Numbers:Natural Numbers

Natural numbers are the most basic form of numbers. These arise from the fact that in our physical universe there exist objects that can be counted. These objects must have the following two properties to enable counting:

1. Distinguishablity: these objects have some sort of identity i.e. they are different from others
2. Persistence: the objects are stable and exist for a sufficient amount of time.

Example: Let us take the example of apples in a box. Each apple has a boundary which is separated from the others. Thus we can easily distinguish between two apples. Moreover, each apple is stable and exists for a sufficiently long period of time. An apple will not suddenly disintegrate in front of our eyes, or change form etc.

One can see that in a universe where the physical objects did not posses such qualities the concept of counting numbers would not occur naturally.

What are these Natural Numbers?

These are written as $\mathbb{N}=\{0,1,2, \cdots \}$, where $0$ is a modern inclusion. Whenever, in mathematical notation we see the symbol $\mathbb{N}$ it is used to represent the natural numbers. The ancients Greek had not discovered it. The concept of zero was introduced by Indian mathematicians and was used in their works. The most famous of these were

• Brahmagupta    $7^{th}$ century
• Mahavira           $9^{th}$ century
• Bhaskara            $12^{th}$ century.

 The Arab scholars borrowed this concept from India and popularized it in the Islamic world. From there it permeated into European and western mathematics. This was possible as the Arabs world, due to its geographical location, enjoyed trade contacts with both India and Europe.

Can the Natural numbers be defined independent of the physical universe?

It can be seen from the definition of the natural numbers that they are independent of the geometry of the physical world. What we mean is that $\mathbb{N}=\{0,1,2, \cdots\}$ do not require any notion of lines, lengths or angles between them. There have been many attempts to define Natural numbers in an abstract way. By abstract we mean, independently of physical universe i.e. without referencing any of the objects in the physical realm.

One of the methods is based on the work of Cantor. Cantor developed ideas from the work of Giuseppe Peano. This method was promoted by the famous engineer Jon Von Nuemann. These ideas are based on Set theory. A set is a collection of objects that obeys certain rules. These rules are not mentioned here keeping in view the time and space constraints. 

Let us try and illustrate this method. The most basic number $0$ is represented as the empty set $\phi = \{\}$. Then, the number $1$ is constructed as $\{ \phi \}$ i.e. the empty set contained in a set. Here, we must point out that 

$\phi \neq \{ \phi \}$ 

as the conditions of equality of sets require that the number of elements in both of them are equal and each of the elements are also equal. Keeping this in mind one can easily see that the number of elements in $\phi$ is $0$, where as the number of elements in $\{ \phi \}$ is $1$. Here, we introduce an other handy notation. The number of elements in a set $A$ is called its cardinality and is represented as $|A|$.

Extending the process one can construct all the natural numbers as follows:

$0 \Longleftrightarrow \phi$

$1 \Longleftrightarrow \{ \phi \}$

$2 \Longleftrightarrow \{ \phi,\{ \phi \} \}$

$3 \Longleftrightarrow \{ \phi,\{ \phi,\{\phi\}\} \}$

$1 \Longleftrightarrow \{ \phi,\{ \phi,\{ \phi,\{\phi\}\} \} \}$

$\vdots$ 

Here, the symbol $\Longleftrightarrow$ is the equivalence. It means that the left hand side can be written as the right hand side and vise versa. Thus, we get an infinite sequence of mathematical entities which have no relation to the physical world. These entities are analogous to the Natural numbers. 

This points towards one of the most fundamental mysteries of our universe. How are the realities of the physical world so similar to rationality and logic that are constructs of human mind or thought.