Re:was Relativity of Existence

[John Mikes]

Bruno wrote:

--------------

"Provable depends on the theory. If the theory is unsound, what it proves might well be false.

And if you trust the theory, then you know that "the theory is consistent" is true, yet the theory itself cannot prove it, so reality is larger that what you can prove in that theory.

So in any case truth is larger than the theory. Even when truth is restricted to arithmetical propositions. Notably because the statement "the theory is consistent" can be translated into an arithmetical proposition.

Bruno"

--------------

Thanks, Bruno, for the wise words.

 

Your usage of "Theory"  is flexible enough and so is Reality.

The point I take exception is the "translated" which is not "identical" only an excerpt applicable within the other language to be translated into: in this case into arithmetic. My agnosticism allows(?) logic and contemplation BEYOND arithmetic and so such translation is Occam's razor to me. As you expressed it correctly: ("when truth is restricted"). In my views we are not capable (as of yesterday) to state The Truth, the whole Truth and Nothing But The Truth (ha ha). 

Nor, what many call: Reality (of the infinite complexity as my belief system calls it). I dunno.

John M

[John Mikes]

Brent wrote:

 

1. Presumably those true things would not be 'real'.  Only provable things would be true of reality.

 

2. Does arithmetic have 'finite information content'?  Is the axiom of succession just one or is it a schema of infinitely many axioms?

 

Appreciable, even in layman's logic.

 

In '#1' -  I question "provable" since in my agnosticism an 'evidence' is partial only, leaving open lots of (so far?) unknown/able aspects to be covered. In the infinity(?) of the "world" also the contrary of an evidence may be 'true'.

 

#2 is a technically precise formulation of what I tried to express in my post to Bruno.

IFF!!! "anything"  (i.e. everything) can be expressed by numerals, the information included into arithmetic  IS infinite, however as it seems: in our (restricted) view of "the world" (Nature?) there seem to be NO numbers to begin with.

In our human 'translation' we see 1,2, or 145, or a million "OF SOMETHING" - no the (integer?) numerals. 

 

Axioms? in my vocabulary: imagined things, necessary for certain theories we cannot substantiate otherwise. 

In another logic than human, in another figment of a "physical world" different axioms would serve science.  

2+2=4? not necessarily in the (fictitious) "octimality" of the '[Zarathustran' aliens in the Cohen-Stewart books

(still product of human minds).

 

[meekerdb]

On 5/26/2012 9:35 AM, John Mikes wrote:

Brent wrote:

 

1. Presumably those true things would not be 'real'.  Only provable things would be true of reality.

Just to be clear, I didn't write 1. above.  But I did write 2. below.

 

2. Does arithmetic have 'finite information content'?  Is the axiom of succession just one or is it a schema of infinitely many axioms?

 

Appreciable, even in layman's logic.

 

In '#1' -  I question "provable" since in my agnosticism an 'evidence' is partial only, leaving open lots of (so far?) unknown/able aspects to be covered. In the infinity(?) of the "world" also the contrary of an evidence may be 'true'.

As Bruno said, "Provable is always relative to some axioms and rules of inference.  It is quite independent of "true of reality".   Which is why I'm highly suspicious of ideas like deriving all of reality from arithmetic, which we know only from axioms and inferences.

 

#2 is a technically precise formulation of what I tried to express in my post to Bruno.

IFF!!! "anything"  (i.e. everything) can be expressed by numerals, the information included into arithmetic  IS infinite,

I see no reason to suppose that.  Everything ever expressed so far has been done with a finite part of arithmetic. Assuming every integer has a successor is just a convenience for modeling things; you don't have to worry about running out of counters.  There is a book "Ad Infinitum, The Ghost in Turing's Machine" by Rotman that proposes what he calls "non-euclidean arithmetic" which does not assume the integers are infinite.  I can't really recommend the book because most of it is written in the style of French deconstructionist philosophy, but the Appendix has some interesting ideas.

however as it seems: in our (restricted) view of "the world" (Nature?) there seem to be NO numbers to begin with.

In our human 'translation' we see 1,2, or 145, or a million "OF SOMETHING" - no the (integer?) numerals. 

 

Axioms? in my vocabulary: imagined things, necessary for certain theories we cannot substantiate otherwise.

Axioms are just part of a logical, i.e. self-consistent, system. Mathematicians don't even care if they are "true of reality".  They may or may not refer to imagined things; they are just assumed true for some inferences.  I could take "I am typing on a keyboard" as an axiom, which I also happen to think is true, or I could take "I am a projection in a Hilbert space" which might be true, but is much more dubious.

