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.
Bruno Marchal wrote:
>
>>
>>
>> Bruno Marchal wrote:
>>>
>>> But it is not enough. usually people agree with the axiom of Peano
>>> Arithmetic, or the initial part of some set theory.
>> But Peano Arithmetics is not a non relative absolute base. It is
>> relative to
>> the meaning we give it and to the existence of some reality. 1+1=2
>> can have
>> infinite meanings, that all are relative to our interpretation ("If
>> I lay
>> another apple into the bowl with one apple in it there are two
>> apples" is
>> one of them) and there being meaning in the first place.
>
> Hmm... Most people agrees on a standard meaning for the natural
> numbers, like in the Fermat theorem, or any theorem or conjecture in
> number theory, or when you are using numbers in computer science.
> 1+1 = 2 is true in all those interpretations, even if computer science
> we use also some algebra where 1+1=0. That does not contradict that
> the standard integer are all different from 0, except 0.
OK, but I insist that the fact that most people agree on something does not
make it a "non relative absolute base".
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"?
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)).
Bruno Marchal wrote:
>
> Again, we use the
> standard meaning.
Okay, but then 0=1 has no standard meaning in arithmetics. It simply not
included in arithmetics.
But it might still be usefully interpreted as "Try again (to make a true
statement in arithmetics)" 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).
Bruno Marchal wrote:
> But for the
> natural numbers, we do agree on those axioms, and their correspond to
> what has been taught at school.
Yes, but this does not imply the axioms that other axioms or variations of
the axioms are not valid.
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).
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.
Your idea that it is somehow problematic if God proves 0=1 assumes that
"0=1" has a definite meaning that is wrong. 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.
I don't think there is such a thing as absolutely wrong statement. All
falsehood is depended on assuming certain axioms - while truth is not.
Undefinedness is no problem, because it does not say anything about what
exists.
Consistency seems to be a fundamental reality and not something we can even
conceive of not being there. Inconsistency has only meaning in specific
systems. 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.
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.
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.
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