Well, actually, you can, but the idea is that proving the nonexistence of something is at least very difficult. How would one go about proving that no unicorn has ever existed? It seems one would have to demonstrate that at no time or place has there ever been a unicorn, and the problem is that one doesn't have access to every time and place – unless, of course, one is omniscient. Similarly, it is argued, to prove that God does not exist, one would have to know everything. (And as an added twist, it is sometimes claimed that if one did know everything, one would be God – in which case atheism would of course be false.)
A variant on this idea is something we might call the "circle argument," in which a circle is said to represent the totality of facts and the atheist is asked to indicate how much of this circle represents the facts they themselves know. The idea is that anyone reasonable will of course admit that what they know is a very
tiny part of the total, and indicate this by, say, placing a small dot in the circle. But if there is so much one doesn't know, how can one claim God does not exist? What if the information showing that he exists lies in the area outside one's knowledge?
To these arguments, most people who call themselves atheists these days would probably reply that atheism isn't the belief that God does not exist; rather, it's the absence of belief in God. Atheism, on this view, is the same thing as non-theism, and includes both those who claim there is no God and those who merely lack belief in God.
Now, it's fine if one wants to understand atheism this way. However, the principal flaws in the above arguments have nothing to do with how they interpret atheism. To begin with, the arguments would in that case still apply to those atheists – sometimes called "positive atheists" – who, like me, claim that there is no God. Moreover, even if you are not a positive atheist – or an atheist at all – you probably do claim that certain other things don't exist – things like leprechauns, mermaids, and Santa Claus. But if the above arguments made sense, then you would be wrong in making those claims as well. In fact, if anyone presents you with the above arguments, all you need to do is ask them how they can say that Santa Claus isn't real. This already shows that there has to be something wrong with their reasoning. It doesn't, however, tell you what is wrong with it, so let's go a bit further.
The first problem is that the circle argument ignores what the knowledge one does have is knowledge of. For example, I know that the earth is round. But since it's impossible to know a falsehood, if I do know that the earth is round, then it follows that the earth really is round. But in that case there cannot be any fact elsewhere in the circle that contradicts this knowledge. Similarly, if I actually know that God does not exist, it doesn't matter at all that there are a lot of other things that remain unknown to me. If I know God doesn't exist, then he doesn't, and nothing outside my area of knowledge can contradict that fact.
But, the believer will object, how can I really know God doesn't exist? Knowing that the earth is round is one thing; but knowing that something does not exist? How could anyone ever prove the nonexistence of something?
Well, to begin with, there are things whose existence one can disprove – for instance, married bachelors, square circles, and sisters without siblings. If some idea can be shown to be internally inconsistent, then what it supposedly describes cannot be real. Someone cannot be both married and a bachelor, so it is impossible for there to be such an individual anywhere in the universe. It follows that if the concept of God is internally inconsistent, then one can in fact prove that there is no God.
But more importantly, why would one have to prove that God doesn't exist in order to be a positive atheist? There are at least two things wrong with this idea.
In the first place, one can reasonably claim to know things that one cannot prove. I cannot prove – at least not with logical certainty – that there are no leprechauns, but that shouldn't stop me (or you) from claiming to know that there aren't.
In the second place, a positive atheist doesn't have to claim to know that God does not exist; it's enough to believe that he doesn't. And one can reasonably believe in something that one doesn't know. For example, I believe – given the immensity of the universe – that there is life on other planets, but I certainly don't know that there is.
Both of the above arguments therefore fail. One does not have to be omniscient to be an atheist. Perhaps it's just that, compared with some of their critics, it seems like atheists are omniscient!