
Math even mathematicians don't understand (the sequel) - ColinWright
http://www.boston.com/bostonglobe/ideas/brainiac/2012/12/math_even_mathe.html
======
kenjackson
No one will ever bother to read this work. There are millions of claimed
proofs of various conjectures. If you want to prove Goldbach or P=NP (etc...)
you either need a very simple to verify proof (not a 1,000 page proof) or you
need to prove a set of interesting conjectures along the way to give yourself
credibility.

In the pages of this mathematician's proof, there must be a few smaller proofs
he could publish that show genuine insight, which would give him credibility.
Otherwise it's just another sci.math sideshow.

~~~
tinco
I wonder if it's possible for an machine learning algorithm to simplify these
proofs.

I take it because of the rather strict domain specificness of math papers it
should be possible to develop an algorithm that can parse these papers into
lemmas. Proofs normally structured as a tree of lemmas right? Or are there
proofs that have more complex structure?

A single basic thing could be to extract the leaf lemmas and assert their
validity. If they are valid, and the relations between the lemmas are valid
then the eventual conclusion should also be valid.

Has anything like this ever been attempted? A math paper proof verifying
machine learning algorithm?

~~~
jlouis
Automated Therom Proving is ridiculously hard and a guess would be that it is
almost impossible to train an ML algorithm to do this efficiently.

However, _verifying_ that a proof is correct is somewhat simple, if you give
the machine enough information to actually run the formalization details. In
modern mathematics, this may be a necessary thing to do since nobody has time
to verify these 1000 page developments anymore.

The problem is that the level of detail currently needed is very high. And
this slows down the formalization process. But the area is in wild development
these years, so hopefully something good comes out of it.

~~~
jamesaguilar
Just imagine inputting the final character of the last inference step, then
waiting for a few minutes while the computer performs incremental compilation
of the last few rules. Finally, it emits simply these four characters: "QED."

~~~
dandrews
I like this, and it reminds me of the tagline from Asimov's _The Last
Question_.

~~~
sp332
There was a billionaire in the book _Permutation City_ who uploaded his brain
into a supercomputer, then let an optimizer run on it for years. The goal was
to get his brain to run in real-time or better, but eventually it spit out an
empty file and a log message: "This program produces no output."

------
mrdmnd
"Mathematicians who make big claims are obligated both to be right and to make
themselves understood."

I spent most of this academic semester struggling with the material in MIT's
18.404 (Theory of Computation) class taught by Michael Sipser - at the end of
one of his lectures, he mentioned that he keeps a large file of proofs on hand
from people claiming to have resolved the P vs NP conjecture, but has little
time or faith to read them... he calls this the "crank" file. It seems to me
(and I'm just a lowly undergraduate math student, feel free to
discount/discredit this) that solutions to many of the large, imposing
academic challenges of our era might be found in the depths of other such
professor's "crank" files.

Then again, there is definitely something to be said for establishing
credentials - there's only so much time in the schedule of someone capable of
verifying claims to be spent on verification, and it seems obvious that they
should spend that time on solutions that come from people who have a track
record of competence.

~~~
crntaylor
Scott Aaronson summarizes it well when he says, of papers that have signs of
being by 'cranks' --

> _If I read all such papers, I wouldn't have time for anything else._ [1]

Crank papers often take a long time to read, by virtue of some combination of
(i) using non-standard notation and terminology, (ii) lack of familiarity with
the established literature, (iii) tortured writing style.

If you claim to have proved a big new result, and you don't already have the
credentials that will ensure you get taken seriously, then it is up to _you_
to write in such a way that makes it easy for other academics to verify your
work. Part of being an academic is interacting with other academics.

