As you say, the way string theory requires all these extra dimensions and this comes from certain consistency requirements about how string should behave and so on.

Well, gauge theory is very fundamental to our understanding of physical forces these days. But they are also dependent on a mathematical idea, which has been around for longer than gauge theory has.

Do not be afraid to skip equations (I do this frequently myself).

This book is about physics and its about physics and its relationship with mathematics and how they seem to be intimately related and to what extent can you explore this relationship and trust it.

I'm pretty tenacious when it comes to problems.

We have a closed circle of consistency here: the laws of physics produce complex systems, and these complex systems lead to consciousness, which then produces mathematics, which can then encode in a succinct and inspiring way the very underlying laws of physics that gave rise to it.

I was indeed very slow as a youngster.

It may well be there is something else going on in the brain that we don't have an inkling of at the moment.

With thought comprising a non-computational element, computers can never do what we human beings can.

I imagine that whenever the mind perceives a mathematical idea, it makes contact with Plato's world of mathematical concepts... When mathematicians communicate, this is made possible by each one having a direct route to truth, the consciousness of each being in a position to perceive mathematical truths directly, through the process of 'seeing'.

I would say the universe has a purpose. It's not there just somehow by chance.

The basic theory in twistor theory is not to add extra dimensions.

There are considerable mysteries surrounding the strange values that Nature's actual particles have for their mass and charge. For example, there is the unexplained 'fine structure constant' ... governing the strength of electromagnetic interactions.

It is hard to see how one could begin to develop a quantum-theoretical description of brain action when one might well have to regard the brain as observing itself all the time!

People think of these eureka moments and my feeling is that they tend to be little things, a little realisation and then a little realisation built on that.

Consciousness ... is the phenomenon whereby the universe's very existence is made known.

Well, I don’t know if I can comment on Kant or Hegel because I’m no real philosopher in the sense of knowing what these people have said in any detail so let me not comment on that too much.

No doubt there are some who, when confronted with a line of mathematical symbols, however simply presented, can only see the face of a stern parent or teacher who tried to force into them a non-comprehending parrot-like apparent competence--a duty and a duty alone--and no hint of magic or beauty of the subject might be allowed to come through.