"Mathematical knowledge may, just like our scientific knowledge, be deep and broad, it may be subtle and wonderfully explanatory, it may be uncontroversially accepted; but it cannot be certain."

“Our best theories are not only truer than common sense, they make more sense than common sense...”

“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 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 , 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.”

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

“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.”

“Since building a universal virtual-reality generator is physically possible, it must actually be built in some universes.”

“A prediction, or any assertion, that cannot be defended might still be true, but an explanation that cannot be defended is not an explanation.”