On 29 Jan 2012, at 06:22, Pierz wrote:

On Jan 27, 5:55 am, Bruno Marchal <marc...@ulb.ac.be> wrote:
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

No, it is modal logic,

A nice term for speculation! Mind you, that's OK. Where would we be
without speculation? But the term 'modal logic' might be used for
numerology too - it *might* be the case that a 4 in one's birthdate
does signify a practical soul.

Here you are just ignorant of the discovery that mathematical formal belief (proof) has been discovered to be translatable in the language of the theories themselves. And then they obeys modal logical axioms. We have no choice to admit them, and they magically avoid all the notorious philosophical difficulties raised by Quine on Modal logic. So, the modal logic just compactify information which is extracted from the number relation. To be short UDA + AUDA shows that the theory of everything (or a good one among an infinity of equivalent) is already provided by very simple arithmetical axioms: like:

Ax ~(0 = s(x)) (For all number x the successor of x is different from zero). With

AxAy ~(x = y) -> ~(s(x) = s(y)) (different numbers have different successors)

and with different sort of axioms, containing usually the addition laws:

Ax x + 0 = x  (0 adds nothing)
AxAy  x + s(y) = s(x + y)   ( meaning x + (y +1) = (x + y) +1)

and the multiplication axioms

Ax   x *0 = 0
AxAy x*s(y) = x*y + x

The only "speculation" is that the brain, or the generalized brain, is Turing emulable at some level of description. I use arithmetic and Church thesis to just make that statement precise.

although model theory does that too. It is
basically the *magic* of computer science.

Magic, but not numerology then.


relatively to a universal
number, a number can denote infinite things,

I think you're saying that a number can be part of an infinite number
of sets, calculations etc, which is true, but what it denotes is
always purely a matter of logical numerical relationships, unless it
denotes something beyond mathematics itself, such as when I count
oranges. I am saying that to denote qualia, the numbers must be
denoting 'oranges' (or maybe the colour orange as an experience),
things outside of pure logic, not mathematical entities.

Math, even just arithmetic is already outside logic. And assuming comp consciousness is related to computation, which exists in arithmetic. You can prove from the axiom above that prime number exists? OK? Likewise you can prove that universal number exist. Then, relatively to that universal number, all other number get a behavior. The universal number interpret the other numbers like if they where machine, program.

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

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.

Sorry, I cannot parse that sentence. It doesn't seem grammatical.

Sorry, I meant:

Consciousness, seen as a bet in some reality, emerges through (infinitely many) theorems in arithmetic.

They emerge like the prime numbers emerges.

'They'? The theorems? You mean consciousness is a bet on an
arithmetical theorem?

The universal numbers emerge like the prime numbers emerge. Logicians prefer to say that they exist, simply. Consciousness is more delicate, because it emerges from the 1p indeterminacy on all the computations going through my computational states. It is a big complex infinite, which is structured by the modalities of self-reference (which are the same for all correct rich enough machines).

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

Once executed, he's not a hero any more, he's a martyr :)


(in the computer science

oh, right ;)

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

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.

I do appreciate Gödel's theorem and its proof that there are true,
unprovable statements within any given arithmetic, so you are correct.

Yes. And one such proposition is concistency (= I don't prove the false = ~Bf = D~f = Dt).

Gödel's second theorem is a nice modal formula: Dt -> ~BDt; if I am consistent then I cannot prove that I am consistent. If you follow an half-hour course in modal logic, you would see that it is easy to derive from Löb's formula B(Bp->p)->Bp, which can be see as a generalization of Gödel's theorem. Löbian machine are the one which can prove their own Gödel and Löb's theorems.

But my error is one of technical terminology I think. Surely there are
statements that can be made within a certain arithmetic and others
that can't. For instance, within Peano arithmetic it does not make
sense to ask about the truth value of statements involving i (the
imaginary number).

You can define i in arithmetic, and then prove many theorems of complex analysis in arithmetic. To be sure, it is an open problem if there are any theorem in usual mathematics (≠ logic, and category theory) which cannot be proved in elementary arithmetic. PA is already a very strong theory. But the indecidable proposition are always proposition that we can express in the language of the theory under consideration.

Then there are limits to what can be called a
mathematical statement - ie one involving the truth and falsity of
purely logical relations. So I can't, in any arithmetic or system of
mathematics, ask if the number 20 is nice or not.

Unless you succeed in defining "nice" is arithmetic. 'course.

Or if the prime
numbers are pink or blue.

Well color does not apply to numbers, but might apply to the hat of a person is a life related to the arithmetical relation.

Arithmetical truth may be as vast as you
like, but my point is that it is still *arithmetical*, and qualities
don't come into it.

If you assume comp, they cannot not come.

It is the set of sentences that can be made about
numbers and those sentences are limited in their symbols. So Gödel
doesn't help you here I don't think.

It helps because it shows that machine have a rich apprehension of all what they cannot do in a provable way, yet can still do by accepting to be possibly not correct. That ignorance space appears to be structured, and there are obvious candidate for qualia (like immediately accessible undoubtable truth, yet non communicable to others).

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.

Terribly perhaps, but still not beyond the arithmetical, by

It might seems weird, but a bit like the set of all sets cannot be a set, arithmetical truth cannot be defined in arithmetic. Arithmetical truth bears on numbers, but as a concept is far above the possibility of expression of numbers. Sure you can define it in set theory (another Löbian machine) but the definition is a fake one, because its intuition will relate on set-theoretical truth, which is much vaster than arithmetical truth. But from inside arithmetic, things get worse. The 1p of arithmetical theories is above all conceivable mathematics. It is counter-intuitive. Mathematical logic is a road made of counter- intuitive statements.

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,

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

Well I certainly don't dispute the central significance mathematics
must play in any complete scientific or philosophical world view. I
suppose the question is whether that mathematics is ontologically
primary or not.

There is no choice in this matter. Arithmetic has to be primary, if only to define comp, and then can be shown necessarily enough.

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

Like arithmetical truth. I think acw explained already.

Are you saying arithmetical truth is not purely mathematical?

It is not arithmetical. But the 3p arithmetical truth can be defined in set theory, and so can be said mathematical (for some set realist, which I am not). The 1p-arithmetical truth is theological and cannot be purely mathematical indeed.

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

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.

Well, I'm not trying to prevent anyone from dreaming! I'm arguing
whether or not maths can include dreams.

Assuming comp, arithmetic can be shown full of computations. All of them, which includes dreaming creatures.

Simple polynomial arithmetical relation does already simulates *all* the rational approximation of the Milky Way collision with Andromeda, at the quantum level of strings. This includes dreaming persons. The hard things is to justify what we stay in such histories, given that the 1-indeterminacy is very *huge*. So huge that when I was young, most people took UDA as a refutation of mechanism. AUDA is almost just there to show that the machine reality is too complex to refute mechanism so easily, because we have to take into account self- reference, and the math of referring to one's own body, etc.

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.

Hehe. I suppose you have some idea that I can't do that! As noted in
prior post in this thread, these are my attempts to understand, as
completely as I can, this interesting philosophy. I admit I like your
theory better than materialism. I am trying to discover if I like it
enough ('like' in the sense that it satisfies my intellectual
intuition and my logic sufficiently) to entertain it seriously over my
current admission of nearly total ignorance as to what the world 'is'.
I don't need to posit an alternative to make that enquiry, and to do
so by questioning whatever in the theory seems weak (even if it proves
in the end to be my understanding that is the weakness).

OK. Cool.



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