(Sir Roger's own illustration and explanation of what follows can be found in Chapter 1.4 of his book. On Amazon go to page 18 of Look Inside.)

At the start of his book The Road to Reality the mathematician and scientist Sir Roger Penrose proposed the following triple conundrum:

To all appearances, the cosmos, the universe in which we exist, is everything. Whatever exists is part of it, by definition. This of course includes us, and our minds.

But we humans have created mathematics, which appears able to express the natural laws governing the cosmos. It is potentially able to quantify and explain everything that happens everywhere. But it can also describe and explain alternate versions of the cosmos. So in a sense our mathematics includes our cosmos as a subset. Mathematics includes more than everything.

But then again, our minds, which have developed mathematics, also contain ideas that are not included in mathematics, ideas which mathematics cannot capture, such as love and spirit. So mathematics is a subset of what our minds may contain. In that sense, our minds can include a greater everything than mathematics, which already contains the universe. Which contains our minds.

Clearly, this circular line of thought cannot be right. But where is the error?

Sir Roger goes to great lengths to clarify this question, but does not pretend to have a final answer for it where it is posed. 

It does suggest to me, however, that the order might be inverted. The one thing no person can deny is the existence of their own mind. Within that mind one may conceive of the existence of mathematics, or at least the existence of the rules mathematics implies about the fundamental nature of the physical cosmos. The person whose mind presents to him full sensory images of the physical world does not have to consciously question or even be aware of how the mind does this. But doing so might lead to similar conclusions as those Sir Roger considers.

Sir Roger does, at the end of his Road, consider some of these questions more deeply. I suppose this is an invitation for us to do the same.