Computability, consciousness and the limitative results of mathematical logic
anonymous
Below I discuss computability and the related limitative results of
mathematical logic, as well as their relations to the "problem" of
consciousness, and hence indirectly to AI.
The first few sections
can be skipped by those familiar with the technicalities of the
mathematical notions, but those familiar with only their semipopular
versions, should perhaps read them. The latter portions present
my own view on the subject.
What I hope from you is comments of any sorts.
1. The Halting Problem and Goedel's Incompleteness theorems
I recently read a philosophical book on the anthropic principle, and in
the introduction it was stated that "Turing's Halting Problem shows that
a computer can never fully understand itself". Had the statement been
followed by an exclamation mark, I would have vomited of disgust.
The Halting Problem, as well as the closely related incompleteness
results of Goedel have a long history of abuse in the hands of certain
anti-AI philosophers. However, those sufficiently familiar with the
actual arguments establishing these theorems should be aware that
they serve not to show the impossibility of strong AI, but on the contrary
its possibility. This is due to the fact that both offer *effective* means
of construction that allow us to infer that certain machines or theories
are, in some special sense, incomplete.
In case of Goedel's incompleteness argument, this should be very clear.
Goedel's proof (or, preferrably, a more recent variant) shows how,
when gíven a formal axiomatic theory, to derive a sentence that cannot
be proved within the theory. This can give one a feeling of omnipotence,
as one can grasp that given whatever theory, he can find a sentence
found to be "true" by metamathematical means, but nevertheless not
proved by the theory.
The catch is that this omnipotence is indeed established, since Goedel's
proof actually tell us how to *effectively* derive this "undecidable"
sentence.
Indeed, so effective is the proof that we could quite easily (but tediously)
compose a computer program that, given a theory, would produce an
undecidable statement.
So what about the claims that Goedel's proof shows that human mathematical
"intuition" cannot be formalised? The idea is that humans have a complete
idea of,
say, the standard model of arithmetic, in which statements can be
really-really
true. And assumably humans, having some magical form of intuition, can
decide, given enough time and creativity, intuit about any particular
sentence
whether it is really-really true or not.
Now, we *do* have a complete idea of the standard model of arithmetic,
simply because the metalanguage we're using contains a strong form of
set theory, or is second order, which seems prima facie very plausible.
We can *prove* that all structures satisfying the Dedekind axioms are
recursively isomorphic.
Assume now that we try and formalise the metalanguage (in our
meta^2-language). If we use second-order logic we end up, in effect,
stipulating that the second order quantifiers range over the really-really
real powerset of the universe of the model. Or, if we use first-order
logic in our formalisation, we find that even though we can *prove*
within the metalanguage (which is now also an object language) that
two models (of the metalanguage) of the Dedekind axioms are
recursively isomorphic, we find that, working in the meta^2-language,
they aren't necessarily really-really isomorphic.
To make the point clear, assume that we have a first order theory
with the application predicate and "being a second order object" predicare
added to the usual predicates, and certain simple axioms of set
theory are added. Assume further that when we study models of the
theory, we restrict attention to only those models in which the extension
S just happens to be the really-really powerset of the universe of the
model - S. Surprise surprise, we have at our hands a second order
logic after all. Of course, completeness fails, compactness fails and so
forth just as in case of "true" second order logic. And it also just
so happens that in case of arithmetic, the designated special class
of models which contain "systems" satisfying the Dedekind axioms (in
the formalisation of arithmetic) are recursively isomorphic.
The situation is similiar with respect to, say, the law of excluded middle.
In the metalanguage, if we are logical classicist (or platonists), all
formulas of the form (p or ~p) are true, while in the meta^2-language
we find that the elevated form of this principle (|-- p or |-- ~p) fails.
To summarise, model theory provides *no* theory of "semantics" in the
non-technical sense of the term. What it provides is a way to translate the
sentences of the object language to sentences of the metalanguage about
certain
structures. It is possible to "fool" oneself working within a given language
and
theory to believe that *all* can be proved, while this need not be true. The
true
core of Goedel's (1st) incompleteness theorem is that it gives us an
*effective*
way to demonstrate the incompletness of any given (sufficiently strong)
theory.
The incompleteness part comes from the fact that proofs of Goedel's theorem
establish the *essential* incompleteness of the theories, i.e. the fact that
their
incompleteness cannot be repaired.
As a side note, it should be noted that while a global "truth" predicate
cannot
be defined, "local" "truth" predicates can be defined for classes of
sentences
the number of quantifiers in which are below given natural number.
The case with Turing's Halting Problem (or, actually, its generalisations)
is
quite analogous. I have spent considerable space to discussing Goedel's
incompletness result and shall now say a few words on diagonalisation
in closing.
