That's a lot stronger and more interesting that the theories that I was
referring to. Also a lot more complex.
**This is getting way off topic, so the rest should probably be ignored.**
One of the theories that I was referring to contained only 0 and a rule
that given a number allowed you to construct the successor to that
number. It clearly couldn't prove it's own consistency, but given
enough time and effort it could clearly [from external observation]
generate all finite integers. If that were your theory, then just about
all you could say about a number was what it's successor was and
possibly what it's predecessors were...though you couldn't do that
latter within the theory itself.
My point was that numbers defined under that theory don't HAVE any other
characteristics. They may be equivalent in some sense to numbers
defined under Number Theory that were generated in an equivalent way,
but they don't have the extra characteristics that the Number Theory
numbers have. And this is because numbers aren't characteristics of the
universe, but rather particular abstractions from various
characteristics of the universe. And if I adopt a constructivist
stance, then only a finite number of numbers exist, in fact precisely
those numbers that have been generated. I don't happen to think that
this is the best approach, because I think that it arises out of an
attempt to reify numbers, but it has it's value in some cases.
In a way this is like my argument against existentialism. I assert that
a true existentialist couldn't walk across the room, because he couldn't
be sure that the floor would exist when he took a step. Nobody is that
complete an existentialist, and there is merit in the existentialist
stance...but it's not the one that is usually described.
The way this becomes important is that all simple representations of
numbers within a computer are inherently finite. (I'm saying simple to
avoid things like lazily evaluated functions which given a transfinite
amount of time, RAM, and energy could generate transfinite numbers...and
so by existing in an unevaluated state can be said to represent said
transfinite numbers.) I.e., for computers constructivist theories
describe what's actually possible, though typically they don't impose
strict enough restrictions to serve that function, they COULD do so.
But it's not clear to me that this is an appropriate approach, even
though it models simply onto the physicality of the situation. In a way
it reminds me of the arguments that used to be raised between the
assembler programmers and the compiler language programmers. Compiler
languages use a more abstract representation, but they require more
system resources, and you can do anything in assembler that's actually
possible. But there are good reasons why the more abstract and general
choice is more commonly made. But for some purposes there's really no
choice but to actually understand things at an assembler level.
Similarly with the more abstract math and the constructivist approach.
The constructivists are correct about what we can actually know and
count and be certain of. But they aren't right when they claim that
this is generally the appropriate stance to take towards math. Doing
that is like deriving planetary motions from quantum equations.
Sorry for going so off-track.
Abram Demski wrote:
Charles,
defining "formal system" is just as difficult as defining "number"--
in fact, those problems are basically equivalent. Godel made use of
that to make his proof work (or at least that's one way of looking at
it). So, if you claim that the concept "number" is dependent on what
formal system it is defined in, shouldn't you also say the same thing
of "formal system"?
So, for example, we might agree to interpret "number" as
"Peano-arithmatic number" for the purpose of some discussion. But, we
might still disagree on how to interpret Peano Arithmetic. I might
say, "statement X isn't derivable from PA, by transfinite induction",
and you might reply, "Hey, no, you're not allowed to use transfinite
induction." Then we would need to settle which logic to interpret PA
in: maybe you convince me that we've got to stick to robinson
arithmetic. But then, I use some line of reasoning about the
properties of Robinson arithmetic, and we've got to settle on an even
higher system to resolve the argument...
The point is, we've got to make some actual choice of default at some
point. By arguing with anything I want to at every level, I'm using
the classical-type default, which is essentially to use the strongest
system available. Perhaps you would choose one of these logics that
Godel's theorem fails for as your default. If we can figure out what
default is normatively ideal, then in my opinion we've made important
headway.
I think I found the logics you're referring to? Looks *very* interesting.
http://en.wikipedia.org/wiki/Self-verifying_theories
--Abram
On Fri, Oct 31, 2008 at 2:26 AM, Charles Hixson
<[EMAIL PROTECTED]> wrote:
It all depends on what definition of number you are using. If it's
constructive, then it must be a finite set of numbers. If it's based on
full Number Theory, then it's either incomplete or inconsistent. If it's
based on any of several subsets of Number Theory that don't allow
incompleteness to be proven (or even described) then the numbers are
precisely this which is included in that subset of the theory.
Number Theory is the one with the largest (i.e., and infinite number) of
unprovable theories about numbers of the variations that I have been
considering. My point in the just prior post is that numbers are precisely
that item which the theory you are using to describe them says they are,
since they are artifacts created for computational convenience, as opposed
to direct sensory experiences of the universe.
As such, it doesn't make sense to say that a subset of number theory leaves
more facts about numbers undefined. In the subsets those aren't facts about
numbers.
Abram Demski wrote:
Charles,
OK, but if you argue in that manner, then your original point is a
little strange, doesn't it? Why worry about Godelian incompleteness if
you think incompleteness is just fine?
"Therefore, I would assert that it isn't that it leaves "*even more*
about numbers left undefined", but that those characteristics aren't
in such a case properties of numbers. Merely of the simplifications
an abstractions made to ease computation."
In this language, what I'm saying is that it is important to examine
the "simplifications and abstractions", and discover how they work, so
that we can "ease computation" in our implementations.
--Abram
On Thu, Oct 30, 2008 at 7:58 PM, Charles Hixson
<[EMAIL PROTECTED]> wrote:
If you were talking about something actual, then you would have a valid
point. Numbers, though, only exist in so far as they exist in the theory
that you are using to define them. E.g., if I were to claim that no
number
larger than the power-set of energy states within the universe were
valid,
it would not be disprovable. That would immediately mean that only
finite
numbers were valid.
P.S.: Just because you have a rule that could generate a particular
number
given a larger than possible number of steps doesn't mean that it is a
valid
number, as you can't actually ever generate it. I suspect that infinity
is
primarily a computational convenience. But one shouldn't mistake the
fact
that it's very convenient for meaning that it's true. Or, given Occam's
Razor, should one? But Occam's Razor only detects provisional truths,
not
actual ones.
If you're going to be constructive, then you must restrict yourself to
finitely many steps, each composed of finitely complex reasoning. And
this
means that you must give up both infinite numbers and irrational numbers.
To do otherwise means assuming that you can make infinitely precise
measurements (which would, at any rate, allow irrational numbers back
in).
Therefore, I would assert that it isn't that it leaves "*even more* about
numbers left undefined", but that those characteristics aren't in such a
case properties of numbers. Merely of the simplifications an
abstractions
made to ease computation.
-------------------------------------------
agi
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