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*". 

> 
> 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?
> 
> 
> 

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...
Indeed the physical version of P not equal to NP, namely that no natural physical systems can solve NP complete problem is very interesting. There are a few concerns 

1) The progrem seems practically "orthogonal" to both experimental and theoretical physics. 
So it does not really provide (so far) useful insights in physics.

There are some nice arguments how one can deduce from this physical version of the conjecture some insights in physics, but these arguments are fairly "soft" and have loopholes. (And such arguments are likely to be problematic, since they rely on very difficult mathematical conjectures such as NP nonot equal to P, and NP not being included in BQP that we do not understand.) 

(A similar comment apply to the "Church-turing thesis".) 

2) Although the physical NP not equal P is a wider conjecture than the mathematical NP not equal P, we can also regard it as more restricted since the algorithms that occur in nature
(and even the men-made algorithms) seem to be a very restricted class of all theoretically possible algorithms. It will be very interesting to understand such restrictions formally, but in any case any exerimental "proof" as suggested in the question will apply only to these restricted class.

3) In scientific modeling, computational complexity represents a sort of a second order matter where first we would like to model a natural phenomena and see what can be predicted based on the model (putting computational complexity theory aside). Giving too much weight to computational complexity issues in the modeling stage does not seem to be fruitful. In many cases, the model is computationally intractable to start with but it may still be feasible for naturally occuring problems or useful to understand the phenomena.

4) I agree with Boaz that the asymptotic issue is not necessary a "deal breaker". Still it is a rather serious matter when it comes to the relevance of computational complexity matters to real life modeling.