# Re: Numbers, Machine and Father Ted

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Le 07-nov.-06, à 20:10, Tom Caylor a écrit :```
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>
> Brent Meeker wrote:
>> Tom Caylor wrote:
>>> Brent Meeker wrote:
>>>> Tom Caylor wrote:
>>>>> Bruno has tried to introduce us before to the concept of universes
>>>>> or
>>>>> worlds made from logic, bottom up (a la constructing elephants).
>>>>> These
>>>>> universes can be consistent or inconsistent.
>>>>>
>>>>> But approaching it from the empirical side (top down rather bottom
>>>>> up),
>>>>> here is an example of a consistent structure:  I think you assume
>>>>> that
>>>>> you as a person are a structure, or that you can assume that
>>>>> temporarily for the purpose of argument.  You as a person can be
>>>>> consistent in what you say, can you not?  Given certain assumptions
>>>>> (axioms) and inference rules you can be consistent or inconsistent
>>>>> in
>>>>> what you say.
>>>> Depending on your definition of consistent and inconsistent, there
>>>> need not be any axioms or inference rules at all.  If I say "I'm
>>>> married and I'm not married." then I've said something inconsistent
>>>> - regardless of axioms or rules.  But *I'm* not inconsistent - just
>>>> what I've said is.
>>>>
>>>>> I'm not saying the what you say is all there is to who
>>>>> you are.  Actually this illustrates what I was saying before about
>>>>> the
>>>>> need for a "reference frame" to talk about consistency, e.g. "what
>>>>> you
>>>>> say, given your currently held axioms and rules".
>>>> If you have axioms and rules and you can infer "X and not-X" then
>>>> the axioms+rules are inconsistent - but so what?  Nothing of import
>>>> about the universe follows.
>>>>
>>>
>>> Yes, but if you see that one set of axioms/rules is inconsistent with
>>> another set of axioms/rules, then you can deduce something about the
>>> possible configurations of the universe, but only if you assume that
>>> the universe is consistent (which you apparently are calling a
>>> category
>>> error).  A case in point is Euclid's fifth postulate in fact.  By
>>> observing that Euclidean geometry is inconsistent with non-Euclidean
>>> geometry (the word "observe" here is not a pun or even a metaphor!),
>>> you can conclude that the local geometry of the universe should
>>> follow
>>> one or the other of these geometries.
>>
>> No, you are mistaken.  You can only conclude that, based on my
>> methods of measurement, a non-Euclidean model of the universe is
>> simpler and more convenient than an Euclidean one.
>>
>>> This is exactly the reasoning
>>> they are using in analyzing the WIMP observations.
>>
>> The WIMP observations are consistent with a Euclidean
>> model...provided you change a lot of other physics.
>>
>>> Time and again in
>>> history, math has been the guide for what to look for in the
>>> universe.
>>> Not just provability (as Bruno pointed out) inside one set of
>>> axioms/rules (paradigm), but the most powerful tool is generating
>>> multiple consistent paradigms, and playing them against one another,
>>> and against the observed structure of the universe.
>>
>> Right.  As my mathematician friend Norm Levitt put it,"The duty of
>> abstract mathematics, as I see it, is precisely to expand our
>> capacity for hypothesizing possible ontologies."
>>
>
> This quote is basically what I've been trying to get at.  The possible
> ontologies are the multiple self-consistent paradigms that I was
> referring to.  When we keep finding that "using abstract math to
> hypothesize" actually works in guiding us correctly to what to look
> for, then we have to start believing that there's got to be some kind
> of truth to math that is greater than trivial self-consistent logical
> inference.  I think this is what Bruno is getting at with the border
> between G (provable truth) and G* (provable and unprovable truth).
> Math helps us find not just G, but we can also explore the border of G
> and G*.

Yes. Note that a lobian machine M1 can *deduce* the G and the G*
corresponding to a simpler lobian machine M2, but can only "infer" or
"hope" or "fear" ... about its own G*.

Remark: I recall for others that G is the modal logic which axiomatizes
completely the self-referential provable discourse of "sufficiently
powerful classical proving machine", and G* formalize completely (at
some level) the true discourse (the provable one and the inferable
one).
It corresponds to the third person point of view (the second hypostase
of Plotinus). G is the discursive, G* is the "divine" one (true).
The main axiom of G is B(Bp -> p) -> Bp" and its arithmetical
interpretation is lob theorem. Exercise: deduce from Godel's theorem
it (I have already answer it but ask if you don't find the answer).

B represents here Godel's provability predicate: Godel's theorem = ~Bf
-> ~B(~Bf) (If the false is not provable, then that fact itself is not
provable).

>> Agreement would be great.  But the proof of scientific pudding is
>> predicting something suprising that is subsequently confirmed.
>>
>> Brent Meeker
Tom:
> I would like to hear Bruno's thoughts on comp with respect to
> prediction of global aspects such as geometry, as I brought up in the
> above paragraph from a previous post.

A sort of physical geometry should arise from the Bp & Dp (& p) povs.
Mathematical geometry can occur through Minkowski geometry of numbers,
or even just the traditional cartesian mapping between numbers and
geometry.

>
> Also, a thought comes to mind that Bruno once said something about
> reality (physics, sensations?) arising from our ignorance of the
> absolute border between G and G*.

Recall that the gap between G and G*, i.e. between proof and truth (by
and about the machine) is lifted to "intelligible matter (Bp & Dp)" and
"sensible matter (Bp & Dp & p)", that is the "probability 1" logic
define in G. I will say more later.

> This brings me back to the analogy
> of the Mandelbrot set.  We can never know the actual absolute border of
> the Mandelbrot set.  If we were asked to point out even one
> (non-trivial) point on the complex plain that is exactly on the border,
> we wouldn't be able to do it.  However, if we take a finite number of
> iterations of the recursive equation, we get a definite border, which
> is an approximation.  We get an actual instantiation/shape we can
> interact with, something that kicks back, something akin to reality.
> Again, as I originally proprosed, perhaps this is like the measurement
> process in physics, or even the process of setting your coffee cup on
> the table...  In my view the table doesn't kick back any more than the
> Mandelbrot set.

I completely agree with you. Actually I have a precise argument that
the Mandelbrot set is "turing universal", and Blub Shub and Smale give
a proof of this in the frame of the theory of the partial recursive
functions defined on a ring (like R or C).

Bruno

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

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