On Proving Negatives

So where does an atheist start to talk about god? We could start with the classical arguments, but those have already gotten a lot of press. I do have some things to say about them, but I think I'd rather start elsewhere.

How about this? One of the comments I often hear when the subject of atheism comes up in a conversation is that atheism is an untenable position because, as everyone knows, "you can't prove a negative". Atheism's central claim is, of course a negative - "There is no god." So the argument is that atheism cannot be proven. This puts it at a disadvantage to both theism and agnosticism. Simple agnosticism makes no claim about god's existence. As theism's central claim is positive - god exists - it is at least possible that it might be proved. Thus the theist can offer positive arguments for his position while the atheist is reduced to merely finding fault with the theist's arguments rather than offering positive proof.

Or so the argument goes. The most glaring problem with this argument is that it rests on a false premise. It turns out that it is possible to prove negatives. Not only that, it's not even especially difficult. Let me start out by noting an oddity in the premise - if it were true, it would be unprovable! That's right the premise, "You can't prove a negative" is itself a negative statement and hence unprovable by it's own lights. We have to be a bit careful here. The fact that the premise is unproveable if true, or false if provable, does not show that the premise is false. Goedel made a lot of his fame in mathematics on the basis of just such a sentence - one that was unprovable if true. But it is a weird to use a negative premise as the central claim in criticizing atheism for having a negative as its central claim. The the extent that the argument works against atheism, it undercuts itself.

Luckily there is a more direct way to show the argument doesn't work. We simply note that proofs of negatives are readily available. For this we can turn to that most proof-laden of disciplines, mathematics. Here's an old one: "there is no greatest prime number." This is Euclid's theorem which can also be stated as "there are infinitely many primes." The proof is pretty straightforward. Or how about "there is no triangle whose internal angles sum to more than 180 degrees." It turns out that we routinely rquire high school students and undergraduates to prove negative mathematical statements. Ok, what about statements outside of math. Here's an easy one "There are no elephants in this room." It seems pretty easy to prove that, just have a quick look around. Elephants in rooms are the sort of thing that are easily discoverable with a quick visual inspection. If you look earnestly for an elephant in your room and don't find one, that proves there isn't one.

The bigger issue here is that statements don't divide neatly into negative and positive one except on purely grammatical grounds. Any claim can be made by using either a positive or a negative statement. Sometimes one is more natural, but there is always someway to state the claim positively and some way to state it negatively. We saw this with Euclid's theorem. The formulation I initially gave was negative, but the second one is positive. It's not hard to show that the two statements are equivalent. Try it. I'll try to show you the answer in a later post. So how would we state the atheist's central claim positively? Well, one way to do it is to divide things into to groups, the divine and the mundane. All gods are in the divine group, everything else is in the mundane group. Now atheism amounts to the claim that everything that exists is mundane. Theism can be expressed as the negation of atheism, i.e. as the claim that not everything is mundane. There are other ways too, but they often depend on the specific concept of god being employed, and we haven't gotten that far yet.

There, I think that's enough for now. In my next posts I want to talk about universal and existential claims, and also about the notion of proof.