Bruno Marchal wrote:
> 
> 
> On 27 Feb 2011, at 00:25, benjayk wrote:
> 
>>
>>
>> Bruno Marchal wrote:
>>>
>>>
>>> On 23 Feb 2011, at 17:37, benjayk wrote:
>>>
>>>>
>>>>
>>>> Bruno Marchal wrote:
>>>>>
>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> Bruno Marchal wrote:
>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> Brent Meeker-2 wrote:
>>>>>>>>>
>>>>>>>>>> The easy way is to assume inconsistent descriptions are merely
>>>>>>>>>> an
>>>>>>>>>> arbitrary
>>>>>>>>>> combination of symbols that fail to describe something in
>>>>>>>>>> particular and
>>>>>>>>>> thus have only the "content" that every utterance has by
>>>>>>>>>> virtue of
>>>>>>>>>> being
>>>>>>>>>> uttered: There exists ... (something).
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> But we need utterances that *don't* entail existence.
>>>>>>>>
>>>>>>>> If we find something that doesn't entail existence, it still
>>>>>>>> entails
>>>>>>>> existence because every utterance is proof that existence IS.
>>>>>>>> We need only utterances that entail relative non-existence or  
>>>>>>>> that
>>>>>>>> don't
>>>>>>>> entail existence in a particular way in a particular context.
>>>>>>>
>>>>>>> You need some non relative absolute base to define relative
>>>>>>> existence.
>>>>>> The absolute base is the undeniable reality of there being
>>>>>> experience.
>>>>>
>>>>> But this one is not communicable. It does play a role in comp,
>>>>> though.
>>>> But we can say "there is an undeniable reality of there being
>>>> experience".
>>>> Isn't this communicating that there is the undeniable reality of
>>>> there being
>>>> experience?
>>>
>>> OK. I was using communicating in the sense of a provable
>>> communication. You cannot convince someone that you are conscious. If
>>> he decides that you are a zombie, you might better run, probably, but
>>> there is no way you could prove the contrary.
>> OK this makes sense. But is there any provable communication, then?  
>> After
>> all we can never prove the axioms needed for a provable communication.
> 
> All axioms are provable in one line. Just say "provable by axioms".
Sorry, I don't understand you here. How does saying "provable by axioms"
prove anything? It seems to be a  description of charateristic that can
either be true or false (provable or not provable by axioms).
Probably you mean something else, but I don't know what.


