I think quantum theories imply that the universe is in reality nothing more than a probability field.
ultrafilter, doesn’t that then answer the posted question?
Does GIT apply to the universe if we envision the universe as a mathematical model of itself?
Not all number theoretical systems are restrained by GIT. Quantum theory is a mathematical system that is one such example. A mathematical model of the universe that is based on such a system would not be constrained by GIT.
Unless one wants to spin it as that quantum states are undecidable and therefore prove the point that the universe is subject to GIT … “strict determinism is impossible”.
Envisioning the universe as a mathematical model of itself doesn’t make it a mathematical model of itself.
No, it’s not. In fact, it’s not even a number theory as required by GIT. GIT is not some very broad, vague statement. It only applies under very specific conditions. If you want to know what those are, check Elliot Mendelson’s Introduction to Mathematical Logic, 4e.
On the other hand, yes, it’s obvious that not every possible state of existence will be realized. It’s possible that I could go home tonight and find Tara Reid in my bed rarin’ to go, but that’s just not gonna happen, and I don’t need any profound mathematics to realize that.
My question is, can the universe express Peano’s axioms? If this is the case, aren’t we compelled to accept GIT? Why not? (since you’ve been against it the whole time I feel rather comfortable in assuming your answer ;))
I’m not arguing against GIT. Like Math Geek, I’m arguing that you’re applying out of context. Unless you can show that the universe meets the conditions under which GIT applies (see the book I recommended in my last post for a list), I’m not gonna buy your argument.
Well, gee, let me see if I can hunt it down.
In my clumsy way, that is what I am trying to say … I think. I was responding to erl’s statement,
with the answer that the universe might be expressable with a mathematical description that is not subject to GIT constraints. One such attempt to do so is quantum theory … etc.
Actually, QM probably is subject, just cause it’s based on complex number theory, and that does fall under the umbrella of GIT. My error.
So, wait… there are systems which can express Peano’s Axioms but which don’t come under GIT? (and I never said you were against GIT, FTR)
No, there aren’t. What did I say to give that impression?
I’ve been doing some vigorous hunting in the past few days about trying to link the halting problem, Turing machines, and Godel’s theorems together and come up with results which seem to support my idea. I have been searching for a detailed examination of what GIT requires, but I have run into several problems as it seems that for many, many systems where GIT itself wouldn’t hold, a similar result does. I found an interesting paper on expressing GIT in terms of information theory, for example, where it sort of falls out in an intuitive manner rather like the halting problem does.
For example, “Axiomatic systems are equivalent to abstract computers, to Turing machines, of which our computers are (approximate) realizations. (True.) Since there are true propositions which cannot be deduced by interesting axiomatic systems, there are results which cannot be obtained by computers, either. (True.)”
You try and make this result seem trivial (that such an such a person won’t be waiting to sleep with you) when in fact I think it is fundamnetally important. For example, dealing with the topic of gaining knowledge from incomplete induction in scientific fields. Some people look to scientific theories as “making do” with what we can get, when in fact it might simply be that there is not only nothing else to do, but that complete knowledge of the universe from within the universe is impossible.
You are correct that I should be able to take GIT’s formulation and demonstrate how it applies to the universe for a “good” debate/argument; however, I would appreciate any help you can offer in this pursuit since you are, IIRC, a math major. I’m not asking you to make my case for me, of course, since you seem to disagree with it, but rather to help me locate the requirements of GIT which are proving difficult to find.
Try searching for it on the web and you come to a thousand pages that are either debunking broad use of the theorem, using the theorem broadly, or discussing it in passing as it pertains to the rest of mathematics. No one is quick to offer the requirements, and if you do have them available I would greatly appreciate it.
This website seems to offer something similar to my OP’s offering.
Somehwere in my searchings I found a comment about how GIT required a specific sense of disjunction where PV~Q with the assertion P means Q or something similar (my brain is becoming addled trying to keep all these “what if’s” everyone offers for different theories apart).
Your rejection, apparently, that the universe can’t express Peano’s axioms.
Once I get my internet access at home set up, and I’ve had a chance to get settled in otherwise, I’ll list the requirements for Godel’s theorem. It’s going to be a long, highly technical post, so be forewarned. It’s gonna be a few days, though, so be patient.
Well, I will keep searching (as I actually have been the whole time). Something is bound to turn up!
A new thought occurred to me, which might hopefully span the gaping chasm of mutual confusion here.
We all seem to agree that the universe, either the real one or the hypothetical automaton universe, can’t do certain things. For example, the automaton universe can’t simulate a theorem-proving program which proves an undecidable proposition.
The new thought is this: that this inability is not an inherent property of the automaton alone, but instead is a property of the automaton and our interpretation of its states.
This means in particular that the statement “Godel’s theorem implies that there exists a state X which the automaton can never enter” is in fact false. For if the statement were true, then some state X which corresponds to a proof of an undecidable proposition would exist hypothetically, and that alone contradicts Godel’s theorem! It’s not that there’s some possible state corresponding to a proof of an undecidable proposition, which the automaton will never enter. It’s that no possible state corresponding to a proof of an undecidable proposition can exist at all, even hypothetically.
The correct conclusion would seem to be, “Godel’s theorem implies that it is impossible to interpret the states of the automaton in such a way that some possible state corresponds to the proof of an undecidable proposition”.
By the same token it’s not the case that there’s some possible state of the universe, corresponding somehow to a proof of an undecidable proposition, which the universe will never enter. Because no matter how we try to interpret the states of the universe…no matter what correspondence we try to construct between states of the universe and number-theoretic proofs…no state corresponding to a proof of an undecidable proposition can possibly exist. Even the hypothetical existence of such a state implies that a proof exists, which is impossible
In short, Godel’s theorem doesn’t set limits on the possible states of the universe. It sets limits on the ways we can interpret those states and give them meaning.
I’m not sure I like this. for it wouldn’t matter, I don’t think, whether or not anyone was actually checking the output of a Turing machine generating theorems. It still can’t generate all true statements, regardless of whether we interpret those or not, no?
But I am perhaps out of touch with what a true statement would mean with respect to a Turing machine, and especially so with respect to the universe. In fact, this is my OP in another guise. What does it mean to say etc? For Godel’s theorem should be true whether or not we had ever proved it, so an interpreter shouldn’t be necessary… should it?
But wait, perhaps you are completely correct. In fact, yes, hmm. Because the problem here isn’t in finding truth at all, is it, but finding truth portrayed in formal systems (or as the result of computation, etc— really, there are so many guises of this paradox that it goes on forever).
So the fundamental problem here is that we must interpret the universe through the language of the universe… namely, through its laws with respect to physical science. But our interpretation of the laws through the symbols of the universe is where the problem lies. The means of expression and understanding of the underlying structure is always hidden from view… obstructed by the symbols themselves.
I got sidetracked mentally by thinking about GIT in terms of what can be demonstrated, and turned to processes in terms of what is a possible state versus what is a state that will happen (when the algorithm is running). But that is a false analogy. The machine will prove all theorems, and in fact those are its only possible states anyway! By definition.
Delightful.
Has it been decided that the universe is an axiomatic deduction system?
That was the hypothesis.
Or rather, the necessary assumption. I make no claims as to its validity. 
But, hey, if we agree on that interpretation, then we could certainly shift the debate over and think of ways the universe isn’t an axiomatic system or a Turing machine…
But I do not accept that the universe is an axiomatic system of deduction. I believe, as I stated, that the universe is a probability field.