Hi Ben and others,
After some more thinking, I decide to try the virtual credit approach afterall.
Last time Ben's argument was that the virtual credit method confuses
for-profit and charity emotions in people. At that time it sounded
convincing, but after some thinking I realized that it is act
Charles,
Interesting point-- but, all of these theories would be weaker then
the standard axioms, and so there would be *even more* about numbers
left undefined in them.
--Abram
On Tue, Oct 28, 2008 at 10:46 PM, Charles Hixson
<[EMAIL PROTECTED]> wrote:
> Excuse me, but I thought there were subs
Excuse me, but I thought there were subsets of Number theory which were
strong enough to contain all the integers, and perhaps all the rational,
but which weren't strong enough to prove Gödel's incompleteness theorem
in. I seem to remember, though, that you can't get more than a finite
number
If not verify, what about falsify? To me Occam's Razor has always been
seen as a tool for selecting the first argument to attempt to falsify.
If you can't, or haven't, falsified it, then it's usually the best
assumption to go on (presuming that the costs of failing are evenly
distributed).
Eric,
I highly respect your work, though we clearly have different opinions
on what intelligence is, as well as on how to achieve it. For example,
though learning and generalization play central roles in my theory
about intelligence, I don't think PAC learning (or the other learning
algorithms pro
Ed,
Since NARS doesn't follow the Bayesian approach, there is no initial
priors to be assumed. If we use a more general term, such as "initial
knowledge" or "innate beliefs", then yes, you can add them into the
system, will will improve the system's performance. However, they are
optional. In NARS
Eric:The core problem of GI is generalization: you want to be able to
figure out new problems as they come along that you haven't seen
before. In order to do that, you basically must implicitly or
explicitly employ some version
of Occam's Razor
It all depends on the subject matter of the generali
===Below Ben wrote===
I suspect that the fact that biological organisms grow
via the same sorts of processes as the biological environment in which
the live, causes the organisms' minds to be built with **a lot** of implicit
bias that is useful for surviving in the environment...
===M
Pei> Triggered by several recent discussions, I'd like to make the
Pei> following position statement, though won't commit myself to long
Pei> debate on it. ;-)
Pei> Occam's Razor, in its original form, goes like "entities must not
Pei> be multiplied beyond necessity", and it is often stated as "A
--- On Tue, 10/28/08, Ben Goertzel <[EMAIL PROTECTED]> wrote:
> What Hutter proved is (very roughly) that given massive computational
> resources, following Occam's Razor will be -- within some possibly quite
> large constant -- the best way to achieve goals in a computable environment...
>
> Tha
Matt,
The "currently known laws of physics" is a *description* of the
universe at a certain level, which is fundamentally different from the
universe itself. Also, "All human knowledge can be reduced into
physics" is not a view point accepted by everyone.
Furthermore, "computable" is a property o
Matt,
Interesting question re the differences between mathematics - i.e.
arithmetic, algebra - and logic vs language.
I haven't really thought about this, but I wouldn't call maths a language.
Maths consists of symbolic systems of quantification and schematic patterns
(geometry) which can on
"well-defined" is not well-defined in my view...
However, it does seem clear that "the integers" (for instance) is not an
entity with *scientific* meaning, if you accept my formalization of science
in the blog entry I recently posted...
On Tue, Oct 28, 2008 at 3:34 PM, Mark Waser <[EMAIL PROTEC
>> Any formal system that contains some basic arithmetic apparatus equivalent
>> to http://en.wikipedia.org/wiki/Peano_axioms is doomed to be incomplete with
>> respect to statements about numbers... that is what Godel originally
>> showed...
Oh. Ick! My bad phrasing. WITH RESPECT TO NUMBERS
Any formal system that contains some basic arithmetic apparatus equivalent
to http://en.wikipedia.org/wiki/Peano_axioms is doomed to be incomplete with
respect to statements about numbers... that is what Godel originally
showed...
On Tue, Oct 28, 2008 at 2:50 PM, Mark Waser <[EMAIL PROTECTED]> wro
Au contraire, I suspect that the fact that biological organisms grow
via the same sorts of processes as the biological environment in which
the live, causes the organisms' minds to be built with **a lot** of implicit
bias that is useful for surviving in the environment...
Some have argued that thi
That is thanks to Godel's incompleteness theorem. Any formal system
that describes numbers is doomed to be incomplete
Yes, any formal system is doomed to be incomplete. Emphatically, NO! It is
not true that "any formal system" is doomed to be incomplete WITH RESPECT TO
NUMBERS.