In another logic than human, in another figment of a "physical world" different axioms would serve science. 

Logic is about the relations of propositions, statements in language.  Humans already have invented different logics.

2+2=4? not necessarily in the (fictitious) "octimality" of the '[Zarathustran' aliens in the Cohen-Stewart books

(still product of human minds).

2+2=11

Brent
"The world consists of 10 kinds of people.  Those who think in binary and those who don't.

[Bruno Marchal]

On 27 May 2012, at 00:06, meekerdb wrote:

> On 5/26/2012 9:35 AM, John Mikes wrote:
>>
>> Brent wrote:
>>
>> 1. Presumably those true things would not be 'real'. Only provable
>> things would be true of reality.
>
> Just to be clear, I didn't write 1. above. But I did write 2. below.

Ah OK. Sorry. I have been wrong on that.

>
>>
>> 2. Does arithmetic have 'finite information content'? Is the axiom
>> of succession just one or is it a schema of infinitely many axioms?
>>
>> Appreciable, even in layman's logic.
>>
>> In '#1' - I question "provable" since in my agnosticism an
>> 'evidence' is partial only, leaving open lots of (so far?) unknown/
>> able aspects to be covered. In the infinity(?) of the "world" also
>> the contrary of an evidence may be 'true'.
>
> As Bruno said, "Provable is always relative to some axioms and rules
> of inference. It is quite independent of "true of reality". Which
> is why I'm highly suspicious of ideas like deriving all of reality
> from arithmetic, which we know only from axioms and inferences.

We don't give axioms and inference rule when teaching arithmetic in
high school. We start from simple examples, like fingers, days of the
week, candies in a bag, etc. Children understand "anniversary" before
"successor", and the finite/infinite distinction is as old as humanity.
In fact it can be shown that the intuition of numbers, addition and
multiplication included, is *needed* to even understand what axioms
and inference can be, making arithmetic necessarily known before any
formal machinery is posited.

Bruno

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[meekerdb]

On 5/27/2012 5:02 AM, Bruno Marchal wrote:

As Bruno said, "Provable is always relative to some axioms and rules of inference.  It is quite independent of "true of reality".   Which is why I'm highly suspicious of ideas like deriving all of reality from arithmetic, which we know only from axioms and inferences.

We don't give axioms and inference rule when teaching arithmetic in high school. We start from simple examples, like fingers, days of the week, candies in a bag, etc. Children understand "anniversary" before "successor", and the finite/infinite distinction is as old as humanity.
In fact it can be shown that the intuition of numbers, addition and multiplication included, is *needed* to even understand what axioms and inference can be, making arithmetic necessarily known before any formal machinery is posited.

But only a small finite part of arithmetic.

Brent

[John Mikes]

Thanks, Brent and Bruno. You are kind to respond.

The point I wanted to approach (far approach, indeed) is that whatever we derive (mentally) about Nature comes from our human mind, be it  binary or not. And: it is not BINDING (restricting?) upon Nature, there may be more we cannot even imagine within our limited capabilities.

We think in our 'model of knowables' and it is incredible how far we got.

A figment of a physical world, an 'almost' perfect technology with a reductionist (conventional) science and I don't even mention: math.

 

I read your discussions with awe and keep my agnostic indeterminism.

 

JohnM

 

 

[Bruno Marchal]

I don't think so. Our arithmetical intuition is already not formalizable. If it was, we would be able to capture it by a finite number of principle, but then we would be persuade that such finite theory is consistent, and that intuition is not in the theory.

I suspect that our intuition is full second order arithmetic, which is not axiomatizable. In fact it is the very distinction between finite and infinite that we cannot formalize.  Like consciousness, we know very well what finite/infinite means, but we cannot defined it, without using implicitly that distinction. The natural numbers are *the* mystery, and it has to be like that: no machine will ever been able to define what they are. Assuming comp, neither will we. Arithmetical truth per se, as no corresponding complete TOE. It is inexhaustible.

Bruno

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

[Bruno Marchal]

On 27 May 2012, at 23:56, John Mikes wrote:

Thanks, Brent and Bruno. You are kind to respond.

The point I wanted to approach (far approach, indeed) is that whatever we derive (mentally) about Nature comes from our human mind, be it  binary or not.

We don't know that. We believe that.

I might be a butterfly only dreaming that he is human.

I might be an amnesic God, just playing to himself that he is a human.

Nor do we know if something like Nature exist.

We do know that we are conscious, but not much more. We believe more, and that's OK, if we grant that those are beliefs, which means that we are aware that they might be wrong.

