

Proofs By Contradiction And Other Dangers - jackfoxy
http://rjlipton.wordpress.com/2011/01/08/proofs-by-contradiction-and-other-dangers/

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sz
"Since is formally a for-all/there-exists ($\Pi_2$) statement of arithmetic,
there is a principle that such a proof should be convertible to one without
contradiction"

Where can I find more information about this sort of thing?

~~~
pohl
Is it Constructivism in mathematics that you would like more information
about? If so, try this...

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

Or are you looking for more about transforming a proof into a constructive
one?

~~~
sz
I was wondering what subfield coined the label $\Pi_2$, specifically. I
haven't looked much into the structure of propositions and proofs per se, and
I'm curious about what can be said about it.

~~~
pohl
That is an excellent question. I've only seen capital pi used for repeated
multiplication. I, too, would like to know what it means in this context.

I don't know if you missed it, but the author links to a PDF that mentions
this notation in the abstract. I haven't had a chance to digest it, though:

<http://www.cs.cmu.edu/~crary/819-f09/Murthy91.pdf>

Edit: Behold...

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

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

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

------
joe_the_user
_"The difficulty is what if the contradiction comes not from the assumption
that P=NP, but rather from some error in the proof of one of the lemmas or
theorems? A problem. A serious problem. Then the proof is wrong."_

For a proof to be correct, every step of the proof has to be correct.

I can't see this as a problem specific to proof by contradiction. I suppose
long proofs are more dangerous than short proofs because long proofs have more
steps that need to be taken.

If someone really thought they had a P/=NP proof, it would be prudent for them
to break it up into Lemmas which stood on their own and only at the end point
out what they'd proved. But these considerations seem completely orthogonal to
the method of proof.

Am I missing something (seriously, I am only an amateur myself)?

~~~
mjw
I wondered this too as an amateur. Although how about this:

Normally the emergence of a contradiction in the course of a proof serves as a
red flag that something's wrong. When proving by contradiction, you're
operating without the safety of these warning signs as you progress through
the proof. If you reach a contradiction, you have to question whether or not
you got there suspiciously easily.

If you're aiming for something more specific than just falsehood -- "not X",
say, as in the contrapositive approach -- you might be less likely to get
there incorrectly. If you reach a contradiction along the way, you don't go on
to say "and I'm done since contradiction implies everything including not-P",
(even though technically speaking this would be perfectly valid) instead you
question where you went wrong in the proof, because this is too easy.

But yes this certainly feels like more of a fuzzy psychological argument
doesn't it. I don't know how (or if) you could formalise a notion of "how
susceptible is this proof tactic to accidents".

~~~
gwern
If we were discussing a machine-checked proof, then this would basically not
be a problem. The correctness of an accepted proof would have little to
nothing to do with its length.

Unfortunately, any proof about NP would be very gnarly and likely to be a
mammoth task to formalize into a machine-checkable proof and so likely not
done for a long time, if ever - the proof-checkers would be other human
mathematicians. And humans are empirically unreliable. Invalid proofs have
been accepted for years, decades, and even millennia (Bertrand Russell, IIRC,
found a number of gaps and hidden assumptions in Euclid).

If you were looking through a program someone written, wouldn't you assume
that there will be a certain bug rate per KLOC? Wouldn't you assume there is a
certain bug-I-will-not-catch-reading-through rate per KLOC?

Now imagine you are looking at thousands of lines of the most fragile program
ever written...

From a philosophical standpoint, I am reminded of Quine's famous attack on
Popperian falsificationism - when we observe that Uranus is not on the orbit
Newton would predict, we could reject Newton's laws of motion, _or_ we could
postulate some additional unobserved physical fact like the existence of a
seventh planet called Neptune. Our observation has only forced us to reject
the consolidated theory Newton+6-planets, it doesn't tell us which of the 2 to
spare. Similarly, when our potential proof finally terminates in a
contradiction, all it tells us is that somewhere we went wrong; it doesn't
tell us which axiom or theorem we ought to throw away as false.

~~~
btilly
I believe it is David Hilbert you are thinking of, not Bertrand Russell. He
found that Euclid's traditional 5 axioms needed to be 20. See
<http://en.wikipedia.org/wiki/Hilbert%27s_axioms> for more.

