On 19 Jan 2015, at 21:09, David Nyman wrote:
On 19 January 2015 at 18:37, meekerdb <[email protected]> wrote:
On 1/19/2015 6:01 AM, David Nyman wrote:
There's an effective riposte to this, I believe, but it might be a
bit subtle, so I ask you to bear with me. I think, in the first
place, that it's beside the point to get hung up on the
'concreteness' or otherwise of arithmetic. Bruno's intent is rather
to enquire into the possibility that every relation necessary to
explain both observers and what is observed can be reduced to those
of basic arithmetic or its equivalent. Such an admittedly remarkable
possibility is itself suggested in the first place by the
computational theory of mind and the universality of the digital
machine.
Further axioms relating to the emulation (or embedding) of
computation in arithmetic and that of various modal logics in
computation are also included at the outset, but remain to be
justified by their effectiveness. This has important consequences,
as we shall see. The question then is whether these assumptions lead
in the right direction. According to Bruno (and I don't claim to
follow him on all the detail of this) they lead in the direction of
self-referential computations that simultaneously emulate or embody
two distinct logical modalities (1-person and 3-person). The
intersection of these distinct but mutually entangled logics
presents novel possibilities of resolving previously intractable
mutual reference issues since mind and body need no longer be seen
as categorically orthogonal.
That said, as you point out, it might still seem open to a doubter
to say so what. So we have computations whose complexities
purportedly embody 3-personal entities, complete with the detailed
appearance of their physical environments. So these computations may
simultaneously entail the putatively 1-personal points of view of
such entities. These two logics may even be related in an analytic
or logically necessary way. All this may be remarkably suggestive
but are we forced to accept that actual conscious experience arises
as a necessary consequence of all this merely arithmetical
*construction*?
This is where the subtlety comes into play. Remember that
consciousness is here modelled as *truth*. When you really come to
think about it, truth is *the* defining characteristic of
consciousness. As Descartes realised (though his insight is often
misconstrued) it can make no sense to doubt the truth of doubt
itself. When we apply this to the mutual reference problem something
truly remarkable occurs. Take the question of Smolin's claiming to
'see red'. This claim is now seen as occurring at the intersection
of two logics: one 'observable', the other 'private'. However,
although this entanglement may explain their co-variance and mutual
reference, neither of these logics fully captures the *truth* of the
claim, or if you prefer, what it would actually be *like* if the
expressed belief were true. Each of them is still, as it were, a
mere epistemological possibility, abstractly lurking somewhere in
the infinitely extended ontology of arithmetic.
But if these logics can't definitively *capture* the truth of the
claims they emulate, they do point to where it might be found. It
comes down to this: Is Smolin, the putative experiencer of the truth
of the claim to 'see red', being *truthful*? Given the hypothesised
mutual consistency of the entangled logics, this is analytically
certain. Smolin is incapable of being other than truthful in this
regard; ergo he does in fact 'see red'. We can, of course, deny that
there is any such analytic compulsion to truth. But this is self-
defeating, in exactly Descartes' sense. If there is no truth of the
matter, then there is equally no red, no Smolin, no belief, no
logic. The 'epistemological' assumptions have been ineffective and
must be discarded. The only remainder is arithmetic itself, since
that is the ontology we assumed at the outset.
An interesting explication. If Smolin can't be mistaken when he
says or thinks "I see red" - and I agree that he can't - then it
must correspond to (or be entangled with) a specific third person
computation (i.e. physics of his brain). But we can ask why is this
entanglement, this 3p point-of-view, even needed? Isn't just
Smolin, i.e. his thoughts, already realized in the infinity of
computation, and even realized infinitely many times? If we ask for
a simulation of a lot of people then it may be more efficient to
simulate a physics that gives them consistent 3p points-of-view. But
I'm not sure efficiency has any relevance in arithmetical infinity.
I'm not sure I'm well equipped to answer your question, assuming
I've understood it. My understanding of Bruno's schema is that it
leads to a kind of Computational Library of Babel. You could say
that it's the way a 'creator' might set about producing a lawful
universe despite knowing nothing beyond simple arithmetic. My kind
of god, in fact ;-) However the dissimilarities with Borges's
alphabetic library are perhaps more striking than the parallels. For
one thing, the hypothesis of the dovetailer (i.e. a very simple
computation that happens to generate all other computations,
including itself) must result in a fractal-like explosion of
redundancy that threatens to defeat the imagination.
Yes. Note that assuming Church thesis, the existence of the universal
machine and of the universal dovetailing (the execution of all
programs, in the original mathematical sense of execution) are theorem
of elementary (even very elementary, like RA) arithmetic.
Then, you are right, not only all execution of programs exists in
arithmetic, but they exist in an extreme redundant way, and they
appears as highly structured from the possible points of view
accessible to the machine, which is useful to delineate the truth of
the relations, and the different mode in which such truth can appear
to the machines.
Consequently the question of differential measure, of classes of
similar computations, becomes absolutely central in determining
which outcomes predominate and the consequences of this are far from
obvious, to say the least.
Indeed, this has to explain why the physical (necessarily observable)
looks physical (as inferred), indeed.
So as you imply, what would emerge from this is less a matter of
efficiency and more one of what would tend to get the upper hand in
the ensuing measure battle. Producing a detailed a priori argument
for this is unfortunately well beyond my capabilities.
In other words, it is ultimately only the level of truth that
validates, or redeems, the epistemological assumptions; otherwise
they remain mere 'free-floating' abstractions,
The epistemological assumptions are about the 3p POV.
