
Number theorist fears many proofs widely considered to be true are wrong - lelf
https://www.vice.com/en_au/article/8xwm54/number-theorist-fears-all-published-math-is-wrong-actually
======
Hasz
Headline is waaaay overblown, but not all of it is total hyperbole -- checkout
out arxiv and try to read the abstract from just about any paper.

It's basically totally impermeable. I'm almost done with an undergraduate
degree in math and basically have no idea what ~90% of the research is about
at anything other than a topical level. This is fine, it's written for
specialists in the field (hopefully), but damn, for most fields, it's not a
whole lot of people.

Compared to physics or chemistry, math gets very specialized, very quickly.
Depending on the field, there's no easy real world isomorphisms either, making
it even more difficult.

~~~
knzhou
That doesn’t mean that it’s wrong, it means that you need to read more. In a
lot of fields, undergraduate material doesn’t qualify you to read papers, a
PhD does.

If all papers actually were written to undergraduate level, many would be 500
pages long. Who has time to write a book-length exposition of the basics of
their field every time they publish? I mean, have you ever published a
technical work? If so, did you go out of your way to make it completely
accessible to high schoolers? If you didn’t, you get why professors don’t do
the same for undergrads.

~~~
nemonemo
Suppose a narrow field where only two human beings can understand. How could
the society judge the progress of the field or the merit of a new project? I
don't think it is possible to avoid collusion or groupthink in such a field
and it would be difficult to differentiate the two.

So does it mean we should curb some superhuman thinking and let them explain
better to other human being? I'd think so. It is all about balance or trade-
off, but at some point, a field would become too esoteric for a society and it
might need to slow down the short-term progress to ensure consistent long-term
progress.

It seems the math field as a whole (or more specifically the number
theorists?) needs more of good communicators who can link the cutting edge and
more general public. The math academia could give incentives to such
activities in addition to pushing the boundaries.

~~~
knzhou
Scientists and mathematicians already spend an extreme amount of energy trying
to think of the shortest, cleanest possible route for people to learn their
fields. The resulting documents take years to create, and are called
textbooks. (At least, the good textbooks are like this.) Often people will
work for thousands of hours on their books, essentially for free. Students go
from outsiders to participants by reading them, all the time.

The problem is that things genuinely are hard, and there's only so much
compression you can do before you're reduced to meaningless handwaving.
Textbooks are already compressed by a factor of 100 relative to the original
research literature, sometimes even oversimplified. Demanding something a
factor of 100 better than a textbook is going to get you something like one of
those useless "learn C++ in 24 hours" paperbacks.

------
ukj
Sounds like a sales pitch for Lean (Microsoft's theorem prover), even though
Coq [1] is the standard tool for the UniMath [2] project.

[1] [https://en.wikipedia.org/wiki/Coq](https://en.wikipedia.org/wiki/Coq)

[2] [https://github.com/UniMath/UniMath](https://github.com/UniMath/UniMath)

~~~
krapht
Well, having used both, Lean really is a lot more ergonomic. Unimath is mostly
category theory; other parts of mathematics are woefully underdeveloped.

~~~
ukj
That's fair.

I actually find Coq rather painful to use. It gives me the distinct impression
that it has been designed/implemented by somebody who has never written human-
computer interfaces before.

When it comes to adapting humans to Mathematics, or adapting Mathematics to
humans - I prefer the latter.

------
Ceezy
"I believe that no human, alive or dead, knows all the details of the proof of
Fermat’s Last Theorem. But the community accept the proof nonetheless,"
Buzzard wrote in a slide presentation[...]

That s 100% clickbait. People knows the demonstration and the demonstration
helped to create entire new fields. By the way there are new and shorter proof
of the theorem.

~~~
hyperbovine
Also, that is quite a remarkable and provocative statement given that Andrew
Wiles is still very much alive.

~~~
Stasis5001
Sure, Wiles obviously understands the main thrust of his proof. But one could
argue that Wiles' result depends on lots of other results, which in turn
depend on other results, and so on through decades and decades of work,
ultimately going back to the foundations of mathematics. Neither Wiles nor
anybody else can claim to rigorously understand all of it. You can imagine
this as a tree, with Wiles' work as root, and his dependencies as ancestors,
and so on. An error at a lower level of the tree could, in theory, invalidate
the root node.

I do agree with Buzzard that it's hard to be sure. I've definitely read papers
where a critical argument isn't well written or what is written seems wrong.
However, if there are low-level errors, I suspect that with some work things
could be patched up.