It is entir
What Hutter proved is (very roughly) that given massive computational
resources, following Occam's Razor will be -- within some possibly quite
large constant -- the best way to achieve goals in a computable
environment...
That's not exactly "proving Occam's Razor", though it is a proof related to
It appears to me that the assumptions about initial priors used by a self
learning AGI or an evolutionary line of AGI's could be quite minimal.
My understanding is that once a probability distribution starts receiving
random samples from its distribution the effect of the original prior
becomes ra
Hi guys,
I took a couple hours on a red-eye flight last night to write up in more
detail my
argument as to why uncomputable entities are useless for science:
http://multiverseaccordingtoben.blogspot.com/2008/10/are-uncomputable-entities-useless-for.html
Of course, I had to assume a specific form
All of math can be done without any words ... it just gets annoying to read
for instance, all math can be formalized in this sort of manner
http://www.cs.miami.edu/~tptp/MizarTPTP/TPTPProofs/arithm/arithm__t1_arithm
and the words in there like
v1_ordinal1(B)
could be replaced with
v1_1234(B)
Hutter proved Occam's Razor (AIXI) for the case of any environment with a
computable probability distribution. It applies to us because the observable
universe is Turing computable according to currently known laws of physics.
Specifically, the observable universe has a finite description length
Ben,
What are the mathematical or logical signs for "normal number"/ "rational
number"? My assumption would be that neither logic nor maths can be done
without some language attached - such as the term "rational number" - but I'm
asking from extensive ignorance.
Ben:yes
MT:MW:"Pi is a normal
--- On Tue, 10/28/08, Mike Tintner <[EMAIL PROTECTED]> wrote:
> MW:"Pi is a normal number" is decidable by arithmetic
> because each of the terms has meaning in arithmetic
>
> Can it be expressed in purely mathematical terms/signs
> without using language?
No, because mathematics is a language.
Mark,
That is thanks to Godel's incompleteness theorem. Any formal system
that describes numbers is doomed to be incomplete, meaning there will
be statements that can be constructed purely by reference to numbers
(no red cats!) that the system will fail to prove either true or
false.
So my questi
2008/10/28 Ben Goertzel <[EMAIL PROTECTED]>:
>
> On the other hand, I just want to point out that to get around Hume's
> complaint you do need to make *some* kind of assumption about the regularity
> of the world. What kind of assumption of this nature underlies your work on
> NARS (if any)?
Not
Numbers can be fully defined in the classical sense, but not in the
constructivist sense. So, when you say "fully defined question", do
you mean a question for which all answers are stipulated by logical
necessity (classical), or logical deduction (constructivist)?
How (or why) are numbers not f
Mark,
Thank you, that clarifies somewhat.
But, *my* answer to *your* question would seem to depend on what you
mean when you say "fully defined". Under the classical interpretation,
yes: the question is fully defined, so it is a "pi question". Under
the constructivist interpretation, no: the ques
Abram,
I agree with your basic idea in the following, though I usually put it
in different form.
Pei
On Tue, Oct 28, 2008 at 2:52 PM, Abram Demski <[EMAIL PROTECTED]> wrote:
> Ben,
>
> You assert that Pei is forced to make an assumption about the
> regulatiry of the world to justify adaptation.
In that case, shouldn't
you agree with the classical perspective on Godelian incompleteness,
since Godel's incompleteness theorem is about mathematical systems?
It depends. Are you asking me a fully defined question within the current
axioms of what you call mathematical systems (i.e. a pi que
>> The question that is puzzling, though, is: how can it be that these
>> uncomputable, inexpressible entities are so bloody useful ;-) ... for
>> instance in differential calculus ...
Differential calculus doesn't use those individual entities . . . .
>> Also, to say that uncomputable entiti
On Tue, Oct 28, 2008 at 3:01 PM, Ben Goertzel <[EMAIL PROTECTED]> wrote:
>
> I believe I could prove that *mathematically*, in order for a NARS system to
> consistently, successfully achieve goals in an environment, that environment
> would need to have some Occam-prior-like property.
Maybe you ca
We can say the same thing for the human mind, right?
Pei
On Tue, Oct 28, 2008 at 2:54 PM, Ben Goertzel <[EMAIL PROTECTED]> wrote:
>
> Sure ... but my point is that unless the environment satisfies a certain
> Occam-prior-like property, NARS will be useless...