And: it is not BINDING (restricting?) upon Nature, there may be more we cannot even imagine within our limited capabilities.

And here computationalism, the theory or hypothesis, makes it possible to say more, like the fact that Nature is necessarily, in that theory, a sort of surface emerging from the vaster volume of a sort of mind, itself emerging in a precise way from arithmetic or alike.

We think in our 'model of knowables' and it is incredible how far we got.

Except that I can argue that if COMP is true, then we have regressed since +500. We have made some progress in technology, and even, I think, in politics (at least conceptually), but we have transformed science into religion, and religion into superstition. As long as we oppose mysticism and rationalism, we can only regress. We are hiding the data since 1500 years. Modernity has existed from -500 to +500, in some limited circle. Since then we are in the obscurantist era. The most fundamental science, theology, is still abandoned to authoritarians. 

A figment of a physical world, an 'almost' perfect technology with a reductionist (conventional) science and I don't even mention: math.

 

I read your discussions with awe and keep my agnostic indeterminism.

That is the genuine scientific attitude.

Bruno

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

[meekerdb]

Our intuition is that space is euclidean, the earth is stationary and flat, and that there is only one world.  It seems to me that the infinity of arithmetic is just the intuition we should always be able to add one more.  But intuition fails us in precisely in questions like Hilbert's hotel. Why should you be so trusting of your intuition is just this particular instance.

Brent

[Evgenii Rudnyi]

On 28.05.2012 18:02 meekerdb said the following:

...

> Our intuition is that space is euclidean, the earth is stationary and
> flat, and that there is only one world. It seems to me that the
> infinity of arithmetic is just the intuition we should always be able
> to add one more. But intuition fails us in precisely in questions
> like Hilbert's hotel. Why should you be so trusting of your intuition
> is just this particular instance.
>

On a side note. Has science has proved something on existence of
Hilbert's hotel? If yes, where does it exist?

Evgenii

[Bruno Marchal]

Not really. I think there are complete theories of (N, successor). But we have an intuition of adding and multiplying and this makes that intuition inexhaustible.

Intuition is not entirely a given, it is something which develop with the familiarity and life working. It is different for all of us, so it nice that we can share some big initial segment of the arithmetical truth. Comp does not need more than the sigma_1 intuition, at the ontic level.

But intuition fails us in precisely in questions like Hilbert's hotel.

Why? Not sure, but it does not concern us, as comp builds on the intuition of the finite things.

Why should you be so trusting of your intuition is just this particular instance.

Do you doubt elementary arithmetic?

Bruno

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

[Bruno Marchal]

In the model of set theories (called "universe" usually, but of course
they are not physical universe, a priori).

It exists also as object in naive set theory. Well "Hilbert hostel" is
just a trick to explain some statements about sets.

Bruno


>
> Evgenii

[meekerdb]

I doubt infinities.

Brent

[Bruno Marchal]

I can doubt actual infinities. Not potential infinities, which gives sense to any non stooping program notion.

Comp is ontologically finitist. As long as you don't claim that there is a biggest prime number, there should be no problem with the comp hyp. Infinities can be put in the epistemology, or at the meta-level: they are mind tool, souls attractor etc.

Bruno

Brent

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[meekerdb]

On 5/29/2012 12:27 AM, Bruno Marchal wrote:

I doubt infinities.

I can doubt actual infinities. Not potential infinities, which gives sense to any non stooping program notion.

Comp is ontologically finitist. As long as you don't claim that there is a biggest prime number, there should be no problem with the comp hyp. Infinities can be put in the epistemology, or at the meta-level: they are mind tool, souls attractor etc.

Bruno

But diagonalization arguments assume realized infinities.

Brent

[Bruno Marchal]

Set theoretical diagonalizations, à-la Cantor, assume realized infinities (like analysis, by the way). I don't use them, if only to explain diagonalization.

Computer science or "arithmetical" diagonalization does not assume realized infinities, only potential. Kleene second theorem is constructive. Gödel's diagonalization is constructive: for each effective theory, it provides the undecidable sentences. 

The intensional diagonalization, leading to reproduction, self-generation and self-reference are all constructive concepts.

The theory of everything is really just logic and

Ax ~(0 = s(x))  (For all number x the successor of x is different from zero).

AxAy ~(x = y) -> ~(s(x) = s(y))    (different numbers have different successors)

Ax x + 0 = x  

AxAy  x + s(y) = s(x + y)   ( meaning x + (y +1) = (x + y) +1) = laws of addition

Ax   x *0 = 0

AxAy x*s(y) = x*y + x    laws of multiplication

The observer is the same + the induction axioms. To define it in the theory above is of course a very long subtle and tedious exercise.

