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From: Bruno Marchal <marchal.domain.name.hidden>

Date: Tue, 28 Oct 2003 16:23:59 +0100

To avoid mailbox explosion I give short and related answers

to many post in one post.

Joao Leao (jleao.domain.name.hidden ) 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" <jackogreen.domain.name.hidden> 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" <ncsamish.domain.name.hidden> 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" <hal.domain.name.hidden> 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" <stephenk1.domain.name.hidden> 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.

Bruno

http://iridia.ulb.ac.be/~marchal/

Received on Tue Oct 28 2003 - 10:34:39 PST

Date: Tue, 28 Oct 2003 16:23:59 +0100

To avoid mailbox explosion I give short and related answers

to many post in one post.

Joao Leao (jleao.domain.name.hidden ) wrote:

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" <jackogreen.domain.name.hidden> wrote (to Joao Leao)

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" <ncsamish.domain.name.hidden> wrote:

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" <hal.domain.name.hidden> wrote

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" <stephenk1.domain.name.hidden> wrote:

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.

Bruno

http://iridia.ulb.ac.be/~marchal/

Received on Tue Oct 28 2003 - 10:34:39 PST

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