On 28 Feb 2011, at 21:37, benjayk wrote:
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.
I meant "provable by the fact of being an axiom".
A proof is a sequence of formula, each of which are either an axiom or
a result from a previously proved formula by the means of the
inference rules.
Let me give an example:
The theory T has the following axioms:
1) p
2) p -> r
3) r -> u
And the (common) modus ponens inference rule: it says that from a
formula A and a formula A -> B, you can derive the formula B
In the theory T, it is easy to prove u.
The proof is the sequence of formula, (I add justification alongside,
but formally they don't belongs to the formal proof)
p (by axiom 1)
p -> r (by axiom 2)
r (by modus ponens and the two preceding formula in
this proof)
r -> u (by axiom 3)
u (by modus ponens and the two preceding formula in this
proof)
So the theory T proves the formula u.
Now, suppose someone ask me for a proof of p, in the theory T. I will
just write the following (rather short) sequence of formula:
p (by axiom 1).
To proves p in one line, consisting simply in remembering one axiom.
That's what I meant by "provable in one line".
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?
It might depend what you mean by "follows from". usually, in classical
logic or in classical mathematics "A -> B", also written "If A then
B", means only that A is false or that B is true.
You might have seen the semantic of "A -> B". It is false only in the
"worlds" where A is true and B false:
A B
1 1 in this case (A -> B) is true
1 0 in this case (A -> B) is false
0 1 in this case (A -> B) is true
0 0 in this case (A -> B) is true
In classical logic, the false implies everything. That's why it is a
catastrophe if your theory proves false, because it will prove any
proposition.
How is that equal to "not-A or B?"
not A or B has the same semantic than A -> B. I let you verify. The
"or" is the inclusive or. For example the unique proposition "((1+1=2)
or (2+2 = 4))" is true.
See? not A or B is really ((not A) or B)
A B
1 1 in this case (not A v B) is true
1 0 in this case (not A v B) is false
0 1 in this case (not A v B) is true
0 0 in this case (not A v B) is true
So from provable('0=1') it does not follow 0=1, even if we assume
consistency and don't mean a provable consequence?
"provable(f) -> f" is a true formula about a consistent theory. But
Gödel's incompleteness makes such a formula, although true, not
provable by the theory itself (in case the theory is as rich to be
able to prove the elementary addition and multiplication laws).
How can something be provable in a consistent system and what is
proven does
not follow?
If the theory proves p, it will automatically prove Bp -> p. Indeed
for any q, it will prove q -> p. This comes from the fact that the
formula p -> (q -> p) is a tautology of classical propositional logic.
But for Löbian theories (like all the classical extension of PA) the
reverse is astonishingly true also, by a theoreme due to Löb, a Deutch
logician in 1955, if PA proves Bp -> p then PA proves p. In particular
PA cannot proves Bf -> f , because by Löb it would prove f.
A machine/theory is sound if Bp -> p is true about it. This is far
more demanding than being just consistent. Many consistent machines
can be unsound, and 'believes' false arithmetical proposition. Those
are satisfied by non standard models of those arithmetical theories.
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.
Ah? All right. Just that logician makes everything explicit. There is
no guess needed, except in the process of finding a proof, not in
verifying that a sequence of formula is a proof. In particular formal
proof have to be automatically checkable. It can be proved that they
are so in the first order axiomatizable theories.
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?
In arithmetic, "p is wrong" means that it is not the case that p, or
that it is not the case that the usual structure (N, +, *) satisfies p.
Just saying something is wrong does not really give this a
meaning.
It gives its classical meaning. We accept the classical propositional
calculus, including (A v ~A). All proposition in arithmetic are true
or false. A machine stops or does not stops when running on some
input, for example.
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?
Well, a lot can be said here, but it might be very long, and I suggest
you to read some book in logic. Now, to help you shortly, to say that
1 = 0 is wrong should be intuitively obvious. The number 1 is not
equal to the number 0. 1 is the successor of zero, and no natural
number is equal to its successor. This is taught in high school,
mainly by the use of many examples. To eat 1 apple is not the same as
eating 0 apple. 1 $ is not the same as 0 $, .... up to the more
abstract statement that 0 is not equal to 1. See below for a
formalization of this.
Bruno Marchal wrote:
But it might still be usefully interpreted as "Try again (to make a
true
statement in arithmetics)"
That would be a higher level heuristic. If you allow "0 = 1" to mean
this, then we are no more sure what we are talking about,
So what does "'0=1' is false" mean? If something is false
intuitively it
means it is not valid in the context of a system.
Which system? The understanding of the intuitive notion comes before
we build a system. In the case of capturing the (admittedly mysterious
intuition of numbers) we ask any system to recover the most intuitive
propositions on which we are agreeing. The intuition comes from living
among a great number of examples. To say that "0 = 1" is false means
that we have the intuition that one thing is not the same as zero
thing, for most of the usual thing we can distinguish. The fact that 0
is different of 1 should not be dependent on any system. if a system
says that 1 = 0, we should just think that the system is talking about
something else.
