Re: arithmetic truth

[Bruno Marchal]

On 26 Nov 2012, at 02:00, Platonist Guitar Cowboy wrote:




On Sun, Nov 25, 2012 at 5:10 PM, Bruno Marchal   
wrote:

Hi Cowboy,


On 24 Nov 2012, at 19:52, Platonist Guitar Cowboy wrote:

Hi Everybody,

At several points the discussions of the list led us to hypothesis  
of arithmetic truth. Bruno mentioned once that the basis for this  
hypothesis was quite strong, requiring studies in logic to grasp.


You might quote the passage. Comp (roughly "I am machine", with the  
3-I, the body) is quite strong, compared to "strong AI" (a machine  
can be conscious).
Although the comp I use is the weaker of all comp; as it does not  
fix the substitution level. But logically it is still stronger than  
strong AI.


But arithmetical truth itself is easy to grasp. Even tribes having  
no names for the natural numbers get it very easily, and basically  
anyone capable of given sense (true or false or indeterminate, it  
does not matter) to sentences like
"I will have only a finite number of anniversary birthdays", already  
betrays his belief in arithmetical truth (the intuitive concept). So  
I would say it is assumed and know by almost everybody, more or less  
explicitly depending on education.



I still have difficulty with intuition as "ability to understand  
something no reasoning" in this loose linguistic sense and how  
mathematicians frame that. When Kleene makes this precise in "The  
Foundations of Intuitionistic Mathematics"... this is a bit too much  
for cowboys with guitars, but for some reason I am intrigued.


Intuitionists just avoid the use of (P v ~P). In fact when we program  
a computer to compute a total computable function in a way we are  
assured that this is the case, we do intuitionist mathematics, and  
"conventional programming" is a form of intuitionistic math. The same  
when an engineer build machines.
But like modal logics, there are a lot of intuitionistic system, and a  
lot of different philosophies associated to it.
For an intuitionist a statement like ExP(x) presuppose a mean of  
constructing the n such that P(n), and this is guarantied if you don't  
use P v ~P. Classical mathematics, which use freely P v ~P, do  
unconventional programming, like in artificial intelligence, where we  
allow to a machine to explore some infinite realm, if she wants too  
(but this is a 3p crashing, and you have to unplug the machine if you  
want to use it again).









But as a non-logician, I have some trouble wrapping my brain around  
Gödel and Tarski's Papers concerning this.


Well, this is quite different. It concerns what machine and theories  
can said about truth. This is far more involved and requires some  
amount of study of mathematical logic. I will come back on this,  
probably in the FOAR list (and not soon enough, as we have to dig a  
bit on the math needed for this before).





What I do see is that Tarski generalizes the notion and its  
difficulties to all formal languages: truth isn't arithmetically  
definable without higher order language. Post attacking the problem  
with Turing degrees also resonates with this in that no formula can  
define truth for arbitrarily large n.



My question as non-logician therefore is: don't these results weaken  
the basis for such a hypothesis or at least make it completely  
inaccessible for us?


No, it is totally accessible to us, but by intuition only. You can  
be sure that music is very similar. We are all sensible to it, but  
to explain this is beyond the formal method; neither a brain nor a  
computer might ever been able to do that.



That is so strange and amazing. Especially that weird parallel to  
music. And "might" is a very large word there to me because don't  
composers or mathematicians of, I'll say vaguely, "similar  
approaches to their craft" already agree on certain facts about  
objects and their properties already? I know, I do with certain  
musicians/composers, without total certainty though.


The "might" is provable for sound machine, even provable by  
themselves. Somehow Gödel's theorem justifies that our intuition of  
arithmetical truth is deeper than our explicit belief we can have.  
Someone ideally immortal and interested in proving as much  
arithmetical truth as possible is condemn to add infinitely often new  
axioms and/or new inference rules, and realized that no finite theory/ 
machine can ever embrace the complete arithmetical truth, the  
intuition will precede the formalization, and always been a bit  
dissatisfied, with respect to that totality.
But "here" to be sure "intuition" is not directly related to the  
intuitionist one (but still indirectly).






