There are several questions here, and we need to distinguish them.
1) Does belief require objects? Beliefs are "about" in a very broad sense, but I believe (as should you) that there is no ratio of two integers that equals 2 when squared. There is no object that answers to my belief, but it's clear that my belief is really a belief, indeed a true one, and in some sense is about something. In this case, we could perfectly sensibly say it's about numbers, though not about some specific number. This might sound superficially paradoxical, but the paradox persists only if we insist on treating the word "about" in a way that rides roughshod over how we actually use it.
2) Is it possible to have a belief about "nothing" or about a negative, as opposed to affirmative proposition? Two different issues here. I can believe that Barack Obama is not seven feet tall. That's a negative proposition, and I clearly can believe it; any analysis of belief that claims otherwise wold be silly. But it's not a belief about nothing; it's a belief about Barack Obama.
So what about beliefs about "nothing"? Depends on what you have in mind. The way to go about settling the matter is to offer some cases that look superficially like beliefs about nothing. Then examine them to see if they really are things someone could believe. Thoughtful common sense is the place to start here. If they are things someone could believe, try to decide in what sense, if any, they are beliefs about nothing. Ask yourself: what's the content of the belief? Whether or not a belief can be about "nothing," it's a mark of beliefs that they have content. Roughly: they can be true or false.
Or do you mean beliefs about nothingness? Depending on how that's understood, there might well be such beliefs, even if nothingness isn't an object or thing. I'm not willing simply to assume that Sartre's Being and Nothingness didn't express beliefs about nothingness, though there's a perfectly good question about what that amounts to. Making progress here would call for figuring out what Sartre meant by "nothingness" (or "le néant") though that's not germane to our topic. In any case, the idea that a belief must always be about some object—some thing—is a rookie mistake. It doesn't begin to do justice to how we actually use the word "belief."
3) Finally, atheism. An atheist, as most philosophers use the word, is someone who believes that there is no God. If someone claims that they don't understand this, I'd be suspicious. They may have gotten themselves in the grip of a bad idea, but believing that there are no pink elephants, or no even primes larger than 2 or no gods are perfectly familiar cases of belief that any account of belief has to make room for. And these beliefs are only puzzling if you squint the wrong way. In ordinary contexts, we understand claims like this without any difficulty.
A lack of belief in the existence of a God is not automatically atheism. An atheist lacks a belief that there is a God, but so does an agnostic; an agnostic neither believes nor disbelieves. Again, this is no puzzle, I neither believe nor disbelieve that the President of the USA in 2030 will be a Democrat. I don't have a belief about it.
A more general comment: there's lots of worthwhile philosophy to be done in analyzing the logic of belief and trying to sort out just what distinguishes beliefs from other mental states. And since beliefs are in some sense "about" or, to use the $64 word, are intentional states, there's interesting and worthwhile work to be done sorting all of that out. But even if our account of these things ends up going beyond what we'd normally be inclined to say, what we'd normally say is the place to start and the place to keep returning to.
There are several questions
There are several questions here, and we need to distinguish them.
1) Does belief require objects? Beliefs are "about" in a very broad sense, but I believe (as should you) that there is no ratio of two integers that equals 2 when squared. There is no object that answers to my belief, but it's clear that my belief is really a belief, indeed a true one, and in some sense is about something. In this case, we could perfectly sensibly say it's about numbers, though not about some specific number. This might sound superficially paradoxical, but the paradox persists only if we insist on treating the word "about" in a way that rides roughshod over how we actually use it.
2) Is it possible to have a belief about "nothing" or about a negative, as opposed to affirmative proposition? Two different issues here. I can believe that Barack Obama is not seven feet tall. That's a negative proposition, and I clearly can believe it; any analysis of belief that claims otherwise wold be silly. But it's not a belief about nothing; it's a belief about Barack Obama.
So what about beliefs about "nothing"? Depends on what you have in mind. The way to go about settling the matter is to offer some cases that look superficially like beliefs about nothing. Then examine them to see if they really are things someone could believe. Thoughtful common sense is the place to start here. If they are things someone could believe, try to decide in what sense, if any, they are beliefs about nothing. Ask yourself: what's the content of the belief? Whether or not a belief can be about "nothing," it's a mark of beliefs that they have content. Roughly: they can be true or false.
Or do you mean beliefs about nothingness? Depending on how that's understood, there might well be such beliefs, even if nothingness isn't an object or thing. I'm not willing simply to assume that Sartre's Being and Nothingness didn't express beliefs about nothingness, though there's a perfectly good question about what that amounts to. Making progress here would call for figuring out what Sartre meant by "nothingness" (or "le néant") though that's not germane to our topic. In any case, the idea that a belief must always be about some object—some thing—is a rookie mistake. It doesn't begin to do justice to how we actually use the word "belief."
3) Finally, atheism. An atheist, as most philosophers use the word, is someone who believes that there is no God. If someone claims that they don't understand this, I'd be suspicious. They may have gotten themselves in the grip of a bad idea, but believing that there are no pink elephants, or no even primes larger than 2 or no gods are perfectly familiar cases of belief that any account of belief has to make room for. And these beliefs are only puzzling if you squint the wrong way. In ordinary contexts, we understand claims like this without any difficulty.
A lack of belief in the existence of a God is not automatically atheism. An atheist lacks a belief that there is a God, but so does an agnostic; an agnostic neither believes nor disbelieves. Again, this is no puzzle, I neither believe nor disbelieve that the President of the USA in 2030 will be a Democrat. I don't have a belief about it.
A more general comment: there's lots of worthwhile philosophy to be done in analyzing the logic of belief and trying to sort out just what distinguishes beliefs from other mental states. And since beliefs are in some sense "about" or, to use the $64 word, are intentional states, there's interesting and worthwhile work to be done sorting all of that out. But even if our account of these things ends up going beyond what we'd normally be inclined to say, what we'd normally say is the place to start and the place to keep returning to.