What does knowing really mean? Suppose Mary (or some other creature or machine) can answer certain questions about the world — without the need to do any actual experiments. Then we'd agree that Mary knows those things about the world. But what would it mean, to you or to me, to hear Jack say that Mary knows geometry? For all we know, Mary might believe that circles are squares and it happens that Jack agrees! Jack's statement tells us more about Jack than about Mary.
When Jack says, Mary knows geometry, this indicates to us that Jack would probably be satisfied by Mary's answers to the questions about geometry that he would be disposed to ask.
The meaning of Mary knows geometry depends on who is saying it. After all, no one knows everything about geometry; that statement would not mean the same to us as to a mathematician, whose concepts of geometry are different from those of ordinary persons. In the same way, the meanings of many other terms depend upon the speaker's role. Even an apparently unambiguous statement like This is a painting of a horse shares this character: you can be sure of little more than that it displays a representation that in someone's view resembles a horse in some respects.
Then why, when we talk about knowledge, don't we have to say who all those speakers and observers are? Because we make assumptions by default. When a stranger says that Mary knows geometry, we simply assume that the speaker would expect any typical person who knows Mary to agree that she knows geometry. Assumptions like that allow us to communicate; unless there is some reason to think otherwise, we assume that all the things involved are typical. It does not bother us that a professional mathematician might not agree that Mary knows geometry — because a mathematician doesn't fit our stereotype of a typical person.
You might maintain that none of this applies to you, since you know what you know about geometry. But there's still an observer on the scene, only now it is hiding inside your mind — namely, the portion of you that claims you know geometry. But the part of you that makes this claim has little in common with the other parts that actually do geometry for you; those agencies are probably incapable of speech and probably devoid of thoughts about your knowledge and beliefs.
Naturally, we'd all prefer to think of knowledge as more positive and less provisional or relative. But little good has ever come from trying to link what we believe to our ideals about absolute truths. We always yearn for certainty, but the only thing beyond dispute is that there's always room for doubt. And doubt is not an enemy that sets constraints on what we know; the real danger to mental growth is perfect faith, doubt's antidote.