
Gödel Incompleteness for Startups (2013) - samuel2
https://skibinsky.com/godel-incompleteness-for-startups/
======
lordgeek
As far as I recall, this was written in the specific historical context almost
a decade ago. Back then "Lean Startup" and "4 Steps to Epiphany" were all the
rage in Silicon Valley. Shitty copy cats of YC were springing up left and
right in the Valley and other locations. A number of people (with a lot of
financial self-interest at stake) were pushing _hard_ the narrative that
creating new unicorn can be industrialized like a factory and they already
figured out the magic formula - no risk, all reward - just sign here under the
dotted line and give me your funds.

The essay publication was quite impactful in middle of that insanity. It uses
Gödel analogy to demonstrate that no amount of formalization (that helps
reduce human/team/org risks) will in any meaningful way reduce the systemic
risk of a new startup. It placed Gödel formalism straight on the path of
hustler/carpetbaggers who were trying to launch new funds and incubators at
that time "well, go ahead and prove to your LPs you have Gödel-like formalism
that detects great startups at an early stage and you do it better than YC".

Right now most of the essay conclusions are self-evident, "lean startup" is an
insult, so it does seem like the author spending a lot of pages and energy on
proving something plain obvious.

That wasn't that clear and that obvious many years ago when it was written.

~~~
balfirevic
> "lean startup" is an insult

Wait, what?

~~~
lordgeek
[https://twitter.com/garrytan/status/1074392721912000513](https://twitter.com/garrytan/status/1074392721912000513)

------
wsxcde
Godel's incompleteness is not that complicated. The main insight Godel had was
that proofs as well as provable statements can be encoded in numbers. This is
obvious to everyone reading HN -- all computing works by encoding things into
numbers. And it's easy to see how you'd encode proofs, proof rules etc. into
numbers. (hint: as flattened ASTs!)

The thing is, Godel didn't live in an era of pervasive computing so he came up
with a wonky encoding based on products of powers of primes. This makes the
proof technically challenging, but the fundamental idea is not that complex.
What he was able to eventually do was encode a recursive statement of the form
"this statement is not provable" where the "this" is kind of like a pointer
back to the full statement. The rest is straightforward.

~~~
gnulinux
This is true, obviously, but I can reassure you that the actual proof is very
technical and long. Back in college I took a logic class that spent almost the
entire semester going over this proof, in its shortened form i.e. using
Rosser's trick. By using Rosser's trick one can prove a more general
Incompleteness theorem (based on Q) and more elegantly. It ends up stating
very concisely that a theory cannot be all 3 of (any 2 or 1 or 0 is fine):

* axiomatizable extension of Q

* consistent

* complete

(where Q is a minimalistic arithmetical theory that can do addition and
multiplication:
[https://en.wikipedia.org/wiki/Robinson_arithmetic](https://en.wikipedia.org/wiki/Robinson_arithmetic))

It's true that the "main idea" of the proof is "everything is a natural
number", which is obvious to us programmers (it possibly wasn't obvious to
anyone in Godel's time). However, this is by no means the only trick that's
used in the proof.

~~~
stepbeek
> This is true, obviously,

Is this a dig at Hilbert?

------
fanzhang
The article strikes at the right general idea: that a startup that grows large
quickly must have executed an idea that was not widely known (e.g. being
exploited well, or in the natural adjacent competency of a current large
company).

Peter Thiel's analogy of this is "what's true but few people agree with you
on" and pg's version is "the best ideas look like jokes / bad on first
glance". There's a popular venn diagram that's almost analogous to the one in
the article: the best startups are at the intersection of true and not
obvious.

This article maps "obvious good ideas" to "provable in a formal system", which
makes the Godel analogy work, but the Godel analogy seems like a worse
analogy.

Analogies are supposed to put a new concept (startup ideas) in terms of more
accessible concept (Godel statements?). The analogy target (Godel statements)
was so obtuse that the author needed to spend pages explaining it.

Also, the Godel analogy is misleading. The canonical Godel sentence is self
referential and also refers to the formal logic system -- a good startup
doesn't need to be self referential or refer to the formal system. Besides
generating Godel's incompleteness theorem, I don't think Godel statements are
"interesting", unlike the most recent successful startups. (In fact, what's
interesting are simple yet difficult theorems that were totally provable
within Peano arithmetic to begin with, like Fermat's Last Theorem.)

~~~
auggierose
A Godel sentence does not have to be self-referential or to refer to the
formal system. It just has to be true in the formal system, but not derivable
in it.

~~~
fanzhang
Right, that's the theory, but what's another concrete Godel sentence that's
not a twist on the standard Godel sentence?

(As an analogy, in theory there are tons of different ways to construct non-
measurable sets, but in practice all examples come down to some quotient of a
nonzero measure set and a measure zero set, like [0,1]/Q.)

~~~
auggierose
Searching on the web reveals this:
[https://en.wikipedia.org/wiki/Paris–Harrington_theorem](https://en.wikipedia.org/wiki/Paris–Harrington_theorem)

------
auggierose
Very interesting article and I enjoyed reading it. Obviously, it is complete
bullocks :-) If a startup would have to rely on finding interesting Gödel
statements, there would be no successful startup. Even in the much cleaner
world of mathematics, basically all interesting work is done within the realm
of formally provable statements.

Also, the ultimate monopolist would not play by the formal rules of the game.
He would make the rules so simple, that now he indeed can formally derive all
truths and exploit these.

