----------
>
From: George
> Subject: RE: [03]: "Objective" and
"Subjective"
> Date: Tuesday, July 22, 1997 11:33 PM
>
> On Tue, 22 Jul 1997, Leigh wrote:
> re Gödel's
proof
>
>
> > I have yet to read an
English-language summary which
> > conveys the 'grounds,'
as you put it, for the proof, while leaving
> > out the
formulae. Maybe Rob can provide such a bite-sized
>
>explanation, but I think you'll probably have to study
something
> > a little more elaborate than anything he or
the other mathematicians
> > here can conveniently post
before you'd be satisfied that you were
> > not being
asked to accept a counterintuitive assertion
> > merely on
someone's say-so.
I concur with Leigh here and I doubt that a really satisfactory answer can be achieved much short of going to more technical stuff than I'm prepared to provide. However, below, I will say something more about it.
> I may have
learned it in school, and forgotten it. In any case, it sounds
>
interesting to me, and it seems to have a lot to say about the idea
of
> truth; for example, that statements can exist within a
logical system
> that are neither true or false.
Well, there are logical systems in which some statements are neither true nor false, but that's not an issue Gödel's theorem speaks to. Gödel's theorem is formulated and proved in the bivalent logic of Principia Mathematica. Every statement in the proof is either true or false.
To get some notion of what the theorem shows, you need the notion of "well-formedness." A statement is well-formed if and only if it obeys the "grammatical" rules for putting together a statement in a given logical system. For example, in the language of arithmetic, "3 + 2 = 6" is well-formed--false, but well-formed. It has the kind of structure of elements that lets it be used to make a statement that is either true or false. By contrast, "3 + =" is not well-formed. It's neither true nor false, but that's not because you need some other truth values in arithmetic than true and false, but because it hasn't managed to make a statement in arithmetic at all.
Now, running roughshod over some important mathematical details (for which you will have to find other sources), what Gödel did was to show how to make a formal system that was powerful enough to express basic arithmetic talk about itself. In effect, he showed that, in such a system, we could construct a well-formed sentence that says of itself that it can't be proven in that system. So, if it's true, it can't be proven and if it can be proven, it's not true (and the axioms of the formal system are inconsistent). Hence, the system is incomplete in the sense that if it is consistent, some true statements in it can't be proven from the axioms.
Now, you might be tempted to think that this just means that the system doesn't have enough axioms. Instead of the five postulates of Peano arithmetic, we really need Peano+ with six or seven postulates. But that would be to misunderstand Gödel's result. It applies to any formal system poweful enough to handle basic arithmetic. The six- or seven-postulate system would also be incomplete in the same way. No matter what you add to the system, there will still be well-formed statements in it that are (if you haven't made the system inconsistent) both true and unprovable.
> What I was
questioning was the idea (seemingly not even asserted, but
>
assumed, that the truth of the "methods of proof from outside the
system"
> is somehow established by their necessity to the
system's consistency.
> This seems a rather wrong-headed
notion of "truth."
No, that's not what was meant. The point (one of the points) of the Gödel proof is that the only way to offer a proof of a given system's consistency (again, if it's at least powerful enough to handle basic arithmetic) is by appealing to a more powerful system which itself can't be proved consistent without appealing to a more powerful system which ....
Of course, that
doesn't mean that there's any reason at all to suppose that
mathematics or logic are inconsistent, just that we can't
have a proof of a certain sort that they are not.
Rob
----------
> From: George
> On Fri, 18
Jul 1997, Chris wrote:
>
> >
> > Well, you
have to be a bit more precise here; "true" means with
> >
respect to a given language and axiom system. There are things
>
> you can express in (e.g.) Peano arithmetic that you cannot
>
> prove true in that system (with associated proof rules); but
>
> you can prove them true in a stronger language. [Of course,
>
> there are then statements in the stronger language that cannot
> > be proved within that stronger language.]
>
>
That's the conclusion I got from the discussion on Gödel's
theorem; the
> limitations on being able to prove truth within
a system; not the
> existence and (by definition or some
equally tautological standard) proven
> correctness of "true
propositions that can't be deductively derived"
> within
that system.
