Good questions. You are right, a theorem
is a statement that some domain is
structured in a particular way.
The Princeton companion to mathematics
lists 35 major theorems, from the ABC
conjecture and the Atiyah-Singer Index
Theorem to the Weil Conjectures.
Theorems are based on connections in
the structure of mathematical systems:
to find a new theorem is like revealing
a hidden structure. Some connections are
shortcuts between different points, others
are bridges between different areas.
Why do so many structures have hidden
internal structures? Interesting question.
It is the reason why we do Mathematics,
otherwise it would be boring. I would say
because there are systems where simple
elements and rules can produce complex
structures.
The basic mathematical elements and axioms
allow a whole universe of combinations
and connections which is consistent and
complex at the same time. Algebraic and
geometric systems seem to contain an
infinite number of complex structures.
The integers 0,1,2,3,4,.. may be simple,
but there is an infinite number of them.
If we consider only the numbers of the
finite Group with 4 elements, then Number
Theory becomes less interesting.
In general the patterns and structures
which can emerge in a system depend on the
basic axioms, elements and operations,
and on the size of the system. A kind
of emergence again, perhaps..
-J.
----- Original Message -----
From: Russ Abbott
To: The Friday Morning Applied Complexity Coffee Group
Sent: Sunday, April 25, 2010 6:47 AM
Subject: [FRIAM] Why are there theorems?
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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