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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.

To say that prediction is the purpose of a scientific theory is to confuse means with ends. It is like saying that the purpose of a spaceship is to burn fuel. … Passing experimental tests is only one of many things a theory has to do to achieve the real purpose of science, which is to explain the world.

Reality contains not only evidence, but also the means (such as our minds, and our artefacts) of understanding it. There are mathematical symbols in physical reality. The fact that it is we who put them there does not make them any less physical.

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Thus we can see that if we take solipsism seriously - if we assume that it is true and that all valid explanations must scrupulously conform to it - it self destructs. How exactly does solipsism, taken seriously, differ from its common-sense rival, realism? The difference is based on no more than a renaming scheme. Solipsism insists on referring to objectively different things (such as external reality and my unconscious mind, or introspection and scientific observation) by the same names. But then it has to introduce the distinction through explanations in terms of something like the 'outer part of myself'. But no such extra explanation would be necessary without its insistence on an inexplicable renaming scheme. Solipsism must also postulate the existence of an additional class of processes - invisible, inexplicable processes which give the mind the illusion of living in an external reality. The solipsist, who believes that nothing exists other than the contents of one mind, must also believe that that mind is a phenomenon of greater multiplicity than is normal supposed. It contains other-people-like thoughts, planet-like thoughts and laws-of-physics-like thoughts. These thoughts are real. They develop in a complex way (or pretend to), and they have enough autonomy to surprise, disappoint, enlighten or thwart other classes of thoughts which call themselves 'I'. Thus the solipsist's explanation of the world is in terms of interacting thoughts rather than interacting objects. But those thoughts are real, and interact according to the same rules that the realist says govern the interaction of objects. Thus solipsism, far from being a world view striped to its essentials, is actually just realism disguised and weighed down by additional baggage, introduced only to be explained away.

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.

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.

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.