Now, comp needs only the sigma_1 truth, which is machine definable,  
to proceed. I use the non-definability of truth only to see the  
relation with God, and for the arithmetical interpretation of  
Plotinus, as the numbers themselves will have to infer more than  
Sigma_1 truth (actually much more).


But it is clear that co

Re: arithmetic truth

[Platonist Guitar Cowboy]

On Sun, Nov 25, 2012 at 5:10 PM, Bruno Marchal  wrote:

> Hi Cowboy,
>
>
> On 24 Nov 2012, at 19:52, Platonist Guitar Cowboy wrote:
>
>  Hi Everybody,
>>
>> At several points the discussions of the list led us to hypothesis of
>> arithmetic truth. Bruno mentioned once that the basis for this hypothesis
>> was quite strong, requiring studies in logic to grasp.
>>
>
> You might quote the passage. Comp (roughly "I am machine", with the 3-I,
> the body) is quite strong, compared to "strong AI" (a machine can be
> conscious).
> Although the comp I use is the weaker of all comp; as it does not fix the
> substitution level. But logically it is still stronger than strong AI.
>
> But arithmetical truth itself is easy to grasp. Even tribes having no
> names for the natural numbers get it very easily, and basically anyone
> capable of given sense (true or false or indeterminate, it does not matter)
> to sentences like
> "I will have only a finite number of anniversary birthdays", already
> betrays his belief in arithmetical truth (the intuitive concept). So I
> would say it is assumed and know by almost everybody, more or less
> explicitly depending on education.
>
>
I still have difficulty with intuition as "ability to understand something
no reasoning" in this loose linguistic sense and how mathematicians frame
that. When Kleene makes this precise in "The Foundations of Intuitionistic
Mathematics"... this is a bit too much for cowboys with guitars, but for
some reason I am intrigued.


>
>
>> But as a non-logician, I have some trouble wrapping my brain around Gödel
>> and Tarski's Papers concerning this.
>>
>
> Well, this is quite different. It concerns what machine and theories can
> said about truth. This is far more involved and requires some amount of
> study of mathematical logic. I will come back on this, probably in the FOAR
> list (and not soon enough, as we have to dig a bit on the math needed for
> this before).
>
>
>
>
>  What I do see is that Tarski generalizes the notion and its difficulties
>> to all formal languages: truth isn't arithmetically definable without
>> higher order language. Post attacking the problem with Turing degrees also
>> resonates with this in that no formula can define truth for arbitrarily
>> large n.
>>
>>
>> My question as non-logician therefore is: don't these results weaken the
>> basis for such a hypothesis or at least make it completely inaccessible for
>> us?
>>
>
> No, it is totally accessible to us, but by intuition only. You can be sure
> that music is very similar. We are all sensible to it, but to explain this
> is beyond the formal method; neither a brain nor a computer might ever been
> able to do that.
>
>
That is so strange and amazing. Especially that weird parallel to music.
And "might" is a very large word there to me because don't composers or
mathematicians of, I'll say vaguely, "similar approaches to their craft"
already agree on certain facts about objects and their properties already?
I know, I do with certain musicians/composers, without total certainty
though.


> Now, comp needs only the sigma_1 truth, which is machine definable, to
> proceed. I use the non-definability of truth only to see the relation with
> God, and for the arithmetical interpretation of Plotinus, as the numbers
> themselves will have to infer more than Sigma_1 truth (actually much more).
>
> But it is clear that consciousness is also not definable, yet we have all
> access to it, very easily. It is the same for arithmetical truth. The
> notion is easy, the precise content is infinitely complex, non computable,
> unsolvable, not expressible in arithmetic, etc.
>
> Only "philosophers" can doubt about the notion of arithmetical truth. In
> math, both classical and intuitionist, arithmetical truth is considered as
> the easy sharable part (even if interpreted differently). COMP is strong
> because "yes doctor" involves a risky bet, and the Church thesis requires a
> less risky bet but is still logically strong, but the Arithmetical realism
> is very weak: it is assumed by every scientists and lay men, and disputed
> only by philosophers (and usually very badly).
>

I have never heard about something like a student abandoning school and
> thinking his teacher is mad when he heard him saying that there is no
> bigger prime number. It *is* a bit extraordinary, when you think twice, but
> we are used to this.
>
>
But isn't this like informally stating that Euclid proof "there will always
be larger prime". So it's more like a proof than intuition? Like you have
to know what prime is, natural numbers are infinite etc., if natural
numbers are infinite than there will always be one more? So this you can
fomalize and state loosely in language, but what the student dreamed at the
last concert he enjoyed is not. It is not clear to me why the prime
statement is intuition.

