Mathematical knowledge may, just like our scientific knowledge, be deep and broad, it may be subtle and wonderfully explanatory, it may be uncontrove… - David Deutsch

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.

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

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.