To avoid mailbox explosion I give short and related answers to many post in one post.
Joao Leao ([EMAIL PROTECTED] ) wrote:
By no means does this translate to the identification you suggest between what is empirical is what is... "incomplete", If anything physical reality sees mathematical reality "from the outside", and it is this "map" that is incomplete and likely to remain so, not because of the sparseness of empirical data, but due to the limited resources of physicists. As far as nature is concerned what is weird is not what cannot be mapped to math but what can (as in Wigner's famous "unreasonableness"paper)!!
I agree with your answer, and I guess I have not been enough clear, because you repeat what I said, except for my use of Church thesis. I was (implicitely) assuming the computationalist hypothesis in the cognitive science. Knowing that from comp physics is reduced into a measure on the self consistent extension, empiricalness and all contingencies is justified by incompleteness. Now that is not obvious. Look in my URL for links to the proof. Note that by Godel second theorem a consistent machine cannot prove the existence of even just one consistent extension, but yet can prove a lot about the geometry of those consistent extensions, once they exist.
====================== "CMR" <[EMAIL PROTECTED]> wrote (to Joao Leao)
I would tend to agree with Chaitin that your apparent confidence in the "precise accessibility" of Mathematics as opposed to that of physics may be misplaced; I would also agree that Leibniz's insights are probably more useful than Plato's on the ultimate "nature" of reality:
>From "Should Mathematics Be More Like Physics? Must Mathematical Axioms Be Self-Evident?"
Just to be clear on that point, I agree totally with Chaitin. Now, some truth (like 317 is a prime) are clearer than any physical proposition. But, as I said, there will be more and more mathematical propositions, in the future, which will have "experimental" status, and this follows from incompleteness phenomena.
"Norman Samish" <[EMAIL PROTECTED]> wrote:
What are the philosophical implications of unsolvable mathematical problems? Does this mean that mathematical reality, hence physical reality, is ultimately unknowable?
It is not because some question are unsolvable in one domain, that all the question are unsolvable in that domain. Reality is most plausibly *partially* knowable. With Godel we know that our uncertainties are proportional to our certainties. This is perhaps related to the free-will question btw.
====================================== "Hal Finney" <[EMAIL PROTECTED]> wrote
If, from a set of axioms and rules of inference, we can produce a valid proof of a theorem, then the theorem is true, within that axiomatic system.
I'd suggest that this notion of provability is analogous to the "reality" of physics. Provable theorems are what we know, within a mathematical system.
Now, one problem with this approach is that it focuses on the theorems, which are generally "about" some mathematical concepts or objects, but not on the objects themselves. For example, we have a theory of the integers, and we can make proofs about them, such as that there are an infinite number of primes. These proofs are what we know about the integers, the "mathematical reality" of this subject.
But what about the integers themselves? They are distinct from the theorems about them. Maybe we would want to say that it is the integers which are "mathematically real", rather than proofs about them.
Yes. Important point. Why do you ask? I am afraid you are confusing levels. Let me be precise on this because a lot of people are wrong or confused. So, roughly speaking. Number theory is the study of numbers. That is, a number theorist works on numbers, and its main methodology is informal proof, conjectures, and nowadays : computer experimentation, and tomorrow perhaps physical experiment. Geometry is the study of spaces. That is a (mathematical) geometer works on spaces (Euclidian, non euclidian, riemannian, hilbert spaces, projective spaces, etc.). and its main methodology is informal proof, conjectures, etc. Proof theory (a branche of logic) is the study of formal proofs. That is, a proof theorist works on proofs and proofs system (quantum like, intuitionnist, monotonic, modal, classical, non standard, etc.). And its main methodology is, like any other mathematicians, *informal proof*, conjecture, etc. No mathematician do formal proofs. All proof in mathematical books and papers are informal (although rigorous). To formalise a proof is made only by proof theorist, because formal proof are their object of study, but then they make informal proofs *about* those formal proofs. So, the number theorist learn about numbers, the geometer learns about spaces, the proof theorist learns about proofs. And then they exist bridges between those branch, and applications, etc. But a mathematician does not learn about proofs per se (of course by practice he develops skills for proving things), but he learns about its subject matter (numbers, spaces, etc.) *through* proofs.
====================================== "Stephen Paul King" <[EMAIL PROTECTED]> wrote:
This is at the heart of my argument against proposals such as those of Bruno Marchal. The "duration" required to instantiate a relation, even one between a priori "existing" numbers can not be assumed to be zero and still be a meaningful one.
I don't want to look presumptuous, but then, I want to be clear about
what I have done (or at least about what I think, and apparently some other
people thinks I have done). In particular, I don't think I have made a
*proposal*. I provide a proof. A proof that *if* we postulate the COMP hyp., *then* it
follows that physics is a branch of number theory. See my URL for the proof, and
don't hesitate to ask question if some steps are unclear.
Note that the proof assumes almost nothing in mathematics. Only the attempt
of an actual derivation of the logical skeleton of the physical propositions
assumes a lot of mathematical logics. But the proof itself of the necessity of the
reduction assumes nothing more that the comp hyp, and of course
some reasoning abilities, if this should be mentioned at all.
I am completely open to the idea that comp will be refuted, and that what
I have done could lead to such a refutation. But currently comp seems to be
confirmed by the facts. For example, the most obvious and oldest fact
which follows from comp is the Many-World-like ontology, and through
physics, the idea is slowly made seemingly reasonable.