The foundations.

I thought it might be useful to touch on a subject I pikced up from several good reads and a very helpful mathematics professor. There are different levels of understanding involved when solving scientific problems. These are levels I generally consider when pondering a physical problem (simple or complex):

1. Philosophy
2. Mathematics
3. Physics
4. Theory and Imagination

Interestingly enough though, the levels seem to form a Cartesian circle.

1. Philosophy
Without some basic axioms and foundations, mathematics would not exist. We take for granted to the scale to which our assumptions mold our ways of thinking.

2. Mathematics
A good read for a description of what I'm about to quickly cover is a book by Roger Penrose called The Road to Reality. A must read for any physics buff!
There are really three seperate 'worlds' that mathematics lives in.
a. The 'real world' - the world we live in
b. The 'mathematical world' - the world where all mathematics resides
c. The 'abstract world' - the world where mathematics lives, but has no relation to the real world. This would be the universe of ideal mathematical models and complex systems which do not and will never exist
All three of these 'worlds' are connected. The mathematical world lives in us, in our brains. We can come up with real math solutions for the 'real world', and abstract and ideal solutions to things which do not exist in 'real world'.

3. Physics - well, you know, you gotsta use da maths.

4. Theory and Imagination
These are the powers which drive scientific progress, and are the powers that are essential in making up your own science (mine is going to be part of the 'abstract world' :) )