
Gödel's Theorem - bookofjoe
http://bactra.org/notebooks/godels-theorem.html
======
jeromebaek
For some reason it's fashionable to bash philosophical implications of Gödel's
Theorem(s). Not only is the internet filled with articles like this, there is
at least one book dedicated to bashing on philosophical implications of the
theorem, the one by Torkel Franzen (whose most notable work is that book).

I find this sort of allergic reaction to be interesting in its own right. Yes,
there are pseudointellectuals who abuse the theorem, but they're just that,
pseudointellectuals, and academics usually don't engage with
pseudointellectuals. I suppose the situation is similar with Aaronson's
despair at the layperson's misunderstanding of quantum computers, but so far
as I can tell Godel's theorem is virtually unknown among the masses, in any
case much, much less known than quantum mechanics. So it's interesting why
academics feel the need to have this sort of allergic reaction - in the case
of quantum mechanics it's more understandable.

So the more interesting because Godel, the man himself, was preoccupied with
philosophical implications of his theorem. This is the man who "proved" God's
existence with modal logic, the man who spent decades after publishing his
most famous theorems in a dusty study writing philosophical (and extremely
rigorous) implications of his theorems that hardly anyone read, the man who
eventually starved to death because he thought - his philosophy led him to
believe - he was being poisoned.

In fact, Godel's theorem ought to be used more, not less, in philosophy. Some,
like me, believe it has profound implications to moral philosophy. See:
[https://arxiv.org/abs/1805.08347](https://arxiv.org/abs/1805.08347)

But this is a frustrating position to be in for someone like me, because
philosophers don't understand Godel's theorem, and mathematicians/computer
scientists don't care for philosophy. Instead they spend their time writing
pieces like this that justify them for _not_ caring about philosophy. It's
pretty lame.

~~~
duckerude
> In fact, Godel's theorem ought to be used more, not less, in philosophy.
> Some, like me, believe it has profound implications to moral philosophy.
> See: [https://arxiv.org/abs/1805.08347](https://arxiv.org/abs/1805.08347)

I've only skimmed it, and I'm not very familiar with moral philosophy, but
this seems bizarre. If I understand it correctly, your claim is:

1\. Free will is uncomputability (undecidability)

2\. A good action is a free action (and therefore uncomputable)

3\. A bad action is an action that pretends an uncomputable system is
computable, thereby denying agency and humanity

I think 1. is suspect because people are not proper Turing machines. They have
limited storage and limited running time. They are not currently practically
computable, but it seems physically possible to simulate them. I don't think
Gödel's theorem applies, although practical uncomputability might be a working
substitute.

If you predict that some person will take some action, that's almost always a
prediction that they will take that action within the next ten
seconds/hours/decades. Actually simulating them is impossible for various
reasons, but Gödel's theorem isn't one of them. You can't prove whether an
arbitrary program will halt, but you can prove whether an arbitrary program
will halt within the next hundred thousand execution steps.

2\. seems like a really bizarre way to define goodness. I think morality is
subjective enough that you could define it like that, but I don't know why
you'd want to. I think the conventional view is that free will is the
requirement for both good and bad actions, not that free actions are by
definition good.

I think 3. might cover almost every thought we have about other people.
Internal computable models of other people are how we operate, as far as I
know. Again, you _could_ define badness like that, but I don't see the point.

> [page 8] Now the question paraphrases to: does there exist an algorithm for
> me to love my child? Of course not. So the distinction between [shallow
> benefits] and [deep] benefits is that: there is an algorithm to bring about
> [shallow benefits], whereas there is no algorithm to bring about [deep]
> benefits.

Doesn't that conflict with the idea that we are Turing machines? If we are
Turing machines, then everything we do is the output of some (possibly
uncomputable) algorithm, including loving people.

I don't think this engages with Gödel's theorem in a useful way, or even
depends on it.

~~~
ngcc_hk
Does these include any arithmetic statement? Any recursive self referral
statement involved?

