Question: I'm always intrigued by the lack of numerical evidence from experimental mathematics for or against the P vs NP question. While the Riemann Hypothesis has some supporting evidence from numerical verification, I'm not aware of similar evidence for the P vs NP question. 

Additionally, I'm not aware of any direct physical world consequences of the existence of undecidable problems (or existence of uncomputable functions). Protein folding is NP-complete problem but it appears to be taking place very efficiently in biological systems. Scott Aaronson proposed using the NP Hardness Assumption as a principle of physics. He states the assumption informally as "*NP-complete problems are intractable in the physical world*". 

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> Assuming NP Hardness Assumption, Why is it hard to design a scientific experiment that decides whether our universe respects the NP Hardness Assumption or not?
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Also, Is there any known numerical evidence from experimental mathematics for or against $P\ne NP$?

**EDIT:** Here is a nice presentation by Scott Aaronson titled [Computational Intractability As A Law of Physics](http://www.scottaaronson.com/talks/colloq2.ppt)

Answer: 
The definitions of "polynomial time" and "exponential time" describe the limiting behavior of the running time as the input size grows to infinity. On the other hand, any physical experiment necessarily considers only inputs of bounded size. Thus, there is absolutely no way to determine experimentally whether a given algorithm runs in polynomial time, exponential time, or something else.

Or in other words: what Robin said.
The study of real-world situations from a computational perspective is quite hard due to the continuous-discrete "jump". While all events in the real world (supposedly) are run in continuous time, the models we usually use are implemented in discrete time. Therefore, it is very tricky to define how small or large a step should be, what should be the size of the problem, etc.

I have written a summary on an Aaronson's paper on the subject, however it is not in English. See [the original paper](http://www.scottaaronson.com/papers/npcomplete.pdf). 

Personally, I have heard of another example of a real world problem modeled into computation. The paper is about control-systems models based on bird flocking.It turns out although it takes a short time in real life for birds, it's intractable ("a tower of 2s") when analyzed as a computational problem. See [the paper by Bernard Chazelle](http://arxiv.org/abs/0905.4241) for details.

[Edit: Clarified the part about the Chazelle paper. Thanks for providing precise information.]