Thanks as always -- for the attention and patience. I'm re-attaching my
little diagram from this morning, since -- along with getting Douglas
Hofstadter's first name wrong, and confusing David Hilbert with Richard
Dedekind, I also spelled "hierarchy" wrong. (01)
For me -- starting as early as 1968 -- the "algebraic Holy Grail" vision of
an absolutely elegant utterly simple universal ontology has glimmered just
out of reach. My instinct today -- is that a couple of extremely simple
processes actually control the entire thing -- something like "pure linear
recursion within a framework of oneness" -- and that's the entire framework
-- something like that. The point would be -- that "the one" is the
absolute container of everything -- and that this one "absolute" framework
supports and contains an infinite number of "relative" frameworks within
itself. And "what it's made of" (supposedly) is also kind of mind-blowingly
simple -- something like "it's made out of itself" -- also linearly
recursive, across an infinite number of fractal levels -- that descend
across levels of scale, I think -- from "the one general universal infinite
to the particular local infinitesimal." "Words" are little labels for the
way we chop up this space -- and its basic "levels of abstraction" range
across all the academic disciplines -- from the absolute hard ground of
physics (or maybe algebra is really the foundation/ground, and even physics
is dependent on algebra) up to the broad and ungrounded abstractions of
metaphysics and holism. (02)
Eric Temple Bell did title one of his books "Mathematics: Queen and Servant
of Science". So maybe, in some sense, mathematics is prior to physics, and
physics is built on it. (03)
This thing about "what IS it that is recursive?" -- goes to this question of
a "primitive". What is this "object"? How could the entire structure of
cognition be made out of a bottomlessly recursive nothingness that is
defined in terms of itself? But for me, a hierarchical cascade of
abstractions is exactly that -- "a cut on a cut on a cut on a cut on a
cut...." (04)
In 1988, I started saying "everything is made out of dimensions, and
dimensions are made out of dimensions". And the concept of "dimension" and
"cut" are intimately related -- maybe they are complementary -- or some kind
of "holon" -- alternative sides of one extremely abstract and
nearly-incomprehensible thing. (05)
So what IS a "cut"? It's space between two things. Then when you ask "how
wide is the cut" -- "how thick is it" -- the algebra starts to get out of
control -- at least that's what happens to me. There's too much going on in
there to map it all accurately... (06)
But this is how I tend to generalize the Aristotelian genus/species theory
of category formation. That IS the general form. But we got to define the
internal connecting links or layers with higher precision (all those
horizontal and vertical levels and divisions are "cuts") -- because they
have been traditionally left muddy -- on the popular presumption that
there's nothing we can do to bridge these gaps. I'd say we just got to get
utterly demanding -- on what is happening in those "database rows and
columns" -- what are the boundaries of those cells, and what is that
content? How "thick" are those boundaries? How complex can "something
inside a cell" really be? Maybe the cell is just a container or location
for a well-defined but infinite recursion contained within it. (07)
Another point -- just to tack this in here -- and I think this is
consistent with everything we've been saying about the relationship of a
concept or variable to reality -- is that "hierarchies do not exist in
reality". They are constructions of the human mind -- which we build as
accurately as we can -- and then validate and authenticate through a process
of correlation and testing -- thus "proving our theories" -- much as we
might argue that the success of engineering is the proof of a scientific
theory. (08)
* (09)
Bruce:
> Is it naïve of me to even consider that it might be possible to ground
> "natural language" in some fundamental universal foundation? (010)
John:
That is a very important question. Short answer, which leads to many more
questions that are better discussed in the other thread: (011)
1. No, if you mean a finite universal set of formal axioms. (012)
2. Maybe, if you allow for an infinite set in the mind of God. (013)
3. Yes, if you mean a foundation based on every child's genome
and experience. (014)
I think my starting point is something like 2). " Infinite set in the mind
of God." I've worked through several books by physicist and Templeton
Prize winner Paul Davies, including God and the New Physics, The Cosmic
Blueprint, and The Mind of God -- and much related stuff, from people like
David Bohm and David Peat -- plus everything I could even slightly
comprehend on the nature of fractals. (015)
After all these years -- and starting about 1966 with something very much
like Ramon Lull's Wheels -- I do tend to suppose that we are starting to get
our minds around the "integral unit" that contains everything we are talking
about. Could it really be that simple? (016)
Well, I am just now looking at Paul Davies' "The Cosmic Blueprint", p 61,
where he is talking about fractals in a section entitled "The nearest thing
to nothing known to man", and he shows something like the decomposition in
my diagram (he divides his construction into thirds -- maybe the middle part
is the mysterious "cut"). On p. 62 his section heading is "The most complex
thing known to man", which he concludes with the sentence "The fact that the
universe is full of complexity does not mean that the underlying laws are
also complex." (017)
In that thought, I think, there is much hope for a revitalization of
clarity... (018)
- Bruce (019)
-----Original Message-----
From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx
[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of John F Sowa
Sent: Wednesday, April 23, 2014 12:25 PM
To: ontolog-forum@xxxxxxxxxxxxxxxx
Subject: Re: [ontolog-forum] axiom (020)
Pat C and Bruce, (021)
These issues can be discussed better under the more general thread "Toward
Human-Level AI". But just a couple of points: (022)
JFS
>> Debates about how to define the word 'axiom' are as pointless as
>> trying to define the word 'primitive'. That's a related question
>> that raises its ugly head from time to time. (023)
PC
> I am cut to the quick! I think primitives are beautiful. (024)
I'll take back the word 'ugly'. And I'll add that primitives are, for many
special cases, useful. But the debate about whether it's possible to have a
universal set of axioms is as pointless as the debate about a universal set
of primitives. (025)
Bruce
> Is it naïve of me to even consider that it might be possible to ground
> "natural language" in some fundamental universal foundation? (026)
That is a very important question. Short answer, which leads to many more
questions that are better discussed in the other thread: (027)
1. No, if you mean a finite universal set of formal axioms. (028)
2. Maybe, if you allow for an infinite set in the mind of God. (029)
3. Yes, if you mean a foundation based on every child's genome
and experience. (030)
John (031)
TaxonomicStructure3.PNG
Description: PNG image
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