>
> ben
>
> On Tue, Oct 28, 2008 at 1
Most certainly ... and the human mind seems to make a lot of other, more
specialized assumptions about the environment also ... so that unless the
environment satisfies a bunch of these other more specialized assumptions,
its adaptation will be very slow and resource-inefficient...
ben g
On Tue,
I believe I could prove that *mathematically*, in order for a NARS system to
consistently, successfully achieve goals in an environment, that environment
would need to have some Occam-prior-like property.
However, even if so, that doesn't mean such is the best way to think about
NARS ... that's a
Ben,
It seems that you agree the issue I pointed out really exists, but
just take it as a necessary evil. Furthermore, you think I also
assumed the same thing, though I failed to see it. I won't argue
against the "necessary evil" part --- as far as you agree that those
"postulates" (such as "the u
Sure ... but my point is that unless the environment satisfies a certain
Occam-prior-like property, NARS will be useless...
ben
On Tue, Oct 28, 2008 at 11:52 AM, Abram Demski <[EMAIL PROTECTED]>wrote:
> Ben,
>
> You assert that Pei is forced to make an assumption about the
> regulatiry of the wo
Ben,
You assert that Pei is forced to make an assumption about the
regulatiry of the world to justify adaptation. Pei could also take a
different argument. He could try to show that *if* a strategy exists
that can be implemented given the finite resources, NARS will
eventually find it. Thus, adapt
On Tue, Oct 28, 2008 at 10:00 AM, Pei Wang <[EMAIL PROTECTED]> wrote:
> Ben,
>
> Thanks. So the other people now see that I'm not attacking a straw man.
>
> My solution to Hume's problem, as embedded in the experience-grounded
> semantics, is to assume no predictability, but to justify induction a
Ben,
Thanks. So the other people now see that I'm not attacking a straw man.
My solution to Hume's problem, as embedded in the experience-grounded
semantics, is to assume no predictability, but to justify induction as
adaptation. However, it is a separate topic which I've explained in my
other pu
Hi Pei,
This is an interesting perspective; I just want to clarify for others on the
list that it is a particular and controversial perspective, and contradicts
the perspectives of many other well-informed research professionals and deep
thinkers on relevant topics.
Many serious thinkers in the a
Mark,
Yes, I do keep dropping the context. This is because I am concerned
only with mathematical knowledge at the moment. I should have been
more specific.
So, if I understand you right, you are saying that you take the
classical view when it comes to mathematics. In that case, shouldn't
you agre
Triggered by several recent discussions, I'd like to make the
following position statement, though won't commit myself to long
debate on it. ;-)
Occam's Razor, in its original form, goes like "entities must not be
multiplied beyond necessity", and it is often stated as "All other
things being equa
yes
On Tue, Oct 28, 2008 at 8:46 AM, Mike Tintner <[EMAIL PROTECTED]>wrote:
> MW:"Pi is a normal number" is decidable by arithmetic
> because each of the terms has meaning in arithmetic
>
> Can it be expressed in purely mathematical terms/signs without using
> language?
>
>
>
> --
MW:"Pi is a normal number" is decidable by arithmetic
because each of the terms has meaning in arithmetic
Can it be expressed in purely mathematical terms/signs without using
language?
---
agi
Archives: https://www.listbox.com/member/archive/303/=now
Mark,
The question that is puzzling, though, is: how can it be that these
uncomputable, inexpressible entities are so bloody useful ;-) ... for
instance in differential calculus ...
Also, to say that uncomputable entities don't exist because they can't be
finitely described, is basically just to
Hi,
We keep going around and around because you keep dropping my distinction
between two different cases . . . .
The statement that "The cat is red" is undecidable by arithmetic because
it can't even be defined in terms of the axioms of arithmetic (i.e. it has
*meaning* outside of arit
Abram,
I could agree with the statement that there are uncountably many *potential*
numbers but I'm going to argue that any number that actually exists is
eminently describable.
Take the set of all numbers that are defined far enough after the decimal point
that they never accurately describe
Mark,
You assert that the extensions are judged on how well they reflect the world.
The extension currently under discussion is one that allows us to
prove the consistency of Arithmetic. So, it seems, you count that as
something observable in the world-- no mathematician has ever proved a
contrad
*That* is what I was asking about when I asked which side you fell on.
Do you think such extensions are arbitrary, or do you think there is a
fact of the matter?
The extensions are clearly judged on whether or not they accurately reflect
the empirical world *as currently known* -- so they aren'
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