

Does Gödel Matter? - mlLK
http://www.slate.com/id/2114561/

======
alan-crowe
Trouble is, there are completeness results. You can prove everthing that is
true in all interpretations of first order logic. The statements that cannot
be proven are unproveable because they are false in a non-standard
interpretation of arithmetic. It is the existence of non-standard
interpretations that is the big surprise, and the fact that statements are, in
a sense, false that blocks proving them.

The equation whose roots are the Godel numbers of the proofs that it have no
solutions has non-standard solutions. That is why there is no proof that it
has no solutions. But the non-standard solution is the Godel number of, err,
well what exactly? The trouble with the extra, non-standard numbers is that
you cannot reach them by counting, so the proof they number is only finite in
the non-standard interpretation. You cannot write it out. It doesn't count
because we require that proofs be finite, and by finite we mean finite in the
ambient meta-theory, which takes the standard model of the natural numbers.

It would be much better to sumarise Godel's theorem as showing that first
order logic fails to tie down the nature of infinity.

Do we care about non-standard arithmetic? If memory serves, there are proofs
that non-standard arithmetic is not computable. I can see why Bernie Madoff
might find that convenient, but the rest of us can regard the uselessness of
non-standard arithmetic as mathematically proven.

Godel's proof shows that first order logic is limited by the existence of
mathematical monstrosities that we cannot use (because their arithmetic is not
computable) and never encounter (because you cannot reach them by counting)

~~~
harshavr
Isn't this broader than the limitations of first order logic? After Turing we
know that no matter what formal system of axioms and rules of inference one
choses(as long as these can be mechanically done) then there will be some true
statements the system wont be able to see. (does the nth turing machine stop?,
does a particular diophantine eqn have integer solutions?)

------
ionfish
I've read the book the article ostensibly reviews, and to be honest it's not
much cop. It's full of, frankly, a load of rubbish about post-modern thought
and the extent to which Gödel's work might vindicate it. There are certainly
interesting aspects to it, but it's neither a good biography nor a
particularly good introduction to incompleteness, and is liable to merely
frustrate anyone possessed of a passing acquaintance with either subject.

~~~
socratees
I read the book, and it was sometimes boring, still there's a lot of positive
things makes you continue with the book. I got to learn about the works of
Godel, Escher and Bach. And then about the patterns which exist in nature, or
any work, where we never thought it could be a pattern. Anyways, I'm not a
mathematician or a physicist to comment on Godel's work. I think the book is
worth reading, whether it contributed to computer science or not, i do not
know.

~~~
andreyf
_I got to learn about the works of Godel, Escher and Bach..._

You're thinking of the wrong book...

~~~
Haskell
I think he is a spam bot that uses Markov chains.

------
Tichy
Didn't he put forward two theorems? One about the undecidable statements as
described in the article, but the other was much "worse": the consistency of
mathematics can not be proven. That is for any axiom system with a certain
minimal strength (like strong enough to derive natural numbers) it can not be
proven that the axioms don't contradict each other.

Hopefully I am citing it right, as I am too lazy to look it up. But I think
that is more of a ticking bomb sitting in the guts of mathematics. It could
happen that one of these days the whole building of mathematics collapses
because a contradiction in the axioms emerges. Apparently the common way to
deal with it is to just deem it very unlikely since things went fine for
centuries, and there is not much that can be done about it anyway.

Must finally read Gödel's proofs again, have been meaning to do so for ages
:-/

~~~
arakyd
The second theorem says that the consistency of a system cannot be proved
_within that system_ (sufficiently strong, etc).

If a contradiction is discovered in set theory, either it will be fixed by a
change to the axioms, or another set of foundational axioms altogether will
become the standard foundation of mathematics. Perhaps the canonical example
of this is naive set theory itself - paradoxes were discovered, but set theory
was interesting enough that mathematicians looked for alternate definitions
that did not contain the paradoxes, and they succeeded.

There are two things to note here. First, axiomatization was a useful tool in
distinguishing various types of set theory and defining the ones that did not
contain the (known) paradoxes. Second, the full formal treatment of these set
theories came relatively late to mathematics, as did widespread use of formal,
axiomatic methods in general. Rigorous axiomatic formalisms are very useful
mathematical tools, but mathematics got along without them for a long time
(not even Euclid counts, by modern standards) and can do so again if
necessary. It is unlikely that this will happen, not because axiomatics can be
proved to always work or because mathematics cannot get along without it, but
because it is too useful a tool to abandon easily.

This was really Godel's point: mathematics is not identical with formalism.
They stand and fall separately. This is not to say that mathematics could
never collapse for any reason, only that it would take a lot more than finding
a paradox at the center of ZFC to make it happen.

~~~
Tichy
Hm, I thought set theory already contains arithmetic, so wouldn't Goedel's
proof show that such axioms without paradoxes can't be found? I thought that
is why he is so famout - for years it had been the goal of mathematics to find
the ultimate axioms, and then Goedel showed that it can't be done, ever.

~~~
cchooper
The second theorem tells us that we can't prove the consistency of set theory
without using an even more powerful system. This doesn't mean that set theory
is inconsistent, it just means that we can't prove its consistency in any
meaningful way. So yes, Gödel indeed shows that there are no 'ultimate axioms'
that contain everything of interest in mathematics.

~~~
mstoehr
Although it does lead to a mathematically uninteresting paradox: if you let A
be the axioms of set theory and you add an axiom P which states that A proves
x and not x, (i.e. set theory is inconsistent) then A' = A and P is still
consistent.

~~~
d0mine
For clearity: A' = (A and P)

~~~
cchooper
Ah, that makes _a lot_ more sense now!

