

Is a Proof a Proof If No One Can Check It?  - kqr2
http://www.nytimes.com/1988/12/20/science/is-a-math-proof-a-proof-if-no-one-can-check-it.html

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J_McQuade
Well, the proof is perfectly valid, as on some abstract level a computer ==
Turing machine == something you can reason about. If you can prove that your
program works in the required way, then your proof holds.

However, brute force solutions are ugly and shed little (if any) new light on
the problem, and in the words of G.H.Hardy - "There is no permanent place in
the world for ugly mathematics". With things like this I just can't help but
think that an elegant and enlightening proof will unexpectedly pop up some
time in the distant future when we're all off researching some sort of new-age
hypermaths or something...

... well, I suppose a man's got to dream, right?

~~~
andrew1
I'd agree with your dislike of ugly solutions. I guess the benefit of this
being accepted as a proof is that all the conjectures that begin 'Assuming
that x is true...' can now be converted to theorems saying 'x is true,
therefore...', which might tidy up a lot of loose ends.

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shib71
There are proofs (by humans) that are too large for all but a rare few to
check. Wiles' proof of "Fermat's Last Theorum" was 100 pages. Courtesy of
Wikipedia:

"Because Wiles had incorporated the work of so many other specialists, it had
been suggested in 1994 that only a small number of people were capable of
fully understanding at that time all the details of what Wiles has done."
[[http://en.wikipedia.org/wiki/Wiles%27_proof_of_Fermat%27s_La...](http://en.wikipedia.org/wiki/Wiles%27_proof_of_Fermat%27s_Last_Theorem)]

No one can argue that an unconfirmed computer proof is more trustworthy than
an unconfirmed proof from a human.

~~~
telegraph
I think you're overlooking the real thrust of the situation. Wiles' proof was
relatively lengthy and involved, but it can be examined and studied by humans
in an extremely reasonable amount of time. It only took three days for Wiles
to present his original proof. There aren't many people in the world who can
understand it, and there are fewer who are knowledgeable enough to confirm its
validity, but they exist.

Conversely, the proof described in the NYTimes article is of such length that
no single mathematician could confirm its validity -- rather than deducing the
non-existence of the object in question by a logical argument, it examined a
huge number of possible cases. In that respect, it is far more similar to the
proof of the four color map theorem. The issue is not so much whether or not
we trust the computer's result, but moreso what it means for mathematics to
proceed with results that we do not, in a traditional sense, understand.

~~~
skermes
This question came up in some of my more abstract classes in college. A few
professors in my department were working on slightly different problems in the
same general domain of automated mathematical problem solving and proof
construction. The general consensus as I remember it was that the simplest way
to get around the problem of no human being able to survey these proofs was to
do something like this:

1\. Define a machine read- and writable logic that can express your theorem
and the steps you think it'll take to get there. 2\. Write an automated proof-
checker that can verify that proofs in this language are correct. 3\. Prove
the correctness of the checker. 4\. Write a program that starts with your
axioms and writes out a proof that ends in your theorem. 5\. Verify it with
the checker.

Now the only proof that needs to be human-surveyable for us to be certain that
everything is correct is the one in step 3. The proof created by step 4 can
fill up a skyscraper full of hard disks, and as long as the proof checker
verifies it we know that it must be correct. Given a simple enough proof
language (FOPL, for example) and a suitable programming language (the choice
at the time was lisp, I believe) step 3's proof is easily short enough for a
human to verify.

The only hole left is the possibility of a subtle machine malfunction that
causes the checker to falsely categorize a proof as correct. On modern
hardware this possibility is remote enough that once a proof has been verified
a number of times by independent researchers on their own hardware we can
safely ignore it.

~~~
vilya
This looks like a good time to quote Knuth: "Beware of bugs in the above code;
I have only proved it correct, not tried it."

~~~
vinutheraj
Yes, exactly, what if the thing they proved is not what they coded in the end,
there is a good enough probability of bugs if the code is sufficiently big !

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enum
If you're willing to understand the program behind the proof, I think it's
more acceptable than a handwritten proof.

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russell
Cray 1S? Isn't that about the power of today's desk top? The Cray could do 160
to 250 MFLOPS depending on the problem (Wikipedia). That seems pretty slow. I
answered my own question
<http://forums.techpowerup.com/showthread.php?t=94721>. There are processors
doing up to 60 GFLOPS. Must be an error in the story.

EDIT: I have to start paying attention to bylines. There are archeologists
among us.