~~~
fnrslvr
> Even in the much cleaner world of mathematics, basically all interesting
> work is done within the realm of formally provable statements.

More-or-less this. I'm going to take this as an opportunity to drop one of my
favourite quotes, because I can't help it:

 _" The view that machines cannot give rise to surprises is due, I believe, to
a fallacy to which philosophers and mathematicians are particularly subject.
This is the assumption that as soon as a fact is presented to a mind all
consequences of that fact spring into the mind simultaneously with it. It is a
very useful assumption under many circumstances, but one too easily forgets
that it is false. A natural consequence of doing so is that one then assumes
that there is no virtue in the mere working out of consequences from data and
general principles."_

\-- Alan Turing, _Computing Machinery and Intelligence_

~~~
wolfram74
I like to think about the 4th quadrant of the known-unknown punnet square: the
unknown Knowns, things definitively implied by what you already know but
haven't realized yet. Latent knowledge like that is often fun to play with,
especially with students, getting them to realize they understand more than
they think they do.

------
andi999
Actually I love reading an article about Gödel and I like perspectives on
startups, I think this article would be best if it were not one but two (only
focussing on one subject). I really think using Gödels theorem as a metapher
is by far overstretching it.

~~~
ratmice
I think a good counterexample to the idea of startup consistency is youtube
(or any other) system which extracts money by both playing ads and not playing
ads.

------
frodetb
Curious. I've noticed a number of people using Gödel's incompleteness theorem
as a ridiculously stretched metaphor for some idea they're trying to get
across. At this point it's gotten utterly cringey to me. It reeks of someone
who only recently heard about it, got a conceptual understanding for what it's
about, and now want to show off and force the comparison in the first
conversation they have. Seems like a more obscure version of the meme that
Heisenberg's uncertainty principle has become.

This one is the most egregious case I've seen though. It seems like it would
be a waste of time to read, so I closed out fairly quickly. Have I misjudged
the article?

------
samuel2
"Most startup ideas are bad - Paul Graham empirically classified these as
“good ideas that look like bad ideas initially”. From Gödel’s model we can
draw even more precise distinction. These ideas look “bad” because they are
unprovable ideas in composite formal system “everything we know so far”. Bad
ideas that are actually bad are usually provably bad even in current system. "

~~~
seesawtron
Reminds me of the fragile [0] nature of entrepreneurs and startups that take
the risk to create services or products to keep the global economy antifragile
which eventually benefits from the good ideas while the bad ideas and their
creators perish.

[0]
[https://en.wikipedia.org/wiki/Antifragile](https://en.wikipedia.org/wiki/Antifragile)

------
katzgrau
I think this was a very long and drawn out way of saying that there are an
infinite number of ideas, most of which will fail, but some will be
successful. The successful ones typically aren't created at big companies
because there's no way to truly know which ones are going to wind up being the
extreme moneymakers, and at some point, the estblished systems of those big
companies make them inherently selective in taking on new things.

You need a sea of risk takers and tinkerers (entrepreneurs and hobbyists) to
bring the best ideas to life.

------
bonoboTP
It would be really good to have some TLDR, abstract or summary.

Ever since Gödel proved his theorem, people have used it as a metaphor for all
kinds of things, from new age mysticism to psychology, biology, quantum
physics, AI etc.

It's a very worn metaphor and mostly used by people who don't actually
understand the theorem precisely, just the imagined "gist", often with
fundamental misunderstandings. A bit how people use Eistein's relativity
theory to then derive moral relativism because "everything's relative".

This blog post may actually provide insight, but it's too long to see.

Given the prior that Gödel is often used just to borrow the prestige and aura
of mathematics and to sell insight porn, I cannot justify reading it without
seeing a summary.

I start to appreciate the rigid form of scientific articles more and more.
People often say it's too rigid, too contorted, we'd be better if people just
wrote blog posts in plain language, but then you get unstructured page after
page, where you don't know where to look. With some training one can quickly
assess the importance/relevance of scientific articles, exactly due to the
rigid format. Here I have no idea where I can find the main idea. It is
important to be effective in getting ideas across.

~~~
normalnorm
Here we go. The obligatory "you don't understand Gödel" post.

> Ever since Gödel proved his theorem, people have used it as a metaphor for
> all kinds of things, from new age mysticism to psychology, biology, quantum
> physics, AI etc.

Including Gödel himself, who was a mystic. It is true that people make all
sorts of silly claims vaguely based Gödel and other famous results, but it is
also true that there are indeed deep philosophical insight to gain from
Gödel's theorems, which are also dismissed without a proper argument.

> I start to appreciate the rigid form of scientific articles more and more.