[1] <http://www.scottaaronson.com/blog/?p=304>

~~~
jseliger
_If you claim to have proved a big new result, and you don't already have the
credentials that will ensure you get taken seriously, then it is up to you to
write in such a way that makes it easy for other academics to verify your
work. Part of being an academic is interacting with other academics._

This reminds me of the introduction to Thomas Kuhn's _The Road Since
Structure_ :

"Kuhn develops further the theme, which again goes back to Structure, that
science is fundamentally a social undertaking. This shows up especially in
times of trouble, with the potential for more or less radical change. It is
only because individuals working in a common research tradition are able to
arrive at differing judgments concerning the degree of seriousness of the
various difficulties they collectively face that some of them will be moved
individually to explore alternative (often—as Kuhn likes to
emphasize—seemingly nonsensical) possibilities, while others will attempt
doggedly to resolve the problem within the current framework" {Conant and
Haugeland "Road"@3}.

------
merraksh
Related read: Apostolos Doxiadis' novel "Uncle Petros and Goldbach's
conjecture", an entertaining account of a mathematician's effort to prove the
result.

~~~
edanm
Very heavily seconded. One of my favorite novels, it does a really good job of
giving a "feeling" of mathematics, without including any actual mathematics.

This is the same author as Logicomix, which is aslo _very_ highly recommended
- a comic about logic, Godel, and the foundations of mathematics.

<http://en.wikipedia.org/wiki/Logicomix>

[http://www.amazon.com/Logicomix-An-Epic-Search-
Truth/dp/1596...](http://www.amazon.com/Logicomix-An-Epic-Search-
Truth/dp/1596914521/ref=sr_1_1?ie=UTF8&qid=1357459643&sr=8-1&keywords=Logicomix)
(this might be with a refferal link, I'm not sure - you have been warned :)

------
jmount
This type of behavior was well documented in Underwood Dudley's "A Budget of
Trisections." The problem is obsession and a deliberate cultivation of the
ticks the public associates with genius are much more common than actual
breakthroughs.

------
CurtMonash
So many of these proofs are of the form:

1\. Assume the desired conclusion is false. 2\. Make an error. 3\. Derive a
contradiction.

And by the way, most mathematicians have made plenty of bad proofs. I told my
thesis adviser I had a "proof" months before I actually did. Years before
that, I found a short proof of Fermat's Last Theorem (which was however too
long for the margin of my book ...).

Generally, the way "big" proofs go is that there's a whole lot of new
mathematics, and some handwaving at various points along the way. Then the
work starts of checking whether the handwaves are actually accurate. The
better the mathematician, the more likely it is that they actually are ...

~~~
raverbashing
Yes

Because proofs (or disproofs) by contradiction are "the easy way out"

Proving something is right is much harder if not impossible (hi Mr. Goedel)

~~~
CurtMonash
That wasn't my point. Rather, I was suggesting that proofs by contradiction
can easily be in error (because if you make an error, you have a good chance
of eventually getting to your contradiction).

Anyhow, when I did mathematics, I didn't run into people caring about proof by
contradiction vs. direct proof. However, I did occasionally run into a
preference for constructive vs. non-constructive proofs. E.g., my thesis used
a prior result that included a funky existence proof, and my adviser pushed me
to find an alternate proof that was directly constructive.

~~~
raverbashing
Oh yes, I had understood that "proofs by contradiction can easily be in error"
but I suppose that's because people try the contradiction proof first, because
it "should be easier"

And of course, constructive proofs are usually more complicated, in terms of
proof structure as well as 'where to go and how to get there'

~~~
CurtMonash
My comment was based on a more precise point, namely that #3 in the process
below is very natural:

1\. Assume the theorem is false. 2\. Make some calculations based on that
assumption. 3\. Eventually, make an error. 4\. Pursue that error to a
contradiction. 5\. Claim victory.

------
balsam
you know what this means? arxiv.org needs a cryogenics section, for potential
resuscitation of mathematical campaigns using future technology. it'll be
challenging to keep out the crackpots, though --

~~~
sillysaurus
_for potential resuscitation of mathematical campaigns using future
technology_

What does this mean?

~~~
TeMPOraL
To freeze the brains of our best and brightests in hope that future tech could
let them live much longer, and possibly make some more discoveries.

~~~
sillysaurus
I doubt the author meant "cryogenics" literally.

