Knowledge arises from our interaction with reality (including ourselves and our previous knowledge). Reality is a proteon, i.e. it cannot be described completely. So at any given time, our knowledge is incomplete. There are gaps in it, there are inconsistencies, there are errors and artefacts, there is vagueness, and there are different degrees of justification. But our knowledge can be extended, revised, corrected, and made more exact and more explicit. It is developing. It is dynamic.
It is of little practical use to restrict the concept of knowledge to “justified true belief”, as one tradition of philosophy does. I am using the term “knowledge” here in a more general sense that is closer to the everyday meaning of the word.
Knowledge can be viewed as consisting of analytical spaces, i.e. limited chunks of consistent knowledge, together with the objects this knowledge refers to. Each analytical space only refers to some aspect of reality. It is sometimes possible to unify several analytical spaces into one larger one, at other times it is necessary to divide an analytical space or to create a new one. Due to the protean nature of reality, it is not possible to unify all analytical spaces into a single one covering all of reality. It is, therefore, not possible, to abstract knowledge completely from the objects it refers to. The real world objects the knowledge refers to always have more properties than can be derived in the theories we develop about them. So the knowledge has to remain in touch with the reality it refers to, so that it can be revised and can evolve. This is why the knowledge and the objects it refers to should be viewed as a unity.
If reality is a proteon, there cannot be a single formal framework from which to judge bits of our knowledge, there is no single complete system of logic which we can use to gauge which theories are right or wrong. This does not mean we have to give up rationality. What is needed, instead, is the interaction with reality, combined with critical assessment of our methods. We have to submit our thoughts, and those of others, as well as our concepts and methods, to critical reflection and, as far as possible, practical experimentation. This critical reflection and inductive verification must in turn be critically reviewed.
This is where I see the role of philosophy. Philosophy is not just another science. Philosophy is about facing proteons. Philosophical thinking occurs when we reach the border areas of our analytical spaces. We step out of the known methods of doing “normal science”. In this area, we do not have fixed methods yet. We are in the realm of technê, not epistêmê. We have to challenge our methods and concepts. There is no complete method or algorithm to do so. In this border area of our knowledge, systematic methods do not yet exist. New systematic methods may be the result of such processes, resulting in a new area of science, a new systematic analytical space. But a formalization of such creative thought processes is only possible in hindsight, and the resulting new methods are going to be incomplete again.
(The picture is from http://commons.wikimedia.org/wiki/File:Collapsed_Platonic_Solids.jpg)