The Halting Problem, as well as Goedel's theorem, can be seen actually
establishing the possibility of strong AI in the following sense. Imagine
you
had never heard of recursion theory (computability theory), and you attempt
to give formulations of effectiveness. Assuming your formulations be
effective,
which certainly seems intuitive, you're immediatedly faced with the dilemma
of diagonalisation:
Arrange your "effective" functions in a sequence f_1, f_2, ..., f_n, ...
and define the function d(n) = f_n(n) + 1. d should be clearly effective
as it is just adds 1 to the result of a computable function, therefore
d = f_i for some i. But now f_i(i) = f_i(i) + 1 by the definition
of d, which is of course absurd.
The Halting Problem shows that the general recursive functions (Turing
computable functions) are actually *closed* under diagonalisation, that
is the above diagonalisation procedure does not lead out of the class.
This is because d(n) turns out to be non-effective, or partial, whichever
you prefer. If you choose to treat d(n) as a partial function (which is
certainly natural) the diagonalisation argument simply establishes
by reductio ad absurdum that d(i), where d = f_i, is not defined.
And, in light of the above discussion, we see once again that in attempt
to show limitations in our "effective" notions, we do this too
"effectively"!
For surely one can actually compute the i for d. (This is not actually a
strong enough claim, but to plunge deeper would require more logical
machinery, and noting that the sequence f_i of general recursive functions
contains an infinite number of repetitions of each f_i).
2. General philosophical musing - the subject and the object
At the very heart of the problem of cognition lies the division between the
object and the subject, or the Self. This can be seen if we consider the
following argument (taken from the critics of sociobiology), which I take
to be fallacious:
"If the motive of a murdered to commit murder was somehow genetically
programmed into him, he cannot be held responsible for his acts. He
could
not help acting out his genetic programming."
The fallacy lies in that we assume that in addition to the "genetically
programmed"
structure of his brains, and thus, assumably, his movites, we assume that
there
is a distinc subject in him, a Self. This Self would then just be rendered
powerless
by the genetically programmed traits. The argument itself is as abrsurd as
the following:
"If the motives of humans are somehow genetically programmed, then
someone
who acts in accordance to his conscience cannot be held virtuous,
because
these good acts were just genetically programmed into his brain, and not
results of intentional volition from his part".
If the motives for human behaviour are, in certain restricted sense,
programmed
by the genotype, then they do not cease to be the *motives* of that
behaviour.
There is no distinct Self to go about being directed by these motives, as if
passively observing them.
As more and more information is gathered about the structure, functioning,
evolutionary "purpose" or the brain, more and more of the Self are being
pushed from the subject side to the object side. If we had no reason to
suppose that there was a biological organ responsible for human behaviour,
includig sense of cognition and consciousness, we could freely attribute
these
to the Self. But if the Self is deprived of its "functions", it becomes the
empty
vessel of the identity of the person, and must be disposed of. (The Self
with
a capital letter is a metaphysical philosophical conception, and not the
ordinary
common usage self).
The identity of the person, however, cannot be restricted
to those areas that are currently not mapped by science. If the motives are,
and it is difficult to see how they could not be, results of evolution, they
*are*
part of the identity of the individual human being, and not something
external
to it.
Another example of an obviously absurd argumentation based on the sharp
object/subject dichotomy is solipsism. Solipsism starts from the assumption
that there is a core Self, the subject, which is independent of the Object,
say, the body, which is just the vessel of the Self, and from this deduces
that
sinse the Self knows directly itself, nothing else can be known. (If nothing
but the Self is known, it is impossible to assign any sort of sensible
"probabilities"
to such statements as "the external world exists" or "other people think").
Also, since the Self is, so to say, hidden, it cannot be directly observed.
This, of course, entails that there is no reason to suppose that "other
people"
have Selves, that is are subjects. This is in crying contradiction to the
common usage of the word self, which is, I take, being explicated.
If we abondon the sharp distinction between the Self and the Object,
this problem dissolves into a trivial empirical problem.
Philosophical disclaimer.
The words Self, Object and Subject are used in the above in metaphysical
sense, and are attributed properties commonly attributed to them in such
discourse. I by no means suggest that the concepts self, object and subject
should be abandoned, only that they not be assigned spurious philosophical
connotations that result in silly absurdities.
Of course, my philosophy is rather boring and dull, since it reduces most
classical philosophic questions either meaningless (not literally, just
meaningless
in the sense the concepts used correspond to no interesting real life
concepts), empirical, technical (this category includes certain ontological
problems) or trivial to answer (the fact that these questions can be
given trivial answer is not, however, trivial, as is shown by their
repeated occurance in philosophy).
I use a principle that might be called Principle of Linguistic
Reductio ad Absurdum, which can be stated as follows:
If a concept A is being explicated (or defined) and is proposed to
be equivalent to a concept B, then if replacing A by B causes certain
statements that are obviosuly true about A to become false,
B should be abandoned as an explication (or definition) for A.
I also employ the following weak verificationist principle:
If situations A_1, ..., A_n cannot be distinguished, they should be
identified.