I'm not sure quite what you're saying here. Of course, the truth of the Gödel-sentences (the ones that are true but not derivable from the postulates) for a system representing Peano-arithmetic are relative to that logical system, but they are "relative to the system" in just the same sense that the truth of "2+2=4" is relative to the system.
Rob
George –
Chris's
response on my behalf (if that's what it was) was very near to
what I would have said and far more concise. I'm responding
because you
indicated enduring puzzlement as to what I thought
the lessons of Gödel's
theorem were.
----------
> From: Chris
>
> George wrote:
>
> [re: Gödel's
Incompleteness Theorem]
>
> > As for what I mean,
not much; I'm just trying to find out more about
> > the
meaning and relevance of Gödel's proof.
>
> Be
careful here; its relevance is minimal to anything outside
>
mathematics. People like to draw philosophical conclusions from
>
it, just as they do for Heisenberg's Uncertainty Principle, but
>
these conclusions are seldom if ever justified.
I think there're some ways in which it has philosophical significance. This may be a matter of professional obligation: Philosophers are expected to "see philosophical significance" where others do not. Sometimes the obligation can also be discharged by not seeing philosophical significance where others do, e.g., in sonorous tautologies like "existence exists." :-)
I agree with Chris (I think) that several prominent claims made on behalf of Gödel's theorem are unwarranted, for example, that it vindicates mathematical Platonism (Gödel himself seems to have thought that) or that it proves the falsehood of determinism or mechanism (in philosophy of mind) or that, in the hysterical-sounding words of Frege (he was talking about something else), "arithmetic totters!"
One conclusion of philosophical significance that I think is warranted by Gödel's theorem (though it may be of little general interest here) is that some positions in the philosophy of mathematics are ruled out, such as the view that math is just the manipulation of tautologies (held at times by Wittgenstein, Russell and, lately, by Doris, here) and what was called "finitism" (espoused by David Hilbert)--which sought just the kind of self-contained consistency proofs that Gödel showed were unavailable.
Another may be of more general interest. In mathematics, for most purposes, knowability and provability are equivalent. What you know is what you can prove. But Gödel's theorem shows that even in a rigorously deductive arena like math, truth outruns knowability: there are truths we can't prove (without going outside the system)--and we can prove that! If it weren't already obvious, I'd say this strongly suggests that in other domains where conclusions are warranted in ways that do not logically imply the falsity of their alternatives, that truth also outruns knowledge or knowability. But in fact I think it is obvious and would hardly be worth stating if it weren't for persistent attempts to "reduce" truth to what we know or accept or are warranted in accepting or, in a Rortian vein, what our cultural peers will let us get away with. Truth is not what's intersubjectively accepted, but what should be.
> > As I
read (about) the proof,
> > I got the idea that it was
talking of the impossibility of "absolute"
> > truth;
that things are true only wrt a given system, and not always
>
> provably so within that system.
I think you need to be careful about "things are true only with respect to a given system." One question is: is it true? If it is, then it's a truth with regard to a different system than the original truth(s). (And is it true about that system that things within it are only true with regard to it?)
In some sense, there are no truths that are completely "free floating" apart from any conceptual scheme or system for representation. But the demand for that kind of free-floating "truth" is absurd and incoherent. The fact that we don't have it isn't a limitation on the kinds of truths we have because the alleged alternative is unreal. What we have are truths or what we take to be truths by our best lights and there's a complicated interplay between what we take to be true and what we take to be the best standards for judging claims about what is true. We improve our stock of truths in the light of our standards; we improve our standards in light of what we take to be true.
[One thing that I think leads to confusion here is failure to realize that if something is true "relative to a system" that doesn't mean that we just make up truths because we, in some sense, create the systems in which they are expressed. An example may help. Once we have the system of number theory and the ability to talk and reason about properties like primeness, it is true relative to that system that there is no largest prime number. But it's not true because we decide to make it true. We didn't. We created the system and in the system, it is true: If we thought otherwise, we'd be mistaken.]
