,*) is Turing universal, which is not entirely obvious to show, but was shown implicitly in Gödel 1931, and explicitly by Church, Post, Turing, Kleene and others in the following decade. Bruno > > > -- > > ---------------------------------------------------------------------------- > Dr Russell Standish Phone 0425 253119 (mobile) > Principal, High Performance Coders > Visiting Senior Research Fellow hpco...@hpcoders.com.au > 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 everything-list+unsubscr...@googlegroups.com. > To post to this group, send email to everything-list@googlegroups.com. > Visit this group at https://groups.google.com/group/everything-list. > For more options, visit 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. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
