On 09 Nov 2012, at 01:54, Craig Weinberg wrote:

On Thursday, November 8, 2012 2:57:35 AM UTC-5, Bruno Marchal wrote:

On 07 Nov 2012, at 19:04, Craig Weinberg wrote:

> On Wednesday, November 7, 2012 10:49:35 AM UTC-5, Bruno Marchal wrote:
> On 07 Nov 2012, at 13:42, Craig Weinberg wrote:
> > Can anyone explain why geometry/topology would exist in a comp
> > universe?
> The execution of the UD cab be shown to be emulated (in Turing sense)
> by the arithmetical relation (even by the degree four diophantine
> polynomial). This contains all dovetailing done on almost all possible
> mathematical structure.
> This answer your question,
> It sounds like you are agreeing with me that yes, there is no reason
> that arithmetic would generate any sort of geometric or topological
> presentation.

"Generating geometry" is a too vague expression.

Create? Discover? Utilize?

OK. That is more precise. Numbers will obviously develop dreams of many geometries. Please note that this is a shorthand for "numbers have relation which correspond to computation supporting person dreaming to ...". I do assume comp, and the knowledge that arithmetic is Turing universal.

Keep in mind that if comp is true, the idea that there is more than
arithmetical truth, or even more than some tiny part of it, is
(absolutely) undecidable. So with comp a good ontology is just the
natural numbers. Then the relation with geometry is twofold: the usual
one, already known by the Greeks and the one related to computer
science, and its embedding in arithmetic.

If the idea of comp is that the origin of consciousness can be traced back to digital functions, I am saying that lets start with an even simpler example of why that isn't true by trying to trace the origin of geometry back to digital function. What specifically does geometry offer that the raw arithmetic behind geometry doesn't?

This is a good question. you can consult the literature. Even Descartes's discovery of analytical geometry is considered by many as a proof of a reduction of geometry to arithmetic or algebra. But the fact that a theory reflect another does not eliminate the interest of the first theory. Like some other logicians, I tend to believe that the whole of the known human math can be obtained by reasoning in arithmetic. This is confirmed up to now/ many analytical constructions have been "reduiced" to arithmetical expression. The famous Riemann hypothesis has been reduced to a PI_1 arithmetical sentence by Turing for example, despite it looks like a statement in complex analysis. Most analytical proof have been reduced into elementary (in arithmetic) proof. Only logician can diagonalize such proofs and find mathematical statements not reducible to arithmetic, but those are ad hoc and made only for that purpose.

Why the redundancy to begin with? What is functional about geometry?

The redundancy can be helpful for the stability of the ideas.
There is also a question of efficiency. All you can do with a high level language, can be done in assembly language, and some earlier computer scientist believed that high level programming was just for the babies, and would never succeed. But you laptop would not exists, if there were no layers and layers of languages and application using those reduction.

> Or are you saying that because geometry can be reduced to arithmetic
> then we don't need to ask why it exists? Not sure.

Geometry is a too large term. I would not say that geometry is reduced
to arithmetic without adding more precisions.

Can't any computable geometry be stored as numerical codes in digital memory locations rather than points or lines in space?

Then a Youtube video will look like


With a very long length. Are you sure you will enjoy it as much as looking to quickly moving pixels that your brain can decode much more easily?

> but the real genuine answer should explain
> why some geometries and topologies are stastically stable, and here
> the reason have to rely on the way the relative numbers can see
> themselves, that is the arithmetical points of view.
> In this case it can be shown that the S4Grz1 hypostase lead to typical
> topologies, that the Z1* and X1* logics leads to Hilbert space/von
> Neuman algebra, Temperley Lieb couplings, braids and hopefully quantum
> computers.
> No need to go that far. Just keep in mind that arithmetic emulates
> even just the quantum wave applied to the Milky way initial
> conditions. And with comp, the creature in there can be shown to
> participate in forums and asking similar question, and they are not
> zombies (given comp, mainly by step 8).
> The question though, is why is arithmetic emulating anything to
> begin with?

Because arithmetic (the natural numbers + addition and multiplication)
has been shown Turing complete. It is indeed not obvious. In fact you
can even limit yourself to polynomial (of degree four) diophantine
relation.  But you can use any Turing complete system in place of
arithmetic if you prefer.

Why would a Turing complete system emulate anything though?

It does not do this necessarily, unless you program it to do it, like the UD, or like the set of arithmetical theorems of a Turing complete theory.

It is what it is. Where does the concept that it could or should be about something else come from?

Because once we bet on comp, we attach consciousness to computations. To be short. Those computations exists independently of us in arithmetic, and that explains where the dreams come from. Then the mind-body problem, to be solved, needs to explains where the coherent sharable dreams come from, and what makes them relatively stable. if comp is true, the reason can be shown in the existence of some relative measure.

I will give a proof of arithmetic Turing universality on FOAR, I will
put it here in cc.

My point is precisely that this kind of universality invalidates Comp. If you have a universal machine, you don't need geometry, don't need feels and smells and hair standing on end...you just need elaborately nested sequences which refer to each other.

Not in the relative situation. Even a program can prefer, or need to use, adapted representation; like when you look at YouTube. This is a key point: it is not because some high level reality emerges from a lower level reality that it can be said not existing, in some genuine, even if not primitive, sense. This is provably so for arithmetic: a complete theory of arithmetical truth cannot be finite, it will needs an infinity of axioms. Higher and higher level exists, and escape necessarily the low levels, for their behavior, despite the existence of them is reducible to the lower realm. That is why humans are NOT bunch of interacting molecules. That's why souls are not bodies.



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