If not I recall that you have borrowed to me 0$ today, so that you
have to give me back 1,004 $ tomorrow. The 0,04 is for the interest.
The same is true for 2 = 0. This is just wrong, even if many algebraic
system satisfies 1 + 1 = 0, because that is very useful in boolean or
electric circuits, for example, but then 0 and 1 means something else.
"2 = 0" is wrong about the natural numbers because we have the
intuition that 2 things are different from 0 things.
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?
Well, if you tell me that Obama is the king of Belgium, I will have
some doubt about the truth of that sentence.
Now in science, we are NEVER sure. I might wake up and remember living
in a world where Obama is indeed the king of Belgium.
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.
? (simpler? I don't even understand what you mean! Ah you explain
below)
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").
You can't send a man on the moon with such a theory, and you can't
interview machine with such a theory. I can accept such a theory as an
example of theory, but, sorry, we need more rich theory to progress in
the study of machine's theology (the local goal here).
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).
You just illustrate that we can make any sentence true by changing the
theory. But the goal is the contrary: we want that our sentence
reflects statements about simple structure that we can define, and
that our axioms are understood by a majority. Concerning the natural
numbers (and not tension in circuit) it is better to have 2 different
from 0, and the same for 3. You force me to give the axioms below.
Congratulation :)
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,
Which is the case in classical mathematics (on which quantum
mechanics, general relativity, computer science, etc. are based).
but it may express something
Anything might express something. but if we don't succeed in being
clear that our simplest expression expresses ONE thing, or the least
number possible of things, we cannot learn to discover finer nuances
in the sequence.
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).
You just do another theory about something else.
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.
Sure. But that is not relevant for the points we were discussing,
although I am getting lost here.
Actually if we don't get lost with arbitrary transformations it is
no worse
than standard arithmetics.
But only a good understanding of standard theory can help and
encourage the study of others, less standard theory. But in applied
math, we have to use the theory which are the most promising with
respect to the goal.
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.
I meant, I don't do philosophy publicly.
As soon as we state some "truth" we are doing philosophy.
If we state that something is true, we do (bad) philosophy. Note
that I have never said that some statement is true. Only that some
statement follows from some statements. I might use those concept for
illustrating some point, but I have never, and will never say that
some statement are true.
The concept of truth is very important though. (But that's different).
Mathematics has no meaning without truth.
That's true, when we *assume* some kind of platonism. Many philosopher
of mathematics are criticizing the idea that mathematics has no
meaning without truth, and I could defend the idea that a big part of
math does not need that concept, even implicitely. I need the truth
(by assumption) that a machine stops or doesn't stop.
There is a constructive part of math which does not need the notion of
{0, 1} classical truth, but they will use more subtle representation
of truth (like subject classifier in category theory, but that's
demanding in abstract algebra).
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.
Yes. That's the point. We are in the context of saying yes to a
digitalist doctor.
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.
I would say that with respect of the natural numbers, assuming you do
remember a bit what is taught in school, "0 = 1" has a clear meaning/
The meaning is that the natural number 0 is equal to the natural
number 1. I would say that this is rather clear, and rather obviously
wrong. About other type of numbers, or if we talk on something else,
it will have a clear or unclear meaning, depending on what you are
supposed to talk about.
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).
If God proves (by the usual means) that 0=1, it means that God is
inconsistent, from which provably nothing would makes sense. Oviously,
if God proves that 0 = 1, that can make sense only if we use "0
=1" (or proof) in some other sense that usual.
I think everything does make sense, including all falsehoods.
But if "0 = 1" is provable, then all propositions are provable. Or,
you are working in a private theory, and not in the public theory PA
(say) which capture an intuition we have about the natural numbers.
False is
relative lack of clarity or usefulness.
I think you are confusing classical logic, with the billion of other
non classical logics. You might develop a logic, like the relevent
logic in which the notion of truth (and falsity) corresponds to what
you say here. But keep in mind this: classical logic is the clearest
of all logic, so that when you want to describe a non classical logic
(usually more complex and subtle) you will have to use classical logic
to describe it. All books on fuzzy logics have theorems proved in
classical logic. Now, as mathematical logic illustrates, despite the
simplicity of its notion of truth, classical logic is already a very
complex subject. And classical logic allows our ignorance to be
manifest. It is the logic of the theologian, not necessarily of the
engineers, but then they can collaborate.
But I make it clear that it is a
phliosophical standpoint (at least now I did ;) ).
Not really. You were just suggesting some non classical logic. They
have many use in AI, engineering, etc. But they are build on the top
of classical logic.
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.
No. Only philosopher, who believes that the presence of term like
consciousness, mind, reality, dreams, etc. means that we are doing
philosophy. It is a bit like I am saying "E = mc^2" (thinking about
atomic bomb) and that philosopher replies "you cannot really say that
energy is the mass times the square of the speed of light", for
philosophical reason. He might be right, in philosophy, but might be
obstructive obstructive in "science-and-technic".
Of course not so many scientists understand well that science is not a
question of domain (like matter = science, mind = philosophy), but of
attitude and methodology.
