they are not a description of reality.
So yeah I got sidetracked by life: university, work, breakdancing :D and what not. Thus, I have been unable to really work on my projects the past month. I have more free time now however, having settled into things and stuff. I'm still working on those articles I talked about before in my spare time.
However, what has been occupying most of my spare thought is deep stuffTM. I do not know why, but it makes me uncomfortable to think on the foundations of physics and reality. I mean, who am I? Oh well, I will trudge on, and hope I get somewhere, at the very least what is important is that I am enjoying myself at this. Id like to take this time to state my admiration of work by GDNET members like dgreen02 and all the others, who work on that MMORPG anyways no matter how impossible they are told it is. At the least they learn something, even if it is that they did not know enough for what they were attempting.
So here it is: it is very unlikely that one would be able to put Physics on a completely axiomatic basis. Thus, physics can never be fully rigorous and will always have illogical attributes and contradictions. There will always be ambigious aspects (vacuum vs matter, space vs time) contained therein.
However, Godel's theroem states that any strongly axiomatic system will always contain unprovable statements. Not impossible, simple unprovable one way or another. So we see that although physics resists axiomization, by doing so, it retains full ability to talk about nature and make predictions albeit with some slight hiccups. It should be noted also, that models with limited applications can in fact be made fully rigorous and placed on a proper axiomatic basis (QFT for example). But not those which wish to maintain full generality and applicability.
So, since we can never fully rigourize the physics and the fact that many of the mathematical structures used to describe nature (and do so quite well) do not actually exist in nature (even numbers) imply that any deep and final theory will likely not be very mathematical! At least so it seems. Contrary to what might be supposed, for physics, it is the intuitionisic logic and not the axiomatic that is the key requirement.
An important question as well is what relationship the mathematical has with nature. What is the tiein, where does one begin and the other end? Obviously the two are both "man made", we are limited to what we can and how we perceive, the limits and manner of our thought processes and the workings of the brain. I will end that there and continue that line of reason, later. But yeah, I have some ideas, which I will note is me just trying to make sense of stuff for myself. Im not trying to make some cranky new theory or whatever.
"It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite amount of logical operations to figure out what goes on in no matter how tiny a region of space and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of time/space is going to do? So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the chequer board with all its apparent complexities."
--R. P. Feynman, The Character of Physical Law
Anyways, doing some searching I stumbled recently upon the work of Gergory Chaitin. His stuff leaves me unsettled although they can be said to lend some support to some of my ideas (barely though)...
So yeah, he studied the Universal Turing machine and the problem of computability or solvability (the so called halting problem). Using research where some people utilized the 2000 year old Diophantine equations to model a Univeral Turing Machine, he studied Godel's incompleteness thereom. How is that for some deep underlying connections?!
To continue, he finds that mathematics is random, there exists sets of problems, those as related or following from a given thereom, the solution form is random. There exists a far greater number of unnconnected and unconnectable mathematical concepts than connected. Those that we have found were more by luck than any deep mathematical truth. There are the also the so called undiscoverable numbers, he claims. I must say this leaves me unsettled to say the least and if true has many implications. Seems pretty shady though. More later after discussing with my logic professor.
A Random Walk in Arithmetic
"Now my own suspicion is that the universe is not only queerer than we suppose, but queerer than we can suppose."
--J. B. S. Haldane quotes (British geneticist 1892-1964)