3. AI and mathematical logic
We now come to my third issue, namely that of the relevance of the
of modern mathematical logic and recursion theory to the study of
consciousness
and AI.
Section 1. should have shown, at the very least, that the Halting problem
and Goedel's incompleteness theorem can be seen as not-opposing to the
thesis of strong AI. I see these limitative results as sort of "guards"
against
arguments against AI, and the fact that they actually hold is intuitively
somewhat miraculous (I am surprised to find that they actually hold!).
Now, a machine can construct questions to another machine that it
cannot answer. It can even do this for itself. But then what? So can humans,
just have a look at the paradoxes. I can even construct a yes-no question I
can
easily answer correctly but *you* cannot, namely: "Will you answer this
question to the negative?". Does this establish that you do not think or
have consciousness?
An important point in discussing, say, Goedel's incompleteness theorems
and their relations to the nature of human mathematical insight is that this
begs the question whether there is at all such a thing as human mathematical
insight. Whose mathematical insight? Whose thinking? I believe it is an
oversimplification to assume that all mathematicians share the same
"subconscious" theory that represents their mathematical intuition.
Another point is that humans are not recursive functions floating about in
a logical vacuum. Humans interact with their environment, and it can be
argued that without this interaction there would be no consciousness at
all. Human beings are not closed systems.
4. AI and ontology
Classical ontology deals with the really-real essence of the way or ways
objects (or processed, or whatever) really-really go about existing.
If the thesis of strong AI is true, ontology in the above mentioned sense
cannot be meaningfully pursued. This is because if the "human intellect"
of the ontologist is recursive, it cannot distinguish between different
"models" of its internal representation of the external word. And according
to the weak verificationist principle, this means that the different
"models"
need and should not be distinguished as distinct, since such speak is,
in the very end, just idle jump to metalevel, from which we proceed to
meta^2-level, to meta^3-level, ..., meta^n-level and so forth, indefintely,
never reaching a level the "absoluteness" we could not refute in a yet
higher level.
In a certain important sense the above is an invalid picture. Even though
we may conceptually jump to a meta^n-level, this is really not essential,
since the *jump itself* is contained already on the object level, since
we must *always* work in the first level, by virtue of its being the level
we work. The meta^n-language collapses to a single plane, although
the process described above can be carried out. To illustrate this,
assume that I speak of the English language in English language
and say that "'P' is true iff P", now this is surely a sentence of the
English language and thus we get "''P' is true iff P' is true iff 'P' is
true iff P".
Of course, this is just a restatement of the fact that the object/meta-
language distinction is relative, not absolute.
What does the abandonment of ontology actually mean?
Well, for
a start it means that, just as getting rid of the Object / Subject
dichotomy, we get rid of the endless Idealism vs. Materialism
disputes.
It also provides an interesting answer to the old illusion arguments
of Descartes. The key to the solution of this problem lies in that
when getting rid of ontology, we get rid also of ontology of the
*Self*. For example, to say that this reality could be all dream and we
could wake up and notice this is to assume that *we* are not
dream, and thus to invoke ontology of Self.
Giving up ontology does not entail claiming that "nothing exists".
That is, after all, an ontological statement. It does, however, prevent
us from saying "the external world exists" in most contexts. I, personally,
have never quite understood what such a claim is supposed to mean.
Things exist in the external world, i.e. are located somewhere in it,
or are located on the structures instantiated in the external world and so
forth, but what about the external world itself? What does existence
mean here?
What about the possible worlds?
Since we have no ontology, and there is no sense of saying that the
world exists, it seems difficult to see how we should distinguish our
world from the other logically possible worlds.
I would suggest that there is hardly any need to do so. The weak
anthropic principle takes care of explaining our existing in just this
world out of all possiblities.
Recursiveness of the universe
It can be quite convincingly argued that the world is recursive. Quantum
mechanics is quantum computable, which in turn is, modulo true random
number generation, equivalent to general recursiveness.
This is a much stronger thesis than that of strong AI, but it has much to
say in favour. Of course, it is empirically unfalsifiable, since proving
that
something is not recursive requires showing that there is no recursive
function of which the (infinite) extension corresponds to that of the
proposed
non-computable function. Alas, empirical science is bound to finite
sampling.
The thesis is, however, theoretically falsifiable, as it could be shown
that a given physical theory rules out the possibility of the universe being
recursive.
The recursiveness of the universe follows from the strong AI thesis and
the weak verification principle, since a recursive being could not recognise
a non-recursive world even if it lived in such.
----
Aatu Koskensilta (aa...@mediaclick.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Torkel Franzen
> This is somewhat long a post, and I apologise for that.
It's too long. It's not realistic to expect considered comments
on posts this long.
> The first few sections
> can be skipped by those familiar with the technicalities of the
> mathematical notions, but those familiar with only their semipopular
> versions, should perhaps read them.
