On Sat, Jun 4, 2011 at 3:12 PM, Stephen Paul King <[email protected]
>
wrote:
Hi Jason,
Very interesting reasoning!
Thank you.
From: Jason Resch
Sent: Saturday, June 04, 2011 1:51 PM
To: [email protected]
Subject: Re: Mathematical closure of consciousness and computation
On Sat, Jun 4, 2011 at 12:06 PM, Rex Allen <[email protected]>
wrote:
On Sat, Jun 4, 2011 at 12:21 PM, Jason Resch <[email protected]>
wrote:
One thing I thought of recently which is a good way of showing how
computation occurs due to the objective truth or falsehood of
mathematical
propositions is as follows:
Most would agree that a statement such as "8 is composite" has an
eternal
objective truth.
Assuming certain of axioms and rules of inference, sure.
Godel showed no single axiomatic system captures all mathematical
truth,
any fixed set of axioms can at best approximate mathematical
truth. If
mathematical truth cannot be fully captured by a set of axioms, it
must
exist outside sets of axioms altogether.
[SPK]
I see two possibilities. 1) Mathematical truth might only
exist in our
minds. But an infinity of such minds is possible...2) Might it be
possible
that our mathematical ideas are still too primitive and simplistic
to define
the kind of set that is necessary?
**
1) More is answered by:
A: "Math -> Matter -> Minds" (or as Bruno suggests "Math -> Minds ->
Matter") than by
B: "Matter -> Minds -> Math", or
C: "Minds -> (Matter, Math)".
Compared to "B", "A" explains the unreasonable effectiveness of
math in the
natural sciences, the apparent fine tuning of the universe (with the
Anthropic Principle), and with computationalism explains QM.
"C" has the least explanatory power, and we must wonder why the
experience
contained within our minds seems to follow a compressible set of
physical
laws, and why mathematical objects seem to posses objective
properties but
by definition lack reality.
Those who say other universes do not exist are only adding baseless
entities
to their theory, to define away that which is not observed. It was
what led
to theories such as the Copenhagen Interpretation, which postulated
collapse
as a random selection of one possible outcome to be made real and
cause the
rest to disappear. Similarly, there are string theorists which
hope to find
some mathematical reason why other possible solutions to string
theory are
inconsistent, and the one corresponding to the the standard model
is the
only one that exists. Why? They think this is necessary to make
their
theory agree with observation, but when the very thing is
unobservable
according to the theory it is completely unnecessary.
The situation is reminiscent of DeWitt and Everett:
In his letter, DeWitt had claimed that he could not feel himself
split,
so, as mathematically attractive as Everett's theory was, he said,
it could
not be true. Everett replied in his letter to DeWitt that,
hundreds of years
ago, after Copernicus had made his radical assertion that the
Earth revolved
around the sun instead of the reverse, his critics had complained
that they
could not feel the Earth move, so how could it be true? Recalling
Everett's
response to him decades later, in which he pointed out how
Newtonian physics
revealed why we don't feel the Earth move, DeWitt wrote, "All I
could say
was touché!"
2) I don't know. Godel proved that any sufficiently complex
axiomatic
system can prove that there are things that are true which it
cannot prove.
Only more powerful systems can prove the things which are not
provable in
those other axiomatic systems, but this creates an infinite
hierarchy.
Whether or not there is some ultimate top to it I don't know.
But isn't that true of nearly anything? How many axiomatic
systems are
there?
Likewise the statement: the Nth fibbinacci number is X.
Has an objective truth for any integer N no matter how large.
Let's
say
N=10 and X = 55. The truth of this depends on the recursive
definition
of
the fibbinacci sequence, where future states depend on prior
states,
and is
therefore a kind if computation. Since N may be infinitely
large, then
in a
sense this mathematical computation proceeds forever. Likewise
one
might
say that chaitin's constant = Y has some objective mathematical
truth.
For
chaintons constant to have an objective value, the execution of
all
programs
must occur.
Simple recursive relations can lead to exraordinary complexity,
consider the
universe of the Mandelbrot set implied by the simple relation Z(n
+1)=
Z(n)^2
+ C. Other recursive formulae may result in the evolution of
structures
such as our universe or the computation of your mind.
The fractal is just an example of a simple formula leading to very
complex
output. The same is true for the UDA:
for i = 0 to inf:
for each j in set of programs:
execute single instruction of program j
add i to set of programs
That simple formula executes all programs.
Is extraordinary complexity required for the manifestation of
"mind"?
If so, why?
I don't know what lower bound of information or complexity is
required for
minds.
[SPK]
Why are we sure that a “lower bound of information” or
“complexity” is
required? Seriously, there seems to be a bit of speculation from
too few
facts when it comes to consciousness!
**
I should clarify, I don't know what the lower bound is or if there
is one.
That said I do believe information and computation are importantly
related
to consciousness.
Is it that these recursive relations cause our experience, or are
just
a way of thinking about our experience?
Is it:
Recursive relations cause thought.
OR:
Recursion is just a label that we apply to some of our
implicational
beliefs.
The latter seems more plausible to me.
Through recursion one can implement any form of computation.
Recursion is
common and easy to show in different mathematical formulas, while
showing a
Turing machine is more difficult. Many programs which can be
easily defined
through recursion can also be implemented without recursion, so I
was not
implying recursion is necessary for minds. For example,
implementing the
Fibonacci formula iteratively would look like:
Fib(N)
X = 1
Y = 1
for int i = 2 to N:
i = X + Y
X = Y
Y = i
print Y
This program iteratively computes successive Fibonacci numbers,
and will
output the Nth Fibbonaci number.
Jason
There was a bug in that program, replace the last two "i"s with "j",
otherwise it breaks out of the loop too early. :-)
--
[SPK]
The existence of such Numbers could be a telltale sign that
numbers
require an eternal computation to define them.
I'm not sure, I can define Pi without an infinite description or
computation. Pi = circumference of a unit circle / 2
I would agree that determining Pi from that definition probably
does require
an eternal/infinite amount of computation though.
Jason
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