Editing The Road to Reality Study Notes (section)

===3.4 Do natural numbers need the physical world?===
Progressing this philosophical train of thought to the natural numbers, Penrose notes that the operations of addition and multiplication of natural numbers are independent of the nature of geometry of the world.  Perhaps our notion of these numbers depends upon our universe specifically, but he stresses that it is hard to imagine that there would not be an important role for such fundamental entities.

To support this, Penrose introduces us to the idea of a set, which is an abstraction that does not appear to be tied to the specific structure of the universe.  He then describes a method by which an [https://en.wikipedia.org/wiki/Empty_set#:~:text=In%20mathematics%2C%20the%20empty%20set,its%20existence%20can%20be%20deduced. empty or null set] can be used to conjure the natural number system, out of nothing but the abstract idea of a set itself, and states that this technique can be used for the real number system as well.

Penrose ends by referring us back to Fig 1.3 (depiction of the mental, physical, and platonic worlds) and notes the mysterious nature of the fact that natural and real numbers having no reliance on the physical world for their existence, yet they seem to have direct relevance in describing the structure of the world.