On 25 Jan 2010, at 07:52, Brent Meeker wrote:

Bruno Marchal wrote:

Now, having postulated the natural numbers with addition and multiplication, they organized themselves, independently of our whishes, in a way which escapes *any* attempt of *complete* unification. They defeat all our theories, in a sense. Once we postulate them, they get a life of their own. To understand them, we have literally no choice, in general, than to observe them and infer laws. We can prove that they have definite behaviors, but we can prove (assuming mechanism) that we cannot predict them, in general.

ISTM that can be read as a reductio against the reality of arithmetic.

On the contrary. It shows that arithmetical reality kicks back. We may also know greater and greater portion of it. We may discover new interesting properties, and we progress indeed since a long time. From Diophantus to Matiyasevitch, to mention a beautiful line.

Are you alluding to fictionalism? Do you defend the idea that "3 is prime" is a false proposition?

No, I just don't think it's truth implies the existence of "3".

So you believe that the proposition "9 is not prime" is false?
To say that "9 is not prime" is the same as saying that It exits a number different from 1 and 9 which divides 9. To believe that "9 is not prime" you need to believe that Ex[x = s(s(s(0)))]. i. e "3 exists" (and divides 9).

I have no real clue of what that could seriously mean.

Of course I would never expect that someone who doesn't believe that 3 is prime can say anything about the consequence of DIGITAL mechanism. Such a move cut the uda (and the auda) at their roots, and everything becomes infinitely mysterious. Frankly I would not ask him to compute my taxes either.

So why not suppose that the natural numbers are just a model of perceptual counting; and their potential infinity is a convenient fiction whereby we avoid having to worry about where we might run out of numbers to count with?

You can do that. But assuming you are not fictionalist, if you say that the infinity of natural numbers is a fiction, you are lead, ITSM, to ultrafinitism.

What's the difference between finitism and ultrafinitism? Doesn't postulating the integers plus ZF also commit you to existence of the whole hierarchy of infinite cardinals?

Finitist believes in all finite numbers or things. And nothing else. A finitist believes in 0, and in s(0), and in s(s(0)), etc ... But he/ she does not believe in the whole set {0, s(0), s(s(0)), ...}. He/She does not believe in infinite objects.

An ultrafinitist believes in 0, s(0), s(s(0)), ..., but he/she does not believe in all finite numbers. He believes that the set of all positive integers is a finite set. I think that Tholerus argued that there is a bigger natural number. This makes sense for some strong form of physicalism: a number exists if and only if it is instantiated in the physical reality (which has to be postulated, then, and assumed to be finite).

With fictionalism, I think that you can say "yes" to the doctor, and reject the reversal consequences. This leads to a matter problem, a mind problem, and the usual mind/matter problem. I would take this as a defect of fictionalism.

Brent, I am not saying that ultrafinitism and fictionalism are false. I am just saying that IF you say yes to your doctor's proposal to substitute the brain for a computer, and this with a reasonable understanding of what a computer is (and this asks for a minimal amount of arithmetical realism) then the laws of physics are necessarily a consequence of the (usual, recursive) definition of addition and multiplication. Indeed it is the global coupling consciousness/realities which emerges from + and * (and classical logic). (or from K and S and the combinators rules, + equality rules (this is much less)).

A sentence like "naturals numbers are just a model of perceptual counting" already assumes (postulates) arithmetic. And with digital mechanism you can explain why universal number can use natural numbers as "model of their perceptual counting".

You should not confuse the numbers as thought by the philosophical humans (what are they? does they exists?) with the numbers as used by mathematicians, physicists or neurophysiologists, like in "this flatworm has a brain constituted by 2 * 39 neurons" or "all positive integers can be written as the sum of *four* integers squares.

(Then the number takes another dimension once you say "yes" to the doctor, because in that case, relatively to the (quantum) environment, you say "yes", not for a "model", but because you bet the doctor will put in your skull the actual thing "you", yet through "other matter", and all what counts is that he put the right number, relatively to the current environment. That other dimension is somehow the object of all our discussions).

May be I can ask you a question, which I asked to Peter Jones, and which is this. Do you see that NON-COMP + arithmetical realism entails the existence of a realm full of zombies?

No, I don't see that.

Probably my fault. Let us call "WEAK AI thesis" the thesis that we can build a machine which behaves like a human, but that such a machine has no consciousness, so it can be seen as a zombie. This is plausibly true concerning some actual japanese androïds, or for some "fake policeman" that some states put on the road as preventing measure.
I find plausible that the following shows a zombie:

I rephrase the question:

Do you see that NON-COMP + WEAK AI thesis entails the existence of zombies in arithmetic?

Of course they would exist in the same sense that numbers exists (and programs or digital machine, together with their computations, exist in arithmetic, except that a zombie androïd will be more tedious to define). Well, for the Japanese zombie (plausibly) you may ask for the (digital) code source.

