Necessary truth is merely the subject-matter of mathematics, not the reward we get for doing mathematics. The object of mathematics is not, and cannot be, mathematical certainty. It is not even mathematical truth, certain or otherwise. It is, and must be, mathematical explanation.

Any theory about improvement raises the question: how is the knowledge of how to make that improvement created? Was it already present at the outset? The theory that it was is creationism. Did it ‘just happen’? The theory that it did is spontaneous generation.

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.

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The misconception that knowledge needs authority to be genuine or reliable dates back to antiquity, and it still prevails. To this day, most courses in the philosophy of knowledge teach that knowledge is some form of justified, true belief, where ‘justified’ means designated as true (or at least ‘probable’) by reference to some authoritative source or touchstone of knowledge.

Think of all our knowledge-generating processes, our whole culture and civilization, and all the thought processes in the minds of every individual, and indeed the entire evolving biosphere as well, as being a gigantic computation. The whole thing is executing a self-motivated, self-generating computer program. More specifically it is, as I have mentioned, a virtual-reality program in the process of rendering, with ever-increasing accuracy, the whole of existence.

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.