If not, Godel theorem is irrelevant

~~~
duckerude
It does:

> [page 5] Just as humans can have free will because we can look at our own
> mental activities from a distance, so a program can be uncomputable because
> it can take as input its own self in data form.

But I don't think it holds up, because we don't have full access to our own
mental activities, only to a grossly simplified model of our own mental
activities.

------
dvt
Godel's First Incompleteness Theorem isn't particularly interesting, can be
proved fairly easily (to an average high-schooler)[1], and, therefore, doesn't
really have much "meat" on its bones. The _Second_ Incompleteness Theorem, on
the other hand, is another beast altogether[2]. It's much more interesting,
much more beautiful, and much more difficult to grasp.

At the end of the day, I agree with the essay (too many people misuse Godel --
starting with GEB, and onward), but I don't like the justification behind the
post. After all, the First Theorem is used to drive towards the Second. It's
intellectually lazy to just cover the first without touching the second: "Any
consistent formal system S within which a “certain amount of elementary
arithmetic” can be carried out cannot prove its own consistency."

In other words, I think the essay falls prey to exactly what it's criticizing:
a purposefully neutered explanation of Godel to arrive to some unrelated
conclusion.

[1] [https://dvt.name/2018/03/12/godels-first-incompleteness-
theo...](https://dvt.name/2018/03/12/godels-first-incompleteness-theorem-
programmers/)

[2] [https://dvt.name/2018/04/11/godels-second-incompleteness-
the...](https://dvt.name/2018/04/11/godels-second-incompleteness-theorem-
programmers/)

~~~
joe_the_user
"too many people misuse Godel -- starting with GEB, and onward."

Assuming this is Hofstader's Godel Escher Bach, I'd like to see support for
this claim. GEB is just a popular exposition of Godel and a paean to self-
reference. It's not a rigorous work but I don't know of any terrible
violations it involves.

~~~
earthicus
GEB not just a popular exposition of mathematical logic, the book is quite
explicitly about philosophy of mind. In particular, he develops the self-
reference stuff so he can draw an analogy to brains and tell us his theory of
consciousness. As for support for that claim, he clarified the point of the
book in an elaborate preface to the 20th anniversary edition, and that still
wasn't enough to get the point across, so he wrote an entire second book ('I
am a Strange Loop') to explain his theory of mind again.

~~~
Zanni
How is that a "misuse" of Godel's Theorem? He doesn't draw conclusions from it
about consciousness. He only points to it as an example of a strange loop.
Godel's Theorem is an analogy, not an explanation.

~~~
earthicus
That's not at all an accurate description of the book. Goedel is not just used
as an example of a strange loop, he bases his entire model of mind on it. This
is developed in chapter 20. Particularly noteworthy are the sections of that
chapter titled "Undecidability is inseparable from a high-level viewpoint",
immediately followed by "consciousness as an intrinsically high-level
phenomenon".

It's true that he uses it as an analogy, but not a loose or vague one. For
example, "But it is important to realize that if we are being guided by
Goedel's proof in making such bold hypotheses, we must carry the analogy
through thoroughly [...] If our analogy is to hold, then, 'emergent' phenomena
would become explicable in terms of a relationship between different levels in
mental systems." p708

He's literally taking the logic of Goedel's proof, transferring it to brains
step by step, to argue for a particular theory of consciousness.

------
undershirt
This seems to throw more shade on the problem for me. There is a profound
sense that there is no mode of reasoning (nevermind just the mechanical
deductivism) that can completely capture all expressible truths without
reaching contradiction.

The nature of knowing what is true is to capture a slice of reality in a
model, and a model by its essence must leave out details, considering only
elements which are important to its function. It seems that all of language,
including anything we create with it, is an incorrect but useful model of
Truth. Different models describing different things are expected to produce
contradictions, but this math thing was talking about itself and it couldn't
even do THAT consistently, right?

If Godel's theorem seems to be blown out of proportion as a synecdoche for the
hundreds of years of collective malaise about unattainable truth, it is at
least appreciable (and relevant to this wider domain) that Godel devastated
the world by showing how logic fails at proving the consistency of a simple
mathematical system (correct me if I'm wrong—it's not just axiomatic systems,
"there ain't no such animal" of a consistency proof).

Anyway, there is a popular distinction between lowercase-t truth and
uppercase-T Truth. The uppercase form represents knowledge we can never
understand (outside our Circumference), and the lowercase form represents what
we understand now. Since our understanding constantly changes over time, I
imagine truth being a function of time, t. As t approaches infinity, truth(t)
approaches this limit of Truth.

I highly recommend this poem (analyzed by Nerdwriter) on the expressibility of
truth as well: [https://youtu.be/55kqNg88JqI](https://youtu.be/55kqNg88JqI)