~~~
hyperbovine
Right, speaking as a lapsed mathematician, I definitely see errors or gaps in
published work. Wiles’ original FLT proof had one. And yes these can generally
be patched up. I’m not quite as alarmed as the author is, because generally a
major false result would have all sorts of alarming ripple effects and
implications which would be pretty easy to spot. FLT is an extreme example
where literally anyone with a calculator could in theory disprove it. The fact
that no one has suggests to me that it’s likely true.

------
colorincorrect
This short pdf which the article cites is more useful than anything the
article itself contains:

[http://wwwf.imperial.ac.uk/~buzzard/one_off_lectures/msr.pdf](http://wwwf.imperial.ac.uk/~buzzard/one_off_lectures/msr.pdf)

~~~
enjoyyourlife
Discussion:
[https://news.ycombinator.com/item?id=20909404](https://news.ycombinator.com/item?id=20909404)

------
raymondh
Summary: Motherboard interviewed an attendee at the Interactive Theorem
Proving conference who says that he thinks Interactive Theorem Proving is
important. Also, professors specializing in proof theory think their tooling
may become important but "the technology still needs improvement".

------
madez
I'm currently working on the formalization of Catalan's conjecture (or
Mihăilescu's theorem) in Isabelle/HOL/Isar.

My personal impression is that more formal verification is necessary in math,
and it should be tought to every student of Mathematics in introductionary
courses because it gives an unforgiving but also rewarding learning experience
with formal proofs.

Feel free to ask questions.

~~~
joker3
How much research is being done on making these tools more user friendly? I
spent a little bit of time using Coq about a decade ago, and while I agree
that it's very important to make things like that widely used, there was a
pretty steep learning curve.

~~~
madez
I can't answer how much research or work is done to make the tools more user-
friendly as I'm rather exclusively on the user-side. There exists extensive
documentation in form of multiple PDFs, but they are only of little help to me
as they lack useful examples. Sadly Google thinks every possible answer about
Isabelle/HOL is answered by these PDFs as they are often the only relevant
search result.

However, Isabelle/HOL/Isar allows proofs to be written in a very human
readable way, so the "documentation" I use is the existing library and how
things are done there. There is a convenient Ctrl+left click action on terms
to jump to their definition. That is very helpful to navigate unknown
libraries. In comparison, I couldn't read and make sense the proofs I've seen
in Coq.

There is a learning curve with Isabelle/HOL/Isar. It can be extremely
frustrating to not know how to express "simple" arguments or definition. But
for someone familiar with manual "formal" proofs in mathematics, most can be
learned in some hours under guidance, like in a seminar on a weekend with a
couple of hours on each day.

I'm learning rather autodidactly by trial and error. I think there would be
value in a series of blog posts about "This is how you define a group" or
"This is how you reason in nonclassical logic" or "This is how to proof the
fundamental theorem of Algebra" all applied to Isabelle/HOL/Isar. There are
often multiple ways to define things, and to be honest, I don't know the
specific advantages and disadvantages. For example, I think you can introduce
rings as type classes or as locales or using axiomatizations, and possibly
more. This is one example of stuff I don't know how to manage in the system,
but there is already a lot of structure defined, and I can reuse that.

------
repolfx
To mathematicians,

Welcome to our world!

[https://xenaproject.wordpress.com/2019/09/27/does-anyone-
kno...](https://xenaproject.wordpress.com/2019/09/27/does-anyone-know-a-proof-
of-fermats-last-theorem/)

 _Actually working through some of [the details], when formalising them, made
me understand that some of these issues were far bigger and more complex than
I had realised._

 _In the perfectoid project there is still an issue regarding the topology on
an adic space, which I had assumed would be all over within about a page or
two of pdf, but whose actual proof goes on for much longer_

Yeah, that's how software developers feel, all day, every day. That's why we
all moan about being asked to give estimates for things. You don't really
_understand_ a thing until you've formalised it in code, and often things that
look simple and easy turn out to be unexpectedly deep.

Mathematics is a field that's been doing the equivalent of writing pseudo-code
for thousands of years. There are endless reams of pseudo-code that _looks_
like it might work, but as any programmer who's ever written pseudo-code
knows, whether it actually runs when you try it out for real is anyone's
guess.

So now mathematics is starting to experience the pain of multiple incompatible
programming languages/ecosystems. There's Lean, Coq, Isabelle/HOL and some new
JetBrains language too. Soon they'll start to discover the value of 'foreign
language interop', rigorous code review, comments, style guides, design
documents, package management, automatic optimisation etc.

Have you ever looked at a proof written in one of these languages? All code in
these languages is a hot mess judged by the standards of professional software
developers: tons of abbreviations, single letter variables, uncommented, ad-
hoc module structure etc. I'm sure we'll eventually see mathematicians having
raging flamewars about what clean style is and tabs vs spaces.

Luckily for mathematicians though they work at their own pace and are under no
deadline pressure. They'll figure it out.

------
adamnemecek
Vice really does a disservice to the author. "All" math is not wrong, but I
sure wish theorem provers were the preferred method of formalizing
mathematics. But theorem provers really aren't quite ready for prime time
either. But it sure feels like a lot of verification of mathematics is
somewhat repetitive.

I think that programming/CS is to mathematics what neolithic revolution was to
human culture. It allows for specialization in the stack. People can
obsessively optimize different parts of the stack to everyones benefit.

------
higherkinded
Suggestion to rely on AI for proof verification is just laughable. Neuron
weights instead of formal definitions. So reliable.

