
How Math’s Most Famous Proof Nearly Broke (2015) - dnetesn
http://nautil.us/issue/67/reboot/how-maths-most-famous-proof-nearly-broke-rp
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drej
I can recommend Simon Singh's book on this topic, his writing is rather
captivating (The Big Bang is one of the best books I've read, so is the Code
Book). He also made a BBC documentary on the topic.

[https://simonsingh.net/books/fermats-last-
theorem/](https://simonsingh.net/books/fermats-last-theorem/)

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ddoran
+1 for the movie. It used to be on youtube but I can't find it right now.
Perhaps the BBC had it removed. This is the finest moment in the movie, for
me, when Wiles describes the moment when everything clicked. The whole movie
is well worth watching.

[https://www.youtube.com/watch?v=SccDUpIPXM0](https://www.youtube.com/watch?v=SccDUpIPXM0)

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soegaard
Look at Vimeo instead.

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oxymoron
Is it really fair to say that Wiles proof is the most famous in mathematics?
Seems to me like something like the Euclid’s proof of the infinitude of the
primes might be more well known? Or the standard geometrical proof of
Pythagoras theorem?

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skh
It’s certainly true that far more people understand and know how to prove the
infinitude of the primes than understand/know how to prove Wiles’ proof.
Wiles’ result has received far more press coverage in the last 30 yeas though.
In this sense it is more famous.

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heinrichf
An amazing (word of a number theorist) documentary from the BBC's Horizon
program:
[https://www.dailymotion.com/video/x223gx8](https://www.dailymotion.com/video/x223gx8)
You can see Wiles crying when he talks about his work.

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jeff76
This brings up a question I have. Is there a way we can code up proofs in a
computer language and have it checked for correctness? If so, how? What
resources can I follow up on? Can current technologies check really
complicated proofs like this one?

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gnulinux
This is not as easy as some comments here imply. Basically, there are 3
popular languages used for this in a academia and the industry: Coq,
Isabelle/HOL and Agda. They're all extremely different languages doing
different things (but their end goal is the same: formalizing and automating
proving). Coq and Agda are intuitionistic by design, you need assume lem to be
classical. Agda has a Haskell heritage so it's purely functional other two are
more "procedural". Coq has 2 languages built into it, a tactic language and a
language go write types in.

Anyway this is one of the hottest research topics in CS. I know we keep seeing
deep learning papers all the time, one might think this is the only thing
happening in CS world. Currently it is nowhere near easy to formalize a
complex enough theorem in a system like this. Good news is there is a large
chunk of certain theories (like homotopy theory) already formalized for you,
so you can go ahead and use them as libraries.

The biggest challenge is mathematics is written for humans, it is not meant to
be formal. When you formalize theorems you start proving smallest shit you'd
never have thought that would need a proof. You even need to prove things like
what it means x == y and that for all x, x==x. There is no AI involved in this
currently, so it's a very manual, intellectually complicated and long process.
These systems have a proof finder built into them which performs a tree search
with heuristic to help you, this usually makes manual proofs easier, but
usually only marginally easier and you still have to micromanage your proofs.

Keywords you can search: automated theorem proving, dependent type theory and
homotopy type theory.

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grumdan
To be clear though, all the mentioned languages are built on purely functional
programming. Coq and Isabelle are more "procedural" in that they include
special proof languages that express proofs in a step-by-step way that feels
procedural, but the programs and definitions one proves properties of are
still expressed in a functional style.

It's true that one needs to prove a lot of trivial things in all these
systems, but the extent of this varies dramatically based on how much
automation they provide. For example, Agda includes only minimal, barebones
proof automation, while both Isabelle and Coq allow using more sophisticated
automation that can take care of a lot of trivial steps automatically (such as
"x == x").

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zozbot123
> they include special proof languages that express proofs in a step-by-step
> way that feels procedural

Serious proof developments don't even use the languages in a procedural way.
They use "declarative" idioms/patterns that are structured more like a very
detailed human proof. This is because "procedural" interaction with the proof
state is extremely fragile (it's like raw assembly code, the slightest change
in any part of the proof can break basically everything else in it) plus it
actually requires "replaying" the proof in order to have any chance of
understanding how it works. Declarative idioms can be read and understood
directly, at least to some extent.

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xiaoxiae
That was quite an enjoyable read.

It's so interesting that a problem so simple to explain would require hundreds
of pages of proof.

I'm really interested in whether the Collatz conjecture will be proven too.

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kzrdude
Maybe you'll like this FLT related story too, it's the most worthwhile
numberphile video:

[https://m.youtube.com/watch?v=nUN4NDVIfVI](https://m.youtube.com/watch?v=nUN4NDVIfVI)

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waynecochran
What are the practical ramifications of FLT? It sounds like there may be
crypto implications w elliptical curves.

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claudiawerner
What's interesting about FLT is that in the sense of proving a hypothesis,
although the attempts themselves have lead to failures in their objective,
they often illuminated whole new areas of mathematics, but this realization
only comes somewhat later than when the proof is realized. It was obvious that
a particular attempt had failed relatively quickly, but the lessons learned
and their applications required a keen (creative mathematical) eye to spot.

Badiou (a continental political philosopher interested in the philosophy of
mathematics) uses FLT as an example in political contexts, to what extent we
can say that a utopian vision has failed even if it results in disaster, we
should be (and some have been) looking out for the lessons which are learned
after failure.

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waynecochran
That last sentence about Badiou, whoever that is, was a sharp turn to the
left.

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MaxBarraclough
Interesting read. Wikipedia recounts the tale as well -
[https://en.wikipedia.org/w/index.php?title=Fermat%27s_Last_T...](https://en.wikipedia.org/w/index.php?title=Fermat%27s_Last_Theorem&oldid=871657366#Wiles's_general_proof)

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nyc111
A 200 page proof even if it is correct may not be the most elegant proof for
this problem since Fermat noted that his proof was too long to fit on the
margin. This suggests at most a one page proof.

Also I'm curious if it can be proved that only one proof is possible. There
are several independent proofs of the Pythagoras theorem.

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coraltime
Fermat could have been mistaken about his proof, however.

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nyc111
That's a possibility. But I still think there must be a more direct proof. A
shorter proof that can be understood by anyone who is familiar with Euclid. I
believe a proof by using only mathematical methods known to Fermat must be
possible.

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Retra
You can make any proof as short as you want. Length of proof means almost
nothing; it is a function of the language you choose to use.

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nyc111
> You can make any proof as short as you want.

If so why did Wiley choose to write a 200 page proof while he could have
stated it in one sentence?

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Retra
Because he was writing for a specific class of predetermined parsers, rather
than an arbitrary one which was optimized for his proof.

