# Role of statistical estimation in formal proof

Consider the following scenario: There is some mathematical constant $c$ that you want to compute. You don't have a formal proof for any particular value of $c$, but you have some sound statistical procedure that will determine $c$ (up to a certain error, perhaps) with high probability. For example, $c$ is the value of some integral and you estimate it with Monte-Carlo sampling.

Now, this is not as rigorous as a Hilbertian proof of the value of $c$. But it is also much more rigorous than a heuristic. One can say confidently, for instance, that your statistical procedure determines the correct value with an exponentially high probability.

(I think you need to be careful of Bayesian reasoning; you should not conclude that the value of $c$ is any particular value with high probability, as usually in the Bayesian paradigm we assume that the subject is capable of perfect reason in the presence of data, and a perfectly rational reasoner would probably deduce $c$ from a Hilbertian proof)

Could, or should, use of statistical procedures be considered valid rigorous reasoning? This scenario is not at all far-fetched.

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There is always the problem with probabilistic or statistical methods, that you may not ignore knowledge. In mathematical proof you may ignore info, that you don't need. If value c is determined, but someone comes with a good argument that the c is incorrect in this particular case, then you may not ignore that argument. That makes any answer obtained by this method, instable due to future knowledge. –  Lucas K. Dec 30 '12 at 14:36