~~~
not2b
You seem to think "AI" means machine learning. In this case it does not.
Theorem prover systems don't use neural nets.

~~~
higherkinded
Okay but they are formal systems and don't have anything to do with AI either,
do they?

------
joker3
The headline is certainly overblown, but there probably errors out there. The
interesting question is where we should be looking for them. All of the major
results in an undergraduate or introductory graduate textbook have been
studied often and thoroughly enough that we can be reasonably confident there
are no errors in their proofs. There are errors in very new or very obscure
material, but those tend to get corrected as more people start paying
attention to them. It therefore seems like the search should therefore be
concentrated in recent results that are becoming more useful but still aren't
standard.

On the other hand, we do need to build a standard library of results that can
be used to prove those newer results. You can't do much in analysis without
basic results on metric spaces, for instance.

------
std_throwaway
There is a big grey area between "not 100% correct" and "wrong and harmful".
Like a faulty computer program that produces correct results in 99.999% of the
cases a wrong mathematical statement can still yield useful results in
practice. You just don't want to run into those pesky edge cases where it
indeed is wrong.

Having a database of all mathematical proofs where all of them are checked for
validity, however, is a very useful tool to actually find those edge cases
where a proof is wrong and give future mathematicians a solid foundation.

Using AI as an extension of human capabilities is a given to me. Like with
heavy machinery, we don't want to do the heavy lifting. We want to steer it
properly to gain most benefit.

~~~
cyphar
> a wrong mathematical statement can still yield useful results in practice

Yes and no. For some basic conjectures this might be the case, but one of the
greatest utilities of certain theorems is how they can be applied to
completely new fields. If a theorem used in this manner is not actually true,
then the proof is invalid -- and it might be the case that (several proofs
down the line) the conjecture you've ended up proving is almost entirely
invalid.

There's also the slippery slope of "Principle of Explosion"-type problems with
accepting "mostly right" proofs -- it means you will be able to prove any
given statement (even ones which are outright false).

------
MrBuddyCasino
Soo... Unit testing / CI for math proofs? If it can be done in software,
great.

~~~
std_throwaway
To my knowledge the hard part is writing the input for the software. While in
text you have a bit of leeway to glance over some details, with the computer
program you must, indeed, write every last relevant detail down and you must
get it right. This is very heavy work.

~~~
ukj
But it's once-off work. Thereafter the computers can take over.

------
dls2016
I had these fears about my dissertation work, but then realized that the
“elder’s” papers were highly cited because they contained important machinery
which could be modified and refined to prove other results... or the paper
contained an important idea which went on to be proved in a variety of ways.
Maybe it’s different outside of PDEs, but I doubt it.

------
bryanrasmussen
I just can't help feeling that either the true should be correct, or the wrong
should be false. But I guess English is not a strongly typed language.

------
iamaelephant
This just seems like a native advertisement for Lean...

~~~
DoctorOetker
I am not that perturbed by whatever framework they use, tools to translate the
bulk of the majority of proofs from Lean to MetaMath to [...] and back will be
made,

the more important thing is that the number of humans aware of the existence
of this tech and the more actually use it the better society will be off in
the long run

the following may seem totally impossible today but seems a lot less
impossible to people who have been thinking about this for a long time: once
sufficient mathematics has been translated, people can start adding physics
postulates, one most physics has been inserted we can start formally verifying
(up to validity of postulates on top of axioms) not just proper operation of
digital circuits etc but also power plants and so on, and once interacting
with proof systems becomes as widespread as alphabetic literacy, people might
one day be able to formalize and vote on their requirements of society and
policy, see predicted perverse incentives, or decide that policy can only be
enacted when facets of their consequences can be proven. We might one day not
vote on promises of strategy, also not vote on strategy, but rather on
outcomes!

I know it _sounds absurd_.

------
lysium
I’m sure this will help proofing papers. There will still be the problem of
proofing that the software proofs what it is intended to proof.

------
gowld
Headline is of course absurd hyperbole that does not belong here.

> "I think there is a non-zero chance that some of our great castles are built
> on sand," Buzzard wrote in a slide presentation. "But I think it's small."

