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 orthatdon'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?Afterall we can never prove the axioms needed for a provable communication.

All axioms are provable in one line. Just say "provable by axioms".

`Of course a theory will be *interesting* if the axioms are plausible,`

`about their subject matter, and simple, and in few numbers, etc.`

`The axioms needs to be "true" in some "reality" (model). But`

`"provable" is always supposed to mean "provable" in this or that`

`theory. Is a theory true? This is outside the scope of science. That`

`question belongs to philosophy, and IMO is almost a private question.`

`Now I do about that, concerning the usual standard natural numbers (0,`

`1, 2, ...) you agree that for all x 0 ≠ s(x), for example. It means`

`that zero is not a successor of a natural number. Of course zero is`

`the successor of -1, but this concerns another structure (the set of`

`integers (..., -2, -1, 0, 1, 2, ...).`

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 computersciencewe 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 somethingdoes notmake it a "non relative absolute base".

`I agree. Science is not democratic. We don't vote to decide the truth`

`of an arithmetical proposition. We prove it in a theory on which`

`people agrees.`

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 7objects which might be interpreted as aserting the existenceof(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 questionwhattheanything is. All sentences we derive from the inconsistencywouldmean the same (even though we don't know what exactly it is).We could just write "1=1" instead and we would have expressedthesame, 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,Platoniais already close to be worthless due to the consistency of inconsistencyfor machine. This already put quite a mess in Platonia. Byallowingcomplete contradiction, you make it a trivial object.Why? When we contradict ourselves we may simply interpret thisas aexpression 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 isdifferentfrom M proves provable('0=1'). The first implies the second, butthesecond 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 assumeconsistency,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.`

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 ifGodor 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 thelatter is not (directly anyway) a statement about numbers. 0=1 maybe astatement about the two numbers 0 and 1 that is just not formulated instandard arithmetic. This does not imply that it is generally false(notanymore then peano arithmetics show there is no meaning in 0=s(n)).

How could I have guessed that?

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".`

It simply not included in arithmetics.

It is indeed even refutable. Simple theories PA can prove ~(0 = 1).

But it might still be usefully interpreted as "Try again (to make atruestatement 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, and people`

`will not believe you if you say that you can derive Schroedinger`

`equation in the mind of correct machine, in arithmetic. They will just`

`argue that the interpretation is so vague that it is not really a new`

`result, but some triviality.`

or as an alternative expression of a statement in arithmetics (like treating it as an expression where some symbols areomitted) or as expressing that all numbers are equal when we talkabout anobject where quantity does not matter (eg one of nothing is stillnothing /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). We get terrible result; the`

`falsity of Aristotelian theologies (the basic current quasi)universal`

`paradigm). We have to start from ultra clear axioms.`

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 orvariations ofthe axioms are not valid.

That is why there are different theories.

Bruno Marchal wrote:If some my student defend ideas like 0 = 1, I give them a 0/10This is valid in mathematics, because there we agree to assume certainaxioms and not doing this is a communication error. I don't think itisvalid when doing philosophy (and interpreting what statementscorrespond toreality is philosopy).

`Philosphy, for me, is a purely private affair. I don't do philosophy,`

`and my main initial goal was to show that some problem in philosophy`

`can be translated into problem in mathematics, once we accept some`

`hypothesis (like the computationalist one). Of course philosophers`

`don't like this, like they never appreciate when science get on their`

`territory.`

I think you don't see my point. Sure, there are statements thatdon't makesense or that are false in a specific system (or more informal: aspecificcontext). That doesn't mean that these statements have nocorrespondingtruth.

`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.`

Your idea that it is somehow problematic if God proves 0=1 assumesthat"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.`

Even if we say "God proves 0=1in standard arithmetic", we just express standard arithmetic meanssomethingelse 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.`

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. 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.`

All falsehood is depended on assuming certain axioms - while truth is not.

That makes truth even more absolute.

Undefinedness is no problem, because it does not say anything aboutwhatexists.Consistency seems to be a fundamental reality and not something wecan evenconceive of not being there. Inconsistency has only meaning inspecificsystems.

`Inconsistency is just the negation of consistency. In classical logic`

`they have the same amount of sense.`

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 wouldmean ifGod proves 0=1 according to the axioms we use. Or it means simplythat Godstates 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 would not insist on this so much if I would not suspect thatconsideringthe possibility of something being an "ultimate falsehood" or "totallywrong" leads to disregarding some statements instead of seeing theirtruth.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. Only in arithmetic does all mathematicians agree and people`

`have a deep confidence in the elementary arithmetical axioms. That's`

`is actually a good reason to build on arithmetic. Set theory and`

`analysis are already more problematical, and different logics abound.`

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 trueinall 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.`

Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.