~~~
goodside
No error. The story is from 1988.

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justin_vanw
For brute force proofs, the proof basically nothing to do with the output of
the program. I'll explain:

A mathematical proof is just some utterance or depiction that will convince
your audience that you have proven something. The actual utterance that might
'count' is totally dependent upon who is in the audience. If you are a
topologist and you are talking to another topologist working on very similar
stuff, your proofs would be incomprehensible to even other mathematics
researchers, who could not determine if you were giving a valid proof or not,
while the other topologist could easily tell.

If your audience is a really smart physics professor, that same topology proof
would have to suck in many other facts, and explicitly quote accepted
theorems, or you would risk the physicist jumping up and saying "not so fast"
every time he see's a break in the reasoning. At least the physicist will be
comfortable with the existence of theorems.

Now, a proof of the same thing, given to an alien with an IQ of 5000 but who's
race never thought up the concept of a theorem would be exquisitely painful.
You might have to actually give the proof of each theorem you use, and each
theorem each of those theorems use. Otherwise, he would say "your logic is
flawless, but you assume facts you have not proven, so this proof is
incomplete."

If people were way smarter, proofs would be way shorter, since nobody writes
down things that are really obvious. Nobody says something like "since the
real numbers are commutative under multiplication..." because there is no need
to say it. If people were 50 times smarter or something, way more complex
things would just 'go without saying'.

So, what does some kind of brute force proof really consist of? It's a program
that runs on a computer generally. So the proof is really the source code. If
people are convinced the program does what you claim, and then you run it,
that is certainly a proof.

If you can't count that, you also can't count any other non-constructive
proof. I would say that they are basically pointless, since you can use the
result of the proof in other proofs, but you can't build on it very much.

~~~
eru
I agree with your notion of "proof as conversation". That's the way presenting
proofs verbally works in practice. (And I am a mathematician.)

~~~
justin_vanw
I got most of this from conversations with Rich Schwartz.
<http://www.math.brown.edu/~res/>

------
bitdiddle
reminds me of when the four color theorem was proved using a computer program.
They had to convince everyone the proof was valid. There were lots of
questions about the validity of the program.

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ErrantX
I cant help thinking Douglas Adams brought this exact problem up ages ago when
Deep Thought found the Answer to Life, The Universe and Everything.

As Adams pointed out the answer is sometimes useless because the _process_ (or
the question) was incomprehensible to the "beings" who sought the answer.

Then later when Earth is destroyed before the Answer is calculated he makes an
even stronger point: _It doesnt matter_

The answer gave occupation to the philosophers for years before it was
discovered, it provided millions of years of purpose for the "mice" in finding
the answer. And Ultimately the lack of the answer didnt matter anyway.

Perhaps we can just accept the answer (knowing that it could be innacurate and
keeping an eye out for indicators) and make use of it? Surely that is more
worthwhile - it's something of a leap to completely trust the computer, but it
seems more dangerous to refuse to trust it at all :)

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KaiP
Explanation of the proof by the person who implemented it:

[http://www.cecm.sfu.ca/organics/papers/lam/paper/html/paper....](http://www.cecm.sfu.ca/organics/papers/lam/paper/html/paper.html)

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sireat
In some ways this kind of proof is similar to computer generating massive end-
game databases for chess.

The resulting perfect play for some endgames is incomprehensible to a human GM
and can last for more than 100 moves.

<http://www.xs4all.nl/~timkr/chess/perfect.htm>

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pj
As the singularity trots backward in time toward us, these types of computer
found solutions that humans can't comprehend are going to increase.

~~~
xenophanes
Huh? Singularities do not trot or time travel.

~~~
jderick
Try using your imagination

~~~
xenophanes
Try being nice :(

~~~
pj
It was a metaphor, here I'll explain it. The singularity is something in the
future. Some believe the singularity will be an entity smarter than humans,
that can do things, including move, perhaps even move itself in time. Who
knows? Thus the trotting use as an anthropomorphic description of singularity
behavior.

Furthermore, as technology progresses, the date at which the singularity will
arrive is getting closer and closer, thus the singularity is moving backward
in time, from the future to the present.

So humanity and the singularity, in a plane beyond the space time continuum
are moving toward each other, humanity moving forward through time, and the
singularity moving backward through time.

It's just poetry. That's why jderick said to use your imagination.

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hexis
If it's called a "proof" shouldn't it prove something to someone? If it can't
be checked, it seems like it should be called something else.

