On 01 Mar 2014, at 00:16, Richard Ruquist wrote:
bruno: God created the Integers. All the rest came when God added
"Add and Multiply".
richard: I trhink the multiply mat be redundant. Is that a useful
property?
Classical logic + addition is not Turing universal
Classical logic + multiplication is not Turing universal
Classical logic + multiplication + addition IS Turing universal.
Multiplication is not redundant, but exponentiation or any other
computable functions would be, once we have add and multiply.
This is not so easy to prove, but is "well known" (by theoretical
computer scientists)
Bruno
On Fri, Feb 28, 2014 at 2:36 PM, Bruno Marchal <[email protected]>
wrote:
On 28 Feb 2014, at 08:20, Chris de Morsella wrote:
Personally the notion that all that exists is comp & information -
encoded on what though? - Is not especially troubling for me. I
understand how some cling to a fundamental material realism; after
all it does seem so very real. However when you get right down to
it all we have is measured values of things and meters by which we
measure other things; we live encapsulated in the experience of our
own being and the sensorial stream of life and in the end all that
we can say for sure about anything is the value it has when we
measure it.
I am getting into the interesting part of Tegmark's book - I read a
bit each day when I break for lunch - so this is partly influencing
this train of thought. By the way enjoyed his description of
quantum computing and how in a sense q-bits are leveraging the
Level III multiverse to compute every possible outcome while in
quantum superposition; a way of thinking about it that I had never
read before.
Naturally I have been reading some of the discussions here, and the
idea of comp is something I also find intuitively possible. The
soul is an emergent phenomena given enough depth of complexity and
breadth of parallelism and vastness of scale of the information
system in which it is self-emergent.
Several questions have been re-occurring for me. One of these is:
Every information system, at least that I have ever been aware of,
requires a substrate medium upon which to encode itself;
information seems describable in this sense as the meta-encoding
existing on some substrate system. I would like to avoid the
infinite regression of stopping at the point of describing systems
as existing upon other and requiring other substrate systems that
themselves require substrates themselves described as information
again requiring some substrate... repeat eternally.
It is also true that exquisitely complex information can be encoded
in a very simple substrate system given enough replication of
elements... a simple binary state machine could suffice, given enough
bits.
But what are the bits encoded on?
At some point reductionism can no longer reduce.... And then we are
back to where we first started.... How did that arise or come to be?
If for example we say that math is reducible to logic or set theory
then what of sets and the various set operations? What of
enumerations? These simplest of simple things. Can you reduce the
{} null set?
What does it arise from?
Perhaps to try to find some fundamental something upon which
everything else is tapestried over is unanswerable; it is something
that keeps coming back to itch my ears.
Am interested in hearing what some of you may have to say about
this universe of the most simple things: numbers, sets; and the
very simple base operators -- {+-*/=!^()} etc. that operate on
these enumerable entities and the logical operators {and, or, xor}
What is a number? Doesn't it only have meaning in the sense that it
is greater than the number that is less than it & less than the
one greater than it? Does the concept of a number actually even
have any meaning outside of being thought of as being a member of
the enumerable set {1,2,3,4,... n}? In other words '3' by itself
means nothing and is nothing; it only means something in terms of
the set of numbers as in: 2<3<4... <n-1<n
And what of the simple operators. When we say a + b = c we are
dealing with two separate kinds of entities, with one {a,b,c} being
quantities or values and {+,=} being the two operators that relate
the three values in this simple equation.
The enumerable set is not enough by itself. So even if one could
explain the enumerable set in some manner the manner in which the
simple operators come to be is not clear to me. How do the
addition, assignment and other basic operators arise? This extends
similarly to the basic logic operators: and, or, xor, not - as well.
Thanks
Those kind of questions are more less clarified. You cannot prove
the existence of a universal system, or machine, or language, from
anything less powerful, but you can prove the existence of all of
them, from the assumption of only one. I use elementary arithmetic,
because it is already taught in school, and people are familiar with
it.
The "TOE" extracted from comp assumes we agree on the laws of
addition and multiplication, and on classical logic. From this you
can prove the existence of the universal numbers and or all their
computations, and even interview the Löbian numbers, on what is
possible for them, in different relative sense.
So, math comes from arithmetic, and arithmetic can explain why it is
impossible to explain arithmetic from less than arithmetic, making
arithmetic (or Turing equivalent) a good start.
God created the Integers. All the rest came when God added "Add and
Multiply".
Basically.
Bruno
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