But these sections are full of very controversial stuff!
Phil Roberts, Jr.
anonymous wrote:
>
> This is somewhat long a post, and I apologise for that.
>
Its also one of the better posts I've seen in these hier hills
in quite some time.
> Below I discuss computability and the related limitative results of
> mathematical logic, as well as their relations to the "problem" of
> consciousness, and hence indirectly to AI.
>
I haven't had time to digest your post, but your points on Godel
are where I am somewhat familiar and in some agreement. I also
noticed below that you are trying to circumvent ontology, although
I don't currently quite understand how you get to that point. But I
am also of the opinion that the mind/matter problem may focus
around our NEED to think of "things" in terms of "things", i.e.,
that the concept of existence itself may be at the heart of the
problem, my own version of circumventing ontology.
With respect to Godel, perhaps the simplist way to understand your
position would be to simply state my own point of view, and let
me try to understand you in terms of your perspective on it.
I believe that Godel's theorem should not be construed as a
PROOF that "minds are different from machines" (Lucas, 1961)
but that it should indeed be construed as AN ARGUMENT that
"minds are different from machines". I have always looked
on Godel as something analagous to a labratory experiment
in which, given artificially contrived ideal circumstances,
it is possible to observe a feature of nature which is
usually obscured by a cloud of irrelevancies relative to
what one wishes to determine. And, in this repsect, it
does very much look to me like Lucas isn't that far off
the beam, in that, indeed, it looks a lot like we can
"see" a truth not provable in the system, that semantics
becomes essential where truth is concerned when push
comes to shove, and that a semantic system can outrun
a purely syntactic proof system. It also looks like an
argument that rational beings might not be determined.
And, like a controlled experiment, I believe one should
extrapolate from the laboratory results to nature as a
whole, and particularly, one should look about for
additional empirical confirmation of this same structural
feature unearthed in the laboratory.
My own area of interest lies in the domain of trying to
understand the naturalistic/surivalistic function of
'feelings of worthlessness' in homo sapiens.
And, I am currently of the opinion that, much in the
manner Godel discovered that mathematical reasoning
can not be 'constrained within a formalism' (my
terminology), so too, mamma nature can not constrain
rationality within the formalism of maximizing inclusive
fitness. IOW, to me, 'feelings of worthlessness'
constitute intersubjectively reproducible empirical
evidence that mamma nature's most rational species is
beginning to show signs of "standing outside the system"
(Lucas), and questioning the will to survive itself
(assuming the value a rational being places on itself
is synonymous with 'the will to survive').
Another way of putting this, is that nature endows her
sentient creatures with a sense of their own importance
which is RATIONALLY INORDINATE, and that as a species
reaches a certain stage in its rational/cultural/memetic
development its members increasingly come to question this
in ordinancy, and increasingly come to require reasons for
maintaining it (needs for love, acceptance, recognition,
attention, achievement, moral integrity, purpose, meaning
etc.). IOW, they increasingly show signs of "standing outside
the system" (Lucas).
--
Phil Roberts, Jr.
The Psychodynamics of Genetic Indeterminism:
Why We Turned Out Like Captain Kirk Instead of Mr. Spock
http://www.fortunecity.com/victorian/dada/90/
anonymous
Torkel Franzen <tor...@sm.luth.se> wrote in message
news:vcbhfb6...@beta13.sm.luth.se...
> "anonymous" <anonymou...@rea.ch> writes:
>
> > This is somewhat long a post, and I apologise for that.
>
> on posts this long.
I am aware of that. Once again, I apologise for the length of the post,
as well as its somewhat confused style. Perhaps I just should refrain
from posting not having slept for three nights.
> > The first few sections
> > can be skipped by those familiar with the technicalities of the
> > mathematical notions, but those familiar with only their semipopular
> > versions, should perhaps read them.
>
> But these sections are full of very controversial stuff!
I agree if you mean my implied (or well, they are rather explicit) views
on the philosophy of mathematics.
Below are two of the technical points, which, at least, I regard as
non-controversial.
In case you should disagree, please feel free to inform me.
1. Goedel's 1st incompleteness theorem, in contradistinction to, say, the
earlier
incompleteness "result" of Finsler, show that any sufficiently strong formal
theory
is incomplete, *and* that this can be established, for any theory, by
actually
effectively constructing an example of a statement undecided by the theory
in
question.
Of course, what cannot be effectively done is to find out exactly what
the undecided propositions are.
2. Any second (or higher) order theory is equivalent to a first order
theory,
provided we a) provide necessary formalism for expressing applicapibility
in first order terms, i.e. add a predicate to express that an object is
applicable
to another, or, in set theoretic sense, belongs to the other; add predicate
to
represent being applicable (or a set); postulate that to no object that can
be
applied to objects can another object be applied; restrict our models to
those in which the application predicate is taken to be set-membership and
the subset of the universe which consists of those objects which can be
applied is the powerset of the subset of the universe consisting of the
objects
that cannot be applied.
