so i begin

3.4 Do natural numbers need the physical world?

Let us consider one way (developed by Cantor from ideas of Giuseppe Peano, and promoted by the distinguished mathematician John von Neumann) that natural numbers can be introduced merely using the abstract notion of set. It also leads on to what are called 'ordinal numbers'. The simplest set of all is referred to as the 'null set' or the 'empty set', and it is characterized by the fact that it contains no members whatever! The set is usually denoted by the symbol Ø, and we can write this definition

Ø={ }

where the curly brackets delineate a set, the specific set under consideration having, as its members, the quantities indicated within the brackets. In this case, there is nothing within the brackets, so the set being described is indeed the empty set. Let us associate Ø with the natural number 0. We can now proceed further and define the set whose only member is Ø; i.e. the set {Ø}. It is important to realize that {Ø} is not the same as the empty set Ø. The set {Ø} has one member (namely Ø), whereas Ø itself has none at all. Let us associate {Ø} with the natural number 1. We next define the set whose two members are the two sets that we just encountered, namely Ø and {Ø}, so the new set is {Ø, {Ø}}, which is to be associated with the natural number 2. Then we associate with 3 the collection of all the three entities that we have encountered up to this point, namely the set {Ø, {Ø}, {Ø, {Ø}}}, and with 4 the set {Ø, {Ø}, {Ø, {Ø}}, {Ø, {Ø}, {Ø, {Ø}}}}, whose members are again the sets that we have encountered previously, and so on. This may not be how we usually think of natural numbers, as a matter of definition, but it is one of the ways that mathematicians can come to the concept. (Compare this with the discussion in the Preface.) Moreover, it shows us, at least, that things like the natural numbers can be conjured literally out of nothing, merely by employing the abstract notion of 'set'. We get an infinite sequence of abstract (Platonic) mathematical entities—sets containing, respectively, zero, one, two, three, etc., elements, one set for each of the natural numbers, quite independently of the actual physical nature of the universe. In Fig.1.3 we envisaged a kind of independent 'existence' for Platonic mathematical notions—in this case, the natural numbers themselves—yet this 'existence' can seemingly be conjured up by, and certainly accessed by, the mere exercise of our mental imaginations, without any reference to the details of the nature of the physical universe. Dedekind's construction, moreover, shows how this 'purely mental' kind of procedure can be carried further, enabling us to 'construct' the entire system of real numbers, still without any reference to the actual physical nature of the world. Yet, as indicated above, 'real numbers' indeed seem to have a direct relevance to the real structure of the world—illustrating the very mysterious nature of the 'first mystery' depicted in Fig.1.3.

[Fig.1.3 consists of three dots labeled: Physical world; Mental world; Platonic mathematical world, serially connected in a triangular 'loop'.]

—Roger Penrose "The Road to Reality" p.64-5