Writers like Kuhn and Polanyi's subjectivistic approach to science are still popular in some circles, mostly because of the west's fascination with individual consciousness and the existential and phenomenological approaches to reality that grew out of that. While this is understandable historically I believe that this approach is still invalid, so I thought I'd say a little about that. But that will involve my discussing some basic philosophical history, so I hope you don't mind if I wax a little nerdy there.
Basically, the most important concept in epistemology is the split between the philosophies of idealism and empiricism. Idealists believe that ideas about the external world are innate. Kant was the last major philosopher to articulate the classical position on this, and his influence is still being felt by contemporary neo-Kantian theories and philosophers. For example, Kant mantained that the ideas of space and time were so fundamental that they had to be built-in, innate ideas. He argued that the test of this is that if one can't imagine a universe without a certain idea, then that idea couldn't have come from external reality. While this is an interesting contention, and there is some support for it (perceptual psychologist Eleanor Gibson showed that even at 1 year of age babies can perceive depth and space very well, in her famous "visual cliff" experiments), it is unlikely that there are truly innate ideas, although there are probably innate abilities like Kant suggested, since as he pointed out, in order for the mind to be actively involved in organizing and structuring the data of the senses, this could not occur unless there were corresponding mental capabilities and constucts to match.
But getting back to the philosophy definitions, as many people know, Locke, Hume, and most of the British philosophers were empiricists; they believed that ideas come from sense data and from external reality. This philosophical split between idealism and empiricism in thinking goes all the way back to Aristotle and Plato, so if you understand what it was about, you basically understand what most of western philosophy was about since then. The one exception here is the British philosopher Berkeley, who was an extreme subjectivist, and his philosophy is known as solipsism. He actually thought that the external world only existed because we perceived it, making it an extreme form of idealism. He did this by arguing that since we ultimately only know our own minds and its consciousness and internal perceptions, that there is no real way to prove that an objective, external reality even exists. While there is some truth to this, it's obviously an extreme position, and as result of recent research over the last 30 years in the neurophysiology and biophysics of sensation and perception, as in the case of David Marr's mathematical and theoretical work and his followers, we know now just how rigorous and analytical the process of perceiving external reality actually is.
Hence, there is very little reason anymore to insist on the fundamental subjectivity of perception in the Kantian sense. It is true that there are visual illusions at the higher levels of sensory perception, but those are now regarded as special cases, and they are being shown to be explainable in terms of mathematical visual field- distortion theories of these mechanisms that can be quantified just like the basic sensory processes.
But getting back to what I was saying before, Kant's view is still popular in some circles, and actually, he was right about certain things, such as the mind having certain built-in capabilities to understand reality, as I mentioned above in the case of idealism. The linguist, Noam Chomsky, and his ideas about an innate language capability are an example of this neo-Kantian approach, actually, which has been supported by developmental studies and by studies of feral children in regard to a critical period between 6 and 8 years of age, which is required for language developement.
However, most scientists and philosophers since the early 20th century are probably either Logical Positivists or Critical Naturalists rather than Idealists or neo-Kantians in the strict sense. The problem with neo-Kantianism is that a systematic ghost of an illusion pervades even the finest specimens of this theory, since there is no strong connection to external reality anymore. Both Critical Naturalism and Logical Positivism were strongly influenced by scientific theories about reality, and Logical Positivism is really just the philosophy and analysis of scientific method and of the logic of scientific hypothesis and theories rather than traditional philosophy in the usual sense. Some of the famous Logical Positivists were people like Rudolf Carnap, A.J. Ayer, and Reichenbach, whose names many people know. Critical Naturalism does get more into traditional philosophical topics like metaphysics and ontology but again, they tend to take their ideas about reality from what science has discovered in quantum theory and cosmology and what that implies as far as figuring out the metaphysics and ontology of the real world.
Alfred North Whitehead and Bertrand Russell were two famous 20th century philosophers who were examples of the Critical Naturalism school, and both of them were mathematicians as well as philosophers. Whitehead was Russell's math professor, and in fact, they both wrote a famous work on mathematical logic, The Principia Mathematica, in which they show that the basic mathematical operations can be derived from logic.
Since we're on the subject, I thought I'd make several comments specifically on Kuhn's theory as set forth in his famous book, the Structure of Scientific Revolutions. Kuhn's idea qualifies as a psychohistorical explanation of the nature of scientific progress, because scientists must have already made a cognitive shift to a new mindset before acceptance of the new theory can occur.
Other people have commented on similar ideas in the works of Feyerabend, Popper, and Polanyi, so I won't repeat any of that here. What I will say, however, is that this theory, while interesting, makes as little, or as much sense, itself, as the irrational science it purports to explain.
First, Kuhn's explanation of the process seems plausible psychologically but in fact is not supported by the psychological literature itself. People change deeply held convictions and ideas not because of an external paradigm shift, but because they become convinced internally that the new idea is superior to the old. Why? Because it explains the facts better, makes more powerful predictions, or is simpler. In other words, it is a fairly logical, reasonable process. This should surprise no-one but Kuhn.
Second, Kuhn's theory ignores the innumerable scientific hypotheses, theories, and advances that displaced earlier explanations with very little or no resistance.
Third, Kuhn misinterprets the initial resistance to Einstein's Theory of Relativity. The real problem with the acceptance of this theory is that when it made its debut (especially in the case of Einstein's General Theory), few physicists themselves could even understand the mathematics and physics involved. Ignorance should not be confused with scientific irrationalism or just stubborn refusal to accept the truth.
Well, I hope you didn't mind my little philosophy digression, but I thought I'd make a few comments about the evolution of these ideas since Kuhn and Polanyi's ideas are best understood in the context of the development of philosophical ideas over the last several centuries.