

If P=NP, then mathematics as a field would be destroyed. - amichail
http://www.claymath.org/millennium/P_vs_NP/Official_Problem_Description.pdf

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amichail
Stephen Cook writes:

 _For example, it would transform mathematics by allowing a computer to find a
formal proof of any theorem which has a proof of reasonable length, since
formal proofs can easily be recognized in polynomial time. Example theorems
may well include all of the CMI prize problems. Although the formal proofs may
not be initially intelligible to humans, the problem of finding intelligible
proofs would be reduced to that of finding a recognition algorithm for
intelligible proofs. Similar remarks apply to diverse creative human
endeavors, such as designing airplane wings, creating physical theories, or
even composing music. The question in each case is to what extent an efficient
algorithm for recognizing a good result can be found. This is a fundamental
problem in artificial intelligence, and one whose solution itself would be
aided by the NP-solver by allowing easy testing of recognition theories._

~~~
scythe
That ignores the size of the polynomial, doesn't it? If the fastest
polynomial-time proof finding algorithm is O(n^50), it's not going to do much
good.

~~~
sown
I don't think it would be so bad. There's be some constant factor or number
where n^50 would always be less than, say n^n after that constant.

~~~
Retric
Yea, 51, but with n = 40 you are already at the number of atoms in the
universe ~10^80. That would leave around 11^50 calculations per atoms in the
universe, at 1,000,000 Ghz you are talking about 10^32 seconds which ~ 3 *
10^24 years. Or 10^15 = (1,000,000,000,000,000) times the current age of the
universe.

For any calculable n, n^n is less than n^50. Ok, _if_ computing power doubled
every year, for the next 200 years we would still not be able do 50^50
calculations. But in 250 years of doubling each year it would be easy.

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brg
Absolutely not, and for far more reasons than I enumerate below.

The language used to encode the search space will be under constant
development and expansion.

The creation of the axioms of any proof system is not a search problem.

Recognition of "meaningful" proofs will be extremely hard to automate. The
checking of statements does not create mathematical theory, but it is the
collection of related statementswhich do.

The search space is likely to be still much too large for proof systems to
exhaustly enumerate all meaningful statements in any person's lifetime.

If mathematical statements could be easily checked, it would make mathematical
study quite different. But likely much more interesting and fast paced.

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travisjeffery
No, no, no Computer Science would be destroyed, not Mathematics.

A large portion of Computer Science is the study of algorithms including their
complexity.

How can you say something is going to be destroyed by something that isn't
even a question in the field?

If anything Mathematics is going to be improved by getting rid of some grunt
work and giving way for some more creativity. Not to mention the theorems that
if P=NP would come about which could be used for even further results.

All-in-all it's a win-win for Mathematics.

~~~
eru
Mathematics and computer science are intertwined. And they would not be
destroyed --- but merely closed (to a large extent) as a solved problem.

Algorithms and proofs have many interconnections. (You can usually abstract
one out of the other with a bit of creativity. I.e. the classic proof for
infinity of primes gives a basic algorithm for creating new primes.)

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sidmitra
I think it would make things more interesting. So many problems that would
then be solvable in polynomial time.

~~~
lsb
Recall that primality testing was proved to be in P, O(n^6) via the AKS
primality test, but in practice it's much too slow. So if something
interesting is in P, but it's O(n^100), what good is that?

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

~~~
dangoldin
Isn't it O(log n)? I guess computationally it may blow up if you are doing a
significant number of operations.

~~~
swolchok
It depends on how you measure n. The O(log^{6+\epsilon}(n)) quote on Wikipedia
is for n being the number to be tested. This is "cheating"; inputs to an
algorithm need to be measured by their length, and numbers are exponential in
their length (i.e., the length of the number n is order log n for "reasonable
encodings").

As an example, the knapsack problem with n objects and weight W is solvable in
O(n*W) time, but it's known to be NP-hard. This doesn't prove P=NP, because W
is exponential in its length. This is called a pseudopolynomial algorithm.

~~~
dangoldin
Ah got it. Thanks for the clarification.

Coincidental too since I just read about the AKS algorithm on Terrence Tao's
blog. [http://terrytao.wordpress.com/2009/08/11/the-aks-
primality-t...](http://terrytao.wordpress.com/2009/08/11/the-aks-primality-
test/)

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mpk
I didn't read the article because it's on scribd and I haven't installed flash
on the new system yet.

But the headline sounds a bit too dramatic. We've been using and expanding
applied mathematics for thousands of years. A proof of P=NP will not retro-
actively make Roman aqueducts collapse, let alone destroy the infrastructure
we're all using day-to-day. Which in turn means that whatever flawed
assumptions have been made (and there's plenty of gray space to be found), the
field of mathematics will not be destroyed.

QED. (Don't I feel smug).

[edit : I clicked on the scribd link initially but have now saved the PDF link
for tomorrow's reading]

~~~
amichail
Click the title for the pdf.

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dejb
Read a great short story by Charles Stross about this called 'Antibodies'. The
notion was that a discovery like this would rapidly lead to strong AI. Never
connected the two myself like that but now it seems obvious - AI is
essentially an algorithms problem.

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mmmmmmm
There is a guy called 'Musatov' on Sci.Math that is doing this very thing. The
idea of mathematics being a representation of our line and line liberties is
interesting in this regard.

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sown
I don't understand. The link is just the problem statement.

~~~
amichail
<http://news.ycombinator.com/item?id=756761>