Bruno Marchal wrote:
> 
>>
>> Bruno Marchal wrote:
>>>
>>>>
>>>>
>>>> Bruno Marchal wrote:
>>>>>
>>>>>>
>>>>>>
>>>>>> Bruno Marchal wrote:
>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> Brent Meeker-2 wrote:
>>>>>>>>>
>>>>>>>>>> So they don't add anything to platonia because they merely
>>>>>>>>>> assert
>>>>>>>>>> the
>>>>>>>>>> existence of existence, which leaves platonia as described by
>>>>>>>>>> consistent
>>>>>>>>>> theories.
>>>>>>>>>>
>>>>>>>>>> I think the paradox is a linguistic paradox and it poses
>>>>>>>>>> really no
>>>>>>>>>> problem.
>>>>>>>>>> Ultimately all descriptions refer to an existing object, but
>>>>>>>>>> some
>>>>>>>>>> are too
>>>>>>>>>> broad or "explosive" or vague to be of any (formal) use.
>>>>>>>>>>
>>>>>>>>>> I may describe a system that is equal to standard arithmetics
>>>>>>>>>> but
>>>>>>>>>> also
>>>>>>>>>> has
>>>>>>>>>> 1=2 as an axiom. This makes it useless practically (or so I
>>>>>>>>>> guess...) but
>>>>>>>>>> it
>>>>>>>>>> may still be interpreted in a way that it makes sense. 1=2 may
>>>>>>>>>> mean that
>>>>>>>>>> there is 1 object that is 2 two objects, so it simply asserts
>>>>>>>>>> the
>>>>>>>>>> existence
>>>>>>>>>> of the one number "two". 3=7 may mean that there are 3 objects
>>>>>>>>>> that are 7
>>>>>>>>>> objects which might be interpreted as aserting the existence  
>>>>>>>>>> of
>>>>>>>>>> (for
>>>>>>>>>> example) 7*1, 7*2 and 7*3.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> The problem is not that there is no possible true
>>>>>>>>> interpretation of
>>>>>>>>> 1=2;
>>>>>>>>> the problem is that in standard logic a falsity allows you to
>>>>>>>>> prove
>>>>>>>>> anything.
>>>>>>>> Yes, so we can prove anything. This simply begs the question  
>>>>>>>> what
>>>>>>>> the
>>>>>>>> anything is. All sentences we derive from the inconsistency  
>>>>>>>> would
>>>>>>>> mean the
>>>>>>>> same (even though we don't know what exactly it is).
>>>>>>>> We could just write "1=1" instead and we would have expressed  
>>>>>>>> the
>>>>>>>> same, but
>>>>>>>> in a way that is easier to make sense of.
>>>>>>>>
>>>>>>>> This is not problematic, it only makes the proofs in the
>>>>>>>> inconsisten
>>>>>>>> system
>>>>>>>> worthless (at least in a formal context were we assume classical
>>>>>>>> logic).
>>>>>>>
>>>>>>> And it would make Platonia worthless. The "real", genuine,  
>>>>>>> Platonia
>>>>>>> is
>>>>>>> already close to be worthless due to the consistency of
>>>>>>> inconsistency
>>>>>>> for machine. This already put quite a mess in Platonia. By  
>>>>>>> allowing
>>>>>>> complete contradiction, you make it a trivial object.
>>>>>> Why? When we contradict ourselves we may simply interpret this  
>>>>>> as a
>>>>>> expression of the trivial truth of existence. This doesn't change
>>>>>> Plantonia
>>>>>> at all, because it exists either way.
>>>>>
>>>>> The whole point of Gödel's theorem is that M proves 0=1 is  
>>>>> different
>>>>> from M proves provable('0=1'). The first implies the second, but  
>>>>> the
>>>>> second does not implies the first. The difference between G and G*
>>>>> comes from this fact.
>>>> If we know that something can be proven, how is it different from
>>>> taking it
>>>> to be proven?
>>>
>>> By incompleteness "provable(false) -> false" is not provable in the
>>> system.
>> OK. But still "provable(false)->false" is true if we assume  
>> consistency,
>> right?
>> So above you meant implying as in "being a provable consequence of"?
> 
> Not really. By A -> B, I mean ~A v B. Or ~(A & ~B). being a provable  
> consequence would better be captured by B(A -> B), with "B" some  
> provability predicate.
Does "A -> B" mean B follows from A?
How is that equal to "not-A or B?"

So from provable('0=1') it does not follow 0=1, even if we assume
consistency and don't mean a provable consequence?
How can something be provable in a consistent system and what is proven does
not follow?



Bruno Marchal wrote:
> 
>>
>>
>> Bruno Marchal wrote:
>>>
>>>>
>>>>
>>>> Bruno Marchal wrote:
>>>>>
>>>>>>
>>>>>> And why is inconsistency allowed for machine, but disallowed for
>>>>>> other
>>>>>> objects?
>>>>>
>>>>> Because if a machine proves "0=1", she will be in trouble, but if  
>>>>> God
>>>>> or Platonia proves "0=1", then we are *all* in trouble.
>>>> I thought we already established that 0=1 can have a clear meaning
>>>> (equivalent to statements of the form  0*A+B=1*C+D in standard
>>>> arithmetics),
>>>> and so it poses no problem.
>>>
>>> ?
>>> I have no clue what you are saying here. If "0 = 1" means "I love
>>> chocolate", then of course "0=1" might be true.
>> But 0=1 is not plausibly interpreted as "I love chocolate" because the
>> latter is not (directly anyway) a statement about numbers. 0=1 may  
>> be a
>> statement about the two numbers 0 and 1 that is just not formulated in
>> standard arithmetic.  This does not imply that it is generally false  
>> (not
>> anymore then peano arithmetics show there is no meaning in 0=s(n)).
> 
> How could I have guessed that?
Well, if someone says that" X+3=1" makes sense (and you didn't know about
negative integers) you would guess he would mean a statement about numbers
(thereby extending the sense of numbers). Similarily you can guess that 0=1
is statement about numbers. 


Bruno Marchal wrote:
> 
>>
>>
>> Bruno Marchal wrote:
>>>
>>> Again, we use the
>>> standard meaning.
>> Okay, but then 0=1 has no standard meaning in arithmetics.
> 
> ? But it has a standard meaning. It is just plainly false. The meaning  
> is "false".
Well, of course, I expressed this badly. So, the statement '0=1' is false in
standard arithmetic. But what does standard arithmetic say about what being
wrong means? Just saying something is wrong does not really give this a
meaning.

