Hi Mirek,
Long and perhaps key post.
On 04 Aug 2009, at 15:32, Mirek Dobsicek wrote:
>
>
>> Come on Mirek: "Theaetetical" is an adjective I have forged from
>> "Theatetus".
>> "Theatetus" gives 195.000 results on Google.
>> "Theatetus" wiki 4310.
>
> Of course, after all you reference the dialogue Theaetetus in your
> papers thus one can easily match the word Theaetetical agains it.
> Let me quickly summarize the experience I had with "theatetical notion
> of knowledge" while reading one of your papers for the first time.
>
> Maybe I am an ignorant, then shame on me, but I have not read the
> Theaetetus.
There is no shame in being ignorant. Only in staying ignorant :)
I feel a bit sorry with my last post. I hate to look like patronizing,
but it is a professional deformation. Apology.
Note that all the Theaetetus' stuff is really needed just to motivate
the "arithmetical" definition of the knower, alias the first person,
alias the "universal soul", and this concerns AUDA (the "arithmetical
UDA), which should be done normally after getting straight the UDA's
point, ... unless you are mathematical logician, who are the only one
who can find AUDA more "easy" than UDA.
> So I took a look at the Wikipedia and read
>
> "In this dialogue, Socrates and Theaetetus discuss three definitions
> of
> knowledge: knowledge as nothing but perception, knowledge as true
> judgment, and, finally, knowledge as a true judgment with an account.
> Each of these definitions are shown to be unsatisfactory."
Socrates asks Theaetetus to define "knowledge". That is a very
difficult question.
Then Socrates shows that all attempts made by Theaetetus lead to
difficulties. He literally concludes that the problem is open, and
this is debated in the philosophical literature since then.
The remarkable thing is that if you accept to modelize "account" by
"sound machine provability", which can be done for the not too complex
machine, like Peano Arithmetic provers, or Zermelo Fraenkel Set Theory
prover, the definitions of Theaetetus make sense, and can be use to
show, at least, that many philosopher are deductively invalid in
their critics. Actually, even the critics by Socrates have to be
weakened.
All the arithmetical hypostases (in the Plotinus paper) are variant of
Theaetetus definition.
The main one (corresponding to Plotinus "primary hypostases) are the
following one:
p (the truth of p)
Bp (the provability of p, the account of p)
Bp & p (the provability of p when p is true)
The amazing thing is that the incompleteness theorem can be used to
show that, about sound machine, we have
Bp <-> Bp & p.
But this equivalence is true but not provable by the machine, making
the ideal knower already obeying a different logic than the ideal
prover. This introduces a non trivial notion of first person for the
machines.
If you remember G and G*, the equivalence between proof and knowledge
belongs to G* minus G. The corona of the true but unprovable (by the
machine) statements. Yet they prove (know) the same arithmetical "p",
yet, from those points of view, it appears very different.
>
>
> Hmm that really helps .., I told to myself and continued with reading.
> With an uneasy feeling of stepping into the water I eventually settled
> down to conclusion that you likely mean something as "true justified
> belief".
I have found dozen of different translations, in french and in
english, of the greek expressions.
>
> I really wished you wrote it more straightforwardly without turning
> your
> readers quite unnecessarily down to the Theaetetus and inventing new
> words such as "Theaetetical".
In french students are burned alive if they dare to create new
adjective, and I thought that in English we have more freedom, but I
may be wrong. Sorry.
>
>
> Anyway, I'd like to stop discussing this issue :-) since my only point
> was to give you a hint why I said that it is not easy to read your
> papers/letters.
There are other reasons, if only the difficulty of the subject.
>
>
>> Feel free to ask for any clarification, position
>> adjustments, question, at any level ...Do you understand what is the
>> comp hypothesis?
>
> Let us see if I get it right. Your comp hypothesis is
> 1) I'm a machine,
OK. This of course could be interpreted in many ways, and that is why
I have introduce the quasi-operational "yes doctor". It makes clear
hat the "I" is the conscious first person I, not the third person body.
>
> 2) Each possible computation is Turing-computable,
OK. That is Church thesis. Very few people doubt it, but it is a
refutable statement. If a human find a well defined function with an
account of how human can compute it, but no machine can, then CT will
be refuted. Only Kalmar did pretend to have such a function, but
eventually his "function" was not well defined.
>
> 3) Natural numbers and their relations do exist.
This is arithmetical realism. Just a way to prevent infinite
discussion about intutionism and ultrafinitism. It is no more than
the belief of all mathematician that the excuded middle principle is
freely used on arithmetical proposition. Except for some philsopher,
or mathematicians, but only at the pause cafe, or during the week-end,
this is a widespread belief. But given that the result could be
considered as a bit weird, I usually prefer to make it explicit. It
can be retrieved from the (classical) Church thesis. Church thesis
does not make sense without arithmetical realism. This is something I
intend to show precisely in the UDA-7 thread.
>
>
> This should not be confused with other quite common comp hypothesis
> that
> the universe is a big computer.
Yes. It is different in principle. It is different unless I am the
physical universe, which I doubt. Comp is really indexical mechanism:
"I am a machine", or "I see no change after the functional
substitution at the right level", or "I survive classical
teleportation", etc.
> This hypothesis entails the existence of
> a physical computer.
Not necessarily. It could depend by what you mean by "physical".
>
>
>
> Ad 1) I take the position that "I" is only a convenient temporary
> pointer to a part of universe. The pointer "Socrates' thoughts" is of
> the same quality.