> > Yet
you seemed to draw the opposite
> > conclusion, that it
proves both the possibility and certainty of absolute
> >
truth.
>
> I'd be surprised if Rob drew that
conclusion from the theorem.
Chris is right, of course. I have no idea what "absolute truth" is unless it just means the absurd ideal of truth that is completely free-floating, that "exists" apart from any conceptual scheme or representation.
> > Your
example gives me an idea, the best way of which to test is to ask
>
> you: do you see the phrase "true propositions that can't be
deductively
> > derived" within a system as meaning the
fundamental axioms of
> > the system?
>
>
This is misunderstanding what's going on, I'm afraid. It is true
> that axioms cannot be proved within a system (if they're
independent
> of one another), and that they are assumed to be
true within that
> system. However, "true propositions that
can't be deductively
> derived" in the sense involved with
Gödel's Theorem are not
> axioms. They can, of course,
be added as axioms, but this is
> a never-ending process. The
point is that they can be proved
> true outside the system
they can be expressed in; while axioms
> cannot be proved true
at all (unless they're theorems in another
> system; but
that's a different kettle of fish).
I'd add a little here. There's one sense in which the axioms in a system can be proved within the system. If the axioms are really axioms, then they can be used as premises and then they can be derived--from themselves. If you're really in doubt about the axioms, that won't be very satisfying. But the "true propositions that can't be deductively derived" can't be given even that kind of unsatisfying proof.
> > And do
you see them as necessarily true (and possibly proven
> >
true) thereby?
It's necessarily true that there are true but unprovable Gödel-sentences, but that doesn't mean that a particular Gödel-sentence is necessarily true. (Compare: It's necessarily true that I either have blue eyes or do not, but that doesn't mean that it's necessarily true that I have blue eyes (although it is true).
> > That
sounds very much like what my dictionary calls a "coherence theory"
> > of truth: "A proposition is true insofar as it is a
necessary constituent
> > of a systematically coherent
whole." (322). I don't know enough about
> > that idea
to comment on it; all I'm trying to accomplish here is to
>
> find out if that is indeed what you're saying.
>
>
I doubt that it is.
Right. That's not what I was saying.
Rob
----------
> From: Leigh
>
> From: Rob Bass
> Sent: Friday, August 08,
1997 12:54 PM
> >I agree with Chris (I think) that several
prominent claims made on behalf
> >of Gödel's
theorem are unwarranted, for example, that it vindicates
>
>mathematical Platonism (Gödel himself seems to have thought
that) or that
> >it proves the falsehood of determinism or
mechanism (in philosophy of mind)
>
.]
> >Another
may be of more general interest. In mathematics, for most purposes,
> >knowability and provability are equivalent. What you
know is what you can
> >prove. But Gödel's theorem
shows that even in a rigorously deductive arena
> >like
math, truth outruns knowability: there are truths we can't prove
>
> Ooops, don't think you meant to say that. I think you
meant to say, "truth
> and knowability outrun provability."
I did mean that. Remember, a couple of sentences above, I said that in math, "knowability and provability are equivalent."
> >Truth
is not what's intersubjectively accepted, but what should
be.
>
> Perhaps it's a little of both? At least I
find persuasive appeal on both
> sides of the argument. Either
both sides are correct in their own way, or
> I'm just
indecisive. While the latter is certainly possible, it's not
>
terribly usual in my case. "Frequently wrong, but never in doubt,"
that's
> my motto!
Intersubjective agreement is often a good test for truth. You assume that if something were mistaken, lots of people wouldn't have or would be unlikely to have agreed to it. But it's not the same thing because agreement can spread in ways that are insensitive even to easily ascertained facts. When I was in elementary school, it was widely believed by kids that men had one less rib than women. Presumably, this was due to the Genesis myth about God taking a rib from Adam to create Eve combined with adult reluctance to challenge the religious beliefs of others. (I even remember a teacher "agreeing" to this in response to a question! I hope she didn't really believe it.)