If I know something is right, I have to know right in what way (intuitively,
I might not be able to convey it), otherwise I don't really know it to be
right. If I say "dfgxS" is right and I don't know what this means I don't
know whether it is right (other than in a trivial manner like I argued
before). If I say "1+1=2" is right I can imagine two objects coming together
and being counted. "Ahhh, in this way it is right".

Similarily I have to understand wrong in what way? How do I understand "1=0"
is wrong beyond it just being a label?



Bruno Marchal wrote:
> 
>> But it might still be usefully interpreted as "Try again (to make a  
>> true
>> statement in arithmetics)"
> 
> That would be a higher level heuristic. If you allow "0 = 1" to mean  
> this, then we are no more sure what we are talking about,
So what does "'0=1' is false" mean? If something is false intuitively it
means it is not valid in the context of a system. OK, but so what? This
doesn't seem to be a problem? Do I have to care whether its valid?

Can we really be sure what we are talking about if we are talking about
something being wrong?


Bruno Marchal wrote:
> 
>> or as an alternative expression of a statement in
>> arithmetics (like treating it as an expression where some symbols are
>> omitted) or as expressing that all numbers are equal when we talk  
>> about an
>> object where quantity does not matter (eg one of nothing is still  
>> nothing /
>> 0=0*1).
> 
> Yes, we are more simple minded than that. 0 = 1 is really 0 = s(0),  
> not something else like 0 = 0*s(0).
I think if we take the successor of "zero" to be "one of zero", this is just
as simple or simpler.

This already makes sense if we don't really understand numbers and sometimes
conflate them (eg we treat everything above 6 as equal because it is
"many").

We take 0 to be the "implicit factor" in every number so it is true that
3=3*0 and not only 3=3*1 (which implicitly means 3*0=3*1*0).

One is the succesor of zero has a clear meaning, but it zero is the
successor of zero can have clear meaning, too. 0 represents "something (not
necessarily of quanitiy one) " in this case.

Sure, we get a trivial structure if we are just concerned about what is true
and false in this system, but it may express something by treating zero as
placeholder (if we don't reduce the equation to 0=0 but expand it until
another equality emerges, that is, until we get a truth that is not as
trivial as 0=0, but still symmetric like 1=1)
s(0)=0
1=0
1=s(0)
1=1 "Ah, here a symmetry emerges, so this is an interesting statement about
a trivial, yet easy to comprehend truth" ). This structure yields all true
statements in the same formulation that normal arithmetic does (among many
more other formulations of true statements in standard arithmetic).

Certainly it's quite vague, but we could even do normal arithmetic and treat
all statements not true in standard arithmetic as saying 0=0 (which is
interpreted as "falsehood" or triviality), so it's not necessarily useless.
Actually if we don't get lost with arbitrary transformations it is no worse
than standard arithmetics.


Bruno Marchal wrote:
> 
> 
>>
>>
>> Bruno Marchal wrote:
>>>
>>> If some my student defend ideas like 0 = 1, I give them a 0/10
>> This is valid in mathematics, because there we agree to assume certain
>> axioms and not doing this is a communication error. I don't think it  
>> is
>> valid when doing philosophy (and interpreting what statements  
>> correspond to
>> reality is philosopy).
> 
> Philosphy, for me, is a purely private affair. I don't do philosophy,  
I think this is impossible.
As soon as we state some "truth" we are doing philosophy.

Mathematics has no meaning without truth.


Bruno Marchal wrote:
> 
>>
>> I think you don't see my point. Sure, there are statements that  
>> don't make
>> sense or that are false in a specific system (or more informal: a  
>> specific
>> context). That doesn't mean that these statements have no  
>> corresponding
>> truth.
> 
> But all proposition have a corresponding truth, if we allow the  
> context to vary. The idea is that we fix the context, and then make  
> the reasoning.
OK but then we have truths in the context of the fixed of context, no more,
no less.


Bruno Marchal wrote:
> 
>>
>> Your idea that it is somehow problematic if God proves 0=1 assumes  
>> that
>> "0=1" has a definite meaning that is wrong.
> 
> yes. The usual standard meaning. "0 = 1" is a generic proposition to  
> give an example of a clear falsehood.
But 0=1 is unclear, because we only know that it is false, not exactly what
this means.
So a clear falsehood exists only in the sense that it is clear which
statements are false, not that the meaning is clear.



