Russ -
Another great question.
While Doug and I have an awful lot in common, this is probably where we
most notably diverge. You ask "why", he asks "why ask why", I ask
"why ask why ask why". ("Who dat who say who dat?" might ring a bell
for some of the other old timers here).
I don't waste many of my otherwise productive cycles on such questions
as "why theorems?" but I do find myself enjoying such questions quite
thoroughly when not occupied with making a living or preparing the
garden (actually it is a good question to ponder while turning over a
garden bed or raking up last falls detritus from the courtyard.
I *do* like Doug's ancillary point, however, of "who decides what is
interesting?" and my corollary would be "isn't 'hidden' a relative
concept?" I do suspect that your question is as much about human
nature/intelligence as it is about anything intrinsic in theorems
(excepting that theorems are human constructs).
I'll give Rich Murray points for a maximal grandiosity to simplicity
ratio, though I don't find that
Single infinite unity is awesome.
It is identity.
has much explanatory power.
Grant seems to tease out an important and key point. The numbers
themselves seem quite simple (counting, progression) but as we add
relationships (the notion of addition or multiplication), the
complexity explodes. Perhaps an interesting corollary to your
question is why do simple systems exhibit a geometric(?) explosion of
properties and relationships as we seek them out? It seems like there
ought to be a meta-answer based entirely in combinatorics of the
language involved. Kurt Godel would seem to have something to say
about this?
It's a good question, I will contemplate it while I complete the
digging of the footer for my emergent greenhouse. And I look forward
to the mail flurries from this group of deep and broad thinkers.
- Steve
Russ, I apologize for being so terse. Let me try again. Here is my take
on your question...
As we know, systems are more than just components, or elements. A
system must also have relationships among its elements before they it
is worthy being called a system.
But, when you take these component relationships into account, the
possibilities for what characteristics, or properties, a system may
exhibit begins to ramify into a potentially large and surprising
number, due to combinatorics. With so many possible component
relationships, it often becomes non-intuitive as to which potential
properties (true statements) of the system are true.
Thus the need for theorems arises due to a system having relationships
among its components. And we haven't even mentioned emergent properties
yet!
This is simple, of course, because it is elemental, foundational to
systemics.
Take care,
Grant
Grant Holland wrote:
There are theorems because systems have relationships as well as
elements, from which arise emergent properties.
Grant
Russ Abbott wrote:
I have what probably seems like a strange question: why are
there theorems? A theorem is essentially a statement to the effect
that some domain is structured in a particular way. If the theorem is
interesting, the structure characterized by the theorem is hidden and
perhaps surprising. So the question is: why do so many structures have
hidden internal structures?
Take the natural numbers: 0, 1, 2, 3, 4, ... It seems so simple: just
one thing following another. Yet we have number theory, which is about
the structures hidden within the naturals. So the naturals aren't just
one thing following another. Why not? Why should there be any hidden
structure?
If something as simple as the naturals has inevitable hidden structure,
is there anything that doesn't? Is everything more complex than it
seems on its surface? If so, why is that? If not, what's a good example
of something that isn't.
-- Russ Abbott
______________________________________
Professor, Computer Science
California State University, Los Angeles
cell: 310-621-3805
blog: http://russabbott.blogspot.com/
vita: http://sites.google.com/site/russabbott/
______________________________________
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FRIAM Applied Complexity Group listserv
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============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
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