Cowboy regards :)


> Best,
>
> Bruno
>
>
>
> http://iridia.ulb.ac.be/~**marchal/ 
>
>
>
> 

Re: arithmetic truth

[Bruno Marchal]

Hi Cowboy,

On 24 Nov 2012, at 19:52, Platonist Guitar Cowboy wrote:


Hi Everybody,

At several points the discussions of the list led us to hypothesis  
of arithmetic truth. Bruno mentioned once that the basis for this  
hypothesis was quite strong, requiring studies in logic to grasp.


You might quote the passage. Comp (roughly "I am machine", with the 3- 
I, the body) is quite strong, compared to "strong AI" (a machine can  
be conscious).
Although the comp I use is the weaker of all comp; as it does not fix  
the substitution level. But logically it is still stronger than strong  
AI.


But arithmetical truth itself is easy to grasp. Even tribes having no  
names for the natural numbers get it very easily, and basically anyone  
capable of given sense (true or false or indeterminate, it does not  
matter) to sentences like
"I will have only a finite number of anniversary birthdays", already  
betrays his belief in arithmetical truth (the intuitive concept). So I  
would say it is assumed and know by almost everybody, more or less  
explicitly depending on education.





But as a non-logician, I have some trouble wrapping my brain around  
Gödel and Tarski's Papers concerning this.


Well, this is quite different. It concerns what machine and theories  
can said about truth. This is far more involved and requires some  
amount of study of mathematical logic. I will come back on this,  
probably in the FOAR list (and not soon enough, as we have to dig a  
bit on the math needed for this before).





What I do see is that Tarski generalizes the notion and its  
difficulties to all formal languages: truth isn't arithmetically  
definable without higher order language. Post attacking the problem  
with Turing degrees also resonates with this in that no formula can  
define truth for arbitrarily large n.


My question as non-logician therefore is: don't these results weaken  
the basis for such a hypothesis or at least make it completely  
inaccessible for us?


No, it is totally accessible to us, but by intuition only. You can be  
sure that music is very similar. We are all sensible to it, but to  
explain this is beyond the formal method; neither a brain nor a  
computer might ever been able to do that.


Now, comp needs only the sigma_1 truth, which is machine definable, to  
proceed. I use the non-definability of truth only to see the relation  
with God, and for the arithmetical interpretation of Plotinus, as the  
numbers themselves will have to infer more than Sigma_1 truth  
(actually much more).


But it is clear that consciousness is also not definable, yet we have  
all access to it, very easily. It is the same for arithmetical truth.  
The notion is easy, the precise content is infinitely complex, non  
computable, unsolvable, not expressible in arithmetic, etc.


Only "philosophers" can doubt about the notion of arithmetical truth.  
In math, both classical and intuitionist, arithmetical truth is  
considered as the easy sharable part (even if interpreted  
differently). COMP is strong because "yes doctor" involves a risky  
bet, and the Church thesis requires a less risky bet but is still  
logically strong, but the Arithmetical realism is very weak: it is  
assumed by every scientists and lay men, and disputed only by  
philosophers (and usually very badly).


I have never heard about something like a student abandoning school  
and thinking his teacher is mad when he heard him saying that there is  
no bigger prime number. It *is* a bit extraordinary, when you think  
twice, but we are used to this.


Best,

Bruno



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



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Re: arithmetic truth and 1p truth

[Bruno Marchal]

On 03 Nov 2012, at 11:58, Roger Clough wrote:


Hi Bruno Marchal

I think in computationalism you only have to be able
to say that the result is arithmetically or algebraically
true. Arithmetic truth is what you seek.

