Russell Standish wrote:
>My point is that linearity appears to be essential, and that it
>deserves an explanation. I give what might charitably be called a
>speculation that it is related to "causality" of the multiverse (or
>subset of the ensemble (aka plenitude, although I avoid that term in
>the paper). Of course, if anyone else has a better idea, I'd love to
>hear it!
Well, it seems that in my chapter 5 of my thesis at
http://iridia.ulb.ac.be/~marchal
I derive ..., well, trying to be modest, I will say that
I derive the shadow of linearity (in fact the skeleton of the
lattice of subspaces of Hilbert spaces) from comp + the theetetic
approach to belief, knowledge, and observation.
I modelize "scientific third person communication by
self-referentially correct universal machine" by the Godelian
modal logic of provability, known as G.
It happens by the "Solovay" miracle that I inherit also of
what I call privately the machines' Guardian Angels, which is an
extension G* of the logic G.
G proves what the machine can prove about its own provability.
G* proves what is true about the machine's provability.
Read Bp as the machine proves p, and read BT as the machine proves
the universaly true tautology.
Now B-T means the machine is inconsistent.
G proves -B-T -> -B-B-T (this is Godel's second incompleteness
theorem).
G* proves -B-T (the guardian angels proves the machine is
consistent), but
of course the machine doesn't prove that.
In general G is closed for B, and G* is closed for -B-.
B is the modal box, and -B- is the modal diamond.
The axioms of G are
B(p->q)->(Bp->Bq) Kripke Axiom
B(Bp->p)->Bp "Lob's theorem"
The rule of inference are
p p->q and p
------- ---
q B p
Known as modus ponens and necessitation respectively.
G* has as axioms all theorems of G, and as ONLY rule of inference
the modus ponens.
Now the theetetic ideas are to define Kp, the knowledge of p,
by p & Bp. It is the true and justified communication proposed by
Thaetetus to Socrate.
We get a logic of knowledge S4Grz which can also be formalised:
The axiome of S4Grz are
B(p->q)->(Bp->Bq) Kripke Axiom
B(B(p->Bp)->p)->p Grzegorczyk formula
the rule are also modus ponens and necessitation.
G* does not change S4Grz. That is S4Grz* = S4Grz. From the point
of view of the subject, truth is identified with proof, like
intuitionnistic philosophy (Brouwer's philosophy of the subject).
Ultimately this is used to tackle the first person point of view.
To get the logic of observation, I do two things, which are linked
to the thought experiment UDA.
Op is "I observe p".
1) I modelize Op by (Bp & -B-p). This is linked
to the fact that I do not take the world I am dead, or I am not
reconstituted, into account for the computation of the probability
on the domain of reconstitution. I impose consistency of the
possible continuation.
2) I modelise the universal dovetailer by the set of sigma1 proposition.
a sigma1 proposition is a verifiable arithmetical proposition. It has
the form "it exists n such that P(n)" with P decidable.
That gives two logics: Z1 and Z1*. Now the "result" (simplified a
little bit) is that Z1* prove B-B-p if and only if QL
proves p, where QL stands for Quantum Logic. This gives an embryo of
a purely arithmetical interpretation of quantum logic, which, overall,
corresponds to the searched logic of the computational continuations,
which by UDA corresponds to the searched logic of physics.
I have not been able to axiomatize completely Z1 and Z1*, and I suspect
that it is not possible. But thanks to Solovay's powerfull theorem I
have been able to program theorem provers which are complete for these
logics.
Once you have a quantum logic, you begin to smell a basic LINEAR
realm. The difference between Z1 and Z1* (guardian angel point of view)
makes it possible to distinguish the physical measure, which is
communicable, from the physical sensation (observable but uncommunicable
truth).
The logic of (Bp & -B-p & p) (reapplication of the theetetic idea) gives
an unexpected notion of true qualia.
I hope this gives you a raw feeling of my way to "derive" physics from
psychology. I guess it is both too concise and too long, but it can
give you hints for my chapter 5.
The problem now is to derive the explicit measure from a suitable
semantics for the Z1(*) logics.
>> I ask you a question:
>>
>> Do you agree with the following statement:
>>
>> ``(Understanding why we must exclude the white rabbit)
>>
>> is equivalent with
>>
>> (Understanding why me must justify the laws of physics)" ?
>>
>
>Not equivalent per se, but exclusion of the white rabbit appears to be
>necessary in order to justify the laws of physics.
That is indeed the main point. I agree.
Bruno
Received on Wed Nov 17 1999 - 06:18:16 PST