The overwhelming majority of theories are rejected because they contain bad explanations, not because they fail experimental tests. - David Deutsch

The next chapter is likely to provoke many mathematicians. This can't be helped. Mathematics is not what they think it is.

Kuhn's theory suffers from a fatal flaw. It explains the succession from one paradigm to another in sociological or psychological terms, rather than as having primarily to do with the objective merit of the rival explanations. Yet unless one understands science as a quest for explanations, the fact that it does find successive explanations, each objectively better than the last, is inexplicable.

A computation is a physical process in which physical objects like computers, or slide rules or brains are used to discover, or to demonstrate or to harness properties of abstract objects—like numbers and equations. How can they do that? The answer is that we use them only in situations where to the best of our understanding the laws of physics will cause physical variables like electric currents in computers (representing bits) faithfully to mimic the abstract entities that we’re interested in.