So, I start of talking about three criteria that we would like any sort of knowledge to possess. The differ slightly from the usual description of epistemology.
We prefer knowledge that reflects the world in which we live, one that has an application to things outside of our imagination. Saying that a ball is red, that an action is evil, that 2 + 2 = 4 would ideally say something about the apple, the action, or the numbers themselves.
We prefer knowledge that we can show, by some sort of objective process, holds to be true. This allows us to convince skeptical people that the knowledge is valid. Saying that a ball is red, and action is evil, or that 2 + 2 = 4 would mean we had accepted means of demonstrations so that every observer who accepted such means as valid would accept the result.
We prefer knowledge that won't change with time, that stands firm. when something is true, it should stay true and not become partly true or even false later on. A red ball should always be red (assuming the ball itself does not change), an evil action should always be evil, and 2 + 2 should always equal 4.
The next part discusses what I see as three different systems that focus on knowledge using the desired attributes above. Of course, any truly broad system of inquiry will make us of all three of the systems I discuss below, but generally only one of them at time. Like a three-way combination of oil and vinegar, they don't seem to mix together. In the examples, I try to offer samples where the primary dependence seems to match the system I use, and the other systems are of secondary significance.
Empirical systems rely on the investigation of the real word and use inductive processes and assumption to decide what is more generally true about that world. They use demonstrable methods such as taking measurements, running controlled experiments, and making predictions to be verified by further observations and experiments. Because the observations are founded in actual investigations, their reality to the world is a natural consequence. However, the use of the inductive method means that the results can not be certain, that all finding are provisional and possibly erroneous. The best that can be hoped for is beyond a good reason to doubt a finding. Examples of empirical systems would include physics and the results of national polls regarding elections.
Formal systems rely on working from a starting point accepted as being true and using an accepted, objective, deductive means of deriving new truths within that system. The method of deriving new truths within the system is the demonstration of that truth, and the process of deriving the truth would be repeatable regardless of the observer. Also, given a specific starting point, the same truth will always be the result (at least, ideally). However, since the starting point is selected rather than derived, there is no guarantee that such a starting point will have any bearing on the world around us. A formal system may be a useful model of reality, but we can never know whether it matches reality. Examples would include philosophy and law (as practiced at the appellate level).
Belief systems use the trusted, revealed knowledge of an authority of some sort to provide as description of the world and its operation. The truths that we get from this authority are accepted because of the level of trust, not because of any independent test that we run. Belief systems do provide a view of reality and knowledge about the state of affairs in the universe. Also, because the source is some consistent authority, they have the feature of certainty, of not changing within the system itself. However, because our acceptance of this knowledge is based upon trust rather than a means to validate them, these truths are generally not demonstrable to others in a significant or meaningful way. Examples include Shintoism and Objectivism.
Then the question that guarantees a minute of silence and ten more of discussion: which one of these is mathematics?