~~~
brobdingnagians
Like the sister comment by clairity, I agree that you are extrapolating beyond
the meaning of the theorem. You can prove the truth value of the statement,
but not in the axiomatic system, it requires having an extended system. So it
would be more of a moral of the story that sometimes you just don't know
enough, but with more information or context you could prove it is true (but
even that moral is probably off). It states more about provability in the
context of a single axiomatic system for constructed self-referential
statements rather than truth in general.

~~~
smallnamespace
> sometimes you just don't know enough, but with more information or context
> you could prove it is true

Yes, but that larger system itself contains statements that are true but
unprovable in itself, and so on in an infinite regression.

And before one hand-waves it away as a mere 'hole' that we can safely ignore,
Chaitin extended Godel's work to show that there are in fact an infinitude of
'relevant' mathematical statements that are unprovable within any formal
system S:

[https://en.wikipedia.org/wiki/Kolmogorov_complexity#Chaitin'...](https://en.wikipedia.org/wiki/Kolmogorov_complexity#Chaitin's_incompleteness_theorem)

~~~
heavenlyblue
Then if you're discussing the topic in the context of "whether our current
mathematics is useless in the context of Godel's theorem" \- then we can
easily define progress in mathematics as "how many contradictions we had found
today" and thus continue living with an objective till the heat death of the
universe.

------
Kinnard
I was actually discussing this today, especially the first fallacious
conclusion. With a Ph.D. candidate at Stanford no less . . .

I don’t think Gödel was demonstrating the "limitations" of logic. I think he
was demonstrating the limitations of lower logics so that we could wield
higher logic. Just as Cantor broke through into the transfinite.

“From the paradise, that Cantor created for us, no-one can expel us.”

I don’t see Gödel’s theorems as expressing the failure of logic, I think the
holes(incompleteness) it illuminates are doorways.

Gödel prompted people to explore things like plurivalent logic and fuzzy
logic, at least in the “west”.

[https://aeon.co/essays/the-logic-of-buddhist-philosophy-
goes...](https://aeon.co/essays/the-logic-of-buddhist-philosophy-goes-beyond-
simple-truth)

~~~
jeromebaek
Once I had a similar conversation with Scott Aaronson about this. I said the
theorem shows the limits of logic; he said it shows the POWER of logic. My
reply was: what does it matter? It just depends on perspective. As in: "logic
is so powerful, it can even demonstrate its own limitations!" or: "logic is
limited, and we know this for eternity, because logic _proved_ it!"

~~~
empath75
It’s not really a limitation of logic, because logic is not constrained to a
particular set of axioms.

~~~
googlemike
But logic is composed of a particular set of axioms and rules - so is not any
thing composed of a set of such things itself constrained by the very nature
of its parts?

~~~
empath75
Godels theorem has nothing to say about first order logic.

------
speedplane
This article tries to be far more controversial than it actually is. Godel's
theorem's don't negate truth, obviously some thing are clearly true (1+1=2)
and false (1+1=3). His point is that there are some things that are
unknowable.

------
lihaciudaniel
Controversially is this:
[https://en.m.wikipedia.org/wiki/Gödel_machine](https://en.m.wikipedia.org/wiki/Gödel_machine)
\- Gödel machine who supposedly is a seed AI who is based on his theorems?

