On 24 Dec 2012, at 15:35, Roger Clough wrote:
Hi Bruno Marchal
It helps me if I can understand arithmetic as true
constructions of a fictional leggo set.
Why fictional? Immaterial OK, but ffictional?
From what you say, the natural numbers and + and * (nn+*).
What is (nn+*)?
are not a priori members of Platonia (if indeed that makes
sense anyway).
They are. Either as basic citizens, or as existing object if we start
with a universal system different from arithmetic, but in all case all
truth about all digital machines are a priori members in all Platonia
rich enough for comp.
They can simply be invoked and used
as needed, as long as they don't produce contradictions.
Alas, after Gödel that is not enough. In arithmetic you can depart a
lot from truth, and still be consistent.
That being the case, don't you need to add =, - , and
/ to the Leggo set ? Then we have (nn+-*/=).
"= " is there. But "-" and all other computable function and programs
can be defined from the axioms I gave, + a very small amount of
logical axioms. If you want I can give explicit presentation(s) some
day.
I wonder if somebody could derive string theory from this set.
Trivially, in a weak sense of "string theory".
Non trivially, in the stronger sense as deriving string theory, and
only string theory from comp. That should be the case if string theory
is the ultimate correct theory of the physical.
Then we might say that the universe is an arithmetic construction.
Probably an absurd idea.
Actually yes. As comp implies that physics, although derivable in
arithmetic + comp, is not an arithmetical construction. We already
know that arithmetical truth is not an arithmetical notion, so this
should not be so astonishing.
Hi Bruno Marchal
No doubt you are right, except that the brain is physical,
while, as I understand it, a UTM is mental.
But the physical is mental, or immaterial, with comp. So, no problem :)
I have to go for prepare Xmas, I have a lot of nephews and little
nephews ...
Happy Xmas to you Roger, and to everyone,
Bruno
[Roger Clough], [rclo...@verizon.net]
12/24/2012
"Forever is a long time, especially near the end." -Woody Allen
----- Receiving the following content -----
From: Bruno Marchal
Receiver: everything-list
Time: 2012-12-23, 09:17:09
Subject: Re: Can the physical brain possibly store our memories ? No.
On 22 Dec 2012, at 17:05, Telmo Menezes wrote:
Hi Bruno,
On Thu, Dec 20, 2012 at 1:01 PM, Roger Clough wrote:
> The infinite set of natural numbers is not stored on anything,
Which causes no problem because there is not a infinite number of
anything in the observable universe, probably not even points in
space.
Perhaps, we don't know.
It causes no problem because natural numbers does not have to be
stored a priori. Only when universal machine want to use them.
Why do the natural numbers exist?
We cannot know that.
Precisely, if you assume the natural numbers, you can prove that you
cannot derived the existence of the natural number and their + and *
laws, in *any* theory which does not assume them, or does not assume
something equivalent.
That is why it is a good reason to start with them (or equivalent).
Somehow, the natural numbers, with addition and multiplication, are
necessarily "mysterious".
With the natural numbers and + and *, you can prove the existence of
all universal machines, and vice versa, if you assume any other
universal system (like the combinators K, S (K K), (K S), ...) you
can prove the existence of the natural numbers and their laws.
We have to assume at least one universal system, and I chose
arithmetic because it is the simpler one. The problem is that the
proof of its universality will be difficult, but at least it can be
found in good mathematical logic textbook, like Mendelson or Kleene,
etc.
Bruno
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