A more simple reason is that all rational approximations of the quantum state evolution of the Milky Way up to some fixed moment, exist in arithmetic, like any finite pieces of any computations exists in arithmetic (and is executed by the UD).

The laws of arithmetic do emulate all those computations. In particular the proposition "all digital approximations of dynamical movement of Brent's body are emulated in arithmetic". But if you don't attribute a consciousness to such virtual and arithmetical "Brent", then they will be abstract (arithmetical, virtual) zombies, and this despite some of them will write the same mails, and invoke the same thoughts and idea. I assume classical (non relativistic) quantum mechanics, for the sake of the illustration, here.

Yet, like in the empty wave of the Bohmians, those zombie acts and talk like you and me, have thought processes, and asks themselves about mechanism, consciousness, realities, and what constitute their environment (matter), and all this in a genuine way, as defined by the logics of (correct/consistent) (relative) self- references. With NON-COMP, I would be tended toward fictionalism myself, because I would wish those zombies could not exist.

Such zombies seem like an incoherent concept to me.

With comp, the problem of the zombie, or of the movie of a computation is solved by the fact that consciousness cannot really be attributed to anything observable. My current consciousness is not related to my observable body, only to my "real body" which emerges from ALL computations going through my state related to "my consciousness" (which exists assuming comp). But then, with QM, we are described by infinite (or very big) tensor products, and this suggest we may "observe" or infer from the simplest theories of the empirical facts, a part of our unobservable "real body". The "real body" is really an infinite (or big) tensor products of bodies. The fact that quantum indeterminacies are sharable (explained in Everett by the multiplication of worlds, or the contagiousness of the superposition states (entanglement)) may suggest also that our substitution level is determined by the Heisenberg uncertainties. The quantum aspect of reality (by which I mean mainly the many worlds and the constant multiplication (or differentiation) of large populations of universal machine/subject) saves both QM and comp from solipism. It makes possible the notions of first person plural.

In a sense, there *exist* local zombies, because from their own first person points of view, they belong to the projection of the set of all computations. Their first person indeterminacy bears on the whole computational space, and what is observable in any stable way can only belong to the border of that space.

How do you define the border of computational space?

Hmm... Imagine a affine line. You can see it as a point in the projective space. Now consider your current relative computational state, describes at *a* correct level of substitution. In arithmetic it exist an infinity of computations going through that relative state (and the UD generates them all). You don't know in which computation "you are", and by the first indeterminacy, it does not matter, given that the next "moment" is given by a measure on the set of all possible "next states". The math suggests there is no "next state" from the first person view, but more a continuous neighborhood. Now for each computation you are in, you have to predict your "observable reality" by the infinite union of all infinite computations going through that state. Example the Milky way, in the position base, will give a continuum of different computations, distinguished by the position of the electron on any atom. Now many computations stop and many computations don't stop, and the interesting things happen in between, a bit like the border of the Mandelbrot set. If the set M inter Q x Q is creative in the sense of Emil Post, then the border of the M set is (recursively isomorphic to ) the border of the computational space. I have still some doubt, but the M set illustrates the self multiplication, and self complexification (self mixing). (power of two for the M set) look at this beautiful (8 minutes) deep and interesting zoom:

Eventually the border is described by the arithmetical "hypostases". The eigth basic person points of view. I guess there will be opportunity to come back on this. The border of the computable is the boundary between the computable and the uncomputable. It is arithmetical truth minus the Sigma_1 complete set(s).

This is really just a consequence of the impossibility to be aware of the UD delays, or of where "we" are in (Sigma_1)-reality, or comp-reality. The comp supervenience thesis is hard to explain without digging in the details, but consciousness, our consciousness, is related to a big infinite cloud of intricate number relations. The "identity thesis" is partially justified only in a very relative and local way. It is a bit like the appearance of a collapse in the QM without collapse.

I don't find the multiple-worlds interpretation of QM very convincing either. In conventional QM it implies that a single radioactive atom causes a continuous splitting of the world. I suspect that real numbers should not be taken seriously.

I have never understood the one world interpretation of QM.
I think that there is no many worlds interpretation of QM. Linearity of evolution and tensor product multiplies or differentiates the "set closed for interaction".

a(b + c) = ab + ac

Feynman says that the collapse of the wave is a collective hallucination. And Everett showed that any machine capable of memorizing the result of its observation will describe those hallucinations in their diaries.

And I think that once we assume comp then the wave itself appears to be, not really an hallucination, but a sum on all hallucinations, if you want. Technically, this means we have to justify the discourses on the wave itself, by the necessities and possibilities and (arithmetica) truth for the (Lobian) universal machines. Physics has to be invariant for the choice of phi_i bases so that postulate the quantum becomes treachery, with respect of the comp mind body problem. We have to justify the quantum from the stable limit discourse of the universal machine (which observes itself).



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