
Very Long Proofs - pron
https://johncarlosbaez.wordpress.com/2016/05/28/very-long-proofs/
======
ikeboy
From the linked post
[https://johncarlosbaez.wordpress.com/2012/10/19/insanely-
lon...](https://johncarlosbaez.wordpress.com/2012/10/19/insanely-long-proofs/)

>The method is called ‘proof by exhaustion’, because it involves reducing the
problem to 10,000 special cases and then settling each one with detailed
calculations… thus exhausting anybody who tries to check the proof by hand.

Ha.

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ittekimasu
It should be noted that there is some reason to aesthetically dislike
computer-generated proofs.

It's not mentioned by Baez, but one of the first breakthroughs in the Erdos
discrepancy problem was a very verbose (running into a few GBs if I remember
right), computer generated proof, for a subcase. Terry Tao later presented a
general proof which was far far shorter.

Obviously there are cases like the 4-color theorem which have withstood the
test of time and prejudice, but there is something sad about not being able to
make sense of such math.

~~~
drauh
"… some reason …"

Name one? I think I understand that it's unsatisfying that there are instances
where mathematics cannot be done with just pencil and paper. However, it's a
reproducible proof. And folks in science have long gotten over the fact that
there are important results that can only be achieved with lots of computing
power, and not just pencil and paper.

~~~
DavidSJ
A reason to dislike computer-generated proofs:

The point of mathematics is not just to know whether some proposition is true,
it's to understand the mathematical objects under study. A beautiful proof
sheds light, it explains _why_.

~~~
Kenji
That's your view on the point of math. I would be perfectly content to have a
long, computer generated proof of P ?= NP. Of course I prefer beauty and
elegance in proofs, but I think leveraging the computer as a tool to prove
things for us is quite an elegant thing by itself.

~~~
Scarblac
It would be horrible if we had an impossible to understand, but correct, proof
of P = NP that gave no hint of how to actually construct polynomial algorithms
for those problems.

~~~
curryhoward
Interestingly, there are algorithms for solving NP-complete problems which are
provably polynomial time iff P = NP. So the moment P = NP is proven, we
immediately know of polynomial-time algorithms for NP-complete problems. We
have the algorithms today, we just don't know if they are in P.

[https://en.wikipedia.org/wiki/P_versus_NP_problem#Polynomial...](https://en.wikipedia.org/wiki/P_versus_NP_problem#Polynomial-
time_algorithms)

~~~
Scarblac
Huh, thanks! That's completely surprising to me.

Edit: ah ok, a semi-algorithm that just tries all possible programs...

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chipsy
It seems to be part of our habit in mathematics to first make many discoveries
using brute-force and large enumerations, and then later extend the results
with more sophisticated methods if we can find them. For one practical
example, if you were doing a lot of computation in the 1960's you might carry
around a book of trig functions, or a slide rule, but after microprocessors
came about a pocket calculator could do all those functions with better
precision. With proofs, many questions are proven up to some number n, which
is only further extended by feeding the algorithm into a powerful computer.
But occasionally we discover a way of reframing the problem so that it can be
solved with relatively little computation.

I believe programming has some analogous quality to it: It's much easier to
solve just one problem and gradually find ways to generalize it.

~~~
jacobolus
> _if you were doing a lot of computation in the 1960 's you might carry
> around a book of trig functions, or a slide rule, but after microprocessors
> came about a pocket calculator could do all those functions with better
> precision_

The pocket calculator is doing a huge amount of computation to find those
results, an amount which would be impractical for the human to do (hence the
use of slide rules instead of laborious pen-and-paper arithmetic). It’s just a
different flavor of brute force. The calculator is basically going back to the
pre-logarithm method, carrying out elementary school arithmetic algorithms
very fast.

To be honest, the slide rule method – converting multiplication problems to
addition problems via a logarithm lookup table encoded on a stick – is quite a
bit more “elegant” than what the calculators are doing. The invention of
logarithms in ~1600 was one of the most important advances in the history of
science and technology.

* * *

The same is true in many other kinds of mathematical problem solving. In the
past, we only had access to manual effort and limited human time/attention, so
the available brute computation was quite limited and many problems were
entirely intractable, and great cleverness was required to solve others. The
goal of symbolic reasoning was to reframe problems to eliminate as much manual
computation as possible. For that reason, it was necessary to learn how to
manipulate trigonometric identities, solve nasty integrals by hand, etc. We
had to be able to rewrite any problem in a form where each concrete
computation only required a few simple arithmetic steps plus as few table
lookups as possible. Despite such simplifications, actually performing
computations often required teams of people mechanically performing arithmetic
algorithms all day.
[https://en.wikipedia.org/wiki/Human_computer](https://en.wikipedia.org/wiki/Human_computer)

Now that computation is cheap, we can dispense with many of the clever/elegant
methods of the past, and just throw silicon at our problems instead. This lets
us treat a wider variety of problems in a uniform way, and get away from doing
nearly so much tricky algebra.

