Originally posted by Incompleteness
I agree but incompleteness doesn't imply a fractal like nature is present in nature.
But a TOE does. How else would it apply to everything, unless it was literally a single pattern (formula, etc.) that explains all phenomena?
But I disagree with your view that there exists more primitive axioms or systems that logic
I'm not sure what you're disagreeing with. If the universe as science knows it did not exist, logic would not exist. And you already agreed that
we can't figure out everything about the universe from within the universe itself (at least as we know it, the physical
universe I should
say). Even if we could
, we still don't know everything about it right now so it makes no difference at the moment. The universe exists
despite us not knowing the "first cause," the TOE, but we assume it exists anyway. I didn't say whatever else is out there can be axiomatized, I
think it's just the boundary where logic breaks down and one must realize that not everything in existence HAS to be logical (in our current sense of
the word, at least), have space, time, volume, etc., because logic itself can't
be solely responsible for the TOE since our logic can't even
prove itself, and what's worse, Godel's theorem makes a mockery of it.
But certain types of logic and mathematics are complete and thus sound to use without worrying about incompleteness like elementary geometry is
complete and first order logic.
Godel proved that any
formal system that is powerful enough to be recursive can be forced to produce inconsistent results without breaking any
rules. If you could make geometry recursive (I'm sure someone has) then you should be able to produce Godel's theorem with it. I'd actually like
to see that myself. And any system that isn't
powerful enough for recursion is just incomplete, like "elementary" geometry. You can have a
closed system that is consistent, but it won't be very powerful and can't tell you much of anything about the universe.
So logic can be proven using logic.
Where do you get that? Have you ever read the book Godel, Escher, Bach
by Doug Hofstadter? He's the one that introduced Godel to me, and
it's a very informative book. Alfred Whitehead and Bertrand Russell tried
proving every single axiom of our mathematics logically in a book
called Principia Mathematica
, but failed, because there is no way to "prove" an arbitrary function like addition or even the natural sequence
of numbers or many of the other basic, common-sense concepts that make up our "logic" but yet no one usually even feels the need to "prove" them.
They may "make sense" to us but they are apparently just as arbitrary as using base-10 instead of base-16 or any other system, none of which are
more correct than any other, they all work. But they certainly tried to prove logic using logic, and not long afterwards Godel published his famous
Here's a quote relating to it from History of Mathematics
, by Carl Boyer, just so you know I'm not just making all this up:
Gödel showed that within a rigidly logical system such as Russell and Whitehead had developed for arithmetic, propositions can be formulated that
are undecidable or undemonstrable within the axioms of the system. That is, within the system, there exist certain clear-cut statements that can
neither be proved or disproved. Hence one cannot, using the usual methods, be certain that the axioms of arithmetic will not lead to contradictions
... It appears to foredoom hope of mathematical certitude through use of the obvious methods. Perhaps doomed also, as a result, is the ideal of
science - to devise a set of axioms from which all phenomena of the external world can be deduced.
A quote from Godel, Escher, Bach
on the same page:
All consistent axiomatic formulations of number theory include undecidable propositions ...
Gödel showed that provability is a weaker notion than truth, no matter what axiom system is involved ...
One more, from a book called An Incomplete Education
(emphasis mine!) :
You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with
new rules and axioms, but by doing so you'll only create a larger system with its own unprovable statements. The implication is that all logical
system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove
according to its own defining set of rules.
So in the case of figuring out everything in our physical universe, we would have to go outside
of our physical universe, according to Godel's
theorem. If we can't do that then we can never truly know everything about the physical universe.
It lets one wonder if our reality is not some transcendental object which can never be fully described but always allowing new things to be
done within it.
Something very much like that, I agree. Pretty much no different than a dream, with many dreamers.