Sorry, that was me being overly sensitive. I will endeavor to grow up.
Papers, there's not much more I can add; chaos theory was only briefly covered in the introductory differential equations (diffeq) course I took, and I haven't gone further into it since. My major of study, when I choose to accept it, is more focused on applying diffeq in various circumstances, which means using qualitative methods that aren't very interesting from a philosophical standpoint. It is something I'd like to study further at some point, though.
I can, however, go more into the background of it. Differential equations is basically calculus backwards; instead of looking at an equation and solving for its slope (derivative) or area (integral) at any given point, diffeq takes a relationship of one or more variables and the derivatives and integrals thereof. This is complicated. Whereas a derivative may be found for any equation, there are certain equations which are unintegrable, and there are very few differential equations which can actually be solved. As was unintentionally demonstrated many times in the course I took on the subject, simply miscopying a problem out of the book can, and usually will, result in something unsolvable.
When a linear first-order differential equation is solved, the solution comes as a set, or family; you then have to supply initial conditions to get a single meaningful solution. There's a lot of possible equations out there that have the same relationships within themselves, and a differential equation is just a description of those internal relationships. You can graph the tangents of the solution set and then connect the dots to see how a solution will behave at a certain point.
For more information I'd suggest SOS Math's subpage on differential equations. They explain it better.
How chaos theory ties into this is that some equations produce wildly different solutions depending on a very small variance in the initial conditions. In engineering and physics, significant figures are very important; given experimental error, human error, and equipment limitations, there's no such thing as perfect accuracy. And some numbers that have nothing to do with experiments just keep going -- pi and e are two obvious examples. Yet if the exact initial conditions are not plugged into the solution set, but rather are approximated due to experimental and/or computational limitations, the resultant 'solution' will be meaningless. This renders approximation techniques useless and the solution set a cryptic curiosity.
I hope that helps a bit. I've always been interested in how chaos theory could intersect with concepts of free will, as math certainly intersects with philosophy, but as Lurid and Quantum so aptly point out, there's a major problem with the definition (or even existence) of free will that would have to be solved first.
Can't really add anything on the quantum mechanics front as I do not get QM. From what little I can tell, the Hindus were right. |
