On 15 Jan 2014, at 15:05, Edgar L. Owen wrote:

If the fundamental axioms of arithmetic are the fundamental axioms of your UDA then where do those come from?

Russell and Whitehead suggested that they could be derived from logic alone, but that has been refuted, and today, we know that we cannot derive arithmetic from anything less than the first order logic specification of a universal Turing system.

For the TOE I could use another Turing-complete theory, like the combinators, with the two axioms

Kxy = x
Sxyz = xz(yz) (the precise meaning of this being not relevant right now)

From those two axioms, I can derive the arithmetic axioms.
From the arithmetic axioms, I can derive the two combinator axioms.

But from anything less than a Turing-complete theory, I can't derive the existence of arithmetic or any other Turing-complete theory.

I guess you need to study some math to see what happens. I try to explain the matter with enough detail from times to times, so that you might grasp (independently of believing it or not).


Unless you can answer that question you have a gap in your theory that mine doesn't have.....

?

Your theory is fuzzy, and seem to assume quantum mechanics, which assumes arithmetic. The very notion of computations assumes arithmetic too. Your theory assumes also a physical or psychological reality (your present time).

Can you justify the numbers and prove addition and multiplication from less than another choice of universal machine/number/system/language?

Our conscious understanding of {0, 1, 2, 3, ...} is a mystery, but comp illustrates that it might be:
1) necessarily mysterious,
2) and, with the laws of addition and multiplication, the only mystery.

Comp explains, I think, why our understanding of {0, 1, 2, 3, ...} can seem and even *be* obvious, *from* the first person point of view. But that point of view is not communicable, and the (Löbian) machines already know that (in precise rather standard sense).

In mathematics, that "obviousness" is more or less captured by either a much less "obvious" theory (like set theory, or category theory, ...), or by second order logic, which gives non effective theories (proofs are no more checkable).

Once you assume comp, it is just a matter of work to understand that the arithmetical reality is full of life and mysteries, when seen from inside.

I don't know if comp is true, but the point is that by its relation with computer science and mathematical logic, comp, well classical comp if you prefer, is made testable/refutable.

Bruno





Edgar



On Wednesday, January 15, 2014 8:50:44 AM UTC-5, Bruno Marchal wrote:

On 15 Jan 2014, at 13:41, Edgar L. Owen wrote:

Bruno,

Of course it is circular - but it is meaningful.

Without further ado, circular statements are *to much* meaningful.



The fundamental axiom MUST be circular,

Is that anew meta-axiom? Again, that is not obvious at all.




but it must be so in a meaningful way. I already noted that when I said it was 'self-necessitating'.

"self-necessitating" contains two hot complex notions: "self" and "necessitate".

We want to explain the complex from the simple, not the other way round.




So far as I know my Existence Axiom is the most meaningful fundamental axiom.

If that was true, you would not need to say so.




What is YOUR fundamental axiom? 'Arithmetic exists because arithmetic exists' perhaps? Sounds like a similarly circular axiom to me....

You should also never put statements in the mouth of others, especially when they are completely ridiculous, like if I would have said that "arithmetic exists because arithmetic exists".

I am working at two levels: an intuitive meta-level, where the assumption is a precise version of Milinda-Descartes old mechanist assumption. To put it shortly it says that not only I can survive with an artificial heart, kidney, skin, but that the brain is not excluded from that list. It means that my body functions, at some level, like some sort of machine. As far as I understand you, it is implied by your "computational stance".
So my assumption, at that level, is a tiny part of your assumption.

By reasoning at that meta-level (UDA), we get as "meta-theorem" that the TOE does not need to assume more than the usual elementary axioms of arithmetic. One precise theory is classical logic + the axioms, where you can read s(x) by "the successor of the number x".

0 ≠ s(x)
s(x) = s(y) -> x = y
x+0 = x
x+s(y) = s(x+y)
x*0=0
x*s(y)=(x*y)+x

Then, in that theory, all the terms I need are defined. It is in that theory that we define the observers and derive physics (and more). That's AUDA, or "the machine's interview" (in the sane2004 paper). Comp makes the whole thing both mathematical and experimentally testable.

Bruno




Edgar




On Wednesday, January 15, 2014 3:10:30 AM UTC-5, Bruno Marchal wrote:

On 14 Jan 2014, at 19:05, Edgar L. Owen wrote:

> Bruno,
>
> 'Non-existence cannot exist', obviously refers to the existence of
> reality itself,

Then it is circular.




> not to milk in your refrigerator! Existence must exist means
> something must exist, whether it's milk or whatever. Individual
> things have individual localized existences, but existence (reality) > itself is everywhere because it defines the logical space of reality
> by its existence.

That is not intelligible.



>
> The Axiom of Existence means there was never a nothingness out of
> which somethingness (the universe) was created.

Assuming that there is a "universe". But then you do not explain why
there is something. You just assume this. You axiom is "something
exists".



>
> Milk is created by female mammals in case you had some doubt?
> :-)
>
> Next question: Reality IS a computational MACHINE in the general
> sense of machine.

That is digital physics, which is refuted.




> Thus of course consistency applies to it.

That does not follow. Machines can be inconsistent.

Bruno

http://iridia.ulb.ac.be/~marchal/




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