Different ways of acquiring knowledge doesn't mean that the knowledge acquired is necessarily different. Thus I may know that it is snowing because I see the snow, or because someone tells me, but the knowledge that it is snowing is the same in both cses, even if in one case I have an experience that I don't have in the other. But there may also be different types of knowledge. One apparent difference that philosophers have discussed is between knowing a fact, such as knowing that it is snowing, and having an ability, such as knowing how to ski. These seem like different types of knowledge, though the relationship between them is a subject of some philosophical discussion. (As they say, those that can, do; those that can't, teach.)

This reminds me of that non-philosophers model of a philosophical question: If a tree falls in the forest and there is nobody there to hear it, does it make a noise? I've never been entirely sure just what question this was asking, but maybe it is about the nature of sound. According to some philosophers, sound is just an experience. In that case the tree would have made no noise, since there was no experience. Others would identify the sound with the vibrations in the air caused by the tree hitting the ground. In that case, the tree makes a noise whether or not there is someone to hear it (though not if there is no air around it). Anyway, back to your dream. A dream is a course of experience, and its existence doesn't depend on whether you remember it later. So you really can have a dream you don't remember, though if you don't remember it then you presumably won't know that you had it. It is the same with an experience you have while you are awake. If you heard a car backfire yesterday...

The popular dispositional theory of colour that Richard mentions has a curious consequence. If being red is just being such as to tend to produce a certain kind of sensation in us, then it isn't even possible that what tends to look red to us isn't really red but is really say some shade of gray. For on the dispositional view, red just is tending to look red.

Reading these questions might just make you more aware; thinking about them just might make you more intelligent.

Here is an additional (and very ahistorical) reason for studying some history of philosophy. The best philosophers of the past had more philosophical talent than any philosopher now living.

We are a very small part of the universe, and our best shot at figuring out what it is like involves making ourselves as causally sensitive as possible to the rest of it. That is what scientists do, through careful and sophisticated observation and experiment. If we are lucky, this will give us good reason to think that some of our scientific theories about the world are along the right lines. There is a very different way one might go about trying to figure out what the world is like, and this is by consulting an answer book that just tells you. Some people believe that their favored religious text is such a book. But we would need some good reason to believe that their text is reliable answer book, some good reason to trust it, and no such reason is available. Of course that won't dissuade someone who is convinced that they do have a reliable answer book, but they don't have good evidence for their view. To put the matter more externally, practices that depend on careful and...

Bayes' Theorem is very popular among philosophers of science who work on the bearing of evidence on theory. As you say, it has some attractive features. In your formulation, "A" stands for the theory and "E" for the evidence. To keep this straight, I'm going to use "T" and "E" respectively. If we take the P(T given E) to be the probability that theory T has after you observe evidence E and P(T) the probability the theory had before, then the difference between these is naturally taken to be the degree to which the evidence supports the theory, and Bayes theorem plausibly says that this will be greater the greater the probability of P(E given T) -- where this probability peaks at unity if T entails E -- and the smaller P(E), the probability of the evidence before you observed it. In other words, this take on Bayes theorem says that you get the strongest support from surprising evidence which would however have to be true if your theory is true. And that sounds right. Of course Bayesianism...

This is a tantalizing question. On some subjects I do have a good reason to prefer my own opinions, say because I was there at the time and saw it with my own eyes. But consider philosophical opinions. Why do I bother to form my own opinions? Why don't I just agree with everything Hilary Putnam says, since he is such a good philosopher? That would save me a lot of time, and it might well increase the reliability of my opinions. Well, in the case of philosophy, I guess part of the answer is that we don't just care about maximising the chances of having the right answer: we also think there is a particular value to working things out for oneself. One more point among the many that your question raises. Most of our opinions are not just our own opinions anyway, since almost everything we know we know because of what other people tell us. Philosophers argue over whether reason or experience is the primary source of knowledge; but at a certain level the answer is neither. Testimony or...

When you ask for an explanation, you usually already have a justification. You want something more -- understanding -- which is what an explanation provides. Thus when you ask why the same side of the moon always faces the earth, you already know that it does, but you do not understand why. Much of the work in the philosophy of explanation can be seen as attempts to say what takes you from knowledge to understanding. (By the way, the reason the same side of the moon always faces the earth is not because the moon is not spinning. In order for the same side of the moon always to face the earth, the moon has to spin around itself with exactly the same period as the period it takes the moon to orbit around the earth. Spooky, eh?)

I don't much like the expression 'scientific method', because it suggests that scientific practices come down to one unitary thing, and because it suggests that that thing is fundamentally different from everyday forms of investigation. But your question still stands: is pure mathematics a science, alongside physics, biology, psychology, etc, or is it fundamentally different? Mathematics is of course used in other sciences, but that is not distinctive. Physics is used in chemistry, and chemistry is used in biology, to give just two other examples. Nor is mathematics clearly different on the grounds that it does not study real objects, since some philosophers and mathematicians believe that mathematical objects and relations are real, though abstract. But many (though by no means all) philosophers would say that mathematics is nevertheless fundamentally different from the (other) sciences, because it is not empirical: we do not need to test its results by means of observation and experiment.