Bruno

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

[meekerdb]

On 5/29/2012 11:39 AM, Bruno Marchal wrote:

On 29 May 2012, at 19:27, meekerdb wrote:

On 5/29/2012 12:27 AM, Bruno Marchal wrote:

I doubt infinities.

I can doubt actual infinities. Not potential infinities, which gives sense to any non stooping program notion.

Comp is ontologically finitist. As long as you don't claim that there is a biggest prime number, there should be no problem with the comp hyp. Infinities can be put in the epistemology, or at the meta-level: they are mind tool, souls attractor etc.

Bruno

But diagonalization arguments assume realized infinities.

Set theoretical diagonalizations, à-la Cantor, assume realized infinities (like analysis, by the way). I don't use them, if only to explain diagonalization.

Computer science or "arithmetical" diagonalization does not assume realized infinities, only potential. Kleene second theorem is constructive. Gödel's diagonalization is constructive: for each effective theory, it provides the undecidable sentences.

But they do depend on infinity (i.e. the axiom of succession).

The intensional diagonalization, leading to reproduction, self-generation and self-reference are all constructive concepts.

Can you explain "intensional diagonalization"?

Brent

The theory of everything is really just logic and

Ax ~(0 = s(x))  (For all number x the successor of x is different from zero).

AxAy ~(x = y) -> ~(s(x) = s(y))    (different numbers have different successors)

Ax x + 0 = x  

AxAy  x + s(y) = s(x + y)   ( meaning x + (y +1) = (x + y) +1) = laws of addition

Ax   x *0 = 0

AxAy x*s(y) = x*y + x    laws of multiplication

The observer is the same + the induction axioms. To define it in the theory above is of course a very long subtle and tedious exercise.

Bruno

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

[Bruno Marchal]

On 31 May 2012, at 18:13, meekerdb wrote:

On 5/29/2012 11:39 AM, Bruno Marchal wrote:

On 29 May 2012, at 19:27, meekerdb wrote:

On 5/29/2012 12:27 AM, Bruno Marchal wrote:

I doubt infinities.

I can doubt actual infinities. Not potential infinities, which gives sense to any non stooping program notion.

Comp is ontologically finitist. As long as you don't claim that there is a biggest prime number, there should be no problem with the comp hyp. Infinities can be put in the epistemology, or at the meta-level: they are mind tool, souls attractor etc.

Bruno

But diagonalization arguments assume realized infinities.

Set theoretical diagonalizations, à-la Cantor, assume realized infinities (like analysis, by the way). I don't use them, if only to explain diagonalization.

Computer science or "arithmetical" diagonalization does not assume realized infinities, only potential. Kleene second theorem is constructive. Gödel's diagonalization is constructive: for each effective theory, it provides the undecidable sentences.

But they do depend on infinity (i.e. the axiom of succession).

The axiom of succession is not an axiom of infinity. It just says that the numbers have each unique and different (from other's) successors. All the standard numbers are finite.

In ZF there is an axiom of infinity, for you cannot prove infinity from below. Unless you have more powerful axiom like the scheme of reflexion. 

The intensional diagonalization, leading to reproduction, self-generation and self-reference are all constructive concepts.

Can you explain "intensional diagonalization"?

It is when you build an expression involving a formal diagonalization, like writing a program which refer to itself.

For example like with the self-duplicating expression Dx = 'xx', so that DD generates 'DD', i.e. its description. That is what Gödel did to prove the existence of a sentence referring to itself, and notably asserting that she is not provable. And that's what Kleene did to prove the existence of a number e, such that phi_e (x) = T(e, x), or generalization. In that case the program e compute T on itself with parameter x.

Intensional diagonalization concerns codes.

Extensional diagonalization concerns set of functions, like the Cantor one, showing that N^N is not enumerable, or Kleene one showing that comp-N^N is not recursively enumerable. (comp-N^N = the partial computable functions from N to N).

Bruno

The theory of everything is really just logic and

Ax ~(0 = s(x))  (For all number x the successor of x is different from zero).

AxAy ~(x = y) -> ~(s(x) = s(y))    (different numbers have different successors)

Ax x + 0 = x  

AxAy  x + s(y) = s(x + y)   ( meaning x + (y +1) = (x + y) +1) = laws of addition

Ax   x *0 = 0

AxAy x*s(y) = x*y + x    laws of multiplication

The observer is the same + the induction axioms. To define it in the theory above is of course a very long subtle and tedious exercise.

Bruno

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

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