*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
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
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
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
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
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
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?
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 equal,
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
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
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,
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
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,
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 11:52 AM,
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 entities
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
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. Pei
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
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
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
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
--- 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.
--
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
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)
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
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
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
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]
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 should
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 PROTECTED]
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
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 of 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...
That's
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 All
Pei
===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...
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
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,
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
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).
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
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
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