~~~
dredmorbius
Your mention of "invention of logarithms" raises a couple of questions.

First: is there a good, _accessible_ (college calculus, some diff eq, some
linear algebra) history of mathematics you might recommend?

Second: I've been kicking around an ontology of technological dynamics (or
mechanisms) for a few months. In it I classify mathematics under symbolic
representation and manipulation, along with what I see as related tools of
speech, language, writing, logic, programming, and AI. If that sets off any
lights, bells, or whistles, I'd be happy to hear ideas or references.

[https://ello.co/dredmorbius/post/klsjjjzzl9plqxz-
ms8nww](https://ello.co/dredmorbius/post/klsjjjzzl9plqxz-ms8nww)

~~~
jacobolus
Stillwell’s book is pretty good.
[http://www.springer.com/us/book/9781441960528](http://www.springer.com/us/book/9781441960528)
[https://amzn.com/144196052X](https://amzn.com/144196052X) (Unfortunately
recent Springer books are printed on demand, and printing/binding quality can
be iffy; in particular quality of books bought via Amazon seems to be quite
poor.)

~~~
dredmorbius
Thanks. Fortunately there are local library copies:

[https://www.worldcat.org/title/mathematics-and-its-
history/o...](https://www.worldcat.org/title/mathematics-and-its-
history/oclc/19269766&referer=space_alien_cat)

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chaosfox
> This is one of the world’s biggest proofs: it’s 200 terabytes long! That’s
> about equal to all the digitized text held by the US Library of Congress.
> There’s also a 68-gigabyte digital signature—sort of a proof that a proof
> exists—if you want to skim it.

This sort of comparisons always bothers me, they make no sense.

Also according to the Nature post linked, the 68G "digital signature" is just
the 200T compressed, just another sign that talking about the file sizes is
meaningless.

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makmanalp
OK look, running a^2 + b^2 = c^2 a few trillion times and calling that a "long
proof" is disingenuous - it's really long if you include every single
operation, but must you? I just described it here in a few sentences. I guess
what's more interesting is a sort of kolmogorov complexity of the proof,
perhaps.

~~~
Houshalter
Yes, in fact every single mathematical proof can be reduced to a few lines of
code! You just write a program that verifies a proof, and another program that
brute forces through the space of all possible proofs. It should fit onto a
page or two. So the kolmogorov complexity of any proof has a pretty small
upper bound.

~~~
rntz
That isn't a proof, it's a procedure for finding a proof. The procedure might
_fail_ , and whether it will fail or not cannot be checked without running it
- which is not guaranteed to halt! Of course, to guarantee that it halts, you
could include an upper bound on the size of the proofs it searches in your
program. So then at least your "proof" is checkable. However, it is not in
general _efficiently_ checkable - that is, checkable in polynomial time. If it
were, it could only produce polynomial compression!

(Though admittedly, polynomial compression is pretty good, practically
speaking.)

As for Kolmogorov complexity, there are infinitely many distinct proofs, so
their Kolmogorov complexity cannot possibly be bounded.

~~~
Houshalter
> That isn't a proof, it's a procedure for finding a proof.

That's exactly my point. I wasn't being serious.

~~~
tgb
I think you're missing an important point though. The article's proof is
guaranteed to be verifiable in a known finite amount of time. Your example
does not so it's hard to take commentary from it onto the article's subject.

~~~
sparky_z
I think you're missing the important point that Houshalter was being
sarcastically critical of makmanalp's dismissive top-level comment, not the
proof in the article.

~~~
tgb
I don't see how Houshalter's post can possibly be read as sarcastic, it looks
like a technical comment that makes an interesting point that is slightly
tangential. I simply don't know how to read that comment as if there were a /s
at the end. And I don't see how the top level comment can be considered
dismissive, it too is making an interesting point that we should use another
metric.

~~~
marcosdumay
> it looks like a technical comment that makes an interesting point that is
> slightly tangential

That's why I love Math.

But hey, I hope your post is sarcastic too. But don't mind, there's no
procedure you could use to convince me it isn't.

