On 26 Jan 2012, at 07:19, Pierz wrote:

As I continue to ponder the UDA, I keep coming back to a niggling
doubt that an arithmetical ontology can ever really give a
satisfactory explanation of qualia.

Of course the comp warning here is a bit "diabolical". Comp predicts that consciousness and qualia can't satisfy completely the self- observing machine. More below.




It seems to me that imputing
qualia to calculations (indeed consciousness at all, thought that may
be the same thing) adds something that is not given by, or derivable
from, any mathematical axiom. Surely this is illegitimate from a
mathematical point of view. Every  mathematical statement can only be
made in terms of numbers and operators, so to talk about *qualities*
arising out of numbers is not mathematics so much as numerology or
qabbala.

No, it is modal logic, although model theory does that too. It is basically the *magic* of computer science. relatively to a universal number, a number can denote infinite things, like the program factorial denotes the set {(0,0),(1,1),(2,2),(3,6),(4,24),(5,120), ...}. Nobody can define consciousness and qualia, but many can agree on statements about them, and in that way we can even communicate or study what machine can say about any predicate verifying those properties.




Here of course is where people start to invoke the wonderfully protean
notion of ‘emergent properties’. Perhaps qualia emerge when a
calculation becomes deep enough.Perhaps consciousness emerges from a
complicated enough arrangement of neurons.

Consciousness, as bet in a reality emerges as theorems in arithmetic. They emerge like the prime numbers emerges. They follow logically, from any non logical axioms defining a universal machine. UDA justifies why it has to so, and AUDA shows how to make this verifiable, with the definitions of knowledge on which most people already agree.




But I’ll venture an axiom
of my own here: no properties can emerge from a complex system that
are not present in primitive form in the parts of that system.

I agree with that in the logical sense. that is why I don't need more than arithmetic for the universal realm.



There
is nothing mystical about emergent properties. When the emergent
property of ‘pumping blood’ arises out of collections of heart cells,
that property is a logical extension of the properties of the parts -
physical properties such as elasticity, electrical conductivity,
volume and so on that belong to the individual cells. But nobody
invoking ‘emergent properties’ to explain consciousness in the brain
has yet explained how consciousness arises as a natural extension of
the known properties of brain cells  - or indeed of matter at all.


Because the notion of matter prevent the progress. What arithmetic explains is why universal numbers can develop a many-dream-world interpretation of arithmetic justifying their local predictive theories. Then for consciousness, we can explain why the predictive theories can't address the question, for consciousness is related to the big picture behind the observable surface. Numbers too find truth that they can't relate to any numbers, or numbers relations.





In the same way, I can’t see how qualia can emerge from arithmetic,
unless the rudiments of qualia are present in the natural numbers or
the operations of addition and mutiplication.

Rudiment of qualia would explains qualia away. They are intrinsically more complex. A qualia needs two universal numbers (the hero and the local environment(s) which executes the hero (in the computer science sense, or in the UD). It needs the "hero" to refers automatically to high level representation of itself and the environment, etc. Then the qualia will be defined (and shown to exist) as truth felt as directly available, and locally invariants, yet non communicable, and applying to a person without description (the 1-person). "Feeling" being something like "known as true in all my locally directly accessible environments".




And yet it seems to me
they can’t be, because the only properties that belong to arithmetic
are those leant to them by the axioms that define them.

Not at all. Arithmetical truth is far bigger than anything you can derive from any (effective) theory. Theories are not PI_1 complete, Arithmetical truth is PI_n complete for each n. It is very big.




Indeed
arithmetic *is* exactly those axioms and nothing more.

Gödel's incompleteness theorem refutes this.




Matter may in
principle contain untold, undiscovered mysterious properties which I
suppose might include the rudiments of consciousness. Yet mathematics
is only what it is defined to be. Certainly it contains many mysteries
emergent properties, but all these properties arise logically from its
axioms and thus cannot include qualia.

It is here that you are wrong. Even if we limit ourselves to arithmetical truth, it extends terribly what machines can justify.




I call the idea that it can numerology because numerology also
ascribes qualities to numbers. A ‘2’ in one’s birthdate indicates
creativity (or something), a ‘4’ material ambition and so on. Because
the emergent properties of numbers can indeed be deeply amazing and
wonderful - Mandelbrot sets and so on - there is a natural human
tendency to mystify them, to project properties of the imagination
into them.

No. Some bet on mechanism to justify the non sensicalness of the notion of zombie, or the hope that he or his children might travel on mars in 4 minutes, or just empirically by the absence of relevant non Turing-emulability of biological phenomenon. Unlike putting consciousness in matter (an unknown into an unknown), comp explains consciousness with intuitively related concept, like self-reference, non definability theorem, perceptible incompleteness, etc.

And if you look at the Mandelbrot set, a little bit everywhere, you can hardly miss the unreasonable resemblances with nature, from lightening to embryogenesis given evidence that its rational part might be a compact universal dovetailer, or creative set (in Post sense).



But if these qualities really do inhere in numbers and are
not put there purely by our projection, then numbers must be more than
their definitions. We must posit the numbers as something that
projects out of a supraordinate reality that is not purely
mathematical - ie, not merely composed of the axioms that define an
arithmetic.

Like arithmetical truth. I think acw explained already.



This then can no longer be described as a mathematical
ontology, but rather a kind of numerical mysticism.

It is what you get in the case where brain are natural machines.



And because
something extrinsic to the axioms has been added, it opens the way for
all kinds of other unicorns and fairies that can never be proved from
the maths alone. This is unprovability not of the mathematical
variety, but more of the variety that cries out for Mr Occam’s shaving
apparatus.

No government can prevent numbers from dreaming. Although they might try <sigh>.

You can't apply Occam on dreams.
They exist epistemologically once you have enough finite things.

Feel free to suggest a non-comp theory. Note that even just the showing of *one* such theory is everything but easy. Somehow you have to study computability, and UDA, to construct a non Turing emulable entity, whose experience is not recoverable in any first person sense. Better to test comp on nature, so as to have a chance at least to get an evidence against comp, or against the classical theory of knowledge.

Bruno


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



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