At 14:20 -0700 31/05/2002, Wei Dai wrote:
>On Thu, May 30, 2002 at 05:02:06PM +0200, Bruno Marchal wrote:
>> - Computability and Logic, by George Boolos and Richard
>>Jeffrey (Cambridge
>> University Press (third ed. 1989).
>
>I noticed that a fourth edition just came out in March of this year. This
>seems to be THE book for learning metamathematics (and how it relates to
>the computability stuff I learned in my theory of computation class) that
>I was searching for before seeing Bruno's recommendation. Thanks!
I didn't knew about the fourth edition. I will take a look. Note that
Boolos dies recently.
My old textbook on the subject was Kleene's "Introduction to Metamathematics".
I think you know it. Very good but somehow out of date.
> > Unfortunately most mathematicians, including the only
> > local logician, were allergic to Godel's theorem! (not so rare attitude)
>
>
>Can you elaborate on that please? What specificly were they objecting to?
>
Well, at Brussels the logician only told me that Godel's theorem was just
a technical result among others. Like some so-called "pure mathematician"
he was afraid that that piece of pure math could have applications! But he was
a sort of autist or caracterial and he was not representative of the logicians.
French mathematicians were much more ideologically enclined to criticize
logic and metamathematics because math, for them, was analysis and algebra, and
nothing else. Cf the critics of logics by Bourbarki(*) (quite unfounded because
their critics bears on pregodelian logics which indeed was presumptuous because
pre-godelian logicians believed that math should rely on logical foundation but
after Godel everyone knew that logic is just a branch of math on the
same par than
algebra or analysis). The french prefers the esthetical motivation for geometry
than the more metaphysical motivation of the logicians. Look at the
work of Rudy
Rucker: everything which bears on geometry has been translated in french, none
of his book on logic have been.
Still, after the parution of Hofstadter's "Godel, Escher Bach" I have discover
how much the french, including some logician, like to minimize the importance
of Godel. Take Jean-Yves Girard, one of the biggest important
logician of the day
(especially for his invention of LINEAR LOGIC). Not only he likes to ridiculize
Hofstadter's book, but he has invented a new *disease*: the Godelite, which
symptom consists in appreciating too much Godel's theorem. Of course it is true
that many pseudo-philosophers has developped unvalid consequences of
Godels, but
to dismiss the importance of Godel from that is invalid too!
Now let us remember that the french has been unlucky with their
logicians because
the biggest one--Jacques Herbrand--dies at the age of 20 (falling
from a mountain).
Other logicians dies in the defense against nazis, like Jean Cavailles.
(*)Nicolas Bourbaki is the name of a *set* of mathematicians. Most were big
algebraists and geometers. There has been three generations of "Bourbakist".
> > I mentionned often the Boolos 1993 as the classical treatise of
>the (modal)
>> Godelian logics of self-reference (also known as "logics of
>> provability", mainly
>> the modal logics G and G* and their children).
>
>Unfortunately at this point (after reading Boolos's _The Logic of
>Provability_) I still don't get what logics of provability have to
>do with the mind/body problem. Please hurry up with your English paper. :)
Thanks for the encouragement. I will try. But if you have read Boolos,
surely you have seen how Godel manages to build self-referential sentences, and
also how a formal theory is able to prove theorems about itself. See my last
answer to Gordon below which can perhaps help in the meantime. Now a sound
universal machine, with extended logical abilities, can be seen as
being equivalent
to a formal theory so that Godel's trick can be used to study what a
machine can
prove, in principle, about itself. The modal logic G formalises such a
self-referential discourse, and the logic G* generalise it by axiomatising what
*is* true for a machine, including those typical godelian and self-referential
truth which the machine cannot prove. (There is no contradiction
because G* does
not talk about itself but about the machine).
Now the uda reasoning shows that our possible next (first person,
subsjective) states
depends on all our next possible (consistent and third person)
states. And we can
ask the machine about her next possible (consistent) state
(accessible by the UD)
and about the geometry of those continuations.
Such question can indeed be translated in the language
of the machine thanks to the G/G* metamathematics. And then we get
those quantum
looking logic (Z1* mainly) + (re)normalisation problem (too much
states and no simple probability soustraction rule). Happy to meet
Saibal Mitra on this
frontier ;)
Bruno
PS message to Gordon:
----
At 10:36 +0100 3/06/2002, june.shippey.domain.name.hidden wrote:
>Hi all
>Laterly some of us have been talking about Mind and Kicking back of
>Reality etc...
>
>I have one question can you show the 'self'without any context I ll
>tell you it is hard than you think and also shows how self may not be
>the fixed thing we think it is???
>
>The idea can also transfer over to how and why we find Logic and Math is
>on flakey foundation?
>
>from the form presently know as Gordon ;)
If you have any universal machine U you can define a "self" relatively
to U. Indeed in U you can define a duplicator machine D such that,
with respect to U (= implemented in U) you can apply D on any machine X (DX),
giving X apply to X. (XX). That is DX gives XX. Then it is enough to apply the
duplicator on itself and you get DD which, through M, gives DD itself.
In the same way you can generate another D such that DX gives T(XX) for an
arbitrary transformation T. Now this D apply to itself gives T(DD), i.e. T
applied on itself. This is the godelian trick in a nutshell. With Church
thesis this gives a *general* solution for defining a machine-independent
notion of "self".
So such a self can transform itself. No need to believe it is
something statical.
The universal machine on which that self is "run" plays the role of the
context. Going from this abstract "biology" to "psychology" consist in making
the machine talking or proving facts about itself instead of just tranforming
itself.
Received on Mon Jun 03 2002 - 08:19:37 PDT