------
amix
I still don't think we have grasped the importance of Gödel's incompleteness
theorem. For example, indirectly it means that we can't ever solve the halting
problem. Personally, I also think it applies to physics, especially the idea
to find a theory of everything. I think theory of everything won't be possible
simply because the incompleteness theorem states that "given any system of
axioms that produces no paradoxes, there exist statements about X which are
true, but which cannot be proved using the given axioms." (where X can be
numbers, computer programs or ...)

~~~
antiform
This is exactly the kind of thinking that the article argues against. Remember
that Goedel is speaking within the context of formal systems, so there is
relatively little impact to any area outside of the small field of philosophy
of mathematics. Anybody who tries to extrapolate beyond those bounds is either
mistaken or misled as to what Goedel's incompleteness theorems actually imply.
There is a much more thorough and eloquent explanation in the following link.
[<http://www.ams.org/notices/200604/fea-franzen.pdf>]

Hell, even Einstein, who was close friends with Goedel and familiar with the
result, was searching for a more fundamental physical theory than quantum
mechanics, so clearly he did not believe it was a lost cause.

~~~
yters
Do you know of anyone who has rigorously determined how Goedel's theorems
apply to anything outside of mathematics?

While his theorems show the shortcomings of a certain approach to math, I
don't think math is the only area where this approach is applied. For
instance, granting Goedel has shown math is Platonic, i.e. it is about
independently existing entities, not formal structures, then why does it make
sense to say the mind that can grasp math is itself a formal system, i.e. the
assumption behind strong AI?

So, it is more accurate to say Goedel's work is restricted to domains where
formalization (of the correct sort) is applied, which covers more ground than
just the philosophy of mathematics.

~~~
antiform
What you say in your last statement is true and helps to clarify a subtle
point. I tend to shove all of that stuff (logic, set theory, problems of
undecidability, etc) into the nondescript box labeled "philosophy of
mathematics." When I meant a field was "small," it was not meant to be
pejorative. Philosophy of mathematics is a rich and productive field with many
applications within its varied fields. The main problems I see are the ones
described the article I linked, in that people attempt to apply the theorems
to things like physics, politics, law, or postmodern philosophy.

Of course Goedel's theorems do not totally exist in a bubble that only
contains arcane mathematical incantations. For instance, Goedel's theorems
imply (like the halting problem) that you cannot create a compiler that would
be able to determine whether a program will not terminate and reject it, by
considering what happens when you feed it a Goedel statement. However, this
problem is something that I would also classify under "philosophy of
mathematics."

Also, I hope I did not provide any implication that my view (or any view, for
that matter), is the final word on the subject. I'm not dead-set in my own
beliefs on incompleteness and if you can provide a convincing argument of a
rigorous, nonmathematical application of Goedels theorems, I'd be more than
happy to change my mind.

There is considerable discussion as to what exactly the incompleteness
theorems imply, and there are many eminent minds on the many different sides
of the argument. For instance, I believe Stephen Hawking is somebody who
believes that physics cannot ultimately be formulated into a final number of
finite principles. Do I believe this to be a rigorous application of Goedel's
incompleteness theorems? No. Do I believe that this can be applied to fields
outside a relatively restricted problem space? No. Does this mean I'm right?
Of course not.

As to your point about Strong AI, I don't want to open that can of worms in
this thread. I'm not yet convinced that the human mind is equivalent to any
finite state machine, so I'm afraid that we don't even agree upon its the
basic premises. However, I would love to discuss it any time outside of the
thread.

~~~
yters
I checked your profile and you don't have any contact information. I'm curious
about what you think the mind is. Most seem to think it is an "emergent
property" of the brain.

------
noonespecial
Well, it wouldn't be much of a braid with just Escher and Bach, now would it?

~~~
gaius
Escher and Bach, a semi-permanent silver braid.

~~~
noonespecial
No, I think that'd just be a twisted pair! :0

------
zitterbewegung
Yes.

------
Allocator2008
Extending the standard model to include gravity, and thereby creating a "TOE"
(Theory of Everything), is not I think going to be hurt by Godel. Because for
TOE to work, all it need do is make predictions over and above the standard
model, which can be verified by experiment, but also be consistent with all
predictions of the standard model. If it can do this, it will stand as a TOE,
at least until it disagrees with prediction, if and when that would happen. In
other words, a TOE need not be "provable to be consistent in all it axioms".
All it needs to do is include the Standard Model's predictions, then, make a
few new ones to include gravity. Like one nice prediction would be to have the
value of 10^-120 plank units for the energy density of "dark energy" arising
naturally and nicely from someplace. That would be a good prediction, just as
an example. If a TOE (inclusive of the standard model, plus a description of
gravitons which delt with infinities), could regurgitate all standard model
predictions, plus just make one prediction more (like setting the energy
density of dark energy without having to appeal to the anthropic principle,
mind you), then we got a good TOE. "Provable?" No. But it lives to fight
another day, which is the best we can ask of a theory anyway.

~~~
fauigerzigerk
That's interesting, and I do think that something empirical needs to come into
play in order to escape issues with formalism without cheating (Sorry about
the incredible vagueness of this sentence). However, what makes me wary of
proof by prediction is that it has failed in the past and it is all too
obvious why.

At the time, proponents of the Ptolemaic system were able to create a very
complicated model that predicted the movements of planets reasonably well
(don't know exactly who it was). So judging them by their predictions might
have lead to the conclusion that the earth was indeed at the center of the
universe.

Also, just try to write unit tests to "prove" that some piece of code is
correct with regard to concurrency. What if a particular problem occurs so
rarely that it is very unlikely to be caught in any empirical test? What if
its occurrence depends on factors that are not well known or hard to
replicate?

The problem with predictive models is that they sometimes break down
unexpectedly. That doesn't make them useless, but it makes the distinction
between logical proof and empirical proof all the more useful.

