Edit: Let me summarize what this question was meant to ask. Is there a quantitative theory of "approximate" soundness? Arguments are usually either sound or unsound. This is binary. If we don't have access to a complete argument, or are unsure whether to trust parts of an argument, can we come up with a number between 0 and 1 that quantifies how sound the incomplete or untrusted argument is?
Original rambling question below:
Before getting to the question, let me first try to make a rough distinction in what I mean by numerics. There are several types, and I'll begin with what I don't mean by numerics: something like numerical integration, or numerical root finding. I consider this zeroth case uninteresting. Problems like these are completely understood. We are just using computers to do our tedious calculations for us, and I see no reason not to trust the result.
For the first type, I'm thinking of large complex calculations such as those used in the four color theorem, or the recent proof that checkers is a draw, or maybe the recent work on the character table for split E8. Here there are just a finite number of cases to be checked, but the computation itself is extremely elaborate just to compute about one bit of information (in the first two cases). For these types of problems, I'm imagining that there is no known way to generate a certificate that would reduce checking the validity of the calculation to an instance of case 0 numerics.
For the second type, consider numerics of the following flavor. In this scenario, my friend Bernhard has a conjecture that all of the infinitely many non-trivial zeros of a certain function lie on a certain line in the complex plane. Unfortunately, I'm not as good as Bernhard, and I don't understand his insight in proposing the conjecture. Lacking his intuition, and in lieu of a proof, I decide to numerically test the conjecture. I find that the first 1010 zeros (say) all line on the correct line.
Now I can start to get to the question. First of all, there is a fuzzy line between case 0 and case 1. Is there a way to make this line more precise?
Next, can we make precise the notion of "trusting" large calculations of the type 1 variety? How much should we trust them? Also, these types of calculations can be very unsatisfying if they don't lead you to a principle which explains why the answer is what it is. Is there a way to make this notion precise as well? Suppose that the proof of the four color theorem could be reduced to checking only 31 cases, instead of several hundred, but a computer was still necessary. Would we consider this a "good" proof? I would like to try to quantify this.
Finally, consider case 2. Have we really given any support at all to the conjecture if we've left an infinite number of cases unchecked? I'm tempted to answer a knee-jerk "No!" to this, especially in light of things like the disproof of the Mertens conjecture or Skewes' number. But it certainly feels like we've made it more plausible than if we hadn't checked those cases. I'm afraid we might have to resort to Bayesian degrees of belief here, but is there another answer?