
An enormous theorem: the classification of finite simple groups (2006) - ur-whale
https://plus.maths.org/content/enormous-theorem-classification-finite-simple-groups
======
dwheeler
This is why we need to move _away_ from depending on just humans to verify
proofs. The goal shouldn't be "does a human fully understand the whole thing"
but "is it proven to be true?"

There are a number of computer-based tools that can verify proofs, including
HOL Light, Isabelle, Coq, Mizar, ProofPower and Lean. We should be working
towards the day where running a proof through that kind of tool is what's
required for publication:
[http://www.cs.ru.nl/%7Efreek/100/](http://www.cs.ru.nl/%7Efreek/100/)

I do a lot with Metamath. The largest Metamath database, which uses classical
logic & ZFC set theory, can be seen here:
[http://us.metamath.org/mpeuni/mmset.html](http://us.metamath.org/mpeuni/mmset.html)
Every one of its proofs routinely verified by 5 different programs written in
5 different programming languages by more than 5 different people. Humans
still need to check the axioms and definitions (to ensure they are "what is
meant"), check that the claims are what is desired, and in general humans need
to create at least part of the proofs. But _no_ one needs to "understand
everything". Instead, you can see whatever you want to see, and be confident
that every step is thoroughly verified.

A Metamath-specific video overview is here:
[https://www.youtube.com/watch?v=8WH4Rd4UKGE](https://www.youtube.com/watch?v=8WH4Rd4UKGE)

THAT is where we need to go. We aren't ready for it yet, but part of the
problem is that we need to agree that this the proper destination.

~~~
raincom
Human understanding is critical. See what Bill Thurston said:"The rapid
advance of computers has helped dramatize this point, because computers and
people are very different. For instance, when Appel and Haken completed a
proof of the 4-color map theorem using a massive automatic computation, it
evoked much controversy. I interpret the controversy as having little to do
with doubt people had as to the veracity of the theorem or the correctness of
the proof. Rather, it reflected a continuing desire for human understanding of
a proof, in addition to knowledge that the theorem is true." From Bill
Thurston: “On Proof and Progress in Mathematics”, Bulletin of the American
Mathematical Society, 30(2), 161-177
[https://arxiv.org/pdf/math/9404236](https://arxiv.org/pdf/math/9404236)

~~~
dwheeler
Some mathematicians think that human understanding of every step is critical.
That doesn't actually make it critical.

At one time it was very important to have humans weave cloth. Now we generally
expect machines to do that. Computers will not eliminate the need for human
mathematicians. It will merely change what they do. In many ways it will be
more freeing, because no one will need to review a proof to see if its steps
are correct.

Mathematicians will still need to develop definitions and determine what is
important to prove, and at least in the short-term they will need to do much
of the proving since machines are not yet very good at that. Many other fields
have changed with computers, there's no reason to believe mathematics cannot
be changed.

~~~
raincom
Well, if you think knowledge is same as truth, computer verifications can do.
Today, at least the consensus in philosophy of science is not centered on
truth. It is true that a falsity can't be knowledge. There is more to
knowledge than truth.

~~~
dwheeler
But that is the problem: we are claiming that we "know" things are absolutely
true, but we are using methods that are known to be inadequate for that
purpose.

Mathematicians are smart, but they are human. They make mistakes and miss
others'. Math has increasingly become too complex to entrust mere human
verification as being adequately trustworthy. Human verification has already
been repeatedly demonstrated as inadequate.

We must choose between truth or easier-to-understand claims that are sometimes
false because we depend on fragile human verification. I think we should
choose truth.

Where we can get human understandability as well as truth, great! But formal
verification should become the minimum over time, because truth should be the
minimum.

We can't do that all at once right now. But knowing the goal is the first
step.

------
newprint
Friend of mine is older Russian mathematician - algebraist, in his late 70's.
He has small picture of John G. Thompson on his work table. Thompson made an
enormous contribution to the proof. From what I remember reading, part of the
proof was formally verified.

It is intersecting to think, that we have proofs that are impossible to grasp
in a lifetime. Though, it would be possible to use the results !

Imagine, sometime in a future, humanity would loose the original proof due to
some catastrophic event, but it would be know that theorem still holds true.

~~~
raincom
If humanity were to loose the original proof of some theorem X in 3000 AD, the
only fact is known to them is that X is true, how on earth one should believe
it? Just because that theorem X was proved in the West?

Just reverse the case, and replace the West with Africa or China. Everyone in
the West would laugh at claims that "Some Chinese/African proved a theorem in
300 BC, but that proof doesn't exist anymore".

That's why Bill Thurston says this: “There is another effect caused by the big
differences between how we think about mathematics and how we write it. A
group of mathematicians interacting with each other can keep a collection of
mathematical ideas alive for a period of years, even though the recorded
version of their mathematical work differs from their actual thinking, having
much greater emphasis on language, symbols, logic and formalism. But as new
batches of mathematicians learn about the subject they tend to interpret what
they read and hear more literally, so that the more easily recorded and
communicated formalism and machinery tend to gradually take over from other
modes of thinking.

There are two counters to this trend, so that mathematics does not become
entirely mired down in formalism. First, younger generations of mathematicians
are continually discovering and rediscovering insights on their own, thus
reinjecting diverse modes of human thought into mathematics. “

------
infruset
Here is a Coq mechanized proof of the Odd order theorem, a (huge) step towards
the classification: [https://github.com/math-comp/odd-
order](https://github.com/math-comp/odd-order)

The paper is here: [https://hal.archives-
ouvertes.fr/hal-00816699/](https://hal.archives-ouvertes.fr/hal-00816699/)

------
ur-whale
An amazing theorem because - reportedly - no single human holds its overall
proof in his head.

~~~
QuesnayJr
That article has a comment from someone involved in the proof who disagrees:

"This article is very well written, but I must object to the implications of
the statement that "no-one in the world today completely understands the whole
proof." Of course, complete understanding is a high benchmark to set. But I
would argue that Thompson, Aschbacher and Lyons understand the proof at least
as well as Danny Gorenstein did, and Steve Smith and I (and probably several
others) come pretty close. See our book which received the AMS Steele Prize
for Mathematical Exposition. Danny Gorenstein deserves enormous credit for
conceiving and daring to articulate the strategy of the proof back in 1971-72.
But his overall perspective did not die with him.

Ron Solomon"

~~~
newprint
I see his point, but explaining this proof to people outside of this group
becomes a challenge and rightfully so.

~~~
JadeNB
> I see his point, but explaining this proof to people outside of this group
> becomes a challenge and rightfully so.

That's probably true for any interesting proof, maybe all the more so for less
famous proofs than this one where it's not clear the effort invested to
understand it will be worth it!

------
Koshkin
The author's picture at the bottom does not look like a good advertisement for
benefits of mathematical learning.

~~~
_Microft
You may be downvoted because it sounds like an ad hominem.

I agree that it is an unflattering choice and do not understand what makes
people use images like this to represent themselves to the largest audience
possible (i.e. anyone with internet access). There is no easier way to
positively influence what others imagine than by _not_ choosing an
unflattering image. If someone really does not like any image of themselves,
they can still leave it away and let people's imagination run free instead.