~~~
kurlberg
Well, they also quote him as saying:

"I’m suddenly concerned that all of published math is wrong because
mathematicians are not checking the details, and I’ve seen them wrong before,”

~~~
kevinzz
[https://xenaproject.wordpress.com/2019/09/27/does-anyone-
kno...](https://xenaproject.wordpress.com/2019/09/27/does-anyone-know-a-proof-
of-fermats-last-theorem/)

------
ummonk
Extreme clickbait headline. The subheading is '"I think there is a non-zero
chance that some of our great castles are built on sand," he said, arguing
that we must begin to rely on AI to verify proofs.'

~~~
asaph
But, more consistent with the alarmist headline:

> “I’m suddenly concerned that all of published math is wrong because
> mathematicians are not checking the details, and I’ve seen them wrong
> before,” Buzzard told Motherboard while he was attending the 10th
> Interactive Theorem Proving conference in Portland, Oregon, where he gave
> the opening talk.

~~~
parsimo2010
Someone who says that is, ironically, not concerned with the details. _All_ of
published math is wrong? No, a lot of is is understandable and verifiable by
advanced undergraduates.

Most of the fundamental theorems in each field are understandable by humans.
As you get more and more abstract, and get into more and more obscure areas,
you start to see non-understandable proofs built on top of non-understandable
proofs. There's a danger that those are wrong, but when they collapse it just
means that one professor has wasted their life, not that _all_ of mathematics
will suddenly collapse.

~~~
kevinzz
The headline is clickbait. A more measured debate about the issues is here
[https://xenaproject.wordpress.com/2019/09/27/does-anyone-
kno...](https://xenaproject.wordpress.com/2019/09/27/does-anyone-know-a-proof-
of-fermats-last-theorem/)

------
breck
Anyone interested in this topic, I have a prediction that the time is right
for a new math, that starts from just 1 and 0, and builds every symbol up in a
rigorous way.

We're working on the infrastructure to make such a thing easier, but the
project does not currently have a leader:
[https://github.com/treenotation/jtree/issues/83](https://github.com/treenotation/jtree/issues/83)

~~~
lidHanteyk
Disregard cranks, acquire Metamath:
[http://us.metamath.org/downloads/metamath.pdf](http://us.metamath.org/downloads/metamath.pdf)

~~~
drchewbacca
Metamath is awesome if you're willing to slum it :)

[https://jiggerwit.wordpress.com/2018/04/14/the-
architecture-...](https://jiggerwit.wordpress.com/2018/04/14/the-architecture-
of-proof-assistants/)

~~~
lidHanteyk
How long does it take your preferred proof assistant to prove things?
Metamath's standard set.mm takes only about 10s to prove _everything_.

~~~
digama0
Actually if you use the smm verifier that's been cut down to about 800ms. :)
However, most of the theorem provers of today have been built on the
philosophy that performance doesn't matter, or at least is secondary to ease
of use, mathematical cleanliness etc, from the functional programming
community. It turns out that once you make this decision it's difficult to get
that raw speed back, even if you start worrying about performance later, and
the HOL family provers make it worse by defining correctness in terms of the
running of an ML program, which bakes the runtime of the ML system into the
proof checking time.

When you combine this with the fact that these ML programs are not proofs but
proof scripts, that perform a lot of "unnecessary" work like searching for a
proof rather than just going straight for the answer, it suddenly begins to
make sense why these systems take on the order of hours to days to check their
whole libraries.

Coq and Lean are somewhere in the middle, because they have proof terms, but
the logic itself still requires some unbounded computations. Checking a proof
here is often fast, unless you make too much use of computation in the logic.
But people often don't care about proof terms, and still store the proof
scripts, which are as slow as ever.

Metamath is in this setting somewhat unique in eschewing proof scripts
altogether, or more accurately, inlining proof scripts immediately on the
author's machine. The resulting proofs are often comparatively long and
verbose, but I would argue this is only a display matter, since all the other
provers are doing the same thing, they just aren't showing it.

