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".
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 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".
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 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.
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?
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 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, 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 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). 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 or
variations of
the 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 0/10
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,
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 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.
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.
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.
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 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.
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 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 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. 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 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.
Bruno
http://iridia.ulb.ac.be/~marchal/
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