Well, my whole point is that, at the 3p level, I consider the
epistemological assumptions to be *provisional*. That is, whatever
results from mechanism doesn't count as knowledge, in the sense in
question, until it is justified or redeemed at a level that
transcends the bare mechanism itself. This is what I mean by the
level of truth. I expand on this a little below.
It is here that incompleteness implies a sort of miracle. Despite we
limit (and have to limit) ourselves to correct machine (to get the
correct physics by construction), so that []p is true entails that p
is true, the Theaetetus' definition, which attach the box []p to p,
with []p & p, does obey a logic of knowledge (trivially). What is not
trivial is that []p does not obeys to a knowledge logic.
Better, we recover the greek-indian notion of knowledge, coherent with
the dream argument, but also coherent with Brouwer notion of non
formalizable, indeed, non mechanical knower.
So, not only computer science proves that machines have a soul, with
the greek definition of soul, but all proofs they have done are
transparently translated in the discourse of the machine. The machine
can prove that the correct machine have a soul, that the soul is not a
machine, that it is immortal, etc.
But the necessary truth you refer to is about the arithmetical
relations instantiating the 1p POV. So when Smolin thinks "I see
red." it is necessarily true that this is associated with a certain
computation that instantiates "redness". But it isn't necessarily
true that the is a read object in Smolin's view - which is what he
actually means when he thinks "I see red".
Whether there is a red *object* is not material to my point. The
truth in question depends only on whether his statements refer to
something real or true in a primary sense. All other senses, in the
context under discussion, such as secondary interpretation or
decomposition into 'objects' of perception, are derivative on this
primary datum.
conceptually disconnected from a base ontology that has no knowledge
or need of them. If we can accept consciousness as the model (in the
mathematicians sense) of such a truth level,
What does "truth level" mean? I don't see what the levels of truth
are; there are true sentences and false sentences and decidable and
undecidable sentences.
Truth is absolute.
Decidable and undecidable always refer to a machine.
Are you referring to true sentences in a metalanguage? And in what
sense can a consciousness model a "truth level"; sounds like a
category mismatch?
I know that Bruno specifically refers to arithmetical truths in the
senses you describe above. But in my view we need to think about
this somewhat more broadly or generally to get a handle on
'consciousness' in the way I am suggesting. That is, the truth in
question is just what would *have to be the case* to validate, not
only the truth-claims of the relevant statements, but to validate
their very status *as* truth-claims (as opposed to mere machinery).
In other words, truth is modelled here as the criterion that
differentiates epistemology from ontology.
Yes, as truth will be true (or not) independently of any machine
believing it, or knowing it, etc.
That use of truth, when limited to the arithmetical domain, is not
problematical or controversial.
As such, it 'retroactively' validates the ascription of
'epistemological' to those features of the ontology that can then
(correctly) be seen as giving rise to it. This may be circular, but
perhaps virtuously, in a sense you've sometimes encouraged us to
entertain.
There are cricular aspect in the box, but I don't think truth is used
in a circular way.
Of course, our intuition of arithmetical truth, and on arithmetic will
not be entirely explainable, but at least we (and machines) can
understand that it has to be the case that something cannot be
entirely explain. mainly our belief in RA axioms. That is not obvious
for many people, because the axioms of RA seems obvious, but since the
failure of logicism, we know they are not.
I would be interested in Bruno's comments (and indeed your own) as
to the viability of extending the notion of arithmetical truth in
this way.
With mechanism, the extension of the arithmetical truth are
automatically epistemological. In fact I already put the PA induction
axioms in the epistemological, although we are free to put them at the
ontological level. It is more clean, and it helps to distinguish the
"reality" (described by RA) from the observers (describes by the
number which emulates PA, in RA).
At this point, I'm somewhat persuaded that this broader sense of
truth, in approximately Descartes' sense, is in fact highly relevant
to what is special and, so to speak, non-negotiable about
consciousness. It has the virtue that it now makes no sense to say,
as Stathis wants to suggest, that the same scenario could equally
well be describing an 'unconscious' (i.e. untruthful) process.
I am OK, *assuming* mechanism.
But Stathis' suggestion cannot logically be excluded with infinite
machines.
we can justify our attempt to abstract, epistemologically, a
multiverse of dreaming machines complete with their hallucinated
physical environments. If we insist on denying this, however, the
entire epistemological enterprise just collapses back into the heap
of its base ontological components.
The usual way to justify a theory is to have it predict some
otherwise unexpected observable fact.
True enough, but my sense of 'justification' here was more
restrictive and specific. I simply meant that our ascription of an
'epistemology', in the relevant sense, to mechanism should be
conditional on the truth-claims provisionally attributed to such
mechanisms being 'justifiable' or 'redeemable' at some level that
transcends mechanism itself. Ordinarily, this implies some level of
extrinsic interpretation, as for example when you are reading this
text. But in the case of a purportedly self-interpreting mechanism,
there can be no such extrinsic locus.
I understand, and I am OK "philosophically". Now, I am a scientist, so
I took 30 years to make sure I got an empirical test for some
mathematically precise version of computationalism (like
computationalism + the S4 theory of knowledge (accepted by all
philosophers, at least when open to the use of analytical tools).
Consequently, we must seek to anchor 'interpretation' in a manner
that is intimately associated with mechanism itself but somehow
transcends it.
Yes. And tha attachment with mechanism is provide by the (sigma_1
complete) predicate of provability []p, which acts like a rational
sort of belief, and the transcendence of it is given by the link with
truth: []p & p.
In fact, my contention is that denying the effectiveness of such a
level (i.e. truth as it pertains to mechanistic 'knowledge'), is
tantamount to dismantling the whole attempt to ascribe an
epistemology to mechanism in the first place.
OK.
Bruno
David
Brent
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