I always appreciated it, but notice that what Gödel did is not science. He
obtained a deep insight about reality that was not based on the scientific
method or on empiricism.

~~~
sabas123
Not OP, bur Gödel was a mathematician and the comment was not necessarily
about being using the scientific method, but rigor.

------
sumanthvepa
Okay I'll take a stab at the TL;DR: The author posits that great startup ideas
are like Godel's unprovable, but true statements. These startup ideas cannot
be valued using the experience one has with valuing existing successful
companies. They tend to look like bad ideas in any valuation, but are in fact
good ones. The search for such a startup ideas also cannot be formalised into
a procedure (like say 4 Four steps to epiphany, or the Lean startup) Finally,
author posits that the space of great startup ideas is infinite, even as the
technology landscape grows ever more complex. There is no end to innovation.
Godel's theorem in this article is used as a metaphor, and no more.

~~~
andi999
This is a great summary, of how the article should have been. I am not sure
the author really only uses Godel as a metaphor, to me it looks more like he
really believes that it is the same (or a consequence).

~~~
sumanthvepa
I guess he is using Godel's theorem in a somewhat trivial way. One could argue
that the statement 'There are an infinite number of valuable startup ideas' is
a true but unprovable statement in the Godel sense.

------
cellular
Applying Gödel's incompleteness theorem to search for unprovably, would-be,
successful companies requires the searcher to develop axioms and rigorous math
to know "This startup is an unprovable case and should be tested".

I really doubt axioms have even been assigned to this problem.

~~~
cellular
I think psycohistory would also be a requirement!

------
michelpp
If anyone is looking for a relatively simpler explanation of GIT than Gödel's
own work, I suggest the beautiful but sometimes inscrutable book "Gödel,
Escher, Bach" by Douglas Hofstadter. Hofstadter draws lines between the math,
art, and music of these three giants of human creation, and it's a lot easier
to read than anything written directly by Gödel.

Roger Penrose's "The Emperor's New Mind" also has some good material on GIT.

~~~
ProfHewitt
There is a simple explanation why the [Gödel 1931] proof for Russell's system
does _not_ work here:

[https://papers.ssrn.com/abstract=3603021](https://papers.ssrn.com/abstract=3603021)

------
lifeisstillgood
I tried to write a Tl;Dr for this - i don't think I can.

It seems to be good start up ideas exist in the 'one step removed' phase space
of all possible startups - not so far advanced that you need to teach Henry
Ford about computers, but just one extra 'twist'.

This seems to be however leading to startups should iterate through 'Facebook
but for [dogs,cats,fish,mobile phones]" which I am not sure wins.

But I loved the history of Godel etc.

------
cflyingdutchman
for the author: "Weather" \- "Whether"

------
sabas123
I tried getting through this blog but it was just way to much effort to learn
the actual idea.

Can anybody give me an TL;DR?

~~~
sradman
TL;DR: The future success of a startup is unprovable so you have to let it
play out to see if it succeeds.

~~~
sabas123
Thanks.

I'm glad I didn't waste more time on this article.

------
neokantian
The hard part in the proof for Gödel's first incompleteness theorem is what
would later become known as Carnap's diagonal lemma:

[https://en.wikipedia.org/wiki/Diagonal_lemma#Proof](https://en.wikipedia.org/wiki/Diagonal_lemma#Proof)

This proof consists of just 7 lines, but these 7 lines are considered to be
fiendishly unreadable. The lemma itself says otherwise something very
understandable and perfectly relatable:

logicSentence <-> prop(%logicSentence)

Meaning of the lemma: If prop(%s) is a predicate property of any logic
sentence s, then there exists at least one true sentence for which this
property is true and/or one false sentence for which it is false. The
expression %logicSentence is the (numerical) description (encoded as a number)
of the logicSentence. The remainder of Gödel's proof is just endless
bureaucracy to establish completely precisely in/from what type of theory it
occurs/can be derived.

If you leave out the bureaucracy, Gödel's first incompleteness theorem follows
almost trivially from the lemma:

logicSentence <-> isNotProvable(%logicSentence)

Hence, according to the expression above, there exists at least one false
sentence that is not isNotProvable (and is therefore provable) [a] or at least
one true sentence that isNotProvable [b]. Hence, the theory in which this
situation occurs is inconsistent [a] or incomplete [b].

Now, the bureaucracy in Gödel's full proof actually does matter, because his
theorem holds for the Peano and Robinson formalizations of arithmetic theory
but not for the Skolem and Presburger ones. The reasons for that can only be
found in the bureaucracy of Gödel's proof.

~~~
ProfHewitt
Russell's rules for orders on propositions rule out the existence of the
[Gödel 1931] proposition I'mUnprovable (such that I'mUnprovable <=>
~⊢I'mUnprovable)

There is no fixed point (Diagonal Lemma) for the mapping Ψ↦~⊢Ψ because the
order of the proposition ~⊢Ψ is one greater than the order of the proposition
Ψ because Ψ is a propositional variable.

For a correct formal proof that there are true but unprovable propositions in
the most powerful foundations see the following:

    
    
        Physical Indeterminacy in Digital Computation
    

[https://papers.ssrn.com/abstract=3459566](https://papers.ssrn.com/abstract=3459566)