Bruno Marchal wrote:
> 
>> Even if we say "God proves 0=1
>> in standard arithmetic", we just express standard arithmetic means  
>> something
>> else as what we commonly understand as it.
> 
> That was not my use of it. I was just saying that if God proves  
> 0=s(0), we are all in trouble. That would be a catastrophe bigger than  
> the big crunch. Nothing would make sense at all.
See, at this point you do philosophy. You interpret something false as "not
making sense at all". This is simply a philosophical belief (and not one I
would endorse).

I think everything does make sense, including all falsehoods. False is
relative lack of clarity or usefulness. But I make it clear that it is a
phliosophical standpoint (at least now I did ;) ).


Bruno Marchal wrote:
> 
>>
>> I don't think there is such a thing as absolutely wrong statement.
> 
> If by absolutely, you mean wrong in all theories, I agree with you,  
> but the point is a bit trivial.
Indeed, in some way it is. But then existence of numbers seems trivial, yet
many people have problems with it.
The notion of triviality may be not so trivial.


Bruno Marchal wrote:
> 
>  If by absloute, you mean wrong in all  
> the models or intepretations of a theory about natural nulmbers, then  
> "0=1" is indeed wrong in all interpretations of arithmetical theories.
But what are "arithmetical theories"? Can this even be defined? Maybe there
is an arithmetical theory where 0=1 is true.


Bruno Marchal wrote:
> 
>> All
>> falsehood is depended on assuming certain axioms - while truth is not.
> 
> That makes truth even more absolute.
Well, that's good in my mind. Truth seems to be the most absolute thing for
me.



Bruno Marchal wrote:
> 
>> I think it is an error to say "if God or Platonia proves "0=1",
>> then we are *all* in trouble." because we can't say what it would  
>> mean if
>> God proves 0=1 according to the axioms we use. Or it means simply  
>> that God
>> states things that are wrong in certain systems, which also poses no
>> problem.
> 
> Well, if God actually comes by, and says "0 = 1", I will conclude  
> something like "oh! God has some sense of humour", or I will think  
> that's not God, or I will thinks "I must be dreaming", etc.
I meant actually proving. Of course we don't know what it would mean if God
would prove 0=1 according to our axioms, but this is my point.


Bruno Marchal wrote:
> 
>>
>> I would not insist on this so much if I would not suspect that  
>> considering
>> the possibility of something being an "ultimate falsehood" or "totally
>> wrong" leads to disregarding some statements instead of seeing their  
>> truth.
>> We don't have to avoid "wrong" statements. They simply are not as
>> proliferative as (more) true statements.
> 
> I disagree. Most statements in most (non arithmetical) theories are  
> wrong.
It's hard to tell, I think. In most theories there is not even an absolutely
clear distinction between right and wrong. Physical theories for example are
(as far as we now) never completely right, so are they wrong?


Bruno Marchal wrote:
> 
>>
>>
>>
>> Bruno Marchal wrote:
>>>
>>>>
>>>> My suggestion is that every statement has such an interpretation.
>>>> Circles
>>>> with edges makes sense if we allow hyperreal numbers as numbers of
>>>> edges and
>>>> lenght of edges, triangles with four sides may mean such a geometric
>>>> object:
>>>> http://commons.wikimedia.org/wiki/File:Triangle-square-area-dev.png
>>>> and that
>>>> God is omnipotent may mean anything.
>>>
>>> Logic has been invented for avoiding interpretations as much as
>>> possible, and then for studying mathematically what can be
>>> interpretations, and the relations (Galois connection) between formal
>>> deduction and relations on interpretations. We force the
>>> "propositions" which mean anything to be eliminated, for helping the
>>> progress toward genuine truth and meaning.
>>>
>>> We can say that first order logic does succeed in the interpretation
>>> elimination, thanks to a theorem of completeness (not incompleteness)
>>> by Gödel. A formula is a theorem IF and ONLY IF the formula is true  
>>> in
>>> all interpretations.
>>>
>> Again, this is true, but it depends on certain axioms, which can be
>> interpreted to be not right in all contexts.
> 
> I disagree. The completeness theorem of Gödel does not depend on any  
> axioms, just on the definition of what is a first order logical theory  
> and an interpretation (model). Note that such a theorem is no more  
> true for higher order logics, which makes them almost as vague as  
> mathematics, for some logicians.
Maybe we don't need explicit axioms, but we need implicit assumptions.

If we define a first order logical theory and have an interpretation, to
validly prove anything we must at least assume that these
definitions/interpretation are meaningful.
Otherwise we can just reject them by not finding them meaningful and thus
reject any proof derived from it.
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