[MD] Metabiology

[John Carl]

I agree Adrie,

On Thu, Aug 26, 2010 at 1:24 PM, ADRIE KINTZIGER <parser666 at gmail.com>wrote:

> Thx , Andy , this stuff is incredible interesting
>
>
I was just reading some other interesting stuff that made me wonder if I
combined the two, compiled and let it run it's course, you'd end with LIFE,
heh-heh.

It's ALIVE!
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http://www.philosophy.uncc.edu/mleldrid/SAAP/MSU/DP16G.html
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Royce derives a complete Boolean algebra upon a different basis: through the
definition of  a more general and inclusive order system, System S.  S’s
laws and principles may be not be defined in first order logic since Royce
implicitly quantifies over relations and sets.  A second-order definition is
too cumbersome for our purposes, so I will remain within the informal
language of S.  This language uses a, b, c, d … to symbolize collections; a,
b, c, … to symbolize elements in collections. Collections may stand under O,
E, and F relations, relations which are n-ary, or what Royce calls polyadic.

            1.  For any collection a and any collection b, if a is an
“O-collection,”  symbolized by O(a), then O(ab).  This defines an operation
of adjunction.

     2.  For any collection b, for any element bn of b, and for any
collection d, if O(dbn) and O(b), then O(d).  This defines   an  operation
of adjunction.

      3.  There exists an element x.

      4.  For any element x, there exists an element y, such that x ¹ y.

      5.  For any element x, for any element y,  there exists a z, such that
if x ¹ y, then            O(xyz) and ~O(xz) and ~O(yz).

      6.  For any element x, for any collection a, there exists a y, such
that if O(ax), then O(xy) and O(xyan).



The rough structure of the system so defined is that of an infinitely large
O-collection, which by virtue of axioms 5 and 6 is continuous, i.e., it is
dense and includes its limits.  Indeed, Royce nicknames his System the
“logical continuum.”

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John: The "logical continuum" with rationality itself as a subset!
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            Certain “transformations” of this “logical continuum” lead to
promising results. The relational properties of O-collections are identical
to special cases of Boolean multiplication:  O(abc..) = a · b · c · … = 0 .
Because of this analogy with Boolean operations, other relations akin to
those in Boolean algebra may be defined.  If, for example, O(ab), then a
totally excludes b and vice-versa, elements which Royce consequently labels
obverses.  It then follows directly that if ~a, then b, and if ~b, then a.
This is to say that a binary O-relation implies two conditional statements
of the foregoing form.  Further, if O(abc) then, if ~b, then a and c.  In
other words, if a,b, and c exclude one another in their totality, then it
follows that if we replace one of the elements with its negative or obverse,
the other two may “overlap” or “coexist.”  Royce calls ~b in this situation
the “mediator” of a and c, and symbolizes it as follows:  F(~b/ac).
Finally, if one designates an element of this triad the origin, say a, then
the F-relation assumes a binary form, with respect to the origin, and also
becomes asymmetrical and transitive.  Such a relation can therefore be the
basis of partially or totally ordered sets.  Royce symbolizes such relations
as follows:  ~b -<a c .  This appears to be a modification of Peirce’s
symbolization of illation.

            Royce derives the whole of the Boolean operations by converting
the O-collection of S into a partially ordered set, that is an F-collection,
as follows.  He arbitrarily selects an element and designates it the
0-element.  By the axioms S, the obverse of 0 exists, which we may designate
as 1, and the totality of the remaining elements become so ordered by
F-relations that for all elements x, x -<o 1.  In more informal language,
every element of set is implied by 0 and implies 1 so that every x is
“between” 0 and 1.  The usual operations of Boolean algebra are then
verifiable by virtue of the construction of this partially ordered set.
Axioms 5 and 6 ensure that this Boolean algebra is complete.
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John: Now here is where I think it gets especially Moq-worthy - the
introduction of 0, or DQ - not just an axis, but a direction! And a
randomizing one at that.
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            Royce’s derivation of a complete Boolean algebra is unusual in
two respects, both of which he attributes to the work of British
mathematician A.B. Kempe.  First, the 0-element is arbitrarily selected from
the elements of S, and second, the set is ordered not by a binary relation,
but by a triadic relation.  However, the inclusion of 0 in the relation
reduces the triadic relation to binary.   The 0-element consequently
functions as merely an origin, in terms of which the direction of the
asymmetry of the binary relation is defined.  It follows from these unusual
features of Royce’s System, that one may define an infinite number of
complete Boolean algebras from System S, each ordered with respect to the
particular element selected as the origin.  In other words, System S
“contains” the partially ordered set in terms of which the Boolean
operations are defined in an infinite number of ways.

            It is notable that Royce thinks that his System also “contains”
any possible ordered set, and so avers that his System is a statement of a
complete system of categories.  Since all rational activity, as Royce puts
it,  is dependent upon ordering relations, and all ordering relations may
be, so he argues, defined in term of his System, his claim is prima facie
plausible.  It turns on his ability to derive various ordered sets.  Before
he died, he successfully derived the order system of common metric geometry,
and in unpublished  notes attempted to derive projective geometry.  The
definitive mathematical investigation of his System he has left to us to
pursue.

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John:  Catch that?  All rational activity is dependent upon ordering
relations (patterning) .  this is where the ongoing iteration in a
metabiological program could be exposed to DQ!



Sorry about that, you may now go back to your regular programming.


John the delusional