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.

Mathematical knowledge may, just like our scientific knowledge, be deep and broad, it may be subtle and wonderfully explanatory, it may be uncontroversially ac…

Time travel may be achieved one day, or it may not. But if it is, it should not require any fundamental change in world-view, at least for those who broadly sh…

The next chapter is likely to provoke many mathematicians. This can't be helped. Mathematics is not what they think it is. (Ch. 10)

Mathematical knowledge may, just like our scientific knowledge, be deep and broad, it may be subtle and wonderfully explanatory, it may be uncontroversially ac…

I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0,…

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.