Another Random Past Article with some interesting content. I just went through some of this over on MSK's OL.
From Next Level in Post 14:
1) In Mathematics, negative assertions are proven all the time. ...
2) ... his problem had more to do with the standard of proof and the cognitive content of the concept of God than anything else.
3) ... You only believe that one cannot prove negative statements if you think that he cannot prove a universal statement, ...
4) Proof is a concept of intelligence - unless there is some agreed upon standard of proof, there is no way you can prove something to someone else.
As far as I have come with this is that the negative assertions that cannot be proved are empirical claims. In the Basic Principles of Objectivism lectures on the subject of God and religion, Nathaniel Branden offered the example that you cannot prove that the far side of the Moon does not have rose gardens and Coca-Cola vending machines. Just because we have not found them yet, does not mean that they are not there. Etc.
As noted, though, in the case of rational or logical claims, as in mathematics, negatives are proved all the time. The oldest may be the proof that the square root of two is not a rational number. Realize that before that, the Greek thinkers seemed to have all accepted that such a number existed, but had only not been found yet. The reason for that is that from the Egyptians, fractions were expressed as sums of addends whose numerators are 1 (one). 3/4 = 1/4 + 1/2 or 3/7 = 1/3 + 1/14 + 1/42. etc. We accept the existence of irrational numbers is a positive claim: pi, e, and infinitely many more. But it began with the proof of a negative assertion.
Where I stopped thinking (the place often called a "conclusion") is in the unity of the analytic and synthetic claims. Why is it that we can prove a negative analytic, but not a negative synthetic?