The existence of number used in my work is the "trivial" notion of
existence, where the triviality is guarantied by our (certainly
mysterious) intution of the natural numbers. You just need to be able
to solve the following puzzle: -find the simplest next terms in the
sequence:
I,
II,
III,
., ?
., ?
...
By simple I mean, please, the less "cutting the air" next terms. Of
course there are realities where
"IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
" might be the successor of III, but we might suspect a "cutting the
air" notion of simplicity.
I manage to put the non trivial part in the theological jump you could
make by saying "yes to the digitalist doctor".
The notion of triviality may be not so trivial.
It is the less trivial notion. You are right.
But that is why it is a key to see we agree on elementary "trivial"
principle, like "if A is true, and if B is true, then A & B is true",
at least when you talking about numbers and their relations.
We have to start from common sense on simple things for realizing that
we loose control when adding a simple thing (addition for example)
with a simple thing (multiplication).
That already escapes all arithmetical theories, all machines.
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"?
They are set of axioms written in first order calculus predicate
language, with a precise formal language (&, v, ~, ), the quantifiers
"for-all" (A) and it-exists" (E), and the variable x, y, z, ...
They contain the first order predicate calculus with equality, the
classical tautologies, the modus ponens rule, and the first order
logic rules and axioms, ---all that is knwon as "classical logic", and
this together with the arithmetical symbols "0, s, +, *".
Arithmetical theories are then given by the arithmetical axioms:
Ax ~(0 = s(x)) (For all number x the successor of x is different from
zero). With
AxAy ~(x = y) -> ~(s(x) = s(y)) (different numbers have different
successors)
and with different sort of axioms, containing usually the addition laws:
Ax x + 0 = x (0 adds nothing)
AxAy x + s(y) = s(x + y) ( meaning x + (y +1) = (x + y) +1)
and the multiplication axioms
Ax x *0 = 0
AxAy x*s(y) = x*y + x
There is theory Q which has the predecessor axiom, a theory R, which
has some other axioms, and PA which has in addition the infinity of
formula of the shape
(F(0) & Ax(F(x) -> F(s(x))) -> AxF(x), with F(x) being a formula in
the arithmetical language (with "0, s, +, *), and the logical symbols
as said above.
They are many of them, but for the sound one, above some threshold
(mainly given by the induction axioms) they obey to the arithmetical
hypostases, and they can formulate the WR problem.
Can this even be defined? Maybe there
is an arithmetical theory where 0=1 is true.
Yes indeed, there is many one.
Take anyone above, and adds the axiom "0 = 1".
Well, then you have to weaken the logical rules in some ways (among
many) to keep the theory interesting. With most logic, classical,
intuitionist, even quantum logic having that "0=1" will trivialize the
theory (making all statement true). But in the relevant logic "0=1"
can be less damageable, but not yet so interesting. I suspect the
fifth hypostase to be an arithmetical relevant logic, and this could
mean that "0 = 1" can play a role. In fact, in G it does already play
a role to in the consistency of inconsistency. not provable false
implies not provable (not provable false), and take "0 = 1" as meaning
false.
If you are interested, you might have to study some logic books.
Those theories are the *object* of study. They are machine theorem
provers with infinite ressource (that "Platonia"). Logicians study
both the theories/machines and their web of interpretations/semantics.
It is a branch of math.
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.
OK. (I mean we might agree on this).
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.
Then pigs have wings. And here is a white rabbit, and two others there.
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.
It would mean, by definition, that God uses an inconsistent
arithmetical theory. Which one precisely?
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.
For all "enough rich" or Löbian theories, the border between right and
wrong countains the fractal shapes of the border between the provable
and the refutable. Universal numbers are confronted with the whole
richness of the geometry separating right and wrong. It is like the
border of the mandelbrot set.
Physical theories for example are
(as far as we now) never completely right, so are they wrong?
Very difficult question. (without or with comp).
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.
Exactly. Always, even when we talk about machine. But to study this
for machine necessitates to handle first the simplest case of ideally
correct machines, and the surprise is that it leads to a transfinite
ladder of difficulties. That little "universal (Turing) machine", or a
degree four Diophantine equation already put quite a mess in Platonia,
and the sound machine can only acknowledge the difficulties.
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.
Sure, but we have a tons of models, so we find them meaningful. Just
that we discover that they are rich of unexpected structures. It is
because we make the rule clear and simple that we can, by using the
"mathematical logical 'Hubble' to discover the complexity of the
subject.
Otherwise we can just reject them by not finding them meaningful and
thus
reject any proof derived from it.
That happens, and happened many times. Both Curry and Church, and
Frege, have developed theories which have fallen down literally, by
being shown to be inconsistent. New theories emerges from that. That
happens all the times. Sometimes a theory survives by its
interpretation evolving, and vice versa.
There is a 'Galois connection' between theories and their
interpretation, like between equation and surface. That is: the less a
theory has axioms, the more it has interpretations.
When we have to take into account the internal interpretations build
by the objects of the theories themselves, it is even far more
complex, but both comp and the quantum gives non trivial hints.
Bruno
http://iridia.ulb.ac.be/~marchal/
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