To keep this post shorter than the previous one, I shall not touch the point
about recursive functions and diagonalisation.
---
Aatu Koskensilta (aa...@meadiaclick.fi)
anonymous
> > mathematical logic, as well as their relations to the "problem" of
> > consciousness, and hence indirectly to AI.
> >
>
> are where I am somewhat familiar and in some agreement. I also
> noticed below that you are trying to circumvent ontology, although
> I don't currently quite understand how you get to that point. But I
The principle of weak verificationism I employ is as follows:
If situations A_1, ..., A_n cannot be distinguished, they should
not be considered distinct.
Now, if strong AI claims are true, human "understanding" and
other cognitive features are recursive. Therefore it is logically
impossible for us to distinguish between the situation in which
we are, say, virtual creatures within a metacomputer, and the
situation where the real world is really-really-really real.
(Fundamental) ontology is just about such distinctions, such
as whether things really-really exist "out there", or whether
they exist only in the mind, whatever that may be. I just don't
see the difference, and therefore having two different concepts,
that of being really-really real (as opposed to being real in
the ordinary sense of not being illusory, which has nothing to
do with "fundamental" ontology) and that of being "ideal"
in some sense.
Of course, I don't wish to reject, say, elementary particle physics.
Nothing of that sort. Elementary particle physicist are, luckily,
not ontologist in the sense I'm using the word. Ontologist ask
the essential very-very really-really nature of these particles,
or anything existent for that matter, while the physicist ask
simply for the properties of these particles that make sense
in the light of the weak verificationist principle (I'm oversimplifying
everything here).
(In order to get my point trough, I have to recourse to the very
ontological terminology I wish to reject. This does not affect
my point, however.)
> am also of the opinion that the mind/matter problem may focus
> around our NEED to think of "things" in terms of "things", i.e.,
> that the concept of existence itself may be at the heart of the
> problem, my own version of circumventing ontology.
Indeed. I don't see why existence or being is such an obsession to
philosophy. Things exist, that is they are, period. Existence is not
something funny things go about doing. The only use for the very
word "exist" is that by means of it we may convey information
about the external world, and whether something is to be found
in it that correspond to a description.
> With respect to Godel, perhaps the simplist way to understand your
> position would be to simply state my own point of view, and let
> me try to understand you in terms of your perspective on it.
>
> I believe that Godel's theorem should not be construed as a
> PROOF that "minds are different from machines" (Lucas, 1961)
Indeed, without quite strong pre-assumptions on that direction,
it cannot be.
> but that it should indeed be construed as AN ARGUMENT that
> "minds are different from machines". I have always looked
Would you please elaborate on this. I fail to see how Goedel's
1st (or 2nd for that matter) incompleteness theorem could act
as such an argument. My very point was that Goedel's theorem shows
that the construction of the undecidable sentences is itself
recursive, and can be thus done by a machine, even a machine
the functioning of which the theory describes.
> on Godel as something analagous to a labratory experiment
> in which, given artificially contrived ideal circumstances,
> it is possible to observe a feature of nature which is
> usually obscured by a cloud of irrelevancies relative to
> what one wishes to determine. And, in this repsect, it
> does very much look to me like Lucas isn't that far off
> the beam, in that, indeed, it looks a lot like we can
> "see" a truth not provable in the system, that semantics
We can only see this truth, in my opinion, because we act
within an another system, namely our "intuitive" metalanguage
consisting of natural language and its various "extensions"
(i.e. natural language that is not shared by most of its users,
but only by a specialised group).
> becomes essential where truth is concerned when push
> comes to shove, and that a semantic system can outrun
> a purely syntactic proof system. It also looks like an
> argument that rational beings might not be determined.
Not determined or defying prediction? These are two different
assertions, and I'd be interested to know which of these you
mean. Determinism does not imply "predicationism".
> And, like a controlled experiment, I believe one should
> extrapolate from the laboratory results to nature as a
> whole, and particularly, one should look about for
> additional empirical confirmation of this same structural
> feature unearthed in the laboratory.
>
> My own area of interest lies in the domain of trying to
> understand the naturalistic/surivalistic function of
> 'feelings of worthlessness' in homo sapiens.
> And, I am currently of the opinion that, much in the
> manner Godel discovered that mathematical reasoning
> can not be 'constrained within a formalism' (my
> terminology), so too, mamma nature can not constrain
I would strongly object to such formulation of Goedel's
result. Goedel showed that no system rich enough to express
number theory (or a relevant portion thereof, so as to
allow the recursive functions to be defined) can be complete,
period. To infer from this that "mathematical reasoning
can not be constrained within any formalism" requires
additional assumptions.