~~~
balsam
i actually meant an archive of papers where mathematicians from the future
could try to improve or dig gems out of (using computerized theorem provers or
just pen and paper). Papers like ramanujan's notebooks or this guy's proof of
Goldbach. But freezing brains could be worth thinking about --- maybe a
crowdsourced fund like kickstarter to freeze society's favorite brains.

~~~
sillysaurus
_freezing brains could be worth thinking about_

Well, freezing destroys cell membranes, unfortunately. There's no way to
repair every cell. And even if you could, there's no guarantee the overall
system will function the same afterward.

Cryogenics will probably take the form of sustaining life, rather than
freezing what has already expired.

For example, perhaps slowing a physical body's metabolism for decades would
enable the body to last centuries before dying. So it'd be "time travel" to
the future. But that has horrendous logistical issues, e.g. feeding,
maintenance of the chamber, somehow retaining muscle mass of the person,
avoiding bed sores, waste disposal, antibiotics during times of sickness,
regulating body temperature, a failsafe for each system, etc.

------
nnq
How hard would it be to write mathematical proofs in a modular way? And have
someone check a "module" of a proof and someone else another "module" and so
on?

...and for the bits that's possible, write in a subset of mathematical
language that computer can check, or at least partially check and ask the
human reviewer for feedback on the parts that can't be automatically proven.

I always fount it odd that mathematicians use so little of the mind augmenting
potential of computers. Is is true that there aren't any smart enough
programmers to craft the tools they would need, or that the tools end up being
so "user hostile" that they always go back to pen and paper? I know people
tried to express physics proofs in "computer-like language", Sussman et all in
Structure and Interpretation of Classical Mechanics, but I've never heard of
such things really catching on and being actively used and developed by
mathematicians.

~~~
12characters
> How hard would it be to write mathematical proofs in a modular way? And have
> someone check a "module" of a proof and someone else another "module" and so
> on?

To a large extent, this is what we do. But of course, modules by themselves
don't prove a thing about what we're originally interested in so that not one
module can stand by itself as any advance in the proof, that theconclusion of
any module of the proof are surely needed for all the others modules to make
sense and that these conclusions are more often than not formulated in a new
part of language (new words, concepts, and rules to play with them) introduced
in said module. Parallel work of the sort I infer you have in mind might be
possible in some instances, but there is so very few of them in my opinion
that it's just not worth trying.

> ...and for the bits that's possible, write in a subset of mathematical
> language that computer can check, or at least partially check and ask the
> human reviewer for feedback on the parts that can't be automatically proven.

No can't do: in mathematical writing, what is actually written is the tip of a
gigantic iceberg of implied reasoning and background scenery. There might be a
way to make computer understand this, but as far as I know it has yet to be
found. That, and making explicit what is not in mathematical writing would
make the length of any proof grow manymanyfold - and I actually mean
manymany....manyfold.

~~~
nnq
> in mathematical writing, what is actually written is the tip of a gigantic
> iceberg of implied reasoning and background scenery

The programming equivalent of this as I see it is a program depending on lots
of libraries encapsulating that "implied reasoning". And you can manage a
program even with tens of millions of lines of code in libraries. And it
doesn't make the program itself larger because that library code is _reusable_
, you can also use it in other proofs. Yes, it's incredibly hard to manage
such a code base, you also need extensive documentation, something like
"literate programming" - but it's not intractable.

You would have to write a big part of mathematics as library code, probably a
multi-decade length project - but the benefits seem _huge_! "Translating"
mathematics in such a form could augment a mathematicians brainpower
"manymany....manyfold" and could be just as important as, let's say, a cancer
research project taking decades and tens of millions in funding, but it should
worth it! _Imagine that hundreds of "less talented" mathematicians could
handle problems that only a handful of people can understand today (with the
guidance of the "real geniuses", of course), revolutionizing many fields of
science through the consequences of their results._

As an analogy, when people used roman numerals, only a handful of highly
educated people could do complex calculations, but nowadays we all have more
than basic arithmetic knowledge, thanks in large part to a "change of
language". I think a "change of language" could really make mathematics a less
"elite only" field (though I have greater hopes for letting "AI systems" give
a helping hand than for letting "lesser mathematicians" do some of the work,
in the end one can't predict what will make the most difference - hardware or
wetware, it's still computing power, and, as the complicated software problems
are solved, more and more of the computing power can translate into
"mathematical reasoning power" I think).