However, I still have yet to know if  a particular
computation seems true to your 1p. That would be
1p truth. Does the arithmetic truth pass the 1p test ?


Yes. Good question.
That's the purpose of AUDA. The arithmetical UDA.
It is in the second part of sane04. The "interview of the universal  
machine".
Up to now, thanks to the Everett/Feynman formulation of QM, comp  
succeeds the first tests.

Arguably.

Bruno






Roger Clough, rclo...@verizon.net
11/3/2012
"Forever is a long time, especially near the end." -Woody Allen


- Receiving the following content -
From: Bruno Marchal
Receiver: everything-list
Time: 2012-11-02, 13:23:36
Subject: Re: Numbers in the Platonic Realm




On 01 Nov 2012, at 22:50, Stephen P. King wrote:


On 11/1/2012 12:04 PM, Bruno Marchal wrote:



On 01 Nov 2012, at 01:18, Stephen P. King wrote:


On 10/31/2012 12:45 PM, Bruno Marchal wrote:

can stop reading as you need to assume the numbers (or anything  
Turing equivalent) to get them.


Dear Bruno,

   So it is OK to assume that which I seek to explain?





You can't explain the numbers without assuming the numbers. This has  
been foreseen by Dedekind, and vert well justified by many theorem  
in mathematical logic. Below the number, you are lead to version of  
ultrafinitism, which is senseless in the comp theory.


Dear Bruno,

   I disagree with ultrafinitists, they seem to be the mathematical  
equivalent of "flat-earthers'.











*and* having some particular set of values and meanings.


I just assume


x + 0 = x
x + s(y) = s(x + y)


x *0 = 0
x*s(y) = x*y + x


And hope you understand.



   I can understand these symbols because there is at least a way to  
physically implement them.



Those notion have nothing to do with "physical implementation".


   So your thinking about them is not a physical act?



Too much ambiguous. Even staying in comp I can answer "yes" and "no".
Yes, because my human thinking is locally supported by physical  
events.
No, because the whole couple mind/physical events is supported by  
platonic arithmetical truth.









Implementation and physical will be explained from them. A natural  
thing as they are much more complex than the laws above.


   Numbers are meaningless in the absence of a means to define them.  
Theories do not free-float.




Truth is free floating, and theories lived through truth, they are  
truth floating, even when false.











In the absence of some common media, even if it is generated by  
sheaves of computations, there simply is no way to understand  
anything.



Why ?

   Because there is not way to know of them otherwise.


Our knowing as nothing to do with truth. If an asteroid would have  
destroy Earth before the Oresme bishop dicovered that the harmonic  
series diverge, she would have still diverge, despite no humans  
would know it.







Unless you can communicate with me, I have no way of knowing  
anything about your ideas. Similarly if there is no physical  
implementation of a mathematical statement, there is no meaning to  
claims to "truth" of such statements.




To claim, no. To be true is independent of the claim of the apes.










You must accept non-well foundedness for your result to work, but  
you seem fixated against that.




1004.

   Pfft. Nice custom made quip.



You are often escaping answers by inappropriate mathematical  
precision, which meaning contradicts your mathematical super- 
relativism. It is really 1004+contradiction.
















A statement, such as 2 = 1+1 or two equals one plus one, are said  
truthfully to have the same meaning because there are multiple and  
separable entities that can have the agreement on the truth value.  
In the absence of the ability to judge a statement independently of  
any particular entity capable of "understanding" the statement,  
there is no meaning to the concept that the statement is true or  
false. To insist that a statement has a meaning and is true (or  
false) in an ontological condition where no entities capable of  
judging the meaning, begs the question of meaningfulness!

  You are taking for granted some things that your arguments disallow.





Do you agree that during the five seconds just after the Big Bang  
(assuming that theory) there might not have been any possible  
observers. But then the Big Bang has no more sense.


   No, I don't. Why? Because that concept of "the five seconds just  
after the Big Bang" is an assumption of a special case or pleading.  
I might as well postulate the existence of Raindow Dash to act as  
the entity to whom the Truth of mathematical statements have  
absolute meaning. To be frank, I thing that the Big Bang theory, as  
usually explained is a steaming pile of rubbish, as it asks