Democrats respond with an argument that goes back to the Marquis de Condorcet. Even if no individual voter knows much, they are still likely to make the right decision. For example, think of a jury deciding whether to convict someone. Even if every individual on the jury is only a little bit more likely to be right than wrong, when they all vote together, the majority is quite likely to come out on top. And if millions of people were on the jury, then the right decision would almost certainly be made. The famous political scientist Skip Lupia put it like this: unless voters are Dumber Than Chimps, they are likely to make the right decision in the aggregate.
So, by this Condorcet Jury Theorem, democracy should work quite well. It's a bit like one of those stories where a million monkeys (or chimps) bash away at typewriters and one of them eventually produces Hamlet. Voters may not know much, but on average, they'll get to the right decision. The Theorem works for more complicated decisions than simple guilty/innocent choices. Suppose we vote on how much to spend on hospitals. Different people have different views, some of them quite wacky, but on average, the median voter is just right. Half the people want to spend too little, and half want to spend too much. If so, then a proposal to spend exactly the right amount will beat any other proposal. (If the other proposal is lower, at least half the voters will prefer the higher amount; if it's higher, at least half will prefer the lower amount.) And a political candidate who promises to spend the right amount will make a monkey out of any other candidate.
However, I think the Jury Theorem is completely bananas.
First, it depends on the idea that people are right on average. But there is no reason to expect that. Remember, people are wrong because they are uninformed, because informing yourself would be costly. Bob Dylan's The Hurricane is a song about Rubin Carter, a black man wrongly accused of murder:
"To the white folks who watched he was a revolutionary bum
And to the black folks he was just a crazy nigger
Noone doubted that he pulled the trigger"
Almost everybody thought that Carter was guilty. Of course, a little investigation would have uncovered serious flaws in the case against him. But in a big jury vote, nobody has the incentive to do the investigating - it was left up to a few public-spirited journalists and activists. Uninformed people will not, in general, be right on average.
The other gorilla in the room is that even if people's views are right on average, they may not choose the right policy on average. This is a little subtler, so if maths makes you go ape, you can skip the next bit. Suppose we are deciding how much to spend on defence, and we face a national rival. We aren't sure about this rival's political ideology: is it to the left of us like Soviet Russia? Or to the right like Nazi Germany? (Maybe it's modern China.) The farther it is from us, the more we ought to spend.
In fact, our rival has just the same ideology as us. We need to spend almost nothing. And "on average", people know this. But their views vary - some people think that our rival is leftist, some people think its rightist. So, if we vote on military spending, we would end up spending quite a hairy amount. The voter with the middle estimate of the other country's ideology, who correctly thinks we should spend nothing, is right at the bottom in terms of the spending he wants.
Of course, if we voted on the facts of the matter, we would get them right. But in practice, most policy decisions depends on hundreds of facts, and there's no practical way we could vote on them all. So, even if people were miraculously right about all the facts - on average - they wouldn't necessarily choose the right candidate, or make the right policy decisions.
To sum up, if you put a million monkeys in a room, you might end up with Shakespeare. But you might just end up with a banana fight and a lot of faeces throwing.