On Sat, Jun 4, 2011 at 3:12 PM, Stephen Paul King <stephe...@charter.net
Very interesting reasoning!
From: Jason Resch
Sent: Saturday, June 04, 2011 1:51 PM
Subject: Re: Mathematical closure of consciousness and computation
On Sat, Jun 4, 2011 at 12:06 PM, Rex Allen <rexallen31...@gmail.com>
On Sat, Jun 4, 2011 at 12:21 PM, Jason Resch <jasonre...@gmail.com>
One thing I thought of recently which is a good way of showing how
computation occurs due to the objective truth or falsehood of
propositions is as follows:
Most would agree that a statement such as "8 is composite" has an
Assuming certain of axioms and rules of inference, sure.
Godel showed no single axiomatic system captures all mathematical
any fixed set of axioms can at best approximate mathematical
mathematical truth cannot be fully captured by a set of axioms, it
exist outside sets of axioms altogether.
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
that our mathematical ideas are still too primitive and simplistic
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
contained within our minds seems to follow a compressible set of
laws, and why mathematical objects seem to posses objective
by definition lack reality.
Those who say other universes do not exist are only adding baseless
to their theory, to define away that which is not observed. It was
to theories such as the Copenhagen Interpretation, which postulated
as a random selection of one possible outcome to be made real and
rest to disappear. Similarly, there are string theorists which
hope to find
some mathematical reason why other possible solutions to string
inconsistent, and the one corresponding to the the standard model
only one that exists. Why? They think this is necessary to make
theory agree with observation, but when the very thing is
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
so, as mathematically attractive as Everett's theory was, he said,
not be true. Everett replied in his letter to DeWitt that,
hundreds of years
ago, after Copernicus had made his radical assertion that the
around the sun instead of the reverse, his critics had complained
could not feel the Earth move, so how could it be true? Recalling
response to him decades later, in which he pointed out how
revealed why we don't feel the Earth move, DeWitt wrote, "All I
2) I don't know. Godel proved that any sufficiently complex
system can prove that there are things that are true which it
Only more powerful systems can prove the things which are not
those other axiomatic systems, but this creates an infinite
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
Likewise the statement: the Nth fibbinacci number is X.
Has an objective truth for any integer N no matter how large.
N=10 and X = 55. The truth of this depends on the recursive
the fibbinacci sequence, where future states depend on prior
therefore a kind if computation. Since N may be infinitely
sense this mathematical computation proceeds forever. Likewise
say that chaitin's constant = Y has some objective mathematical
chaintons constant to have an objective value, the execution of
Simple recursive relations can lead to exraordinary complexity,
universe of the Mandelbrot set implied by the simple relation Z(n
+ C. Other recursive formulae may result in the evolution of
such as our universe or the computation of your mind.
The fractal is just an example of a simple formula leading to very
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
If so, why?
I don't know what lower bound of information or complexity is
Why are we sure that a “lower bound of information” or
required? Seriously, there seems to be a bit of speculation from
facts when it comes to consciousness!
I should clarify, I don't know what the lower bound is or if there
That said I do believe information and computation are importantly
Is it that these recursive relations cause our experience, or are
a way of thinking about our experience?
Recursive relations cause thought.
Recursion is just a label that we apply to some of our
The latter seems more plausible to me.
Through recursion one can implement any form of computation.
common and easy to show in different mathematical formulas, while
Turing machine is more difficult. Many programs which can be
through recursion can also be implemented without recursion, so I
implying recursion is necessary for minds. For example,
Fibonacci formula iteratively would look like:
X = 1
Y = 1
for int i = 2 to N:
i = X + Y
X = Y
Y = i
This program iteratively computes successive Fibonacci numbers,
output the Nth Fibbonaci number.
There was a bug in that program, replace the last two "i"s with "j",
otherwise it breaks out of the loop too early. :-)
The existence of such Numbers could be a telltale sign that
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
an eternal/infinite amount of computation though.
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