This I do not understand. "I", used by Mirek, is a person, subject of
consciousness. I think you confuse that first person "I", and its
(current) body.
At that stage it is better to be reasonably agnostic about a universe.
"I" is the person who survives teleportation, with a new body. With
mechanism, "I" can be said the owner of the body.
>
>
> Ad 2) Breath taking. While 1) and 3) are assumptions of the kind "OK,
> let's think for a while that ...",
Hmm... Most scientist believes implicitly or explicitly in "1)" and
"3)". Only Penrose happens to be explicitly arguing that "1)" is
false, but note that he defends "3)" eloquently. John Searles, a
philosopher, pretends that he believes that "1)" is false, in case of
digital machine, but then he reasons in most place like if it was
implicitly believing it, or being very fuzzy about what is non
mechanical in biology. Only "fairy tales" kind of religious person
have (fairy tales) theory of the soul, in which case they would say
"no to the doctor" (even in principle: I add this because I would, in
most cases, say "no" to the doctor for technical reasons probably).
> 2) has the status of a thesis. I
> don't have any firm position on what could an objective reality be
> (and
> without a justification I tend to think it is inaccessible to us), but
> if there is any objective reality, 2) could be a statement about it.
You don't need to have a firm position on what could an objective
reality be.
You need only to believe that you can be objective on a part of that
reality. In particular you need to be objective with sentences like:
x divides y if and only if it exists a number z such that y = x*z.
if 4 divides 8 then 2 divides 8
For any natural number x, if 4 divides x then 2 divides x.
Goldbach conjecture is false or true.
Etc.
I think it is reasonable to bet on the objectivity of those
statements. And besides, they are used in the practice of all other
exact sciences all the times. You don't have to believe more than that
for being an arithmetical realist.
>
>
> Ad 3) If natural numbers and their relations are the only entities
> which
> do exist then me, you, everything is a recipe of a Turing-computable
> number.
No. Not at all. Sorry. Gosh, you will be very surprised if you follow
the UDA-7. On the contrary. Arithmetical truth VASTLY extends the
computable domain. Most relations between numbers are not Turing
emulable.
And then, why do you introduce suddenly the idea that numbers and
their relations would be the only entities existing for me?
This is not part of the assumption. The assumption are not more than;
A - There is a level of substitution such that I can survive a digital
functional substitution made at that level.
B - Church thesis (classical Church thesis, not the intuitionist one,
thus I use classical realism, classical logic, mathematical logic)
What you say is in the conclusion of the UDA reasoning (although in a
rough and simplified form(*)).
If only number exists, then it would be like I am proposing a new
theory. I could have done that, but this is not what I have done. What
I give is a constructive proof that if I am machine, and CT is
correct, then the laws of physics have to be reduced and derived from
the laws of number (or any recursively isomorphic structure). The
movie graph argument is what makes this obligatory. The notion of
fundamental matter loose its meaning. Matter becomes a Moiré effect
lived by the number when they infer relations from their point of
views. You can still believe in matter if you want too, but you cannot
use it to explain even the physical observation, which have to emerge
from special number's points of view.
That is what UDA shows.
Now AUDA is only used to illustrate that incompleteness in computer
science and logic provide consistency to such a view. But as I said,
Theaetetus all what the multiple explanations of what could be
knowledge are directly translatable in arithmetic thanks again to
incompleteness. For this you need Solovay theorem, and you need to
read Boolos or Smorynski, or Smullyan's book in logic, or study
sane04. I am surfing on the shoulds of giants here: like Post, Gödel,
Kleene, Löb, and Solovay.
>
>
> OK, that is it. This is how I understand to your starting assumptions.
You may have to revised or I may have ben unclear, or the result is
too much counter-intuitive, and things take time. People are not
accustomed to see a proof in philosophy or theology, but I show, or
try to show, that the very common comp belief is much strangest, and
far less reductionist than what most materialists believe.
The intended result, in a nutshell, is the incompatibility of very
weak forms of materialism with digital mechanism.
By a weak form of materialism I mean any doctrine which posits a
primitive or substantial matter and reduces the mind to it
(computationally or not computationally). Materialism and mechanism,
which are always thought as ally are really just epistemologically
incompatible.
OK? I mean do you see that such a result is far from trivial? UDA1-7
conveys already the basic understanding of what happens. The Movie
graph captures the mind-body problem in the comp frame, and is
responsible for the inadequacy of not just physics, but of any
particular universal machine. What you see is not the product of a
universal machine (be it physical or mathematical), but of an infinity
of them. At first sight the mystery is now: why does the physical
world looks so computable. That is the unavoidable white rabbit
problem. Who knows, it could one day lead to the refutation of comp.
But AUDA, computer science and self-reference logics suggest that comp
could provide a coherent picture after all, capable of explaining
where the laws of physics come from, without eliminating the person.
AUDA lead to identify physics with one of the Theaetetus-like
definition of a form of knowledge, and this is enough to extract a
logic of what is observable, and this makes comp testable. I give a
tool capable of measuring the degree of computability of nature. But
alas, it does not really distinguish comp and many weakening of comp.
Most of the "gods" still pay taxes ("gods" are defined by NON Turing
emulable self-referential entities ...). Up to now the comp logic is
not contradicted by quantum logic.
I have to go. This is an important post. Other people, like Jones
recently, get me wrong on the starting assumption. It is really just
"I am a machine" made precise. Then computer science makes it possible
to reason and prove things with that assumption. I have no original
theory. I propose just a proof, a reasoning or an argument.
Bruno
http://iridia.ulb.ac.be/~marchal/
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