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: 
I don't think that the fact that $P\neq NP$ is an asymptotic statement is an automatic "dealbreaker". One can make concrete conjectures that are consistent with our knowledge but stronger than P vs NP such as "It takes at least $2^{n/10}$ steps to find a satisfying assignment for a random n variable 10SAT formula" (with "random" being e.g., the planted model of [Achlioptas Coja-Oghlan](http://users.soe.ucsc.edu/~optas/papers/barriers.pdf), this is just an example - I don't know what are reasonable concrete numbers offhand). 

Such conjecture can result in a refutable prediction that any natural system that will try to solve this will fail (e.g., get stuck in a local minima), something you can verify with experiments. Indeed, I'm not an expert on this but to my knowledge, as Joe Fitzsimons mentioned, such predictions had been confirmed with Adiabatic computing. (Scott Aaronson also had some entertaining experiments with soap bubbles.) 

Of course you can also see some "empirical evidence" for $P \neq NP$ in the fact that people have been trying to solve optimization problems, cryptoanalyze encryptions, etc.. and haven't been successful so far...
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.