I have a recurring need to understand numbers at a deeper level. The current blog is in addition to the one I have written about numbers before with which I have not been entirely satisfied

So, again, what is a number? I want to approach this question from the following hypothetical scenario. Let us say there is a world inhabited by only two intelligent human beings and call them ${x}$ and ${y}$. Let us also assume ${x}$ and ${y}$ have, until now, developed their language ${L}$ but this language is devoid of any couting terminology.

Now, assume that there is one apple in their world. As ${x}$ and ${y}$ are intelligent beings they come up with a symbol ${q}$ (say) to denote the one apple. Whenever they want to talk about the apple they would write the symbol ${q}$ and immediatley the image of an apple flashes in their brain.

Now, say there are many types of fruits in their world and also that each of these fruits is only one in number. As these beings are intelligent, they come up with a simple method for denoting these :

• orange ${\rightarrow r}$
• pumpkin ${\rightarrow s}$ etc.

So, every time ${x}$ writes ${r}$, ${y}$ remembers the image of one orange, etc. Similarly, for two apples, two oranges, two pumpkins, let us say they come with symbols, ${t,u,v}$ respectively.

Until now, the inhabitants have been perfectly logical about the understanding of their world. Whatever these beings could do can be replicated by any automatic machine (a computer) as these beings are only following an algorithm to represent any new object with symbols already in their language, ${L}$.

A significant leap of thinking / imagination is required on the part of ${x}$ and ${y}$ to start denoting one apple, one orange, one pumpkin as ${1_a}$, ${1_o}$, ${1_p}$ and two apples, two oranges, two pumpkins as ${2_a}$, ${2_o}$, ${2_p}$, etc. The advantage of this representation is that the total number of symbols used has decreased from ${6}$ (q,r,s,t,u,v) to ${5}$ (1,2,a,o,p). The advantage is more evident when one considers three, four or more of those fruits.

How can ${x}$ and ${y}$ come up with this new representation? The main requirement is that both ${x}$ and ${y}$ have to be capable of abstract thinking. That is, being able to think of “a detached fruit” as representing the number one whatever the fruit is. It is extremely important to note here that the number one is an abstract idea which cannot be expressed without the help of the language, ${L}$.

Therefore, I think that numbers came into existence only because of the Human being’s ability to think abstractly. A question that pops up now is : Can an automatic machine discover the concept of a number OR is a computer capable of abstract thought?