ONT Re: Differential Logic A -- Discussion
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
DLOG A. Discussion Note 16
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
HT = Hugh Trenchard
Re: DLOG A Discussion 14. http://suo.ieee.org/ontology/msg05425.html
In: Differential Logic A. http://suo.ieee.org/ontology/thrd1.html#05359
Hugh,
It seems that many of these consternations
are accountable to defects of my exposition
that I do not know how to remedy right now,
but I will try to address some of the less
entangled ones.
HT: Thanks for the succinct exposition. I think I have understood the whole
time that Peircean logic (albeit formalized logic at any level) involves
a set of axioms and procedures which are fundamentally abstract and do
not necessarily allow for the overlap of other or competing formalisms
(although I take your point that Peircean logic is not entirely unique).
I do not think that "lack of overlap" follows from abstraction.
Indeed, one of the reasons for scrambling up the concrete rocks
of experience to some higher level of abstraction is just so we
can better see the commonalities among domains that appear to be
disparate when our nose is to their grindstones. And one of the
reasons that I started this work on differential logic was to get
a better focus on the principles from mathematical systems theory
that might be analogized and generalized to deal with outstanding
issues of change and diversity in logic-based intelligent systems.
Now, I really do understand the impatience to get on to the interesting bits.
The literature that you mention, or the precursors of it, evokes memories of
the very sort of excitements that filled my days in the 70's and 80's of the
late great 1900's. But -- famous first words -- that was only the beginning.
But there is a prerequisite subject that we should have taken in school
in order to sew up this very overlap, to forge the links of necessity
between the quantitative-probabilistic and the qualitative-logical,
but we can't have taken it because it hadn't been invented yet,
and so I've been busy working on that.
HT: And I do think that current studies of emergent phenomena involve
very particular formalisms -- that it isn't just about how and
where we might wish to insert our own interpretation of what
we want to call an emergent phenomenon. One only needs to
review such articles as "Collective Induced Computation"
by Delgado and Sole, or Per Bak or countless others to
see how emergent phenomena may be analyzed mathematically.
The fact that we are bound to use particular formalisms and
systems of concepts to talk and to think about phenomena of
any kind does not entail that "all is vanity and subjective".
But there are general strategies that all formal sciences use
to identify the objective properties that may be lurking under
the bushes of appearance.
As a very typical example, consider the way that the Chomsky-Schutzenberger
complexity of a formal language is defined. A given formal language will
have many formal grammars, or, alternatively, many automata, that accept
or generate it. Now, formal grammars and automata can be classified by
ostensible complexity-like properties that they wear on their sleeves,
that is, you can easily identify a context-free grammar or a pushdown
automaton simply by inspecting the form of its given presentation.
A formal language gets to be called "context-free" if there is
'some' context-free grammar or pushdown automaton that accepts
or generates it, but that only gives an upper bound on the
complexity of the language. In order for it to be called
"properly context-free" one has to prove that there are
'no' finite-state automata or grammars that do the job.
This definition of formal language complexity exemplifies
a very common pattern for defining invariant properties.
If one thinks of a grammar as a "theory" of a language, that is,
as a particular way of describing a language in effective terms,
then the attribution of a property as an invariant of a language,
as with the case of graduated complexity properties, entails the
idea that the property in question can be demonstrated to belong
to the language in question from the standpoint of every relevant
description, grammar, or theory of the language.
Extracting the general pattern, the attribution of a property that
merits being called an "invariant" or an "objective" property of the
object under review will typically involve a logical quantification,
like "all" or "some", ranging over sets of theoretical perspectives.
Have to take a break here ...
Jon Awbrey
HT: I do also realize that unless I can deliver the formal goods
myself on any possible overlap, there isn't much point in my
going on about it.
HT: My only purpose in raising the questions is to see if perhaps you yourself
might see some underlying common mathematical principle which remains to be
identified. I see that you do not believe there is one, nor are you about
to make finding one an objective of yours, and I certainly don't blame you.
HT: In terms of your last question about where does one go once he has
identified an emergent phenomenon, one might conceivably do what,
for example, Delgado and Sole have done, which is to address the
question of whether there is "a tradeoff between the individual
complexity and collective behavior in such a way that complex
emergent properties cannot appear if individuals are too
much complex", after which they proceeded by rigourous
statistical analysis to answer the question.
HT: Or, do as Per Bak did: identify the properties of emergent phenomena
(e.g. self-organized criticality), identify the patterns of order
which characterize emergent phenomena like power laws (such as
Zipf's law or Pareto's law), or fractals (as I mentioned in
a previous email).
HT: Or do as Langland, Bak, Wolfram, and others have done in simulating the
phenomena by computer analysis to show how complex phenomena does indeed
derive from very simple rules. To boot, Wolfram believes practically all
of science can be examined in terms of cellular automata (see "A New Kind
of Science"). (Maybe I should be asking Steven Wolfram this question (?)).
HT: These are simply a few examples.
HT: In terms of power laws, as we all know, they are ubiquitous in
complex dynamic systems (this is what I am doing in my analysis
of bicycle racing -- I am trying to confirm the existence of
a power law relationship between the number and intensity of
"attacks"), and given their relative omni-presence, I don't
think it entirely unreasonble to suggest they might appear
in surprising ways in other formalisms (if such formalisms
describe in some way physical phenomena), as unlikely as
it might seem. Although again, unless I can make some
sort of intelligent connection myself, there isn't
much point in pursuing it.
HT: Having said all that, from this point forward I'll stay inside
the Peircean box, unless we jump to a completely separate box
should we wish, but never the twain shall meet.
HT: p.s. famous last words?
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
http://www.cs.bsu.edu/homepages/mighty/history.html
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o