~~~
jhuni
The QED project was started in 1993 to build a database of all mathematical
knowledge. It died out because formalized mathematics does not yet resemble
traditional mathematics. All existing theorem provers and proof assistants
such as Mizar, HOL, and Cow have major shortcomings. Very few people are
working to improve formalized mathematics because there isn't any compelling
profitable application of fully formalized mathematics.

<http://www.cs.ru.nl/~freek/qed/qed.html>

~~~
nnq
...so it's one of those "someone has to do it to prove it can be done" or
"someone has to first make it work to prove it's useful" problems?

Hopefully there are a few madmen to start working at it and they will be wise
enough to do it all "in the open": open source code and publication of all
developments, "blog stye", instead of the "talking about only after you've
really understood it" approach. I would throw a helping hand on the
programming side and maybe devops, even throw a few bucks at it if one were to
crowd-source and crowd-fund such an effort ...just because I find it "so
cool". And I'm sure there are others that would help any group of "brilliant
madmen" trying to do this, even without any imaginable profitable
applications. But for now, I'm trying to wrap my head around what the QED
project was trying to do and what it did... _thanks for the directions to
them!_

------
tprice7
The validity of a mathematical proof depends on a few very precise rules that
can be checked by a computer, so I think it's interesting that we still rely
so much on humans for the verification of mathematical proofs. As the article
points out it's clearly problematic.

~~~
foobarqux
Machine checked proofs are intractably long and are not fully automatic except
for very simple problems.

~~~
tprice7
What do you mean by not fully automatic? What is the manual part?

I think right now the job of describing a (sophisticated) proof to a computer
is analogous to the job of writing a sophisticated computer program like a
game or Facebook in assembly code. What mathematics needs is a concise
language that can easily be understood by people and computers alike, like
Python.

~~~
foobarqux
The manual part is writing out steps of the proof. Some times these steps can
be automated. Usually they are guided by a user. They are typically not
written out step by step by hand because it is very tedious. I doubt
complicated ideas expressed in advanced mathematical papers can be easily or
succinctly expressed formally. <http://en.wikipedia.org/wiki/Theorem_prover>
<http://www.cs.ru.nl/~freek/100/>

But more importantly, the objective of a mathematical paper is to get readers
to _understand_ the proof, a goal which is poorly achieved with formal
verification.

------
robryan
I think this has an analog in software. The massive enterprise/ government
systems which get bogged down for years and become almost impossible to
replace.

Will be interesting to see if newer generations embrace collaboration and new
technology more or will things still get pushed forward by the lone and
difficult to understand outlier who becomes at least convincing enough for
others to trawl through their complicated texts.

I guess issues of credit would have weighed into this method. In this case, if
intermediate findings were published in a simple to understand way they may
have attracted enough interest to be beaten to the final proof?

------
revskill
Why not having a MathHub (like Github) for math, i believe even the toughest
problem will have the pull request, and many people could understand the
proof.

~~~
ufo
Math doesn work like that. You can't just create a git repo for PvsNP and
expect that solving the problem is just a matter of accumulating enough
commits. Not only does everyone involved have to know how everything works in
order to come up with insight (making it hard to "delegate" subproblems) but
you don't even know what subproblems need to be solved to come up with a
solution!

~~~
revskill
Why not seeing "Theorems" like "Open source projects" ? I think they have the
same meaning. Math theorems are the framework for other to use or extend. And
they should be "open sourced", i think we don't solve the Conjecture with this
"MathHub" but it will get solved and understandable by anyone who're
interested in a time.

------
teeja
For a similar fate in the physics world, WP:
Arthur_Eddington#Fundamental_theory. Which demonstrated that it's not enough
to have an idea, you have to find some kindred souls with the stamina to be
your soundboard - everyone else will be looking for any excuse to avoid the
confrontation.

E.g. dare to suggest that the universe <i>might not</i> be simple, after all.

------
taligent
If the Wikipedia page is correct then British publisher Tony Faber offered $1
million dollars if a proof was submitted before April 2002.

And according to this article it was submitted in March 2002.

So how come this guy didn't come forward to collect the prize ?

~~~
IvyMike
The timing explains a lot.

He almost certainly came forward to collect the prize in March 2002. Along
with the probably hundreds of others who also submitted 'proofs'.

