Re: Is the Universe really that complicated?
- From: PaulHjelmstad <phjelmstad@xxxxxxx>
- Date: Wed, 17 Sep 2008 13:12:55 EDT
I think what you are asking about is the classic
analysis vs abstraction problem (aka the egg/chicken
problem).
Let's reformulate it into "Is the universe a linear
or nonlinear system!?"
Analysis:
The mathematician who believe that the universe is a
nonlinear system and that no problem is really linear
emphasize the fact that every system will experience
a nonlinear behavior at sometime. They tend to dig
deeper in the problem to find these nonlinear
components. Then use the linear systems to
approximate an affordable near optimal solution.
These researchers tend to work with numerical
analysis.
Abstraction:
On the other hand the algebricians who believe in
abstraction tend to emphasize that the universe is
purely a linear system and once we have a nonlinear
behavior of some phenomenon means that we have not
abstracted enough to understand it.
Example:
One very clear example is Polynomials. Although it
plots nonlinear curves and numerical solutions may be
needed to find its roots, the algebricians have
employed abstract algebra to study polynomials
linearly.
Conclusion:
In general algebricains believe that as long as we
always have new operators as repetitive application
of the previous operator such as;
+ is repetitive application of the SUCC(0)
* is repetitive application of the +
^ is repetitive application of the *
4th operator is repetitive application of the ^
5th operator is repetitive application of the 4th
and so on ...
What exactly are 4th, 5th, 6th operators? Higher Power systems?
then any system using these operators is linear.
One odd case would be the matrix algebra where matrix
multiplication is totally different from scalar
multiplication. However, it is still cannot be
interpreted as nonlinear behavior because * is simply
vector inner product (still linear!!!!!).
Thanks this was very helpful. Aren't they typically
referred to as "algebraists" though? I am reading
Not Even Wrong, and I think I am finally getting
what Representation Theory is. I am hooked on Group
Theory anyway, character tables of course use linear
algebra in their representations. Is that what you
meant?
PGH
.
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