On 05 Sep 2012, at 21:36, meekerdb wrote:
On 9/5/2012 8:37 AM, Bruno Marchal wrote:
Put in another way: there is no ontological hardware. The hardware
and wetware are emergent on the digital basic ontology (which can
be described by numbers or combinators as they describe the same
computations and the same object: you can prove the existence of
combinators in arithmetic,
I don't think I understand that remark. Doesn't arithmetic *assume*
combinators, i.e. + and * ?
Combinators are defined by
K is a combinator
S is a combinator
if x and y are combinator, then (x, y) are combinators.
So they are K, S, (K K), (S S), (K S), (S K), (K (K K)), ((K K) K), etc.
The left parenthesis are often not written, for reason of readability.
The axioms are
Kxy = x
Sxyz = xz(yz).
This is Turing universal, and you can define numbers, + and * in that
system. See the lovely book by Smullyan "To mock a mocking bird" for
more, or my little course on them on this list.
Likewise, you can define them, and emulate them, using only 0,
s(0), ... and the laws:
x+0 = x
x+s(y) = s(x+y)
x*0=0
x*s(y)=(x*y)+x
Which is also Turing universal.
Bruno
Brent
and you can prove the existence of numbers from the combinator S
and K. So the basic ontology is really the same and we can "know"
it (betting on comp). It is really like the choice of a base in a
linear space.
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