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: 
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.
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> Also, Is there any known numerical evidence from experimental mathematics for or against $P \neq NP$?
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Not as far as I know. However, one could get numerical evidence against $NP \subseteq P/poly$ of a similar sort of that found for the Riemann Hypothesis or Goldbach conjecture, by using explicit computations to show that, say, SAT up to length 10 does not have circuits of size 20 (where you let the values of 10 and 20 vary). This still runs into the same issue raised in the other answers -- finite evidence cannot give us an asymptotic answer. But essentially the same issue is true for the current "evidence" for the Riemann Hypothesis.

Now, finding a minimum circuit for SAT up to length 10 is currently very difficult. However, some of the ideas in geometric complexity theory allow you to get similar results with a more efficient (I think only exponential instead of doubly-exponential) computational search. One of Mulmuley's conjectures is that in fact this search can be done in polynomial time, but we are a long way from proving anything close to that.