> rationality within the formalism of maximizing inclusive
> fitness. IOW, to me, 'feelings of worthlessness'
This sounds like a very very strong assumption! To assume that rationality
somehow managed to evolve and yet does not increase the inclusive
fitness of its possessors, or may even decrease it, is quite contrary
to the very spirit of sociobiology and evolutionary psychology.
> constitute intersubjectively reproducible empirical
> evidence that mamma nature's most rational species is
> beginning to show signs of "standing outside the system"
> (Lucas), and questioning the will to survive itself
> (assuming the value a rational being places on itself
> is synonymous with 'the will to survive').
I'm not an expert on these issues, but I would imagine that
there are sociobiological explanations as to self-destructive
behaviour, of which suicide is an extreme form. One such
explanation could be that in extreme cases, the feeling
of worthlessness is justified, and thus the individual
contributes more to its inclusive fitness by not consuming
resources that could be used by its close relatives.
I doubt that this model would stand in close scrutiny,
but I would also think that there might be biological
explanations for the feeling of worthlessness and,
say, suicide. No doubt such models have been proposed,
but as said, I am no expert on sociobiology (I am
familiar with the theoretical side, but as to its applications
I am quite ignorant).
Also, questioning the will to survive on the cognitive level does
not mean imply that underlying human motives are changed. You can
question the will to survive on the cognitive level and still go about
living.
> Another way of putting this, is that nature endows her
> sentient creatures with a sense of their own importance
> which is RATIONALLY INORDINATE, and that as a species
> reaches a certain stage in its rational/cultural/memetic
> development its members increasingly come to question this
> in ordinancy, and increasingly come to require reasons for
> maintaining it (needs for love, acceptance, recognition,
> attention, achievement, moral integrity, purpose, meaning
> etc.). IOW, they increasingly show signs of "standing outside
> the system" (Lucas).
This is of course possible, but would constitute an argument
in favour that a) evolution is not directed towards a tangible
"purpose" b) intelligence might not be always contribute to
fitness even in cases where this would intuitively seem the
case.
Of course, there is a strong tendency in, say, introverted and
philosophically oriented persons to question the meaningfullness
of life, and its "inherent purpose". These do, however, remain
meta-level opinions. On the actual level in which humans lead
their lives, even the most devoted nihilist would probably *act*
in some ways.
The feel of need of "reasons" for living ones life, and not committing
suicide might also be of evolutionary significance. For if individuals
with sufficient mental capabilities to reach the conclusion that
from the factual one cannot pass to the imperative, i.e. human
motives are inherently arational and unjustified, would not have
an in-built need to somehow find meaningfullness in their life,
they would probably just commit suicide, which probably does not
serve to increase ones fitness.
Of course, humans do stand outside the (evolutionary) system
in a sense. Humans have culture, language, etc. the specific instances
of which are clearly not the result of biological evolution. To say that,
in the very end these too are "created" by the genes since the
neurological structure of the central nervous system is genetically
programmed, is of course also true, as it is, given our scientific
knowledge, a truism.
Humans as cultural, linguistic and
conscious beings may see the futility of evolution, its random
meaningless nature and so forth, but they do not cease to be
biological beings by this insight. They continue to be governed
by their motives, which in turn are genetically programmed because
the brain in which they "reside" is.
> --
>
> Phil Roberts, Jr.
James Hunter
anonymous wrote:
Since in the eight-fold way, Ontologists are made of particles
It does not compute. It has always been *every* generation's
*philosophers*, not the species, who have always jumped
"outside the system". The rest of species has always not
cared too much one way or the other whether they survive.
Torkel Franzen
> 1. Goedel's 1st incompleteness theorem, in contradistinction to, say, the
> earlier
> incompleteness "result" of Finsler, show that any sufficiently strong formal
> theory
> is incomplete, *and* that this can be established, for any theory, by
> actually
> effectively constructing an example of a statement undecided by the theory
> in
> question.
There are some ambiguities here. What can be established, for any
theory T, is that if T is consistent and "sufficiently strong", then
a certain effectively constructed statement R is undecided by the
theory. The categorical statement "R is undecided by T" can not
necessarily in any sense be established, even if true.
> 2. Any second (or higher) order theory is equivalent to a first order
> theory,
> provided we a) provide necessary formalism for expressing applicapibility
> in first order terms, i.e. add a predicate to express that an object is
> applicable
> to another, or, in set theoretic sense, belongs to the other; add predicate
> to
> represent being applicable (or a set); postulate that to no object that can
> be
> applied to objects can another object be applied; restrict our models to
> those in which the application predicate is taken to be set-membership and
> the subset of the universe which consists of those objects which can be
> applied is the powerset of the subset of the universe consisting of the
> objects
> that cannot be applied.
If you restrict the models as described, there is indeed no
difference between first and second order theories except in syntax.
