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:
http://www.youtube.com/watch?v=WbFFs4DHWys&NR=1&feature=fvwp
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:
http://www.youtube.com/watch?v=Pzg4XvaBnT4
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).
Bruno
http://iridia.ulb.ac.be/~marchal/
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