And for those who say that we need to use fuzzy logic (which is used in neural
networks, Bayesian probability theory) still won't solve paradoxes and what
Gödel showed us.

Here some food for thought, imagine perfectly crafted sentences with logical
induction (list like- where the wrong one gets deleted and the right one goes
one list up). Even if we manage to make all sentences consistent (the one who
are true) there still be one comment who can not be denied or approved. So if
someone wants to build a Gödel machine (philosopher machine) he can use
reddit/hn like Turk Machine (pitch it as a seed AI Gödel machine, this idea is
obviously absurd but interesting nonetheless)

Edit: mathematics , as Russell said, is the most precise tool at reaching
clarity so imagine what language fails at attaining.

------
blauditore
On a related note, I have the impression many people don't really understand
the implications of the halting problem. You often hear the famous sentence,
"it's impossible to build a program which decides whether another program
holds", and use this as an argument that any attempt at building such
developer tooling is futile.

But in fact, the whole proof is based on the self-application problem, where
it's roughly analogous to the liars paradox. But the proof doesn't say
anything about other, non-"reflective" cases. Theoretically, it might be
possible to build a program that makes correct decisions whenever that's
possible, and otherwise ends up in an infinite loop (e.g. in the self-
application case). I wonder if there are any efforts in this direction; so far
I've never heard of it.

~~~
szemet
On the non-reflective cases we at least see that we can forge small but very
hard problems too. The most simple ones are the infinite loops testing some
whole number related property, like: Goldbach conjecture, Fermat's theorem.

Now we know the program of the second case never halts but it was not an easy
proof, the first one is still open.

------
yters
The article seems to say two contradictory things: A) Axioms are not truths,
so G's theorem doesn't mean our ability to grasp truth is limited. B) G's
theorem does not imply the human mind is outside of axiomatic systems.

Isn't it the case that if B is correct, then the mind is contained within some
axiomatic system, therefore A is wrong (i.e. our ability to learn truth is
indeed limited by G's theorem)? I do not see how both A and B can be correct
at the same time.

------
feanaro
Isn't his second point incompatible with the first one? In the second, he
implies that human minds might be Turing machines due to no one proving
otherwise.

But if human minds _were_ Turing machines, then they are certainly axiomatic
systems since Turing machines are axiomatic systems. This contradicts his
initial point which concludes that not all of our sources of knowledge are
axiomatic in nature.

------
cambaceres
One could argue that Gödel's theroem proves that there is no objective truth
at all, since the whole universe can be viewed as an axiomatic system.

------
Kinnard
> The first is that Gödel's theorem imposes some some of profound limitation
> on knowledge, science, mathematics.

?

~~~
throwawaymath
What is your question?

~~~
Kinnard
"some some" Is a typo?

------
alasdair_
Reading articles and books with the subtext of "what Gödel REALLY meant
was..." reminds me a lot of literary criticism, in that it doesn't really
matter what the original author REALLY meant any more - what's far more
interesting is how the author's ideas affected the thinking of others and how
much those people will argue about how _their_ interpretation is the only true
one.

~~~
throwawaymath
What? No, it's math. Theorems are not up to interpretation, emphatically and
by design. That's the purpose of math.

Gödel's theorems are profoundly interesting in their own right. They also
required significant creativity and ingenuity for their time. But if there are
conflicting "interpretations" of what they "actually mean", then that means
most must be incorrect (or more charitably, extremely incomplete).

Frankly I'm flabbergasted you would compare (mis)interpretations of
mathematical theorems to the legitimate subjectivity inherent to literary
criticism...

~~~
alasdair_
Note that I said “reading _articles_ ” about what Godel really meant, not
“reading agodel’s papers” - the point was that once we become that far removed
from the underlying work and start mostly talking about individual people’s
_interpretation_ of that work as the focal point of the article, it reminds me
of literary criticism more than anything else.

Additionally, interpretations are always subjective because interpretation of
something is unique to an individual subjective viewpoint.