But it's a bit misleading to use the phrase "equivalent to a first
order theory", since this suggests a semantic or proof-theoretic
equivalence, assuming the ordinary semantics or proof system of
first order logic.
anonymous
> "anonymous" <anonymou...@rea.ch> writes:
>
> > 1. Goedel's 1st incompleteness theorem, in contradistinction to, say,
the
> > earlier
> > incompleteness "result" of Finsler, show that any sufficiently strong
formal
> > theory
> > is incomplete, *and* that this can be established, for any theory, by
> > actually
> > effectively constructing an example of a statement undecided by the
theory
> > in
> > question.
>
> There are some ambiguities here. What can be established, for any
> theory T, is that if T is consistent and "sufficiently strong", then
> a certain effectively constructed statement R is undecided by the
> theory. The categorical statement "R is undecided by T" can not
> necessarily in any sense be established, even if true.
True. I were implicitly assuming that we're working in a metatheory
sufficiently
strong to establish the consistency of the theory in question.
True, and I would not have used the word "equivalent" in any other
context than that of the above explanation.
Stephen
I find this topic interesting and it is obvious that you
have given the ideas a great deal of thought. There is a
portion I do not agree with which I will get to in this
snipped version of your post. Below is the url for "The
Theory of Everthing, What's Knowable and What's Not" by
John Barrow. This area is quite complex. John Myhill and
and Rudolf Carnap among others have done some exploration.
Ideas involved are listable, computable and prospective.
There's a free registration(sorry to say)to read the article.
http://www.the-scientist.com/yr1990/dec/opin_901210.html
anon:
The principle of weak verificationism I employ is as follows:
If situations A_1, ..., A_n cannot be distinguished, they should
not be considered distinct.
sh: This brings to mind the Turing test. I suspect Turing came
up with this idea from contemplating the Halting problem.
Now about situations that cannot be distinguished. The Turing
test has to do with behavior that is observed in situations or
a series of events that constitutes the test. Passing the test
depends on matching a pattern which is rated by how close it
comes to being indistinguishable from the human norm/ideal. I
think this logical category is described by Myhill's term
"prospective"; which is something which can always be more
closely approximated but never reached.
anon:
Now, a machine can construct questions to another machine that
it cannot answer. It can even do this for itself. But then what?
So can humans, just have a look at the paradoxes. I can even
construct a yes-no question I can
easily answer correctly but *you* cannot, namely: "Will you
answer this question to the negative?". Does this establish that
you do not think or have consciousness?
sh: I think not. Nor can this limitation of logical depth ever
be removed from the Turing test.
anon quoted again:
Now, if strong AI claims are true, human "understanding" and
other cognitive features are recursive. Therefore it is
logically impossible for us to distinguish between the situation
in which we are, say, virtual creatures within a metacomputer,
and the situation where the real world is really-really-really
real.
sh: Suppose AI claims are false. Can one also conclude that
it is logically impossible to distinguish between viruality
and really-really. If the same conclusion can be reached
independently of recursion/strong ai, then I dont see the
relevance of of the "therefore". If it cannot then: if strong
ai is true (human recursive) then one cannot distinguish.
However, this would mean that if strong ai is not true and
human thinking is non-recursive(at least in part)then we
perceive what really-really is. How do we know that?
If human A cannot distinguish input from X as non-human
and agent Z cannot distinguish input from X as non-human
does this make human A and agent Z the same? If a plant
is not a star and an animal is not a star are they the same.
No, because they correspond at one categorical level(atomic)
but not at others. I dont see how an a1,a2..aN correspondence
can be established/shown for a comparison of the mathematical
category of potential human thought in a comparison of finite
evaluation. I think discussing these things is self-referential.
I am not sure we get closer to the truth(prospective)if the
vartiation of potential human thought is infinite whether or
not we only see concrete explications. No finite segment of
a numerical pattern can be proven to be random. If our thinking
is assigned a numerical value for each thought we think how
can any patterned string be shown to be in the same category as
another finitely sampled patterned string we compare it with?
Regards,
Stephen
(Fundamental) ontology is just about such distinctions, such as
whether things really-really exist "out there", or whether they
exist only in the mind, whatever that may be. I just don't see
the difference, and therefore having two different concepts,
that of being really-really real (as opposed to being real in
the ordinary sense of not being illusory, which has nothing to
do with "fundamental" ontology) and that of being "ideal"
in some sense.
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amir
it!!!
Print this page now or you will be sorry!!!
Now you probably have been scanning different news groups to find out which one
is the best, haven’t you? I certainly did. But don’t leave just yet. You’ve
probably seen newsgroups which start “I was looking through a newsgroup, just
like you are now” I was skeptical at first and I’ve spent more than that on
lottery tickets and contest. I have seen many of them around, asking for you to
send $6 and you’ll receive $15,000. Well, I am asking you to send $10 and
receive well over $5,000,000 (5 million bucks). I know it sounds crazy but here
is what you have to do. Follow the instructions CAREFULLY.
