
Proving something exists nonconstructively using probability. - amichail
http://en.wikipedia.org/wiki/Probabilistic_method
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pz
yay! i like seeing math posts on HN, even though the prob method usually
leaves you feeling less than satisfied. the proofs, at least the ones i've
seen, don't usually elicit that AHA moment. but that's just me. i think i
still have the copy of the Spencer book i checked out of the library years
ago.

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stcredzero
Perhaps there are alien minds constructed out there whose organizing principle
for their version of rational thought involves optimizing Bayesian belief
networks, and when they look at these, the deductive reasoning part is
unsatisfying, but the probabilistic part gives them the Ah-Ha!

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edmccaffrey
What's the probability of that being true?

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stcredzero
AH-HA!

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stcredzero
Title seems a little misleading, in that some constructive proof techniques
are used to set-up the probabilistic tools.

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mgreenbe
That's not how constructivity works, though --- it's all or nothing. You can
be careful about the points at which you use classical reasoning (or, here,
probability), but then the whole proof is no longer constructive. The argument
in favor of the constructive approach is that you can very carefully decide
when you depart from it.

For a (Curry-Howard) corresponding intuition, a program is no longer
functional the moment any computational effect is involved (mutable state,
control effects like continuations and backtracking, etc.).

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jdoliner
The best description I was ever given of the probabilistic method: "It's not
like we're searching for a needle in a haystack here. We're searching for hay
in a haystack... But every time we reach in, we find a needle" - Laszlo Babai
This was said of the Ramsey number (Example 1) which is very hard to construct
examples of (AFAIK there isn't an algorithm to do it), even though a randomly
generated graph will be an example with probability tending toward 1.

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thras
The Princeton Companion to Mathematics has a good description of this starting
on page 572.

Basically the proof goes like this: Define a random variable that can only
take on integer values. Show that the expected value is less than one. The
random variable must therefore sometimes take on the value of zero.

If your random variable was defined to mean something interesting at zero,
then the above is an existence proof. What they mean by "non-constructive" is
that you still don't actually have an example of the thing existing -- you
just know it does. Very cool.

