Dear Stephen,
At 10:54 18/06/04 -0400, Stephen Paul King wrote:
Thank you for bringing this book to our attention. I will see if I can
get a copy ASAP. I am very interested in how Changeux deals with the
epiphenomena problem, if he even addresses it.
Not really, Changeux is eliminarivist both on mind and mathematics!
He takes that as high-level pure (implicitely 3-person) description of natural
phenomena.
I have read the Smullyan book and found that it strenghend my insistence
that we need to give a more detailed explanation as to how, at least, the
appearence of physical implementations are necessary; he did not address
physicality at all.
Obviously. It is just an introduction to the Godel-Lob logic G of
Self-Reference.
I would bet he is not aware of my proof that if we are digitalisable
machine then
physics is (re)define as the measure one on the consistent extensions (UDA),
and he is probably still less aware of its arithmetical translation
(the machine interview) and the derivation of QL. Today provability logicians
and quantum logicians seems not really aware of each others, and all logicians
and even probably all mathematicians are not really interested in fundamental
questions.
Still, Smullyan is aware that G could be useful in (theoretical) artificial
intelligence; he even asserts that G has a psychological appeal.
To tell you how "grave" is the situation ;) : Smullyan, like 99,%
of the scientists, and like 99,9% of the laymen since Aristotle,
believes in a physical reality. Actually there is an explicit passage where
Smullyan says so by giving a "typical" platonic dualist, even modal, account
of the difference between math and physics. Let me quote him, but before
let me recall that, in classical (also called boolean) propositional logic, a
tautology is a formula true in all possible worlds, where a world is an
assignement of 1 or 0 to propositional letters. For example,
((NOT A) OR ((NOT B) OR A)
has always the truth value 1, independently of the truth value given to
A or to B, as I hope people can verify by building the truth table.
After having explain along this way what a tautology is, Smullyan
says:
<< This is related to Leibniz's notion of possible worlds. Leibniz
claimed that of all possible worlds, this one was the best. frankly I
have no idea whether he was right or wrong in this, but the
interesting things is that he considered other possible worlds. Out
of this, a whole branch of philosophical logic known as possible world
semantics has developed in recent years---notably by the philosopher
Saul Kripke---which we will discuss in a later chapter. Given any
possible world, the set of all propositions that are true for that world,
together with the set of propositions that are false for that world,
constitute the state of affairs holding for *that* world. A tautology,
then, is true, not only for this world, but for *all* possible worlds.
The physical sciences are interested in the state of affairs that holds
for the *actual* world, whereas pure mathematics and logic study
*all* possible states of affairs. >>
Smullyan confuses physics with what I call "geography"!.
I guess quantum many-worlders have good evidence that physics
is related to the interference of the many worlds. Also modern laws
of physics are most of the time derivable from being invariant for
a (symmetrical) transformation ... among "states of affairs".
And computationalists *know* much more: physics is as
mathematical as mathematics and machine psychology:
physics entirely emerges from the interference between all
states of affairs (actually: maximal sequences of states of
affairs)... as anticipated by sound machines.
Note also that Smullyan uses (but does not study nor exploit)
the Thaetetus trick by defining knowledge by true belief: a reasoner
knows p if he believes p and p is true.
Note also Smullyan can be quite perverse! Indeed, he proposes
a "self-referential" interpretation of G*---which arithmetically never
talks about itself but *about* some machine. It makes look G*
a very queer (although consistent but not self-referentially correct)
machine.
But the book has an infinite charm, and is very much easier
than Boolos 1993 (classical treatise on G) or Smorinski 1985
(classical textbook).
Hopefully your explanation of G and G* will help.
It is really the explanation of my thesis which you need, I am
afraid I can hardly explain better G than Smullyan. But ok I will
try. You can find in my url links to this list where I made
previous attempt. If someone has an idea if we can send mails
mixed with simple drawings, that would help ...
One question to leave you with: If it can be proved that a physical
implementation of quantum computation exist in Nature (in microtubles to be
specific), what effect would this have in your thinking? (I am very aware of
Tegmark's paper
(http://www.consciousness.arizona.edu/hameroff/papers/decoherence/decoherenc
e.html ) but have found a possible