1) get 10 $1 bills/coins and wrap them in separate pieces of paper each saying
“PLEASE PUT ME ON YOUR MAILING LIST.” Also your name and address. (it might be a
good idea to wrap the $1 bill/coin in dark paper to reduce the risk of mail
theft.)
2) put these separate pieces of paper and put them into separate envelopes,
sending each one to the names below you should bow have 10 envelopes with each
$1 in and will your name inside. MAKE SURE THAT THERE ARE ENOUGH STAMPS ON YOUR
ENVELOPES!!!!
3) Now you need to send this message to 200 newsgroups on the internet. What you
do is put your cursor at the top and highlight it all and select. Copy then open
up word or anything to write on and select paste. Now all you need to do is
place your name and address on the number 10 spot and move everybody up a place,
so 10 becomes 9, 9 becomes 8 ect, ect, now knock number 1 off the top, so 2
becomes 1.
4) Here are the names you need to send the money to:
I. W.C. Daniels, 5910 Transit Rd. Depew, NY 14043, USA
II. Chris Adkins, 127 Cupsaw Dr., Ringwood N.J. 07456 USA
III. Simon Belgian, 1212 Viola Ave., Glendale CA, 91202 USA
IV. Creative Consulting, 4302 N. 16th Ave., Phoenix, AZ 85015 USA
V. M L Enterprises 27-1300 King St. E. Box 135 Oshwa Ontario, Canada L 1H-8J4
VI. S.Soudakov Elmshurst house Bromsgrove School Worcs B61 7DuEngland
VII. S. Kerr 21 Serpentine Rd Selly Park Birmingham B29 7HU
VIII. N. Smith 909 S. 7th Ave W., Newton, IA, 50208 USA
IX. C. Law 112 Wild Iris Lane, Chapel hill, NC 27514 USA
X. Amir heydari, 19 calle alamitos, rcho sta marg, CA 92688 USA
NOW THIS IS HOW IT WORKS. As you post this, to 200 newsgroups and people read
and think “Ah I will oin up to this”. And let’s say only 5 people reply (avery
low number) that is $5. Now those 5 people send out 200 news groups and follow
the instructions as your name is at number 9. Lets say 5 people answer to each
200 newsgroups, then that’s another $25. Now if they follow the instructions
carefully, your name is at number 7 and you get $125. Now if those 125 people
send out 200 newsgroups you receive $625. And again your name moves up to 6 and
you receive $3,125. They now post the newsgroups and your name moves up to 5 and
those people reply you get 15,625. Now this is where the newsgroups I was
speak8ng about earlier finish, that’s it. $6, places. But now your name is now
sent out at your name at number4 and you receive $78,125. Your name has still
got 3 more places to go yet. So now with your name at no. 3 you get a whopping
$390,625. Once again your name moves to 2 and you are going to receive
$1,953,125 (1 million nine hundred and fifty thousand, one hundred and twenty
five dollars!!!!!). BUT WAIT, you’ve got one more place to move up. Number 1.
You will receive $9,765,625. Can you grasp how much that is nine million. Oh no,
your name had drooped of the list, oh well just start again and get some more
millions rolling in. there numbers have only been calculated, if only 5 people
respond to each 200 newsgroups, just think of how many people are on the net!!
It has been estimated that 20,000 to 50,000 join the internet every month!
--- DIRECTIONS FOR HOW TO POST TO NEWSGROUPS---
1) COPY THIS ARTICLE FROM Step 1 to down, and save it on your hard drive.
2) Change the info for the #10 spot to your name/address
3) Open Netscape or Internet Explorer, and go to a search engine, and search for
Newsgroups, message Boards, Chatting, Discussions or search for ‘EASY MONEY’.
4) Visit these message boards, open up the file that you saved, and paste this
message onto the newsgroups, message boards, etc… once you get the hang of it,
it only takes about 30 seconds per groups.
REMEMBER, THE MORE NEWSGROUPS YOU POST IN (Massage-boards etc.) THE MORE MONEY
YOU WILL MAKE!!BUT YOU HAVE TO POST A MINIMUM OF 200 that’s it . you will begin
receiving money from around the world within days!. You may eventually want to
rent a P.O. Box due to the large amount of mail you will receive. If you wish to
stay anonymous, you can invent a name to use, as long as the postman will
deliver it.
: PLEASE REMEMBER that this program remains successful because of the HONESTY
and INTEGRITY of the participants and by their CAREFULLY ADHERING to the
directions. Look at it this way . if you are of integrity, the program will
continue and the money that so many others have received will come your way.
(NOTE: you may want to retain every name and address sent to you, either on a
computer or hard copy and keep the notes people send you. This VERIFIES that you
are truly providing a service.)
: THE REASON why you must send off the money in the first place is that
you create a LEGAL SERVICE and this is not fraud. If you did NOT SEND THE MONEY
then you would be committing FRAUD and can be put away, because you are using
other people’s name for your own god.